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1004_(GTM170)Sheaf Theory

# GraduateTexts inMathematics Glen E. Bredon # Sheaf Theory Second Edition Springer Editorial Board S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC ## Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 Oxtoby. Measure and Category. 2nd ed. 3 Schaefer. Topological Vector Spaces. 4 Hilton/Stammbach. A Course in Homological Algebra. 2nd ed. 5 MAC LANE. Categories for the Working Mathematician. 6 Hughes/Piper. Projective Planes. 7 Serre. A Course in Arithmetic. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 Humphreys. Introduction to Lie Algebras and Representation Theory. 10 Cohen. A Course in Simple Homotopy Theory. 11 Conway. Functions of One Complex Variable I. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed. 14 Golubitsky/Guillemin. Stable Mappings and Their Singularities. 15 Berberlan. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 Rosenblatt. Random Processes. 2nd ed. 18 Halmos. Measure Theory. 19 Halmos. A Hilbert Space Problem Book. 2nd ed. 20 Husemoller. Fibre Bundles. 3rd ed. 21 Humphreys. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 Greub. Linear Algebra. 4th ed. 24 Holmes. Geometric Functional Analysis and Its Applications. 25 Hewitt/Stromberg. Real and Abstract Analysis. 26 Manes. Algebraic Theories. 27 Kelley. General Topology. 28 Zariski/Samuel. Commutative Algebra. Vol.I. 29 Zariski/Samuel. Commutative Algebra. Vol.II. 30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 Hirsch. Differential Topology. 34 Spitzer. Principles of Random Walk. 2nd ed. 35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed. 36 Kelley/Namioka et al. Linear Topological Spaces. 37 MONK. Mathematical Logic. 38 Grauert/Fritzsche. Several Complex Variables. 39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras. 40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 Apostol. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 Serre. Linear Representations of Finite Groups. 43 Gillman/Jerison. Rings of Continuous Functions. 44 Kendig. Elementary Algebraic Geometry. 45 Loëve. Probability Theory I. 4th ed. 46 Loève. Probability Theory II. 4th ed. 47 Moise. Geometric Topology in Dimensions 2 and 3. 48 SACHS/WU. General Relativity for Mathematicians. 49 Gruenberg/Weir. Linear Geometry. 2nd ed. 50 Edwards. Fermat's Last Theorem. 51 Klingenberg. A Course in Differential Geometry. 52 Hartshorne. Algebraic Geometry. 53 Manin. A Course in Mathematical Logic. 54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs. 55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis. 56 Massey. Algebraic Topology: An Introduction. 57 Crowell/Fox. Introduction to Knot Theory. 58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed. 59 LANG. Cyclotomic Fields. 60 Arnold. Mathematical Methods in Classical Mechanics. 2nd ed. Graduate Texts in Mathematics 1 / Editorial Board S. Axler F.W. Gehring P.R. Halmos Springer-Science+Business Media, LLC ## Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 Oxtoby. Measure and Category. 2nd ed. 3 Schaefer. Topological Vector Spaces. 4 Hilton/Stammbach. A Course in Homological Algebra. 5 MAC LANE. Categories for the Working Mathematician. 6 Hughes/Piper. Projective Planes. 7 Serre. A Course in Arithmetic. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 Humphreys. Introduction to Lie Algebras and Representation Theory. 10 Cohen. A Course in Simple Homotopy Theory. 11 Conway. Functions of One Complex Variable I. 2nd ed. 12 Beals. Advanced Mathematical Analysis. 13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed. 14 Golubitsky/Guillemin. Stable Mappings and Their Singularities. 15 Berberlan. Lectures in Functional Analysis and Operator Theory. 16 Winter. The Structure of Fields. 17 Rosenblatt. Random Processes. 2nd ed. 18 Halmos. Measure Theory. 19 Halmos. A Hilbert Space Problem Book. 2nd ed. 20 Husemoller. Fibre Bundles. 3rd ed. 21 Humphreys. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 Greub. Linear Algebra. 4th ed. 24 Holmes. Geometric Functional Analysis and Its Applications. 25 Hewitt/Stromberg. Real and Abstract Analysis. 26 Manes. Algebraic Theories. 27 Kelley. General Topology. 28 Zariski/Samuel. Commutative Algebra. Vol.I. 29 Zariski/Samuel. Commutative Algebra. Vol.II. 30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galvis Theory. 33 Hirsch. Differential Topology. 34 Spitzer. Principles of Random Walk. 2nd ed. 35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed. 36 Kelley/Namioka et al. Linear Topological Spaces. 37 Monk. Mathematical Logic. 38 Grauert/Fritzsche. Several Complex Variables. 39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras. 40 KEMENY/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 Serre. Linear Representations of Finite Groups. 43 Gillman/Jerison. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 Loëve. Probability Theory I. 4th ed. 46 Loève. Probability Theory II. 4th ed. 47 MoISE. Geometric Topology in Dimensions 2 and 3. 48 SACHS/WU. General Relativity for Mathematicians. 49 Gruenberg/WEIR. Linear Geometry. 2nd ed. 50 Edwards. Fermat's Last Theorem. 51 KLINGENBERG. A Course in Differential Geometry. 52 Hartshorne. Algebraic Geometry. 53 Manin. A Course in Mathematical Logic. 54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs. 55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis. 56 Massey. Algebraic Topology: An Introduction. 57 Crowell/Fox. Introduction to Knot Theory. 58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed. 59 LANG. Cyclotomic Fields. 60 Arnol.D. Mathematical Methods in Classical Mechanics. 2nd ed. Glen E. Bredon # Sheaf Theory Second Edition Glen E. Bredon Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA Editorial Board S. Axler Department of Mathematics Michigan State University East Lansing, MI 48824 USA F.W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P.R. Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA ## Mathematics Subject Classification (1991): 18F20, 32L10, 54B40 Library of Congress Cataloging-in-Publication Data Bredon, Glen E. Sheaf theory / Glen E. Bredon. - 2nd ed. p. cm. - (Graduate texts in mathematics ; 170) Includes bibliographical references and index. ISBN 978-1-4612-6854-3 ISBN 978-1-4612-0647-7 (eBook) DOI 10.1007/978-1-4612-0647-7 1. Sheaf theory. I. Title. II. Series. QA612.36.B74 1997 \( {514}^{\prime }{.224} - \mathrm{{dc}}{20} \) 96-44232 Printed on acid-free paper. The first edition of this book was published by McGraw Hill Book Co., New York-Toronto, Ont. - London, © 1967. (C) 1997 Springer Science+Business Media New York Originally published by Springer-Verlag Berlin Heidelberg New York in 1997 Softcover reprint of the hardcover 2nd edition 1997 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Jacqui Ashri. Camera-ready copy prepared from the author's LaTeX files. ## Preface This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." Sheaves play several roles in this study. For example, they provide a suitable notion of "general coefficient systems." Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories. The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory. Several innovations will be found in this book. Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces. \( {}^{1} \) Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory. This is not without reason, since cohomology relative to a closed subspace

1004_(GTM170)Sheaf Theory

ology theories. The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory. Several innovations will be found in this book. Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces. \( {}^{1} \) Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory. This is not without reason, since cohomology relative to a closed subspace can be obtained by taking coefficients in a certain type of sheaf, while that relative to an open subspace (or, more generally, to a taut subspace) can be obtained by taking cohomology with respect to a special family of supports. However, even in these cases, it is sometimes of notational advantage to have a relative cohomology theory. For example, in our treatment of characteristic classes in Chapter IV the use of relative cohomology enables us to develop the theory in full generality and with relatively simple notation. Our definition of relative cohomology in sheaf theory is the first fully satisfactory one to be given. It is of interest to note that, unlike absolute cohomology, the relative cohomology groups are not the derived functors of the relative cohomology group in degree zero (but they usually are so in most cases of interest). The reader should be familiar with elementary homological algebra. Specifically, he should be at home with the concepts of category and functor, with the algebraic theory of chain complexes, and with tensor products and direct limits. A thorough background in algebraic topology is also nec- --- \( {}^{1} \) This is not even restricted to Hausdorff spaces. This result was previously known only for paracompact spaces. The proof uses the notion of a "relatively Hausdorff subspace" introduced here. Although it might be thought that such generality is of no use, it (or rather its mother theorem II-11.1) is employed to advantage when dealing with the derived functor of the inverse limit functor. --- essary. In Chapters IV, V and VI it is assumed that the reader is familiar with the theory of spectral sequences and specifically with the spectral sequence of a double complex. In Appendix A we give an outline of this theory for the convenience of the reader and to fix our notation. In Chapter I we give the basic definitions in sheaf theory, develop some basic properties, and discuss the various methods of constructing new sheaves out of old ones. Chapter II, which is the backbone of the book, develops the sheaf-theoretic cohomology theory and many of its properties. Chapter III is a short chapter in which we discuss the Alexander-Spanier, singular, de Rham, and Čech cohomology theories. The methods of sheaf theory are used to prove the isomorphisms, under suitable restrictions, of these cohomology theories to sheaf-theoretic cohomology. In particular, the de Rham theorem is discussed at some length. Most of this chapter can be read after Section 9 of Chapter II and all of it can be read after Section 12 of Chapter II. In Chapter IV the theory of spectral sequences is applied to sheaf cohomology and the spectral sequences of Leray, Borel, Cartan, and Fary are derived. Several applications of these spectral sequences are also discussed. These results, particularly the Leray spectral sequence, are among the most important and useful areas of the theory of sheaves. For example, in the theory of transformation groups the Leray spectral sequence of the map to the orbit space is of great interest, as are the Leray spectral sequences of some related mappings; see [15]. Chapter \( \mathrm{V} \) is an exposition of the homology theory of locally compact spaces with coefficients in a sheaf introduced by A. Borel and J. C. Moore. Several innovations are to be found in this chapter. Notably, we give a definition, in full generality, of the homomorphism induced by a map of spaces, and a theorem of the Vietoris type is proved. Several applications of the homology theory are discussed, notably the generalized Poincaré duality theorem for which this homology theory was developed. Other applications are found in the last few sections of this chapter. Notably, three sections are devoted to a fairly complete discussion of generalized manifolds. Because of the depth of our treatment of Borel-Moore homology, the first two sections of the chapter are devoted to technical development of some general concepts, such as the notion and simple properties of a cosheaf and of the operation of dualization between sheaves and cosheaves. This development is not really needed for the definition of the homology theory in the third section, but is needed in the treatment of the deeper properties of the theory in later sections of the chapter. For this reason, our development of the theory may seem a bit wordy and overcomplicated to the neophyte, in comparison to treatments with minimal depth. In Chapter VI we investigate the theory of cosheaves (on general spaces) somewhat more deeply than in Chapter V. This is applied to Čech homology, enabling us to obtain some uniqueness results not contained in those of Chaper V. At the end of each chapter is a list of exercises by which the student may check his understanding of the material. The results of a few of the easier exercises are also used in the text. Solutions to many of the exercises are given in Appendix B. Those exercises having solutions in Appendix B are marked with the symbol \( \text{⑤} \) . The author owes an obvious debt to the book of Godement [40] and to the article of Grothendieck [41], as well as to numerous other works. The book was born as a private set of lecture notes for a course in the theory of sheaves that the author gave at the University of California in the spring of 1964. Portions of the manuscript for the first edition were read by A. Borel, M. Herrera, and E. Spanier, who made some useful suggestions. Special thanks are owed to Per Holm, who read the entire manuscript of that edition and whose perceptive criticism led to several improvements. This book was originally published by McGraw-Hill in 1967. For this second edition, it has been substantially rewritten with the addition of over eighty examples and of further explanatory material, and, of course, the correction of the few errors known to the author. Some more recent discoveries have been incorporated, particularly in Sections II-16 and IV- 8 regarding cohomology dimension, in Chapter IV regarding the Oliver transfer and the Conner conjecture, and in Chapter V regarding generalized manifolds. The Appendix B of solutions to selected exercises is also a new feature of this edition, one that should greatly aid the student in learning the theory of sheaves. Exercises were chosen for solution on the basis of their difficulty, or because of an interesting solution, or because of the usage of the result in the main text. Among the items added for this edition are new sections on Čech cohomology, the Oliver transfer, intersection theory, generalized manifolds, locally homogeneous spaces, homological fibrations and \( p \) -adic transformation groups. Also, Chapter VI on cosheaves and Čech homology is new to this edition. It is based on [12]. Several of the added examples assume some items yet to be proved, such as the acyclicity of a contractible space or that sheaf cohomology and singular cohomology agree on nice spaces. Disallowing such forward references would have impoverished our options for the examples. As well as the common use of the symbol \( ▱ \) to signal the end, or absence, of a proof, we use the symbol \( \diamond \) to indicate the end of an example, although that is usually obvious. Throughout the book the word "map" means a morphism in the particular category being discussed. Thus for spaces "map" means "continuous function" and for groups "map" means "homomorphism." Occasionally we use the equal sign to mean a "canonical" isomorphism, perhaps not, strictly speaking, an equality. The word "canonical" is often used for the concept for which the word "natural" was used before category theory gave that word a precise meaning. That is, "canonical" certainly means natural when the latter has meaning, but it means more: that which might be termed "God-given." We shall make no attempt to define that concept precisely. (Thanks to Dennis Sullivan for a theological discussion in 1969.) The manuscript for this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the author would have had the energy to complete this project. North Fork, CA 93643 November 22, 1996 ## Contents Preface I Sheaves and Presheaves 1 1 Definitions . 1 2 Homomorphisms, subsheaves, and quotient sheaves 8 3 Direct and inverse images 12 4 Cohomomorphisms 14 5 Algebraic constructions 16 6 Supports . 21 7 Classical cohomology theories 24 Exercises 30 II Sheaf Cohomology 33 1 Differential sheaves and resolutions 34 2 The canonical resolution and sheaf cohomology 36 3 Injective sheaves 41 4 Acyclic sheaves 46 5 Flabby sheaves 47 6 Connected sequences of functors 52 7 Axioms for cohomology and the cup product 56 8 Maps of spaces 61 9 Φ-soft and Φ-fin

1004_(GTM170)Sheaf Theory

this second edition was prepared using the SCIENTIFIC WORD technical word processing software system published by TCI Software research, Inc. This is a "front end" for Donald Knuth's TEX typesetting system and the LATEX extensions to it developed by Leslie Lamport. Without SCIENTIFIC WORD it is doubtful that the author would have had the energy to complete this project. North Fork, CA 93643 November 22, 1996 ## Contents Preface I Sheaves and Presheaves 1 1 Definitions . 1 2 Homomorphisms, subsheaves, and quotient sheaves 8 3 Direct and inverse images 12 4 Cohomomorphisms 14 5 Algebraic constructions 16 6 Supports . 21 7 Classical cohomology theories 24 Exercises 30 II Sheaf Cohomology 33 1 Differential sheaves and resolutions 34 2 The canonical resolution and sheaf cohomology 36 3 Injective sheaves 41 4 Acyclic sheaves 46 5 Flabby sheaves 47 6 Connected sequences of functors 52 7 Axioms for cohomology and the cup product 56 8 Maps of spaces 61 9 Φ-soft and Φ-fine sheaves 65 10 Subspaces 71 11 The Vietoris mapping theorem and homotopy invariance 75 12 Relative cohomology 83 13 Mayer-Vietoris theorems 94 14 Continuity 100 The Künneth and universal coefficient theorems 107 16 Dimension 110 Local connectivity 126 Change of supports; local cohomology groups 134 19 The transfer homomorphism and the Smith sequences 137 Steenrod's cyclic reduced powers 148 The Steenrod operations 162 Exercises 169 III Comparison with Other Cohomology Theories 179 1 Singular cohomology 179 2 Alexander-Spanier cohomology 185 3 de Rham cohomology 187 4 Čech cohomology 189 Exercises 194 IV Applications of Spectral Sequences 197 1 The spectral sequence of a differential sheaf 198 2 The fundamental theorems of sheaves 202 3 Direct image relative to a support family 210 4 The Leray sheaf . 213 5 Extension of a support family by a family on the base space 219 6 The Leray spectral sequence of a map 221 7 Fiber bundles 227 8 Dimension 237 9 The spectral sequences of Borel and Cartan 246 10 Characteristic classes 251 11 The spectral sequence of a filtered differential sheaf . 257 12 The Fary spectral sequence 262 13 Sphere bundles with singularities 264 14 The Oliver transfer and the Conner conjecture 267 Exercises 275 V Borel-Moore Homology 279 1 Cosheaves 281 2 The dual of a differential cosheaf 289 3 Homology theory 292 4 Maps of spaces 299 5 Subspaces and relative homology 303 6 The Vietoris theorem, homotopy, and covering spaces 317 7 The homology sheaf of a map 322 8 The basic spectral sequences . 324 9 Poincaré duality 329 10 The cap product 335 11 Intersection theory 344 12 Uniqueness theorems 349 13 Uniqueness theorems for maps and relative homology 358 14 The Künneth formula 364 15 Change of rings 368 16 Generalized manifolds 373 17 Locally homogeneous spaces 392 18 Homological fibrations and \( p \) -adic transformation groups 394 19 The transfer homomorphism in homology 403 Smith theory in homology 407 Exercises 411 VI Cosheaves and Čech Homology 417 1 Theory of cosheaves 418 2 Local triviality 420 3 Local isomorphisms 421 4 Čech homology 424 5 The reflector 428 6 Spectral sequences 431 7 Coresolutions 432 8 Relative Čech homology 434 9 Locally paracompact spaces 438 10 Borel-Moore homology 439 11 Modified Borel-Moore homology 442 12 Singular homology 443 13 Acyclic coverings 445 14 Applications to maps 446 Exercises 448 A Spectral Sequences 449 1 The spectral sequence of a filtered complex 449 2 Double complexes . 451 3 Products . 453 4 Homomorphisms 454 B Solutions to Selected Exercises 455 Solutions for Chapter I . 455 Solutions for Chapter II 459 Solutions for Chapter III 472 Solutions for Chapter IV 473 Solutions for Chapter V 480 Solutions for Chapter VI 486 Bibliography 487 List of Symbols 491 List of Selected Facts 493 Index 495 ## Chapter I Sheaves and Presheaves In this chapter we shall develop the basic properties of sheaves and pre-sheaves and shall give many of the fundamental definitions to be used throughout the book. In Sections 2 and 5 various algebraic operations on sheaves are introduced. If we are given a map between two topological spaces, then a sheaf on either space induces, in a natural way, a sheaf on the other space, and this is the topic of Section 3. Sheaves on a fixed space form a category whose morphisms are called homomorphisms. In Section 4, this fact is extended to the collection of sheaves on all topological spaces with morphisms now being maps \( f \) of spaces together with so-called \( f \) -cohomomorphisms of sheaves on these spaces. In Section 6 the basic notion of a family of supports is defined and a fundamental theorem is proved concerning the relationship between a certain type of presheaf and the cross-sections of the associated sheaf. This theorem is applied in Section 7 to show how, in certain circumstances, the classical singular, Alexander-Spanier, and de Rham cohomology theories can be described in terms of sheaves. ## 1 Definitions Of central importance in this book is the notion of a presheaf (of abelian groups) on a topological space \( X \) . A presheaf \( A \) on \( X \) is a function that assigns, to each open set \( U \subset X \), an abelian group \( A\left( U\right) \) and that assigns, to each pair \( U \subset V \) of open sets, a homomorphism (called the restriction) \[ {r}_{U, V} : A\left( V\right) \rightarrow A\left( U\right) \] in such a way that \[ {r}_{U, U} = 1 \] and \[ {r}_{U, V}{r}_{V, W} = {r}_{U, W}\;\text{ when }\;U \subset V \subset W. \] Thus, using functorial terminology, we have the following definition: 1.1. Definition. Let \( X \) be a topological space. A "presheaf" \( A \) (of abelian groups) on \( X \) is a contravariant functor from the category of open subsets of \( X \) and inclusions to the category of abelian groups. In general, one may define a presheaf with values in an arbitrary category. Thus, if each \( A\left( U\right) \) is a ring and the \( {r}_{U, V} \) are ring homomorphisms, then \( A \) is called a presheaf of rings. Similarly, let \( A \) be a presheaf of rings on \( X \) and suppose that \( B \) is a presheaf on \( X \) such that each \( B\left( U\right) \) is an \( A\left( U\right) \) -module and the \( {r}_{U, V} : B\left( V\right) \rightarrow B\left( U\right) \) are module homomorphisms [that is, if \( \alpha \in A\left( V\right) ,\beta \in B\left( V\right) \) then \( \left. {{r}_{U, V}\left( {\alpha \beta }\right) = {r}_{U, V}\left( \alpha \right) {r}_{U, V}\left( \beta \right) }\right\rbrack \) . Then \( B \) is said to be an \( A \) -module. Occasionally, for reasons to be explained later, we refer to elements of \( A\left( U\right) \) as "sections of \( A \) over \( U \) ." If \( s \in A\left( V\right) \) and \( U \subset V \) then we use the notation \( s \mid U \) for \( {r}_{U, V}\left( s\right) \) and call it the "restriction of \( s \) to \( U \) ." Examples of presheaves are abundant in mathematics. For instance, if \( M \) is an abelian group, then there is the "constant presheaf" \( A \) with \( A\left( U\right) = M \) for all \( U \) and \( {r}_{U, V} = 1 \) for all \( U \subset V \) . We also have the presheaf \( B \) assigning to \( U \) the group (under pointwise addition) \( B\left( U\right) \) of all functions from \( U \) to \( M \), where \( {r}_{U, V} \) is the canonical restriction. If \( M \) is the group of real numbers, we also have the presheaf \( C \), with \( C\left( U\right) \) being the group of all continuous real-valued functions on \( U \) . Similarly, one has the presheaves of differentiable functions on (open subsets of) a differentiable manifold \( X \) ; of differential \( p \) -forms on \( X \) ; of vector fields on \( X \) ; and so on. In algebraic topology one has, for example, the presheaf of singular \( p \) -cochains of open subsets \( U \subset X \) ; the presheaf assigning to \( U \) its \( p \) th singular cohomology group; the presheaf assigning to \( U \) the \( p \) th singular chain group of \( X \) mod \( X - U \) ; and so on. It is often the case that a presheaf \( A \) on \( X \) will have a relatively simple structure "locally about a point \( x \in X \) ." To make precise what is meant by this, one introduces the notion of a "germ" of \( A \) at the point \( x \in X \) . Consider the set \( \mathfrak{M} \) of all elements \( s \in A\left( U\right) \) for all open sets \( U \subset X \) with \( x \in U \) . We say that the elements \( s \in A\left( U\right) \) and \( t \in A\left( V\right) \) of \( \mathfrak{M} \) are equivalent if there is a neighborhood \( W \subset U \cap V \) of \( x \) in \( X \) with \( {r}_{W, U}\left( s\right) = {r}_{W, V}\left( t\right) \) . The equivalence classes of \( \mathfrak{M} \) under this equivalence relation are called the germs of \( A \) at \( x \) . The equivalence class containing \( s \in A\left( U\right) \) is called the germ of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) . Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit \[ {\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( U\right) \] where \( U \) ranges over the open neighborhoods of \( x \) in \( X \) . The set \( {\mathcal{A}}_{x} \) inherits a canonical group structure from the groups \( A\left( U\right) \) . The disjoint union \( \mathcal{A} \) of the \( {\mathcal{A}}_{x} \) for \( x \in X \) provides information about the local structure of \( A \), but most global structure has been lost, since we have discarded all relationships between the \( {\mathcal{A}}_{x} \) for \( x \) varying. In order to retrieve some global structure, a topology is introduced into the set \( \mathcal{A} \) of germs of \( A \), as follows. Fix an element

1004_(GTM170)Sheaf Theory

m of \( s \) at \( x \in U \) . Thus, for example, one has the notion of the germ of a continuous real-valued function \( f \) at any point of the domain of \( f \) . Of course, the set \( {\mathcal{A}}_{x} \) of germs of \( A \) at \( x \) that we have constructed is none other than the direct limit \[ {\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( U\right) \] where \( U \) ranges over the open neighborhoods of \( x \) in \( X \) . The set \( {\mathcal{A}}_{x} \) inherits a canonical group structure from the groups \( A\left( U\right) \) . The disjoint union \( \mathcal{A} \) of the \( {\mathcal{A}}_{x} \) for \( x \in X \) provides information about the local structure of \( A \), but most global structure has been lost, since we have discarded all relationships between the \( {\mathcal{A}}_{x} \) for \( x \) varying. In order to retrieve some global structure, a topology is introduced into the set \( \mathcal{A} \) of germs of \( A \), as follows. Fix an element \( s \in A\left( U\right) \) . Then for each \( x \in U \) we have the germ \( {s}_{x} \) of \( s \) at \( x \) . For \( s \) fixed, the set of all germs \( {s}_{x} \in {\mathcal{A}}_{x} \) for \( x \in U \) is taken to be an open set in \( \mathcal{A} \) . The topology of \( \mathcal{A} \) is taken to be the topology generated by these open sets. (We shall describe this more precisely later in this section.) With this topology, \( \mathcal{A} \) is called "the sheaf generated by the presheaf \( A \) " or "the sheaf of germs of \( A \) ," and we denote this by \[ \mathcal{A} = \text{ Sheaf }\left( A\right) \;\text{ or }\;\mathcal{A} = \text{ Sheaf }\left( {U \mapsto A\left( U\right) }\right) . \] In general, the topology of \( \mathcal{A} \) is highly non-Hausdorff. There is a natural map \( \pi : \mathcal{A} \rightarrow X \) taking \( {\mathcal{A}}_{x} \) into the point \( x \) . It will be verified later in this section that \( \pi \) is a local homeomorphism. That is, each point \( t \in \mathcal{A} \) has a neighborhood \( N \) such that the restriction \( \pi \mid N \) is a homeomorphism onto a neighborhood of \( \pi \left( t\right) \) . (The set \( \left\{ {{s}_{x} \mid x \in U}\right\} \) for \( s \in A\left( U\right) \) is such a set \( N \) .) Also it is the case that in a certain natural sense, the group operations in \( {\mathcal{A}}_{x} \), for \( x \) varying, are continuous in \( x \) . These facts lead us to the basic definition of a sheaf on \( X \) : 1.2. Definition. A "sheaf" (of abelian groups) on \( X \) is a pair \( \left( {\mathcal{A},\pi }\right) \) where: (i) \( \mathcal{A} \) is a topological space (not Hausdorff in general); (ii) \( \pi : \mathcal{A} \rightarrow X \) is a local homeomorphism onto \( X \) ; (iii) each \( {\mathcal{A}}_{x} = {\pi }^{-1}\left( x\right) \), for \( x \in X \), is an abelian group (and is called the "stalk" of \( \mathcal{A} \) at \( x \) ); (iv) the group operations are continuous. (In practice, we always regard the map \( \pi \) as being understood and we speak of the sheaf \( \mathcal{A} \) .) The meaning of (iv) is as follows: Let \( \mathcal{A}\mathcal{A} \) be the subspace of \( \mathcal{A} \times \mathcal{A} \) consisting of those pairs \( \langle \alpha ,\beta \rangle \) with \( \pi \left( \alpha \right) = \) \( \pi \left( \beta \right) \) . Then the function \( \mathcal{A}\Delta \mathcal{A} \rightarrow \mathcal{A} \) taking \( \langle \alpha ,\beta \rangle \mapsto \alpha - \beta \) is continuous. [Equivalently, \( \alpha \mapsto - \alpha \) of \( \mathcal{A} \rightarrow \mathcal{A} \) is continuous and \( \langle \alpha ,\beta \rangle \mapsto \alpha + \beta \) of \( \mathcal{A}\Delta \mathcal{A} \rightarrow \mathcal{A} \) is continuous.] Similarly one may define, for example, a sheaf of rings or a module (sheaf of modules) over a sheaf of rings. Thus, for a sheaf \( \mathcal{R} \) of abelian groups to be a sheaf of rings, each stalk is assumed to have the (given) structure of a ring, and the map \( \langle \alpha ,\beta \rangle \mapsto {\alpha \beta } \) of \( \mathcal{R}\bigtriangleup \mathcal{R} \rightarrow \mathcal{R} \) is assumed to be continuous (in addition to (iv)). By a sheaf of rings with unit we mean a sheaf of rings in which each stalk has a unit and the assignment to each \( x \in X \) of the unit \( {1}_{x} \in {\mathcal{R}}_{x} \) is continuous. \( {}^{1} \) If \( \mathcal{R} \) is a sheaf of rings and if \( \mathcal{A} \) is a sheaf in which each stalk \( {\mathcal{A}}_{x} \) has a given \( {\mathcal{R}}_{x} \) -module structure, then \( \mathcal{A} \) is called an \( \mathcal{R} \) -module (or a module over \( \mathcal{R} \) ) if the map \( \mathcal{R}\bigtriangleup \mathcal{A} \rightarrow \mathcal{A} \) given by \( \langle \rho ,\alpha \rangle \mapsto {\rho \alpha } \) is continuous, where, of course, \( \mathcal{R}\bigtriangleup \mathcal{A} = \{ \langle \rho ,\alpha \rangle \in \mathcal{R} \times \mathcal{A} \mid \pi \left( \rho \right) = \pi \left( \alpha \right) \} \) . \( {}^{1} \) Example 1.13 shows that this latter condition is not superfluous. For example, the sheaf \( {\Omega }^{0} \) of germs of smooth real-valued functions on a differentiable manifold \( {M}^{n} \) is a sheaf of rings with unit, and the sheaf \( {\Omega }^{p} \) of germs of differential \( p \) -forms on \( {M}^{n} \) is an \( {\Omega }^{0} \) -module; see Section 7. If \( \mathcal{A} \) is a sheaf on \( X \) with projection \( \pi : \mathcal{A} \rightarrow X \) and if \( Y \subset X \), then the restriction \( \mathcal{A} \mid Y \) of \( \mathcal{A} \) is defined to be \[ \mathcal{A} \mid Y = {\pi }^{-1}\left( Y\right) \] which is a sheaf on \( Y \) . If \( \mathcal{A} \) is a sheaf on \( X \) and if \( Y \subset X \), then a section (or cross section) of \( \mathcal{A} \) over \( Y \) is a map \( s : Y \rightarrow \mathcal{A} \) such that \( \pi \circ s \) is the identity. Clearly the pointwise sum or difference (or product in a sheaf of rings, and so on) of two sections over \( Y \) is a section over \( Y \) . Every point \( x \in Y \) admits a section \( s \) over some neighborhood \( U \) of \( x \) by (ii). It follows that \( s - s \) is a section over \( U \) taking the value 0 in each stalk. This shows that the zero section \( 0 : X \rightarrow \mathcal{A} \), is indeed a section. It follows that for any \( Y \subset X \), the set \( \mathcal{A}\left( Y\right) \) of sections over \( Y \) forms an abelian group. Similarly, \( \mathcal{R}\left( Y\right) \) is a ring (with unit) if \( \mathcal{R} \) is a sheaf of rings (with unit), and moreover, \( \mathcal{A}\left( Y\right) \) is an \( \mathcal{R}\left( Y\right) \) -module if \( \mathcal{A} \) is an \( \mathcal{R} \) -module. Clearly, the restriction \( \mathcal{A}\left( Y\right) \rightarrow \mathcal{A}\left( {Y}^{\prime }\right) \), for \( {Y}^{\prime } \subset Y \), is a homomorphism. Thus, in particular, the assignment \( U \mapsto \mathcal{A}\left( U\right) \), for open sets \( U \subset X \), defines a presheaf on \( X \) . This presheaf is called the presheaf of sections of \( \mathcal{A} \) . Another common notation for the group of all sections of \( \mathcal{A} \) is \[ \Gamma \left( \mathcal{A}\right) = \mathcal{A}\left( X\right) \] See Section 6 for an elaboration on this notation. We shall now list some elementary consequences of Definition 1.2. The reader may supply any needed argument. (a) \( \pi \) is an open map. (b) Any section of \( \mathcal{A} \) over an open set is an open map. (c) Any element of \( \mathcal{A} \) is in the range of some section over some open set. (d) The set of all images of sections over open sets is a base for the topology of \( \mathcal{A} \) . (e) For any two sections \( s \in \mathcal{A}\left( U\right) \) and \( t \in \mathcal{A}\left( V\right), U \) and \( V \) open, the set \( W \) of points \( x \in U \cap V \) such that \( s\left( x\right) = t\left( x\right) \) is open. Note that if \( \mathcal{A} \) were Hausdorff then the set \( W \) of (e) would also be closed in \( U \cap V \) . That is generally false for sheaves. Thus (e) indicates the "strangeness" of the topology of \( \mathcal{A} \) . It is a consequence of part (ii) of Definition 1.2. 1.3. Example. A simple example of a non-Hausdorff sheaf is the sheaf on the real line that has zero stalk everywhere but at 0, and has stalk \( {\mathbb{Z}}_{2} \) at 0 . There is only one topology consistent with Definition 1.2, and the two points in the stalk at 0 cannot be separated by open sets (sections over open sets in \( \mathbb{R} \) ). As a topological space, this is the standard example of a non-Hausdorff 1-manifold. 1.4. Example. Perhaps a more illuminating and more important example of a non-Hausdorff sheaf is the sheaf \( \mathcal{C} \) of germs of continuous real-valued functions on \( \mathbb{R} \) . The function \( f\left( x\right) = x \) for \( x \geq 0 \) and \( f\left( x\right) = 0 \) for \( x \leq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mathcal{C} \) . The sheaf of germs of differentiable functions gives a similar example, but the sheaf of germs of real analytic functions is Hausdorff. 1.5. We now describe more precisely the construction of the sheaf generated by a given presheaf. Let \( A \) be a presheaf on \( X \) . For each open set \( U \subset X \) consider the space \( U \times A\left( U\right) \), where \( U \) has the subspace topology and \( A\left( U\right) \) has the discrete topology. Form the topological sum \[ E = \mathop{+}\limits_{{U \subset X}}\left( {U \times A\left( U\right) }\right) \] Consider the following equivalence relation \( R \) on \( E \) : If \( \langle x, s\rangle \in U \

1004_(GTM170)Sheaf Theory

eq 0 \) has a germ \( {f}_{0} \) at \( 0 \in \mathbb{R} \) that does not equal the germ \( {0}_{0} \) of the zero function, but a section through \( {f}_{0} \) takes value 0 in the stalk at \( x \) for all \( x < 0 \) sufficiently near 0 . Thus \( {f}_{0} \) and \( {0}_{0} \) cannot be separated by open sets in \( \mathcal{C} \) . The sheaf of germs of differentiable functions gives a similar example, but the sheaf of germs of real analytic functions is Hausdorff. 1.5. We now describe more precisely the construction of the sheaf generated by a given presheaf. Let \( A \) be a presheaf on \( X \) . For each open set \( U \subset X \) consider the space \( U \times A\left( U\right) \), where \( U \) has the subspace topology and \( A\left( U\right) \) has the discrete topology. Form the topological sum \[ E = \mathop{+}\limits_{{U \subset X}}\left( {U \times A\left( U\right) }\right) \] Consider the following equivalence relation \( R \) on \( E \) : If \( \langle x, s\rangle \in U \times A\left( U\right) \) and \( \langle y, t\rangle \in V \times A\left( V\right) \) then \( \langle x, s\rangle R\langle y, t\rangle \Leftrightarrow (x = y \) and there exists an open neighborhood \( W \) of \( x \) with \( W \subset U \cap V \) and \( s\left| {W = t}\right| W \) ). Let \( \mathcal{A} \) be the quotient space \( E/R \) and let \( \pi : \mathcal{A} \rightarrow X \) be the projection induced by the map \( p : E \rightarrow X \) taking \( \langle x, s\rangle \mapsto x \) . We have the commutative diagram  Recall that the topology of \( \mathcal{A} = E/R \) is defined by: \( Y \subset \mathcal{A} \) is open \( \Leftrightarrow {q}^{-1}\left( Y\right) \) is open in \( E \) . Note also that for any open subset \( {E}^{\prime } \) of \( E \), the saturation \( R\left( {E}^{\prime }\right) = {q}^{-1}q\left( {E}^{\prime }\right) \) of \( {E}^{\prime } \) is open. Thus \( q \) is an open map. Now \( \pi \) is continuous, since \( p \) is open and \( q \) is continuous; \( \pi \) is locally one-to-one, since \( p \) is locally one-to-one and \( q \) is onto. Thus \( \pi \) is a local homeomorphism. Clearly \( {\mathcal{A}}_{x} = {\pi }^{-1}\left( x\right) \) is the direct limit of \( A\left( U\right) \) for \( U \) ranging over the open neighborhoods of \( x \) . Thus the stalk \( {\mathcal{A}}_{x} \) has a canonical group structure. It is easy to see that the group operations in \( \mathcal{A} \) are continuous since they are so in \( E \) . Therefore \( \mathcal{A} \) is a sheaf. \( \mathcal{A} \) is called the sheaf generated by the presheaf \( A \) . As we have noted, this is denoted by \( \mathcal{A} = \) Sheaf \( \left( A\right) \) or \( \mathcal{A} = \) Sheaf \( \left( {U \mapsto A\left( U\right) }\right) \) . 1.6. Let \( {\mathcal{A}}_{0} \) be a sheaf, \( A \) the presheaf of sections of \( {\mathcal{A}}_{0} \), and \( \mathcal{A} = \) Then \( f\left( A\right) \) . Any element of \( {\mathcal{A}}_{0} \) lying over \( x \in X \) has a local section about it, and this determines an element of \( \mathcal{A} \) over \( x \) . This gives a canonical function \( \lambda : {\mathcal{A}}_{0} \rightarrow \mathcal{A} \) . By the definition of the topology of \( \mathcal{A},\lambda \) is open and continuous. It is also bijective on each stalk, and hence globally. Therefore \( \lambda \) is a homeomorphism. It also preserves group operations. Thus \( {\mathcal{A}}_{0} \) and \( \mathcal{A} \) are essentially the same. For this reason we shall usually not distinguish between a sheaf and its presheaf of sections and shall denote them by the same symbol. 1.7. Let \( A \) be a presheaf and \( \mathcal{A} \) the sheaf that it generates. For any open set \( U \subset X \) there is a natural map \( {\theta }_{U} : A\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) (recall the construction of \( \mathcal{A} \) ) that is a homomorphism and commutes with restrictions (which is the meaning of "natural"). When is \( {\theta }_{U} \) an isomorphism for all \( U \) ? Recalling that \( {\mathcal{A}}_{x} = \mathop{\lim }\limits_{{x \in U}}A\left( U\right) \), it follows that an element \( s \in A\left( U\right) \) is in \( \operatorname{Ker}{\theta }_{U} \Leftrightarrow s \) is "locally trivial" (that is, for every \( x \in U \) there is a neighborhood \( V \) of \( x \) such that \( s \mid V = 0 \) ). Thus \( {\theta }_{U} \) is a monomorphism for all \( U \subset X \Leftrightarrow \) the following condition holds: (S1) If \( U = \mathop{\bigcup }\limits_{\alpha }{U}_{\alpha } \), with \( {U}_{\alpha } \) open in \( X \), and \( s, t \in A\left( U\right) \) are such that \[ s\left| {{U}_{\alpha } = t}\right| {U}_{\alpha }\text{for all}\alpha \text{, then}s = t{.}^{2} \] A presheaf satisfying condition (S1) is called a monopresheaf. Similarly, let \( t \in \mathcal{A}\left( U\right) \) . For each \( x \in U \) there is a neighborhood \( {U}_{x} \) of \( x \) and an element \( {s}_{x} \in A\left( {U}_{x}\right) \) with \( {\theta }_{{U}_{x}}\left( {s}_{x}\right) \left( x\right) = t\left( x\right) \) . Since \( \pi : \mathcal{A} \rightarrow X \) is a local homeomorphism, \( \theta \left( {s}_{x}\right) \) and \( t \) coincide in some neighborhood \( {V}_{x} \) of \( x \) . We may assume that \( {V}_{x} = {U}_{x} \) . Now \( \theta \left( {{s}_{x} \mid {U}_{x} \cap {U}_{y}}\right) = \theta \left( {{s}_{y} \mid {U}_{x} \cap {U}_{y}}\right) \) so that if (S1) holds, we obtain that \( {s}_{x}\left| {{U}_{x} \cap {U}_{y} = {s}_{y}}\right| {U}_{x} \cap {U}_{y} \) . If \( A \) were a presheaf of sections (of any map), then this condition would imply that the \( {s}_{x} \) are restrictions to \( {U}_{x} \) of a section \( s \in A\left( U\right) \) . Conversely, if there is an element \( s \in A\left( U\right) \) with \( s \mid {U}_{x} = {s}_{x} \) for all \( x \), then \( \theta \left( s\right) = t \) . We have shown that if (S1) holds, then \( {\theta }_{U} \) is surjective for all \( U \) (and hence is an isomorphism) \( \Leftrightarrow \) the following condition is satisfied: (S2) Let \( \left\{ {U}_{\alpha }\right\} \) be a collection of open sets in \( X \) and let \( U = \bigcup {U}_{\alpha } \) . If \( {s}_{\alpha } \in A\left( {U}_{\alpha }\right) \) are given such that \( {s}_{\alpha }\left| {{U}_{\alpha } \cap {U}_{\beta } = {s}_{\beta }}\right| {U}_{\alpha } \cap {U}_{\beta } \) for all \( \alpha ,\beta \) , then there exists an element \( s \in A\left( U\right) \) with \( s \mid {U}_{\alpha } = {s}_{\alpha } \) for all \( \alpha \) . A presheaf satisfying (S2) is called conjunctive. If it only satisfies (S2) for a particular collection \( \left\{ {U}_{\alpha }\right\} \), then it is said to be conjunctive for \( \left\{ {U}_{\alpha }\right\} \) . Thus, sheaves are in one-to-one correspondence with presheaves satisfying (S1) and (S2), that is, with conjunctive monopresheaves. For this \( {}^{2} \) Clearly, we could take \( t = 0 \) here, i.e., replace \( \left( {s, t}\right) \) with \( \left( {s - t,0}\right) \) . However, the condition is phrased so that it applies to presheaves of sets. reason it is common practice not to distinguish between sheaves and conjunctive monopresheaves. \( {}^{3} \) Note that with the notation \( {U}_{\alpha ,\beta } = {U}_{\alpha } \cap {U}_{\beta } \) ,(S1) and (S2) are equivalent to the hypothesis that the sequence \[ 0 \rightarrow A\left( U\right) \overset{f}{ \rightarrow }\mathop{\prod }\limits_{\alpha }A\left( {U}_{\alpha }\right) \overset{g}{ \rightarrow }\mathop{\prod }\limits_{{\langle \alpha ,\beta \rangle }}A\left( {U}_{\alpha ,\beta }\right) \] is exact, where \( f\left( s\right) = \mathop{\prod }\limits_{\alpha }\left( {s \mid {U}_{\alpha }}\right) \) and \[ g\left( {\mathop{\prod }\limits_{\alpha }{s}_{\alpha }}\right) = \mathop{\prod }\limits_{{\langle \alpha ,\beta \rangle }}\left( {{s}_{\alpha }\left| {{U}_{\alpha ,\beta } - {s}_{\beta }}\right| {U}_{\alpha ,\beta }}\right) \] where \( \langle \alpha ,\beta \rangle \) denotes ordered pairs of indices. 1.8. Definition. Let \( \mathcal{A} \) be a sheaf on \( X \) and let \( Y \subset X \) . Then \( \mathcal{A} \mid Y = \) \( {\pi }^{-1}\left( Y\right) \) is a sheaf on \( Y \) called the "restriction" of \( \mathcal{A} \) to \( Y \) . 1.9. Definition. Let \( G \) be an abelian group. The "constant" sheaf on \( X \) with stalk \( G \) is the sheaf \( X \times G \) (giving \( G \) the discrete topology). It is also denoted by \( G \) when the context indicates this as a sheaf. A sheaf \( \mathcal{A} \) on \( X \) is said to be "locally constant" if every point of \( X \) has a neighborhood \( U \) such that \( \mathcal{A} \mid U \) is constant. 1.10. Definition. If \( \mathcal{A} \) is a sheaf on \( X \) and \( s \in \mathcal{A}\left( X\right) \) is a section, then the "support" of \( s \) is defined to be the closed set \( \left| s\right| = \{ x \in X \mid s\left( x\right) \neq 0\} \) . The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 . 1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -manifold \( {M}^{n} \) . Using singular homology, this can be defined as the sheaf \( {\mathcal{O}}_{n} = {\mathcal{H}}_{\text{heaf }}\left( {U \mapsto {H}_{n}\left( {{M}^{n},{M}^{n} - U;\mathbb{Z}}\right) }\right) \) . It is easy to see that this is a locally constant sheaf with stalks \( \mathbb{Z} \) . It is constant if \( {M}^{n} \) is orientable. If \( {M}^{n} \) has a boundary then \( {\mathcal{O}}_{n} \) is no longer locally constant since its stalks are zero over points of the boundary. More generally, for any space \( X \) and index \( p \) there is the sheaf \( {\mathcal{H}}_{p}\left( X\right) = \) Then \( \ell \left( {U \mapsto {H}_{p}\left( {X, X - U;\mathbb{Z}}

1004_(GTM170)Sheaf Theory

ight| = \{ x \in X \mid s\left( x\right) \neq 0\} \) . The set \( \left| s\right| \) is closed since its complement is the set of points at which \( s \) coincides with the zero section, and that is open by item (e) on page 4 . 1.11. Example. An important example of a sheaf is the orientation sheaf on an \( n \) -manifold \( {M}^{n} \) . Using singular homology, this can be defined as the sheaf \( {\mathcal{O}}_{n} = {\mathcal{H}}_{\text{heaf }}\left( {U \mapsto {H}_{n}\left( {{M}^{n},{M}^{n} - U;\mathbb{Z}}\right) }\right) \) . It is easy to see that this is a locally constant sheaf with stalks \( \mathbb{Z} \) . It is constant if \( {M}^{n} \) is orientable. If \( {M}^{n} \) has a boundary then \( {\mathcal{O}}_{n} \) is no longer locally constant since its stalks are zero over points of the boundary. More generally, for any space \( X \) and index \( p \) there is the sheaf \( {\mathcal{H}}_{p}\left( X\right) = \) Then \( \ell \left( {U \mapsto {H}_{p}\left( {X, X - U;\mathbb{Z}}\right) }\right) \), which is called the " \( p \) -th local homology sheaf" of \( X \) . Generally, it has a rather complicated structure. The reader would benefit by studying it for some simple spaces. For example, the sheaf \( {\mathcal{H}}_{1}\left( \bot \right) \) has stalk \( \mathbb{Z} \oplus \mathbb{Z} \) at the triple point, stalks 0 at the three end points, and stalks \( \mathbb{Z} \) elsewhere. How do these stalks fit together? --- \( {}^{3} \) Indeed, in certain generalizations of the theory, Definition 1.2 is not available and the other notion is used. This will not be of concern to us in this book. --- 1.12. Example. Consider the presheaf \( P \) on the real line \( \mathbb{R} \) that assigns to an open set \( U \subset \mathbb{R} \), the group \( P\left( U\right) \) of all real-valued polynomial functions on \( U \) . Then \( P \) is a monopresheaf that is conjunctive for coverings of \( \mathbb{R} \) , but it is not conjunctive for arbitrary collections of open sets. For example, the element \( 1 \in P\left( \left( {0,1}\right) \right) \) (the constant function with value 1) and the element \( x \in P\left( \left( {2,3}\right) \right) \) do not come from any single polynomial on \( \left( {0,1}\right) \cup \) \( \left( {2,3}\right) \) . The sheaf \( \mathcal{P} = \mathcal{{Meaf}}\left( P\right) \) has for \( \mathcal{P}\left( U\right) \) the functions that are "locally polynomials"; e.g.,1 and \( x \), as before, do combine to give an element of \( \mathcal{P}\left( {\left( {0,1}\right) \cup \left( {2,3}\right) }\right) \) . Important examples of this type of behavior are given in Section 7 and Exercise 12. 1.13. Example. Consider the presheaf \( A \) on \( X = \left\lbrack {0,1}\right\rbrack \) with \( A\left( U\right) = \mathbb{Z} \) for all \( U \neq \varnothing \) and with \( {r}_{V, U} : A\left( U\right) \rightarrow A\left( V\right) \) equal to the identity if \( 0 \in V \) or if \( 0 \notin U \) but \( {r}_{V, U} = 0 \) if \( 0 \in U - V \) . Let \( \mathcal{A} = \mathcal{R} \) heaf \( \left( A\right) \) . Then \( {\mathcal{A}}_{x} \approx \mathbb{Z} \) for all \( x \) . However, any section over \( \lbrack 0,\varepsilon ) \) takes the value \( 0 \in {\mathcal{A}}_{x} \) for \( x \neq 0 \) , but can be arbitrary in \( {\mathcal{A}}_{0} \approx \mathbb{Z} \) for \( x = 0 \) . The restriction \( \mathcal{A}|(0,1\rbrack \) is constant. Thus \( \mathcal{A} \) is a sheaf of rings but not a sheaf of rings with unit. (In the notation of 2.6 and Section 5, \( \mathcal{A} \approx {\mathbb{Z}}_{\{ 0\} } \oplus {\mathbb{Z}}_{(0,1\rbrack } \) .) 1.14. Example. A sheaf can also be described as being generated by a "presheaf" defined only on a basis of open sets. For example, on the circle \( {\mathbb{S}}^{1} \), consider the basis \( \mathcal{B} \) consisting of open arcs \( U \) of \( {\mathbb{S}}^{1} \) . For \( U \in \mathcal{B} \) and for \( x, y \in U \) we write \( x > y \) if \( y \) is taken into \( x \) through \( U \) by a counterclockwise rotation. Fix a point \( {x}_{0} \in {\mathbb{S}}^{1} \) . For \( U \in \mathcal{B} \) let \( A\left( U\right) = \mathbb{Z} \) and for \( U, V \in \mathcal{B} \) with \( V \subset U \) let \( {r}_{V, U} = 1 \) if \( {x}_{0} \in V \) or if \( {x}_{0} \notin U \) (i.e., if \( {x}_{0} \) is in both \( U \) and \( V \) or in neither). If \( {x}_{0} \in U - V \) then let \( {r}_{V, U} = 1 \) if \( {x}_{0} > y \) for all \( y \in V \) , and \( {r}_{V, U} = n \) (multiplication by the integer \( n \) ) if \( {x}_{0} < y \) for all \( y \in V \) . This generates an interesting sheaf \( {\mathcal{A}}_{n} \) on \( {\mathbb{S}}^{1} \) . It can be described directly (and more easily) as the quotient space \( \left( {-1,1}\right) \times \mathbb{Z} \) modulo the identification \( \langle t, k\rangle \sim \langle t - 1,{nk}\rangle \) for \( 0 < t < 1 \), and with the projection \( \left\lbrack {\langle t, k\rangle }\right\rbrack \mapsto \left\lbrack t\right\rbrack \) to \( {\mathbb{S}}^{1} = \left( {-1,1}\right) /\{ t \sim \left( {t - 1}\right) \} \) . Note, in particular, the cases \( n = 0, - 1 \) . The sheaf \( {\mathcal{A}}_{n} \) is Hausdorff for \( n \neq 0 \) but not for \( n = 0 \) . ## 2 Homomorphisms, subsheaves, and quotient sheaves In this section we fix the base space \( X \) . A homomorphism of presheaves \( h \) : \( A \rightarrow B \) is a collection of homomorphisms \( {h}_{U} : A\left( U\right) \rightarrow B\left( U\right) \) commuting with restrictions. That is, \( h \) is a natural transformation of functors. A homomorphism of sheaves \( h : \mathcal{A} \rightarrow \mathcal{B} \) is a map such that \( h\left( {\mathcal{A}}_{x}\right) \subset \) \( {\mathcal{B}}_{x} \) for all \( x \in X \) and the restriction \( {h}_{x} : {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \) of \( h \) to stalks is a homomorphism for all \( x \) . A homomorphism of sheaves induces a homomorphism of the presheaves of sections in the obvious way. Conversely, let \( h : A \rightarrow B \) be a homomorphism of presheaves (not necessarily satisfying (S1) and (S2)). For each \( x \in X, h \) induces a homomorphism \( {h}_{x} : {\mathcal{A}}_{x} = \mathop{\lim }\limits_{{x \in U}}A\left( U\right) \rightarrow \mathop{\lim }\limits_{{x \in U}}B\left( U\right) = {\mathcal{B}}_{x} \) and therefore a function \( h : \mathcal{A} \rightarrow \mathcal{B} \) . If \( s \in A\left( U\right) \) then \( h \) maps the section \( \theta \left( s\right) \in \mathcal{A}\left( U\right) \) onto the section \( \theta \left( {h\left( s\right) }\right) \in \mathcal{B}\left( U\right) \) . Thus \( h \) is continuous (since the projections to \( U \) are local homeomorphisms and take this function to the identity map). The group of all homomorphisms \( \mathcal{A} \rightarrow \mathcal{B} \) is denoted by \( \operatorname{Hom}\left( {\mathcal{A},\mathcal{B}}\right) \) . 2.1. Definition. A "subsheaf" A of a sheaf B is an open subspace of B such that \( {\mathcal{A}}_{x} = \mathcal{A} \cap {\mathcal{B}}_{x} \) is a subgroup of \( {\mathcal{B}}_{x} \) for all \( x \in X \) . (That is, \( \mathcal{A} \) is a subspace of \( \mathcal{B} \) that is a sheaf on \( X \) with the induced algebraic structure.) If \( h : \mathcal{A} \rightarrow \mathcal{B} \) is a homomorphism of sheaves, then \[ \operatorname{Ker}h = \{ \alpha \in \mathcal{A} \mid h\left( \alpha \right) = 0\} \] is a subsheaf of \( \mathcal{A} \) and \( \operatorname{Im}h \) is a subsheaf of \( \mathcal{B} \) . We define exact sequences of sheaves as usual; that is, the sequence \( \mathcal{A}\overset{f}{ \rightarrow }\mathcal{B}\overset{g}{ \rightarrow }\mathcal{C} \) of sheaves is exact if \( \operatorname{Im}f = \operatorname{Ker}g \) . Note that such a sequence of sheaves is exact \( \Leftrightarrow \) each \( {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \rightarrow {\mathcal{C}}_{x} \) is exact. \( {}^{4} \) Since \( \mathop{\lim }\limits_{ \rightarrow } \) is an exact functor, it follows that the functor \( A \mapsto {\mathcal{{Rea}}}^{\prime }\left( A\right) \), from presheaves to sheaves, is exact. Let \( h : A\overset{f}{ \rightarrow }B\overset{g}{ \rightarrow }C \) be homomorphisms of presheaves. The induced sequence \( \mathcal{A}\overset{{f}^{\prime }}{ \rightarrow }\mathcal{B}\overset{{g}^{\prime }}{ \rightarrow }\mathcal{C} \) of generated sheaves will be exact if and only if \( \theta \circ g \circ f = 0 \) and the following condition holds: For each open \( U \subset X \) , \( x \in U \), and \( s \in B\left( U\right) \) such that \( g\left( s\right) = 0 \), there exists a neighborhood \( V \subset U \) of \( x \) such that \( s \mid V = f\left( t\right) \) for some \( t \in A\left( V\right) \) . This is an elementary fact resulting from properties of direct limits and from the fact that \( \mathcal{A} \rightarrow \) \( \mathcal{B} \rightarrow \mathcal{C} \) is exact \( \Leftrightarrow {\mathcal{A}}_{x} \rightarrow {\mathcal{B}}_{x} \rightarrow {\mathcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V}\right) }\right) = 0 \), i.e., that \( \left( {g \circ f}\right) \left( s\right) \) is "locally zero." 2.2. Proposition. If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of sheaves, then the induced sequence \[ 0 \rightarrow {\mathcal{A}}^{\prime }\left( Y\right) \rightarrow \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right) \] is exact for all \( Y \subset X \) . Proof. Since the restriction of this sequence to \( Y \) is still exact, it suffices to prove the statement in the case \( Y = X \) . The fact that the sequence of sections o

1004_(GTM170)Sheaf Theory

thcal{C}}_{x} \) is exact for all \( x \in X \) . It will be used repeatedly. Note that the condition \( \theta \circ g \circ f = 0 \) is equivalent to the statement that for each \( s \in A\left( U\right) \) and \( x \in U \), there is a neighborhood \( V \subset U \) of \( x \) such that \( g\left( {f\left( {s \mid V}\right) }\right) = 0 \), i.e., that \( \left( {g \circ f}\right) \left( s\right) \) is "locally zero." 2.2. Proposition. If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of sheaves, then the induced sequence \[ 0 \rightarrow {\mathcal{A}}^{\prime }\left( Y\right) \rightarrow \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right) \] is exact for all \( Y \subset X \) . Proof. Since the restriction of this sequence to \( Y \) is still exact, it suffices to prove the statement in the case \( Y = X \) . The fact that the sequence of sections over \( X \) has order two and the exactness at \( {\mathcal{A}}^{\prime }\left( X\right) \) are obvious \( {}^{4} \) Caution: an exact sequence of sheaves is not necessarily an exact sequence of pre-sheaves. See Proposition 2.2 and Example 2.3. Categorical readers might check that these definitions give notions equivalent to those based on the fact that sheaves and presheaves form abelian categories. (look at stalks). We can assume that \( {\mathcal{A}}^{\prime } \) is a subspace of \( \mathcal{A} \) . Then a section \( s \in \mathcal{A}\left( X\right) \) going to \( 0 \in {\mathcal{A}}^{\prime \prime }\left( X\right) \) must take values in the subspace \( {\mathcal{A}}^{\prime } \), as is seen by looking at stalks. But this just means that it comes from a section in \( {\mathcal{A}}^{\prime }\left( X\right) \) . 2.3. Example. This example shows that \( \mathcal{A}\left( Y\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( Y\right) \) need not be onto even if \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \) is onto. On the unit interval \( \mathbb{I} \) let \( \mathcal{A} \) be the sheaf \( \mathbb{I} \times {\mathbb{Z}}_{2} \) and \( {\mathcal{A}}^{\prime \prime } \) the sheaf with stalks \( {\mathbb{Z}}_{2} \) at \( \{ 0\} \) and \( \{ 1\} \) and zero otherwise. (There is only one possible topology in \( {\mathcal{A}}^{\prime \prime } \) .) The canonical map \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \) is onto (with kernel being the subsheaf \( {\mathcal{A}}^{\prime } = \left( {0,1}\right) \times {\mathbb{Z}}_{2} \cup \left\lbrack {0,1}\right\rbrack \times \{ 0\} \subset \mathbb{I} \times {\mathbb{Z}}_{2} \) ), but \( \mathcal{A}\left( \mathbb{I}\right) \approx {\mathbb{Z}}_{2} \) while \( {\mathcal{A}}^{\prime \prime }\left( \mathbb{I}\right) \approx {\mathbb{Z}}_{2} \oplus {\mathbb{Z}}_{2} \), so that \( \mathcal{A}\left( \mathbb{I}\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( \mathbb{I}\right) \) is not surjective. Also see Example 2.5 and Exercises 13, 14, and 15. 2.4. Definition. Let \( \mathcal{A} \) be a subsheaf of a sheaf \( \mathcal{B} \) . The "quotient sheaf" \( \mathcal{B}/\mathcal{A} \) is defined to be \[ \mathcal{B}/\mathcal{A} = \mathcal{P}\text{ heaf }\left( {U \mapsto \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) }\right) . \] The exact sequence of presheaves \[ 0 \rightarrow \mathcal{A}\left( U\right) \rightarrow \mathcal{B}\left( U\right) \rightarrow \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \rightarrow 0 \] (1) induces a sequence \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \) . On the stalks at \( x \) this is the direct limit of the sequences (1) for \( U \) ranging over the open neighborhoods of \( x \) . This sequence of stalks is exact since direct limits preserve exactness. Therefore, \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \) is exact. \( {}^{5} \) Suppose that \( 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \) is an exact sequence of sheaves. We may regard \( \mathcal{A} \) as a subsheaf of \( \mathcal{B} \) . The exact sequence \[ 0 \rightarrow \mathcal{A}\left( U\right) \rightarrow \mathcal{B}\left( U\right) \rightarrow \mathcal{C}\left( U\right) \] provides a monomorphism \( \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \rightarrow \mathcal{C}\left( U\right) \) of presheaves and hence a homomorphism of sheaves \( \mathcal{B}/\mathcal{A} \rightarrow \mathcal{C} \), and the diagram \[ \begin{array}{l} 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{B}/\mathcal{A} \rightarrow 0 \\ 0 \rightarrow \mathcal{A} \rightarrow \mathcal{B} \rightarrow \mathcal{C} \rightarrow 0 \end{array} \] commutes. It follows, by looking at stalks, that \( \mathcal{B}/\mathcal{A} \rightarrow \mathcal{C} \) is an isomorphism. 2.5. Example. Consider the sheaf \( \mathcal{C} \) of germs of continuous real-valued functions on \( X = {\mathbb{R}}^{2} - \{ 0\} \) . Let \( Z \) be the subsheaf of germs of locally constant functions with values the integer multiples of \( {2\pi } \) . Then \( Z \) can be regarded as a subsheaf of \( \mathcal{C} \) . (Note that \( Z \) is a constant sheaf.) The polar angle \( \theta \) is locally defined (ambiguously) as a section of \( \mathcal{C} \), but it \( {}^{5} \) Note, however, that \( \left( {\mathcal{B}/\mathcal{A}}\right) \left( U\right) \neq \mathcal{B}\left( U\right) /\mathcal{A}\left( U\right) \) in general. is not a global section. It does define (unambiguously) a section of the quotient sheaf \( \mathcal{C}/Z \) . This gives another example of an exact sequence \( 0 \rightarrow Z \rightarrow \mathcal{C} \rightarrow \mathcal{C}/Z \rightarrow 0 \) of sheaves for which the sequence of sections is not right exact. Note that \( \mathcal{C}/Z \) can be interpreted as the sheaf of germs of continuous functions on \( X \) with values in the circle group \( {\mathbb{S}}^{1} \) . Note also that \( Z\left( X\right) \) is the group of constant functions on \( X \) with values in \( {2\pi }\mathbb{Z} \) and hence is isomorphic to \( \mathbb{Z};\mathcal{C}\left( X\right) \) is the group of continuous real valued functions \( X \rightarrow \mathbb{R} \) ; and \( \left( {\mathcal{C}/Z}\right) \left( X\right) \) is the group of continuous functions \( X \rightarrow {\mathbb{S}}^{1} \) . The sequence \[ 0 \rightarrow Z\left( X\right) \rightarrow \mathcal{C}\left( X\right) \overset{j}{ \rightarrow }\left( {\mathcal{C}/Z}\right) \left( X\right) \xrightarrow[]{\text{ deg }}\mathbb{Z} \rightarrow 0 \] is exact by covering space theory, and so \( \operatorname{Coker}j \approx \mathbb{Z} \) . 2.6. Let \( A \) be a locally closed subspace of \( X \) and let \( \mathcal{B} \) be a sheaf on \( A \) . It is easily seen (since \( A \) is locally closed) that there is a unique topology on the point set \( \mathcal{B} \cup \left( {X\times \{ 0\} }\right) \) such that \( \mathcal{B} \) is a subspace and the projection onto \( X \) is a local homeomorphism (we identify \( A \times \{ 0\} \) with the zero section of \( \mathcal{B} \) ). With this topology and the canonical algebraic structure, \( \mathcal{B} \cup \left( {X\times \{ 0\} }\right) \) is a sheaf on \( X \) denoted by \[ {\mathcal{B}}^{X} = \mathcal{B} \cup \left( {X\times \{ 0\} }\right) \] Thus \( {\mathcal{B}}^{X} \) is the unique sheaf on \( X \) inducing \( \mathcal{B} \) on \( A \) and 0 on \( X - A \) . Clearly, \( \mathcal{B} \mapsto {\mathcal{B}}^{X} \) is an exact functor. The sheaf \( {\mathcal{B}}^{X} \) is called the extension of \( \mathcal{B} \) by zero. Now let \( \mathcal{A} \) be a sheaf on \( X \) and let \( A \subset X \) be locally closed. We define \[ {\mathcal{A}}_{A} = {\left( \mathcal{A} \mid A\right) }^{X} \] For \( U \subset X \) open, \( {\mathcal{A}}_{U} \) is the subsheaf \( {\pi }^{-1}\left( U\right) \cup \left( {X\times \{ 0\} }\right) \) of \( \mathcal{A} \), while for \( F \subset X \) closed \( {\mathcal{A}}_{F} \) is the quotient sheaf \( {\mathcal{A}}_{F} = \mathcal{A}/{\mathcal{A}}_{X - F} \) . If \( A = U \cap F \) , then \( {\mathcal{A}}_{A} = {\left( {\mathcal{A}}_{U}\right) }_{F} = {\left( {\mathcal{A}}_{F}\right) }_{U}{}^{6} \) In this notation, the sheaf \( {\mathcal{A}}^{\prime } \) of 2.3 is \( {\mathcal{A}}_{\left( 0,1\right) } \), and \( {\mathcal{A}}^{\prime \prime } \approx {\mathcal{A}}_{\{ 0,1\} } \) . 2.7. Example. Let \( {U}_{i} \) be the open disk of radius \( 1 - {2}^{-i} \) in \( X = {\mathbb{D}}^{n} \), the unit disk in \( {\mathbb{R}}^{n} \) . Put \( {A}_{1} = {U}_{1} \) and \( {A}_{i} = {U}_{i} - {U}_{i - 1} \) for \( i > 1 \) . Note that for \( i > 1,{A}_{i} \approx {\mathbb{S}}^{n - 1} \times (0,1\rbrack \) is locally closed in \( X \) . Using the notation of 2.6, put \[ {\mathcal{L}}_{1} = {\mathbb{Z}}_{{U}_{1}} \] \[ {\mathcal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}} \] \[ {\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}} \] \[ \text{...} \] \( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal{L}}_{i} \subset \mathbb{Z} \) . Then the stalks of \( \mathcal{L} \) are 0 on \( \partial {\mathbb{D}}^{n} \) and are \( {2}^{i}\mathbb{Z} \) on \( {A}_{i} \) . This is an example of a fairly complicated subsheaf of the constant sheaf \( \mathbb{Z} \) on \( {\mathbb{D}}^{n} \) even in the case \( n = 1 \) . It is a counterexample to \( \lbrack {40} \), Remark II-2.9.3]. ## 3 Direct and inverse images Let \( f : X \rightarrow Y \) be a map and let \( \mathcal{A} \) be a sheaf on \( X \) . The presheaf \( U \mapsto \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) on \( Y \) clearly satisfies (S1) and (S2) and hence is a sheaf. This sheaf on \( Y \) is denoted by \(

1004_(GTM170)Sheaf Theory

cal{L}}_{2} = {\mathcal{L}}_{1} \cup 2{\mathbb{Z}}_{{U}_{2}} \] \[ {\mathcal{L}}_{3} = {\mathcal{L}}_{2} \cup 4{\mathbb{Z}}_{{U}_{3}} \] \[ \text{...} \] \( {}^{6} \) Note that any locally closed subspace is the intersection of an open subspace with a closed subspace; see [19]. Let \( \mathcal{L} = \bigcup {\mathcal{L}}_{i} \subset \mathbb{Z} \) . Then the stalks of \( \mathcal{L} \) are 0 on \( \partial {\mathbb{D}}^{n} \) and are \( {2}^{i}\mathbb{Z} \) on \( {A}_{i} \) . This is an example of a fairly complicated subsheaf of the constant sheaf \( \mathbb{Z} \) on \( {\mathbb{D}}^{n} \) even in the case \( n = 1 \) . It is a counterexample to \( \lbrack {40} \), Remark II-2.9.3]. ## 3 Direct and inverse images Let \( f : X \rightarrow Y \) be a map and let \( \mathcal{A} \) be a sheaf on \( X \) . The presheaf \( U \mapsto \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) on \( Y \) clearly satisfies (S1) and (S2) and hence is a sheaf. This sheaf on \( Y \) is denoted by \( {f}_{\mathcal{A}} \) and is called the direct image of \( \mathcal{A}.{}^{7} \) Thus we have \[ f\mathcal{A}\left( U\right) = \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \] (2) By 2.2 it is clear that \( \mathcal{A} \mapsto f\mathcal{A} \) is a left exact covariant functor. The direct image is not generally right exact, and in fact, the theory of sheaves is largely concerned with the right derived functors of the direct image functor. For the map \( \varepsilon : X \rightarrow \star \), where \( \star \) is the one point space, the direct image \( \varepsilon \mathcal{A} \) is just the group \( \Gamma \left( \mathcal{A}\right) = \mathcal{A}\left( X\right) \) (regarded as a sheaf on \( \star \) ). Consequently, the direct image functor \( \mathcal{A} \mapsto f\mathcal{A} \) is a generalization of the global section functor \( \Gamma \) . Now let \( \mathcal{B} \) be a sheaf on \( Y \) . The inverse image \( {f}^{ * }\mathcal{B} \) of \( \mathcal{B} \) is the sheaf on \( X \) defined by \[ {f}^{ * }\mathcal{B} = \{ \langle x, b\rangle \in X \times \mathcal{B} \mid f\left( x\right) = \pi \left( b\right) \} \] where \( \pi : \mathcal{B} \rightarrow Y \) is the canonical projection. The projection \( {f}^{ * }\mathcal{B} \rightarrow Y \) is given by \( \langle x, b\rangle \mapsto x \) . To check that \( {f}^{ * }\mathcal{B} \) is indeed a sheaf, we note that if \( U \subset Y \) is an open neighborhood of \( f\left( x\right) \) and \( s : U \rightarrow \mathcal{B} \) is a section of \( \mathcal{B} \) with \( s\left( {f\left( x\right) }\right) = b \), then the neighborhood \( \left( {{f}^{-1}\left( U\right) \times s\left( U\right) }\right) \cap {f}^{ * }\mathcal{B} \) of \( \langle x, b\rangle \in {f}^{ * }\mathcal{B} \) is precisely \( \left\{ {\left\langle {{x}^{\prime },{sf}\left( {x}^{\prime }\right) }\right\rangle \mid {x}^{\prime } \in {f}^{-1}\left( U\right) }\right\} \) and hence maps homeomorphically onto \( {f}^{-1}\left( U\right) \) . The group structure on \( {\left( {f}^{ * }\mathcal{B}\right) }_{x} \) is defined so that the one-to-one correspondence \[ {f}_{x}^{ * } : {\mathcal{B}}_{f\left( x\right) }\overset{ \approx }{ \rightarrow }{\left( {f}^{ * }\mathcal{B}\right) }_{x} \] (3) defined by \( {f}_{x}^{ * }\left( b\right) = \langle x, b\rangle \), is an isomorphism. It is easy to check that the group operations are continuous. We have already remarked that if \( s : U \rightarrow \mathcal{B} \) is a section, then \( x \mapsto \) \( \langle x, s\left( {f\left( x\right) }\right) \rangle = {f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( {f}^{ * }\mathcal{B} \) over \( {f}^{-1}\left( U\right) \) . Thus we have the canonical homomorphism \[ {f}_{U}^{ * } : \mathcal{B}\left( U\right) \rightarrow \left( {{f}^{ * }\mathcal{B}}\right) \left( {{f}^{-1}\left( U\right) }\right) \] (4) defined by \( {f}_{U}^{ * }\left( s\right) \left( x\right) = {f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) \) . --- \( {}^{7} \) For a generalization of the direct image see IV-3. --- From (3) it follows that \( {f}^{ * } \) is an exact functor. Note that for an inclusion \( i : X \hookrightarrow Y \) and a sheaf \( \mathcal{B} \) on \( Y \), we have \( \mathcal{B} \mid X \approx {i}^{ * }\mathcal{B} \), as the reader is asked to detail in Exercise 1. 3.1. Example. Consider the constant sheaf \( \mathbb{Z} \) on \( X = \left\lbrack {0,1}\right\rbrack \) and its restriction \( \mathcal{L} = \mathbb{Z} \mid \left( {0,1}\right) \) . Let \( i : \left( {0,1}\right) \hookrightarrow X \) be the inclusion. Then \( i\mathcal{L} \approx \mathbb{Z} \), because for \( U \) a small open interval about 0 or 1, we have that \( i\mathcal{L}\left( U\right) = \mathcal{L}\left( {U \cap \left( {0,1}\right) }\right) \approx \mathbb{Z} \) . Also, \( {\mathcal{L}}^{X} = {\mathbb{Z}}_{\left( 0,1\right) } \) . Therefore, \( i\mathcal{L} ≉ {\mathcal{L}}^{X} \) in general. However, for an inclusion \( i : F \hookrightarrow X \) of a closed subspace and for any sheaf \( \mathcal{L} \) on \( F \), it is true that \( i\mathcal{L} \approx {\mathcal{L}}^{X} \), as the reader can verify. (This is essentially Exercise 2.) 3.2. Example. Consider the constant sheaf \( \mathcal{A} \) with stalks \( \mathbb{Z} \) on \( \mathbb{R} - \{ 0\} \) and let \( i : \mathbb{R} - \{ 0\} \hookrightarrow \mathbb{R} \) . Then \( i\mathcal{A} \) has stalk \( \mathbb{Z} \oplus \mathbb{Z} \) at 0 and stalk \( \mathbb{Z} \) elsewhere, because, for example, \( i\mathcal{A}\left( {-\varepsilon ,\varepsilon }\right) = \mathcal{A}\left( {\left( {-\varepsilon ,0}\right) \cup \left( {0,\varepsilon }\right) }\right) \approx \mathbb{Z} \oplus \mathbb{Z} \) . A local section over a connected neighborhood of 0 taking value \( \left( {n, m}\right) \in {\left( i\mathcal{A}\right) }_{0} \) at 0 is \( n \in {\left( i\mathcal{A}\right) }_{x} \) for \( x < 0 \) and is \( m \in {\left( i\mathcal{A}\right) }_{x} \) for \( x > 0 \) . 3.3. Example. Consider the constant sheaf \( \mathcal{A} \) with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) ; let \( Y = \left\lbrack {-1,1}\right\rbrack \) and let \( \pi : X \rightarrow Y \) be the projection. Then \( \pi \mathcal{A} \) has stalks \( \mathbb{Z} \) at -1 and at 1 but has stalks \( {\left( \pi \mathcal{A}\right) }_{x} \approx \mathbb{Z} \oplus \mathbb{Z} \) for \( - 1 < x < 1 \) . The reader should try to understand the topology connecting these stalks. For example, is it true that \( \pi \mathcal{A} \approx \mathbb{Z} \oplus {\mathbb{Z}}_{\left( -1,1\right) } \), as defined in Section 5? Is there a sheaf on \( Y \) that is "locally isomorphic" to \( \pi \mathcal{A} \) but not isomorphic to it? 3.4. Example. Let \( Z \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) and let \( {Z}^{t} \) denote the "twisted" sheaf with stalks \( \mathbb{Z} \) on \( X \) (i.e., \( {Z}^{t} = \left\lbrack {0,1}\right\rbrack \times \mathbb{Z} \) modulo the identifications \( \left( {0 \times n}\right) \sim \left( {1 \times - n}\right) ) \) . Let \( f : X \rightarrow X \) be the covering map of degree 2 . Then \( {fZ} \) is the sheaf on \( X \) with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) twisted by the exchange of basis elements in the stalks. Also, \( {f}^{ * }{Z}^{t} \approx Z \) since it is the locally constant sheaf with stalks \( \mathbb{Z} \) on \( X \) twisted twice, which is no twist at all. Note that \( {fZ} \) has both \( Z \) and \( {Z}^{t} \) as subsheaves. The corresponding quotient sheaves are \( \left( {fZ}\right) /Z \approx {Z}^{t} \) and \( \left( {fZ}\right) /{Z}^{t} \approx Z \) . However, \( {fZ} ≉ Z \oplus {Z}^{t} \) (defined in Section 5). 3.5. Example. Let \( X \) and \( Z \) be as in Example 3.4 but let \( f : X \rightarrow X \) be the covering map of degree 3 . Then \( {fZ} \) is the locally constant sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} \) twisted by the cyclic permutation of factors. Thus \( {fZ} \) has the constant sheaf \( Z \) as a subsheaf (the "diagonal") with quotient sheaf the locally constant sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) twisted by the, essentially unique, nontrivial automorphism of period 3 . 3.6. Example. Let \( \pi : {\mathbb{S}}^{n} \rightarrow {\mathbb{{RP}}}^{n} \) be the canonical double covering. Then \( \pi \mathbb{Z} \) is a twisted sheaf with stalks \( \mathbb{Z} \oplus \mathbb{Z} \) on projective space, analogous to the sheaf \( {fZ} \) of Example 3.4. It contains the constant sheaf \( \mathbb{Z} \) as a subsheaf with quotient sheaf \( {Z}^{t} \), a twisted integer sheaf. If \( n \) is even, so that \( {\mathbb{{RP}}}^{n} \) is nonorientable, \( {Z}^{t} \) is just the orientation sheaf \( {\mathcal{O}}_{n} \) of Example 1.11. \( \diamond \) ## 4 Cohomomorphisms Throughout this section we let \( f : X \rightarrow Y \) be a given map. 4.1. Definition. If \( A \) and \( B \) are presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions. 4.2. Definition. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is a collection of homomorphisms \( {k}_{x} : {\mathcal{B}}_{f\left( x\right) } \rightarrow {\mathcal{A}}_{x} \) for each \( x \in X \) such that for any section \( s \in \mathcal{B}\left( U\right) \) the function \( x \mapsto {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( \mathcal{A} \) over \( {f}^{-1}\left( U\right) \) (i.e., this function is continuous). \( {}^{8} \) An \( f \) -cohomomorphism of sheaves induces an \( f \) -cohomomorphism of presheaves by putting \( {k}_{U}\left( s

1004_(GTM170)Sheaf Theory

presheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : B \rightsquigarrow A \) is a collection of homomorphisms \( {k}_{U} : B\left( U\right) \rightarrow A\left( {{f}^{-1}\left( U\right) }\right) \), for \( U \) open in \( Y \), compatible with restrictions. 4.2. Definition. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) and \( Y \) respectively, then an "f-cohomomorphism" \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is a collection of homomorphisms \( {k}_{x} : {\mathcal{B}}_{f\left( x\right) } \rightarrow {\mathcal{A}}_{x} \) for each \( x \in X \) such that for any section \( s \in \mathcal{B}\left( U\right) \) the function \( x \mapsto {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) is a section of \( \mathcal{A} \) over \( {f}^{-1}\left( U\right) \) (i.e., this function is continuous). \( {}^{8} \) An \( f \) -cohomomorphism of sheaves induces an \( f \) -cohomomorphism of presheaves by putting \( {k}_{U}\left( s\right) \left( x\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \) where \( U \subset Y \) is open and \( s \in \mathcal{B}\left( U\right) \) . Conversely, an \( f \) -cohomomorphism of presheaves \( k : B \rightsquigarrow A \) induces, for \( x \in X \), a homomorphism \[ {k}_{x} : {\mathcal{B}}_{f\left( x\right) } = \mathop{\lim }\limits_{ \rightarrow }B\left( U\right) \rightarrow \mathop{\lim }\limits_{ \rightarrow }A\left( {{f}^{-1}U}\right) \rightarrow {\mathcal{A}}_{x} \] [where \( U \) ranges over neighborhoods of \( f\left( x\right) \) ]. Then for \( \theta : B\left( U\right) \rightarrow \mathcal{B}\left( U\right) \) the canonical homomorphism and for \( s \in B\left( U\right) \), we have \[ \theta \left( {{k}_{U}\left( s\right) }\right) \left( x\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) \] so that \( \left\{ {k}_{x}\right\} \) is an \( f \) -cohomomorphism of sheaves \( \mathcal{B} \rightsquigarrow \mathcal{A} \) (generated by \( B \) and \( A \) ). For any sheaf \( \mathcal{B} \) on \( Y \), the collection \( {f}^{ * } = \left\{ {f}_{x}^{ * }\right\} \) of (3) defines an \( f \) - cohomomorphism \[ {f}^{ * } : \mathcal{B} ⤳ {f}^{ * }\mathcal{B} \] If \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) is any \( f \) -cohomomorphism, let \( {h}_{x} : {\left( {f}^{ * }\mathcal{B}\right) }_{x} \rightarrow {\mathcal{A}}_{x} \) be defined by \( {h}_{x} = {k}_{x} \circ {\left( {f}_{x}^{ * }\right) }^{-1} \) . Together, the homomorphisms \( {h}_{x} \) define a function \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) . For \( s \in \mathcal{B}\left( U\right) \), the equation \[ h\left( {{f}_{U}^{ * }\left( s\right) \left( x\right) }\right) = h\left( {{f}_{x}^{ * }\left( {s\left( {f\left( x\right) }\right) }\right) }\right) = {k}_{x}\left( {s\left( {f\left( x\right) }\right) }\right) , \] together with the fact that the \( {f}_{U}^{ * }\left( s\right) \) form a basis for the topology of \( {f}^{ * }\mathcal{B} \) , implies that \( h \) is continuous. Thus any \( f \) -cohomomorphism \( k \) admits a unique factorization \[ k : \mathcal{B}\overset{{f}^{ * }}{ \hookrightarrow }{f}^{ * }\mathcal{B}\overset{h}{ \rightarrow }\mathcal{A} \] \( h \) being a homomorphism. --- \( {}^{8} \) Note that an \( f \) -cohomomorphism \( \mathcal{B} ⤳ \mathcal{A} \) is not generally a function, since it is multiply valued unless \( f \) is one-to-one, and it is not defined everywhere unless \( f \) is onto. Of course, cohomomorphisms are the morphisms in the category of all sheaves on all spaces. --- Similarly, for any sheaf \( \mathcal{A} \) on \( X \), the definition (2) provides an \( f \) - cohomomorphism \( f : f\mathcal{A} \rightarrow \mathcal{A} \) . Since \( {f}_{U} : f\mathcal{A}\left( U\right) \rightarrow \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) \) is an isomorphism, it is clear that any \( f \) -cohomomorphism \( k \) admits a unique factorization \[ k : \mathcal{B}\overset{j}{ \rightarrow }f\mathcal{A}\overset{f}{ \rightarrow }\mathcal{A} \] (i.e., \( {k}_{U} = {f}_{U}{j}_{U} \) ), where \( j \) is a homomorphism. Thus to each \( f \) -cohomomorphism \( k \) there correspond unique homomorphisms \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) and \( j : \mathcal{B} \rightarrow f\mathcal{A} \) . This correspondence is additive and natural in \( \mathcal{A} \) and \( \mathcal{B} \) . Therefore, denoting the group of all \( f \) - cohomomorphisms from \( \mathcal{B} \) to \( \mathcal{A} \) by \( f \) -cohom \( \left( {\mathcal{B},\mathcal{A}}\right) \), we have produced the following natural isomorphisms of functors: \[ \operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \approx f\text{-cohom}\left( {\mathcal{B},\mathcal{A}}\right) \approx \operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right) . \] Leaving out the middle term, we shall let \( \varphi \) denote this natural isomorphism \[ \varphi : \operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \overset{ \approx }{ \rightarrow }\operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right) \] (5) of functors. \( {}^{9} \) Taking \( \mathcal{A} = {f}^{ * }\mathcal{B} \), we obtain the homomorphism \[ \beta = \varphi \left( 1\right) : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B} \] (6) and taking \( \mathcal{B} = f\mathcal{A} \), we obtain the homomorphism \[ \alpha = {\varphi }^{-1}\left( 1\right) : {f}^{ * }f\mathcal{A} \rightarrow \mathcal{A}. \] (7) If \( h : {f}^{ * }\mathcal{B} \rightarrow \mathcal{A} \) is any homomorphism, then the naturality of \( \varphi \) implies that the diagram \[ \mathrm{{Hom}}\left( {{f}^{ * }\mathcal{B},{f}^{ * }\mathcal{B}}\right) \;\overset{\varphi }{ \rightarrow }\;\mathrm{{Hom}}\left( {\mathcal{B}, f{f}^{ * }\mathcal{B}}\right) \] \[ \operatorname{Hom}\left( {{f}^{ * }\mathcal{B}, h}\right) \downarrow \operatorname{Hom}\left( {\mathcal{B}, f\left( h\right) }\right) \] \[ \operatorname{Hom}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \;\overset{\varphi }{ \rightarrow }\;\operatorname{Hom}\left( {\mathcal{B}, f\mathcal{A}}\right) \] commutes. That is, \[ \varphi \left( h\right) = f\left( h\right) \circ \varphi \left( 1\right) = f\left( h\right) \circ \beta \] (8) which means that \( \varphi \left( h\right) \) is the composition \[ \mathcal{B}\overset{\beta }{ \rightarrow }f{f}^{ * }\mathcal{B}\xrightarrow[]{f\left( h\right) }f\mathcal{A} \] --- \( {}^{9} \) The existence of such a natural isomorphism means that \( {f}^{ * } \) and \( f \) are "adjoint functors." --- Similarly, if \( j : \mathcal{B} \rightarrow f\mathcal{A} \) is any homomorphism, then the diagram \[ \operatorname{Hom}\left( {{f}^{ * }f\mathcal{A},\mathcal{A}}\right) \overset{\varphi }{ \rightarrow }\operatorname{Hom}\left( {f\mathcal{A}, f\mathcal{A}}\right) \] \[ \operatorname{Hom}\left( {{f}^{ * }\left( j\right) ,\mathcal{A}}\right) \downarrow \operatorname{Hom}\left( {j, f\mathcal{A}}\right) \] \[ \mathrm{{Hom}}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \;\overset{\varphi }{ \rightarrow }\;\mathrm{{Hom}}\left( {\mathcal{B}, f\mathcal{A}}\right) \] commutes, whence \[ {\varphi }^{-1}\left( j\right) = {\varphi }^{-1}\left( 1\right) \circ {f}^{ * }\left( j\right) = \alpha \circ {f}^{ * }\left( j\right) \] (9) is the composition \[ {f}^{ * }\mathcal{B}\xrightarrow[]{{f}^{ * }\left( j\right) }{f}^{ * }f\mathcal{A}\overset{\alpha }{ \rightarrow }\mathcal{A} \] In particular, applying (9) to \( j = \beta : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B} \), we obtain that \( \alpha \circ {f}^{ * }\left( \beta \right) = {\varphi }^{-1}\left( \beta \right) = 1 \) . That is, the composition \[ {f}^{ * }\mathcal{B}\xrightarrow[]{{f}^{ * }\left( \beta \right) }{f}^{ * }f{f}^{ * }\mathcal{B}\overset{\alpha }{ \rightarrow }{f}^{ * }\mathcal{B} \] is the identity. Thus \( {f}^{ * }\left( \beta \right) \) is a monomorphism, and since \( {\left( {f}^{ * }\mathcal{B}\right) }_{x} = {\mathcal{B}}_{f\left( x\right) } \) , it follows that \[ \beta : \mathcal{B} \rightarrow f{f}^{ * }\mathcal{B} \] is a monomorphism provided that \( f : X \rightarrow Y \) is surjective. In the next chapter we shall apply this to the special case in which \( f : {X}_{d} \rightarrow X \) is the identity, where \( {X}_{d} \) denotes \( X \) with the discrete topology. In this case \[ \left( {f{f}^{ * }\mathcal{B}}\right) \left( U\right) = \mathop{\prod }\limits_{{x \in U}}{\mathcal{B}}_{x} \] is the group of "serrations" of \( \mathcal{B} \) over \( U \), where a serration is a possibly discontinuous cross section of \( \mathcal{B} \mid U \) . Then \( \beta : \mathcal{B}\left( U\right) \rightarrow \left( {f{f}^{ * }\mathcal{B}}\right) \left( U\right) \) is just the inclusion of the group of (continuous) sections in that of serrations. 4.3. We conclude this section with a remark on cohomomorphisms in quotient sheaves. Let \( {\mathcal{A}}^{\prime } \) be a subsheaf of a sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -cohomomorphism \[ \mathcal{B}\left( U\right) /{\mathcal{B}}^{\prime }\left( U\right) \sim \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) /{\mathcal{A}}^{\prime }\left( {{f}^{-1}\left( U\right) }\right) \] of presheaves, which, in turn, induces an \( f \) -cohomomorphism \( \mathcal{B}/{\mathcal{B}}^{\prime } \sim \) \( \mathcal{A}/{\mathcal{A}}^{\prime } \) of the generated sheaves. ## 5 Algebraic constructions In this section we shall consider covariant functors \( F\left( {{G}_{1},{G}_{2},\ldots }\right) \) of several variables from the category of abelian groups to itself. (More generally, one may consider covariant functors from the category of "diagrams of ab

1004_(GTM170)Sheaf Theory

sheaf \( \mathcal{A} \) on \( X \) and \( {\mathcal{B}}^{\prime } \) a subsheaf of \( \mathcal{B} \) on \( Y \) . Let \( k : \mathcal{B} \rightsquigarrow \mathcal{A} \) be an \( f \) -cohomomorphism that takes \( {\mathcal{B}}^{\prime } \) into \( {\mathcal{A}}^{\prime } \) . Then \( k \) clearly induces an \( f \) -cohomomorphism \[ \mathcal{B}\left( U\right) /{\mathcal{B}}^{\prime }\left( U\right) \sim \mathcal{A}\left( {{f}^{-1}\left( U\right) }\right) /{\mathcal{A}}^{\prime }\left( {{f}^{-1}\left( U\right) }\right) \] of presheaves, which, in turn, induces an \( f \) -cohomomorphism \( \mathcal{B}/{\mathcal{B}}^{\prime } \sim \) \( \mathcal{A}/{\mathcal{A}}^{\prime } \) of the generated sheaves. ## 5 Algebraic constructions In this section we shall consider covariant functors \( F\left( {{G}_{1},{G}_{2},\ldots }\right) \) of several variables from the category of abelian groups to itself. (More generally, one may consider covariant functors from the category of "diagrams of abelian groups of a given shape" to the category of abelian groups.) For general illustrative purposes we shall take the case of a functor of two variables. We may also consider \( F \) as a functor from the category of presheaves on \( X \) to itself in the canonical way (since \( F \) is covariant). That is, we let \[ F\left( {A, B}\right) \left( U\right) = F\left( {A\left( U\right), B\left( U\right) }\right) , \] for presheaves \( A \) and \( B \) on \( X \) . The sheaf generated by the presheaf \( F\left( {A, B}\right) \) will be denoted by \( \mathcal{F}\left( {A, B}\right) = \mathcal{R} \) has \( \left( {F\left( {A, B}\right) }\right) \) . In particular, if \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \) then \( \mathcal{F}\left( {\mathcal{A},\mathcal{B}}\right) = \mathcal{{Mean}}(U \mapsto F\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) = \) \( F\left( {\mathcal{A}\left( U\right) ,\mathcal{B}\left( U\right) }\right) ){.}^{10} \) Now suppose that the functor \( F \) commutes with direct limits. That is, suppose that the canonical map \( \underline{\lim }F\left( {{G}_{\alpha },{H}_{\alpha }}\right) \rightarrow F\left( {\underline{\lim }{G}_{\alpha },\underline{\lim }{H}_{\alpha }}\right) \) is an isomorphism for direct systems \( \left\{ {G}_{\alpha }\right\} \) and \( \left\{ {H}_{\alpha }\right\} \) of abelian groups. Let \( \mathcal{A} \) and \( \mathcal{B} \) denote the sheaves generated by the presheaves \( A \) and \( B \) respectively. Then for \( U \) ranging over the neighborhoods of \( x \in X \), we have \( \lim F\left( {A, B}\right) \left( U\right) = \lim F\left( {A\left( U\right), B\left( U\right) }\right) \approx F\left( {\lim A\left( U\right) ,\lim B\left( U\right) }\right) = \) \( F\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right) \) so that we have the natural isomorphism \[ \mathcal{F}{\left( A, B\right) }_{x} \approx F\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right) \] (10) when \( F \) commutes with direct limits. More generally, consider the natural maps \( A\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) and \( B\left( U\right) \rightarrow \) \( \mathcal{B}\left( U\right) \) . These give rise to a homomorphism \( F\left( {A, B}\right) \rightarrow F\left( {\mathcal{A},\mathcal{B}}\right) \) of pre-sheaves and hence to a homomorphism \[ \mathcal{F}\left( {A, B}\right) \rightarrow \mathcal{F}\left( {\mathcal{A},\mathcal{B}}\right) \] (11) of the generated sheaves. If \( U \) ranges over the neighborhoods of \( x \in X \) , then the diagram \[ \underline{\lim }F\left( {A\left( U\right), B\left( U\right) }\right) \; \rightarrow \;\underline{\lim }F\left( {\mathcal{A}\left( U\right) ,\mathcal{B}\left( U\right) }\right) \] \[ \downarrow \; \downarrow \] \[ F\left( {\underline{\lim }A\left( U\right) ,\underline{\lim }B\left( U\right) }\right) \rightarrow F\left( {\underline{\lim }\mathcal{A}\left( U\right) ,\underline{\lim }\mathcal{B}\left( U\right) }\right) \] commutes. The bottom homomorphism is an isomorphism by definition of \( \mathcal{A} \) and \( \mathcal{B} \) . The top homomorphism is the restriction of (11) to the stalks at \( x \) . The vertical maps are isomorphisms when \( F \) commutes with direct limits. Thus we see that (11) is an isomorphism of sheaves provided that \( F \) commutes with direct limits. That is, in this case,  naturally. \( {}^{10} \) Our notation in some of the specific examples to follow will differ from the notation we are using in the general discussion. We shall now discuss several explicit cases, starting with the tensor product. If \( \mathcal{A} \) and \( \mathcal{B} \) are sheaves on \( X \), we let \[ \mathcal{A} \otimes \mathcal{B} = \mathcal{H}\text{ half }\left( {U \mapsto \mathcal{A}\left( U\right) \otimes \mathcal{B}\left( U\right) }\right) . \] Since \( \otimes \) commutes with direct limits, we have the natural isomorphism \[ {\left( \mathcal{A} \otimes \mathcal{B}\right) }_{x} \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{x} \] by (10). Since \( \otimes \) is right exact for abelian groups, it will also be right exact for sheaves since exactness is a stalkwise property. The following terminology will be useful: 5.1. Definition. An exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) of sheaves on \( X \) is said to be "pointwise split" if \( 0 \rightarrow {\mathcal{A}}_{x}^{\prime } \rightarrow {\mathcal{A}}_{x} \rightarrow {\mathcal{A}}_{x}^{\prime \prime } \rightarrow 0 \) splits for each \( x \in X \) . This condition clearly implies that \( 0 \rightarrow {\mathcal{A}}^{\prime } \otimes \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } \otimes \mathcal{B} \rightarrow 0 \) is exact for every sheaf \( \mathcal{B} \) on \( X \) . Our second example concerns the torsion product. We shall use \( G * H \) to denote \( \operatorname{Tor}\left( {G, H}\right) \) . For sheaves \( \mathcal{A} \) and \( \mathcal{B} \) on \( X \) we let \[ \mathcal{A} * \mathcal{B} = \mathcal{P}\text{ heaf }\left( {U \mapsto \mathcal{A}\left( U\right) * \mathcal{B}\left( U\right) }\right) . \] We have that \[ \left( {\mathcal{A} * \mathcal{B}}\right) x \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{x} \] since the torsion product \( * \) commutes with direct limits. Let \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) be an exact sequence of sheaves. Then for each open set \( U \subset X \), we have the exact sequence \[ 0 \rightarrow {\mathcal{A}}^{\prime }\left( U\right) * \mathcal{B}\left( U\right) \rightarrow \mathcal{A}\left( U\right) * \mathcal{B}\left( U\right) \rightarrow \left( {\mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) * \mathcal{B}\left( U\right) \] \[ \rightarrow {\mathcal{A}}^{\prime }\left( U\right) \otimes \mathcal{B}\left( U\right) \rightarrow \mathcal{A}\left( U\right) \otimes \mathcal{B}\left( U\right) \rightarrow \left( {\mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) \otimes \mathcal{B}\left( U\right) \rightarrow 0 \] of presheaves on \( X \), where \( \mathcal{B} \) is any sheaf. Now \( {\mathcal{A}}^{\prime \prime } \) is canonically isomorphic to \( \mathcal{P} \) heaf \( \left( {U \mapsto \mathcal{A}\left( U\right) /{\mathcal{A}}^{\prime }\left( U\right) }\right) \) . Thus this sequence of presheaves generates the exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } * \mathcal{B} \rightarrow \mathcal{A} * \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } * \mathcal{B} \rightarrow {\mathcal{A}}^{\prime } \otimes \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \rightarrow {\mathcal{A}}^{\prime \prime } \otimes \mathcal{B} \rightarrow 0 \) (12) of sheaves on \( X{.}^{11} \) Before passing on to other examples of our general considerations, we shall introduce some further notation concerned with tensor and torsion \( {}^{11} \) It should be noted that this is a special case of a general fact. Namely, if \( \left\{ {F}_{n}\right\} \) is an exact connected sequence of functors of abelian groups (as above), then the induced sequence of functors \( \left\{ {\mathcal{F}}_{n}\right\} \) on the category of sheaves to itself is also exact and connected. products. If \( X \) and \( Y \) are spaces and \( {\pi }_{X} : X \times Y \rightarrow X,{\pi }_{Y} : X \times Y \rightarrow Y \) are the projections, then for sheaves \( \mathcal{A} \) on \( X \) and \( \mathcal{B} \) on \( Y \) we define the total tensor product \( \mathcal{A}\widehat{ \otimes }\mathcal{B} \) to be the sheaf \[ \mathcal{A}\widehat{ \otimes }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) \otimes \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right) \] on \( X \times Y \) . Similarly, the total torsion product is defined to be \[ \mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right) \] Clearly, we have natural isomorphisms \[ {\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{y} \] \[ {\left( \mathcal{A}\widehat{ * }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{y} \] Another special case of our general discussion is provided by the direct sum functor. Thus, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) is a family of sheaves on \( X \), we let \[ \oplus {\mathcal{A}}_{\alpha } = \text{ Heaf }\left( {U \mapsto \bigoplus \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) }\right) . \] Since direct sums commute with direct limits, we have that \[ \left( {{\left( {\bigoplus }_{\alpha }{\mathcal{A}}_{\alpha }\right) }_{x} \approx {\bigoplus }_{\

1004_(GTM170)Sheaf Theory

Y \) . Similarly, the total torsion product is defined to be \[ \mathcal{A}\widehat{ * }\mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) * \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right) \] Clearly, we have natural isomorphisms \[ {\left( \mathcal{A}\widehat{ \otimes }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{y} \] \[ {\left( \mathcal{A}\widehat{ * }\mathcal{B}\right) }_{\langle x, y\rangle } \approx {\mathcal{A}}_{x} * {\mathcal{B}}_{y} \] Another special case of our general discussion is provided by the direct sum functor. Thus, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) is a family of sheaves on \( X \), we let \[ \oplus {\mathcal{A}}_{\alpha } = \text{ Heaf }\left( {U \mapsto \bigoplus \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) }\right) . \] Since direct sums commute with direct limits, we have that \[ \left( {{\left( {\bigoplus }_{\alpha }{\mathcal{A}}_{\alpha }\right) }_{x} \approx {\bigoplus }_{\alpha }{\left( {\mathcal{A}}_{\alpha }\right) }_{x}}\right) \] In the case of the direct product, we note that the presheaf \( U \mapsto \prod \left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) \) satisfies (S1) and (S2) on page 6 and therefore is a sheaf. It is denoted by \( \prod {\mathcal{A}}_{\alpha } \) . However, direct products do not generally commute with direct limits, and in fact, \( {\left( \prod {\mathcal{A}}_{\alpha }\right) }_{x} ≉ \prod {\left( {\mathcal{A}}_{\alpha }\right) }_{x} \) in general. [For example let \( {\mathcal{A}}_{i} = {\mathbb{Z}}_{\lbrack 0,1/i)} \subset \mathbb{Z} \) for \( i \geq 1 \) on \( X = \left\lbrack {0,1}\right\rbrack \) . Then for \( U = \lbrack 0,1/n) \) , we have that \( {\mathcal{A}}_{i}\left( U\right) = 0 \) for \( i > n \), and so \( \mathop{\prod }\limits_{{i = 1}}^{\infty }\left( {{\mathcal{A}}_{i}\left( U\right) }\right) = {\mathbb{Z}}^{n} \), whence \( {\left( \prod {\mathcal{A}}_{i}\right) }_{\{ 0\} } \approx \underline{\lim }{\mathbb{Z}}^{n} = {\mathbb{Z}}^{\infty } \), the countable direct sum of copies of \( \mathbb{Z} \) . However, \( \prod {\left( {\mathcal{A}}_{i}\right) }_{\{ 0\} } \approx \mathop{\prod }\limits_{{i = 1}}^{\infty }\mathbb{Z} \), which is uncountable. For another example, let \( {\mathcal{B}}_{n} = {\mathbb{Z}}_{\{ 1/n\} } \) . Then \( \prod {\left( {\mathcal{B}}_{n}\right) }_{\{ 0\} } = 0 \) ; but \( {\left( \prod {\mathcal{B}}_{n}\right) }_{\{ 0\} } \neq 0 \) since the sections \( {s}_{n} \in {\mathcal{B}}_{n}\left( X\right) \) that are 1 at \( 1/n \) give a section \( s = \prod {s}_{n} \) of the product that is not zero in any neighborhood of 0 and hence has nonzero germ at 0 .] For a finite number of variables (or, generally, for locally finite families), direct sums and direct products of sheaves coincide. For two variables (for example) \( \mathcal{A} \) and \( \mathcal{B} \), the direct sum is denoted by \( \mathcal{A} \oplus \mathcal{B} \) . (Note that \( \mathcal{A}\bigtriangleup \mathcal{B} \) is the underlying topological space of \( \mathcal{A} \oplus \mathcal{B} \) .) The notation \( \mathcal{A} \times \mathcal{B} \) is reserved for the cartesian product of \( \mathcal{A} \) and \( \mathcal{B} \), which with coordinatewise addition is a sheaf on \( X \times Y \) when \( \mathcal{A} \) is a sheaf on \( X \) and \( \mathcal{B} \) is one on \( Y \) . Note that \( \mathcal{A} \times \mathcal{B} = \left( {{\pi }_{X}^{ * }\mathcal{A}}\right) \oplus \left( {{\pi }_{Y}^{ * }\mathcal{B}}\right) \) . Note that for \( X = Y \), we have that \( \mathcal{A} \oplus \mathcal{B} = \left( {\mathcal{A} \times \mathcal{B}}\right) \mid \Delta \), where \( \Delta \) is the diagonal of \( X \times X \), identified with \( X \), and similarly that \( \mathcal{A} \otimes \mathcal{B} = \left( {\mathcal{A}\widehat{ \otimes }\mathcal{B}}\right) \mid \Delta \) and \( \mathcal{A} * \mathcal{B} = \left( {\mathcal{A}\widehat{ * }\mathcal{B}}\right) |\Delta . \) Our next example is given by a functor on a category of "diagrams." Let \( A \) be a directed set. Consider direct systems \( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) (where \( {\pi }_{\alpha ,\beta } \) : \( \left. {{G}_{\beta } \rightarrow {G}_{\alpha }\text{is defined for}\alpha > \beta \text{in}A\text{and satisfies}{\pi }_{\alpha ,\beta }{\pi }_{\beta ,\gamma } = {\pi }_{\alpha ,\gamma }}\right) \) of abelian groups based on the directed set \( A \) . Let \( F \) be the (covariant) functor that assigns to each such direct system \( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) its direct limit \[ F\left( \left\{ {{G}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \right) = \underline{\lim }{G}_{\alpha } \] Now let \( \left\{ {{\mathcal{A}}_{\alpha };{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of sheaves based on \( A \) . Then we define \[ \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } = \text{ Sheaf }\left( {U \mapsto \mathop{\lim }\limits_{ \rightarrow }{}_{\alpha }{\mathcal{A}}_{\alpha }\left( U\right) }\right) . \] There are the compatible maps \( {\mathcal{A}}_{\alpha }\left( U\right) \rightarrow {\underline{\lim }}_{\alpha }{\mathcal{A}}_{\alpha }\left( U\right) \) that induce canonical homomorphisms \( {\pi }_{\beta } : {\mathcal{A}}_{\beta } \rightarrow {\underline{\lim }}_{\alpha }{\mathcal{A}}_{\alpha } \) such that \( {\pi }_{\beta } = {\pi }_{\alpha } \circ {\pi }_{\alpha ,\beta } \) whenever \( \alpha > \beta \) . Since direct limits commute with one another, we have that \[ {\left( \underline{\lim }{\mathcal{A}}_{\alpha }\right) }_{x} \approx \underline{\lim }{\left( {\mathcal{A}}_{\alpha }\right) }_{x} \] Now suppose that \( \mathcal{A} \) is another sheaf on \( X \) and that we have a family of homomorphisms \( {h}_{\alpha } : {\mathcal{A}}_{\alpha } \rightarrow \mathcal{A} \) that are compatible in the sense that \( {h}_{\beta } = {h}_{\alpha } \circ {\pi }_{\alpha ,\beta } \) whenever \( \alpha > \beta \) . These induce compatible maps \( {\mathcal{A}}_{\alpha }\left( U\right) \rightarrow \) \( \mathcal{A}\left( U\right) \) for all open \( U \) and hence a homomorphism \( \underline{\lim }\left( {{\mathcal{A}}_{\alpha }\left( U\right) }\right) \rightarrow \mathcal{A}\left( U\right) \) of presheaves. In turn this induces a homomorphism \[ h : \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \rightarrow \mathcal{A} \] of the generated sheaves such that \( h \circ {\pi }_{\alpha } = {h}_{\alpha } \) for all \( \alpha \) . That is, the direct limit of sheaves satisfies the "universal property" of direct limits. In particular, if \( \left\{ {\mathcal{A}}_{\alpha }\right\} \) and \( \left\{ {\mathcal{B}}_{\alpha }\right\} \) are direct systems based on the same directed set, then the homomorphisms \( {\mathcal{A}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \) and \( {\mathcal{B}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) induce compatible homomorphisms \( {\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha } \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) and hence a homomorphism \( \mathop{\lim }\limits_{ \rightarrow }\left( {{\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha }}\right) \rightarrow \mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha } \) . On stalks this is an isomorphism since tensor products and direct limits commute. Thus it follows that this is an isomorphism \[ \lambda : \mathop{\lim }\limits_{ \rightarrow }\left( {{\mathcal{A}}_{\alpha } \otimes {\mathcal{B}}_{\alpha }}\right) \overset{ \approx }{ \rightarrow }\mathop{\lim }\limits_{ \rightarrow }{\mathcal{A}}_{\alpha } \otimes \mathop{\lim }\limits_{ \rightarrow }{\mathcal{B}}_{\alpha }. \] (13) The functor \( \operatorname{Hom}\left( {G, H}\right) \) on abelian groups is covariant in only one of its variables, so that the general discussion does not apply. However, note that every homomorphism \( \mathcal{A} \rightarrow \mathcal{B} \) of sheaves induces a homomorphism \[ \mathcal{A}\left| {U \rightarrow \mathcal{B}}\right| U \] Thus we see that the functor \[ U \mapsto \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) \] (14) defines a presheaf on \( X \) . We define \[ \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) = \mathcal{H}\text{ heaf }\left( {U \mapsto \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) }\right) . \] It is clear that the presheaf (14) satisfies (S1) and (S2), so that \[ \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) . \] It is important to note that the last equation does not apply in general to sections over nonopen subspaces, and in particular that \[ \operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right) \] in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack {0,1}\right\rbrack \) and let \( \mathcal{A} = {\mathcal{B}}_{\{ 0\} } \), which has stalk \( \mathbb{Z} \) over \( \{ 0\} \) and stalks \( 0 \) elsewhere. Then \( \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) = 0 \) for the open sets of the form \( U = \lbrack 0,\varepsilon ) \), and hence \( {\mathcal{{Hom}\left( {A, B}\right) }}_{\{ 0\} } = 0 \), whereas \( {\mathcal{A}}_{\{ 0\} } = \mathbb{Z} = {\mathcal{B}}_{\{ 0\} } \) , whence \( \mathrm{{Hom}}\left( {{\mathcal{A}}_{\{ 0\} },{\mathcal{B}}_{\{ 0\} }}\right) \approx \mathrm{{Hom}}\left( {\dot{\mathbb{Z}},\dot

1004_(GTM170)Sheaf Theory

ns over nonopen subspaces, and in particular that \[ \operatorname{Hom}{\left( \mathcal{A},\mathcal{B}\right) }_{x} ≉ \operatorname{Hom}\left( {{\mathcal{A}}_{x},{\mathcal{B}}_{x}}\right) \] in general. For example, let \( \mathcal{B} \) be the constant sheaf with stalks \( \mathbb{Z} \) on \( X = \) \( \left\lbrack {0,1}\right\rbrack \) and let \( \mathcal{A} = {\mathcal{B}}_{\{ 0\} } \), which has stalk \( \mathbb{Z} \) over \( \{ 0\} \) and stalks \( 0 \) elsewhere. Then \( \mathcal{{Hom}}\left( {\mathcal{A},\mathcal{B}}\right) \left( U\right) \approx \operatorname{Hom}\left( {\mathcal{A}\left| {U,\mathcal{B}}\right| U}\right) = 0 \) for the open sets of the form \( U = \lbrack 0,\varepsilon ) \), and hence \( {\mathcal{{Hom}\left( {A, B}\right) }}_{\{ 0\} } = 0 \), whereas \( {\mathcal{A}}_{\{ 0\} } = \mathbb{Z} = {\mathcal{B}}_{\{ 0\} } \) , whence \( \mathrm{{Hom}}\left( {{\mathcal{A}}_{\{ 0\} },{\mathcal{B}}_{\{ 0\} }}\right) \approx \mathrm{{Hom}}\left( {\dot{\mathbb{Z}},\dot{\mathbb{Z}}}\right) \approx \mathbb{Z}. \) If \( \mathcal{R} \) is a sheaf of rings on \( X \) and if \( \mathcal{A} \) and \( \mathcal{B} \) are \( \mathcal{R} \) -modules, then one can define, in a similar manner, the sheaves \[ \mathcal{A}{ \otimes }_{\mathcal{R}}\mathcal{B},\;{\mathcal{{Int}}}_{n}^{\mathcal{R}}(\mathcal{A},\mathcal{B}),\;\mathcal{{Hom}}{}_{\mathcal{R}}(\mathcal{A},\mathcal{B}). \] ## 6 Supports A paracompact space is a Hausdorff space with the property that every open covering has an open, locally finite, refinement. The following facts are well known (see [34], [53] and [19]): (1) Every paracompact space is normal. (2) A metric space is paracompact. (3) A closed subspace of a paracompact space is paracompact. (4) If \( \left\{ {U}_{\alpha }\right\} \) is a locally finite open cover of a normal space \( X \), then there is an open cover \( \left\{ {V}_{\alpha }\right\} \) of \( X \) such that \( {\bar{V}}_{\alpha } \subset {U}_{\alpha } \) . (5) A locally compact space is paracompact \( \Leftrightarrow \) it is a disjoint union of open, \( \sigma \) -compact subspaces. A space is called hereditarily paracompact if every open subspace is paracompact. It is easily seen that this implies that every subspace is paracompact. Of course, metric spaces are hereditarily paracompact. 6.1. Definition. A "family of supports" on \( X \) is a family \( \Phi \) of closed subsets of \( X \) such that: (1) a closed subset of a member of \( \Phi \) is a member of \( \Phi \) ; (2) \( \Phi \) is closed under finite unions. \( \Phi \) is said to be a "paracompactifying" family of supports if in addition: (3) each element of \( \Phi \) is paracompact; (4) each element of \( \Phi \) has a (closed) neighborhood which is in \( \Phi \) . We define the extent \( E\left( \Phi \right) \) of a family of supports to be the union of the members of \( \Phi \) . Note that \( E\left( \Phi \right) \) is open when \( \Phi \) is paracompactifying. The family of all compact subsets of \( X \) is denoted by \( c \) . It is para-compactifying if \( X \) is locally compact. We use 0 to denote the family of supports whose only member is the empty set \( \varnothing \) . It is customary to "denote" the family of all closed subsets of \( X \) by the absence of a symbol, and we shall also use \( {cld} \) to denote this family. Recall that for \( s \in \mathcal{A}\left( X\right) ,\left| s\right| = \{ x \in X \mid s\left( x\right) \neq 0\} \) denotes the support of the section \( s \) . Now if \( A \) is a presheaf on \( X \) and \( s \in A\left( X\right) \), we put \( \left| s\right| = \left| {\theta \left( s\right) }\right| \), where \( \theta : A\left( X\right) \rightarrow \mathcal{A}\left( X\right) \) is the canonical map, \( \mathcal{A} \) being the sheaf generated by \( A \) . Note that for \( s \in A\left( X\right), x \notin \left| s\right| \Leftrightarrow (s \mid U = 0 \) for some neighborhood \( U \) of \( x \) ). If \( \mathcal{A} \) is a sheaf on \( X \), we put \[ {\Gamma }_{\Phi }\left( \mathcal{A}\right) = \{ s \in \mathcal{A}\left( X\right) \mid \left| s\right| \in \Phi \} \] Then \( {\Gamma }_{\Phi }\left( \mathcal{A}\right) \) is a subgroup of \( \mathcal{A}\left( X\right) \), and for an exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \) \( \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \), the sequence \[ 0 \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime }\right) \rightarrow {\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime \prime }\right) \] is exact. For a presheaf \( A \) on \( X \) we put \( {A}_{\Phi }\left( X\right) = \{ s \in A\left( X\right) \left| \;\right| s| \in \Phi \} {.}^{12} \) 6.2. Theorem. Let \( A \) be a presheaf on \( X \) that is conjunctive for coverings of \( X \) and let \( \mathcal{A} \) be the sheaf generated by \( A \) . Then for any paracompactifying family \( \Phi \) of supports on \( X \), the sequence \[ 0 \rightarrow {A}_{0}\left( X\right) \rightarrow {A}_{\Phi }\left( X\right) \overset{\theta }{ \rightarrow }{\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow 0 \] is exact. Proof. The only nontrivial part is that \( \theta \) is surjective. Let \( s \in {\Gamma }_{\Phi }\left( \mathcal{A}\right) \) and let \( U \) be an open neighborhood of \( \left| s\right| \) with \( \bar{U} \) paracompact. By covering \( \bar{U} \) and then restricting to \( U \), we can find a covering \( \left\{ {U}_{\alpha }\right\} \) of \( U \) that is locally finite in \( X \) and such that there exist \( {s}_{\alpha } \in A\left( {U}_{\alpha }\right) \) with \( \theta \left( {s}_{\alpha }\right) = s \mid {U}_{\alpha } \) . Similarly, we can find a covering \( \left\{ {V}_{\alpha }\right\} \) of \( U \) with \( U \cap {\bar{V}}_{\alpha } \subset {U}_{\alpha } \) . Add \( X - \left| s\right| \) to the collection \( \left\{ {V}_{\alpha }\right\} \), giving a locally finite covering of \( X \), and use the zero section for the corresponding \( {s}_{\alpha } \) . \( {}^{12} \) The \( \Gamma \) notation will be used only for sheaves and not for presheaves. For \( x \in X \) let \( I\left( x\right) = \left\{ {\alpha \mid x \in {\bar{V}}_{\alpha }}\right\} \), a finite set. For each \( x \in X \) there is a neighborhood \( W\left( x\right) \) such that \( y \in W\left( x\right) \Rightarrow I\left( y\right) \subset I\left( x\right) \) and such that \( W\left( x\right) \subset {V}_{\alpha } \) for each \( \alpha \in I\left( x\right) \) . If \( \alpha \in I\left( x\right) \), then \( \theta \left( {s}_{\alpha }\right) \left( x\right) = s\left( x\right) \) . Since \( I\left( x\right) \) is finite, we may further assume that \( W\left( x\right) \) is so small that \( {s}_{\alpha } \mid W\left( x\right) \) is independent of \( \alpha \in I\left( x\right) \) [since \( {\mathcal{A}}_{x} = \mathop{\lim }\limits_{ \rightarrow }A\left( N\right), N \) ranging over the neighborhoods of \( x \) ]. Let \( {s}_{x} \in A\left( {W\left( x\right) }\right) \) be the common value of \( {s}_{\alpha } \mid W\left( x\right) \) for \( \alpha \in I\left( x\right) \) . Suppose that \( x, y \in X \) and \( z \in W\left( x\right) \cap W\left( y\right) \) . Let \( \alpha \in I\left( z\right) \subset I\left( x\right) \cap I\left( y\right) \) . Then \( {s}_{x} = {s}_{\alpha }\left| {W\left( y\right) \text{, so that}{s}_{x}}\right| W\left( x\right) \cap W\left( y\right) = {s}_{y} \mid W\left( x\right) \cap W\left( y\right) \) . Since \( A \) is conjunctive for coverings of \( X \), there is a \( t \in A\left( X\right) \) such that \( t \mid W\left( x\right) = {s}_{x} \) for all \( x \in X \) . Clearly \( \theta \left( t\right) = s \), and by definition, \( \left| t\right| = \left| s\right| \in \Phi \) . Note that \( {A}_{0}\left( U\right) = 0 \) for all open \( U \subset X \Leftrightarrow A \) is a monopresheaf. 6.3. Definition. If \( A \subset X \) and \( \Phi \) is a family of supports on \( X \), then \( \Phi \cap A \) denotes the family \( \{ K \cap A \mid K \in \Phi \} \) of supports on \( A \), and \( \Phi \mid A \) denotes the family \( \{ K \mid K \subset A \) and \( K \in \Phi \} \) of supports on \( A \) or on \( X{.}^{13} \) If \( X, Y \) are spaces with support families \( \Phi \) and \( \Psi \) respectively, then \( \Phi \times \Psi \) denotes the family on \( X \times Y \) of all closed subsets of sets of the form \( K \times L \) with \( K \in \Phi \) and \( L \in \Psi \) . If \( f : X \rightarrow Y \) and \( \Psi \) is a family on \( Y \), then \( {f}^{-1}\Psi \) denotes the family on \( X \) of all closed subsets of sets of the form \( {f}^{-1}K \) for \( K \in \Psi \) . 6.4. Example. For the purposes of this example, let us use the subscript \( Y \) on the family of supports \( {cld} \) or \( c \) to indicate the space to which these symbols apply. (In other places we let the context determine this.) Let \( X = \mathbb{R} \) and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \) If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while \( {cl}{d}_{X} \mid A \) is the family of closed subsets of \( A \) bounded away from 1. Also, \( {c}_{X} \cap A \) is the family of closed subsets of \( A \) bounded away from 0 . If \( X = \mathbb{R} = Y \), then the family \( {c}_{X} \times {c}_{Y} = {c}_{X \times Y} \), while \( {c}_{X} \times {cl}{d}_{Y} \) is the family of all closed subsets of \( X \times Y \) whose projection to \( X \) is bounded (but the projection need not be closed). This is the same as the family \( {\pi }_{X}^{-1}{c}_{X} \) . Also, \( {cl}{d}_{X} \times {cl}{d}_{Y} = {cl}{d}_{X \times Y} = {\pi }_{X}^{-1}{cl}{d}_{X} \) . For any map \( f : X \rightarrow Y \), the family \( {f}^{-1}{c}_{Y}

1004_(GTM170)Sheaf Theory

and \( A = \left( {0,1}\right) \) . Then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {cl}{d}_{X} \mid A = {c}_{A} \) . Also, \( {c}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A}. \) If instead, \( X = (0,1\rbrack \), then \( {cl}{d}_{X} \cap A = {cl}{d}_{A} \) and \( {c}_{X} \mid A = {c}_{A} \), while \( {cl}{d}_{X} \mid A \) is the family of closed subsets of \( A \) bounded away from 1. Also, \( {c}_{X} \cap A \) is the family of closed subsets of \( A \) bounded away from 0 . If \( X = \mathbb{R} = Y \), then the family \( {c}_{X} \times {c}_{Y} = {c}_{X \times Y} \), while \( {c}_{X} \times {cl}{d}_{Y} \) is the family of all closed subsets of \( X \times Y \) whose projection to \( X \) is bounded (but the projection need not be closed). This is the same as the family \( {\pi }_{X}^{-1}{c}_{X} \) . Also, \( {cl}{d}_{X} \times {cl}{d}_{Y} = {cl}{d}_{X \times Y} = {\pi }_{X}^{-1}{cl}{d}_{X} \) . For any map \( f : X \rightarrow Y \), the family \( {f}^{-1}{c}_{Y} \) can be thought of as the family of (closed) "basewise compact" sets. In IV-5 we shall define what can be thought of as the family of "fiberwise compact" sets. 6.5. Proposition. If \( \Phi \) is a paracompactifying family of supports on \( X \) and if \( Y \subset X \) is locally closed, then \( \Phi \mid Y \) is a paracompactifying family of supports on \( Y \) . Proof. For \( Y = U \cap F \) with \( U \) open and \( F \) closed, we have that \( \Phi \mid Y = \) \( \left( {\Phi \mid U}\right) \mid \left( {U \cap F}\right) \), so that it suffices to consider the two cases \( Y \) open and \( Y \) closed. These cases are obvious. --- \( {}^{13} \) Note that \( \Phi \mid F = \Phi \cap F \) for \( F \) closed. --- 6.6. Proposition. Let \( A \subset X \) be locally closed, let \( \Phi \) be a family of supports on \( X \), and let \( \mathcal{B} \) be a sheaf on \( A \) . Then the restriction of sections \( \Gamma \left( {\mathcal{B}}^{X}\right) \rightarrow \Gamma \left( {{\mathcal{B}}^{X} \mid A}\right) = \Gamma \left( \mathcal{B}\right) \) induces an isomorphism \[ {\Gamma }_{\Phi }\left( {\mathcal{B}}^{X}\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{\Phi \mid A}\left( \mathcal{B}\right) \] Similarly, for a sheaf \( \mathcal{A} \) on \( X \), the restriction of sections induces an isomorphism \[ {\Gamma }_{\Phi }\left( {\mathcal{A}}_{A}\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{\Phi \mid A}\left( {\mathcal{A} \mid A}\right) \] Proof. A section \( s \in {\Gamma }_{\Phi }\left( {\mathcal{B}}^{X}\right) \) must have support in \( A \) since \( {\mathcal{B}}^{X} \) vanishes outside of \( A \) . Thus \( \left| s\right| \in \Phi \mid A \) . Moreover, \( s \mid A \) can be zero only if \( s \) is zero. Now suppose that \( t \in {\Gamma }_{\Phi \mid A}\left( \mathcal{B}\right) \), and let \( s : X \rightarrow {\mathcal{B}}^{X} \) be the extension of \( t \) by zero. It suffices to show that \( s \) is continuous. Since \( s \) coincides with the zero section on the open set \( X - \left| t\right| \), it suffices to restrict our attention to the neighborhood of any point \( x \in \left| t\right| \) . Let \( v \in {\mathcal{B}}^{X}\left( U\right) \) be a section of \( {\mathcal{B}}^{X} \) with \( v\left( x\right) = t\left( x\right) = s\left( x\right) \) . We may assume, by changing the open neighborhood \( U \) of \( x \), that \( v\left| {U \cap A = t}\right| U \cap A \) . But \( v \) must vanish on \( U - A \) , so that \( v = s \mid U \) . Hence \( s \) is continuous on \( U \), and this completes the proof of the first statement. The second statement is immediate from the identity \( {\left( \mathcal{A} \mid A\right) }^{X} = {\mathcal{A}}_{A} \) 6.7. In this book the \( \Gamma \) notation will be used only for the group of global sections. Thus the group of sections over \( A \subset X \) of a sheaf \( \mathcal{A} \) on \( X \) is denoted by \( \Gamma \left( {\mathcal{A} \mid A}\right) \) . In the literature, but not here, it is often denoted by \( \Gamma \left( {A,\mathcal{A}}\right) \) . Of course, for the case of a support family \( \Phi \) on \( X \), there are at least two variations: \( {\Gamma }_{\Phi \cap A}\left( {\mathcal{A} \mid A}\right) \) and \( {\Gamma }_{\Phi \mid A}\left( {\mathcal{A} \mid A}\right) \) . ## 7 Classical cohomology theories As examples of the use of Theorem 6.2 and also for future reference we will briefly describe the "classical" singular, Alexander-Spanier, de Rham, and Čech cohomology theories. ## Alexander-Spanier cohomology Let \( G \) be a fixed abelian group. For \( U \subset X \) open let \( {A}^{p}\left( {U;G}\right) \) be the group of all functions \( f : {U}^{p + 1} \rightarrow G \) under pointwise addition. Then the functor \( U \mapsto {A}^{p}\left( {U;G}\right) \) is a conjunctive presheaf on \( X \) . [For if \( {f}_{\alpha } : {U}_{\alpha }^{p + 1} \rightarrow \) \( G \) are functions such that \( {f}_{\alpha } \) and \( {f}_{\beta } \) agree on \( {U}_{\alpha }^{p + 1} \cap {U}_{\beta }^{p + 1} \), then define \( f : {U}^{p + 1} \rightarrow G \), where \( U = \bigcup {U}_{\alpha } \), by \( f\left( x\right) = {f}_{\alpha }\left( x\right) \) if \( x \in {U}_{\alpha }^{p + 1} \) and \( f\left( x\right) \) arbitrary if \( \left. {x \notin {U}_{\alpha }^{p + 1}\text{for any}\alpha \text{.}}\right\rbrack \) Let \( {\mathcal{A}}^{p}\left( {X;G}\right) = \mathcal{R} \) eq \( \left( {U \mapsto {A}^{p}\left( {U;G}\right) }\right) \) . The differential (or "coboundary") \( d : {A}^{p}\left( {U;G}\right) \rightarrow {A}^{p + 1}\left( {U;G}\right) \) is defined by \[ {df}\left( {{x}_{0},\ldots ,{x}_{p + 1}}\right) = \mathop{\sum }\limits_{{i = 0}}^{{p + 1}}{\left( -1\right) }^{i}f\left( {{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p + 1}}\right) , \] where \( f : {U}^{p + 1} \rightarrow G \) . Now \( d \) is a homomorphism of presheaves and \( {d}^{2} = 0 \) . Thus \( d \) induces a differential \[ d : {\mathcal{A}}^{p}\left( {X;G}\right) \rightarrow {\mathcal{A}}^{p + 1}\left( {X;G}\right) \] with \( {d}^{2} = 0 \) . The classical definition of Alexander-Spanier cohomology with supports in the family \( \Phi \) is \[ {AS}{H}_{\Phi }^{p}\left( {X;G}\right) = {H}^{p}\left( {{A}_{\Phi }^{ * }\left( {X;G}\right) /{A}_{0}^{ * }\left( {X;G}\right) }\right) . \] Note that \( {A}_{0}^{p}\left( {X;G}\right) \) is the set of all functions \( {X}^{p + 1} \rightarrow G \) that vanish in some neighborhood of the diagonal. Thus two functions \( f, g : {X}^{p + 1} \rightarrow G \) represent the same element of \( {A}^{p}\left( {X;G}\right) /{A}_{0}^{p}\left( {X;G}\right) \Leftrightarrow \) they coincide in some neighborhood of the diagonal. \( {}^{14} \) Thus Theorem 6.2 implies that if \( \Phi \) is a paracompactifying family of supports, then there is a natural isomorphism  (15) There is a "cup product" \( \cup : {A}^{p}\left( {U;{G}_{1}}\right) \otimes {A}^{q}\left( {U;{G}_{2}}\right) \rightarrow {A}^{p + q}\left( {U;{G}_{1} \otimes {G}_{2}}\right) \) given by the Alexander-Whitney formula \[ \left( {f \cup g}\right) \left( {{x}_{0},\ldots ,{x}_{p + q}}\right) = f\left( {{x}_{0},\ldots ,{x}_{p}}\right) \otimes g\left( {{x}_{p},\ldots ,{x}_{p + q}}\right) \] with \( d\left( {f \cup g}\right) = {df} \cup g + {\left( -1\right) }^{p}f \cup {dg} \) and \( \left| {f \cup g}\right| \subset \left| f\right| \cap \left| g\right| \) . This induces products \[ \cup : {\mathcal{A}}^{p}\left( {X;{G}_{1}}\right) \otimes {\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) \rightarrow {\mathcal{A}}^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) , \] \[ {\Gamma }_{\Phi }\left( {{\mathcal{A}}^{p}\left( {X;{G}_{1}}\right) }\right) \otimes {\Gamma }_{\Psi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) }\right) \rightarrow {\Gamma }_{\Phi \cap \Psi }\left( {{\mathcal{A}}^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) }\right) \] and \[ {H}^{p}\left( {{\Gamma }_{\Phi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{1}}\right) }\right) }\right) \otimes {H}^{q}\left( {{\Gamma }_{\Psi }\left( {{\mathcal{A}}^{q}\left( {X;{G}_{2}}\right) }\right) }\right) \rightarrow {H}^{p + q}\left( {{\Gamma }_{\Phi \cap \Psi }\left( {{\mathcal{A}}^{ * }\left( {X;{G}_{1} \otimes {G}_{2}}\right) }\right) }\right) ; \] i.e., \[ {}_{AS}{H}_{\Phi }^{p}\left( {X;{G}_{1}}\right) \otimes {}_{AS}{H}_{\Psi }^{q}\left( {X;{G}_{2}}\right) \rightarrow {}_{AS}{H}_{\Phi \cap \Psi }^{p + q}\left( {X;{G}_{1} \otimes {G}_{2}}\right) . \] In particular, for a base ring \( L \) with unit and an \( L \) -module \( G,{\mathcal{A}}^{0}\left( {X;L}\right) \) is a sheaf of rings with unit, and each \( {\mathcal{A}}^{n}\left( {X;G}\right) \) is an \( {\mathcal{A}}^{0}\left( {X;L}\right) \) -module. \( {}^{14} \) Note that it is the taking of the quotient by the elements of empty support that brings the topology of \( X \) into the cohomology groups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology. ## Singular cohomology Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \) be the group of singular \( p \) - cochains of \( U \) with values in \( \mathcal{A} \) . That is, an element \( f \in {S}^{p}\left( {U;\mathcal{A}}\right) \) is a function that assigns to each singular \( p \) -simplex \( \sigma : {\Delta }_{p} \rightarrow U \) of \( U \), a cross section \( f\left( \sigma \right) \in \Gamma \left( {{\sigma }^{ * }\left( \mathcal{A}\right) }\right) \), where \( {\Delta }_{p} \) denotes the standard \( p \) -simplex. Since \( \mathcal{A} \) is locally constant and \( {\Delta }_{p} \) is simply connected, \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is a constant sheaf on \( {\Delta }_{p} \) (as \( {\sigma }^{ * }\left( \mathcal{A}\rig

1004_(GTM170)Sheaf Theory

ups, since \( {A}^{ * }\left( {X;G}\right) \) itself is totally independent of the topology. ## Singular cohomology Let \( \mathcal{A} \) be a locally constant sheaf on \( X \) . (Classically \( \mathcal{A} \) is called a "bundle of coefficients.") For \( U \subset X \), let \( {S}^{p}\left( {U;\mathcal{A}}\right) \) be the group of singular \( p \) - cochains of \( U \) with values in \( \mathcal{A} \) . That is, an element \( f \in {S}^{p}\left( {U;\mathcal{A}}\right) \) is a function that assigns to each singular \( p \) -simplex \( \sigma : {\Delta }_{p} \rightarrow U \) of \( U \), a cross section \( f\left( \sigma \right) \in \Gamma \left( {{\sigma }^{ * }\left( \mathcal{A}\right) }\right) \), where \( {\Delta }_{p} \) denotes the standard \( p \) -simplex. Since \( \mathcal{A} \) is locally constant and \( {\Delta }_{p} \) is simply connected, \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is a constant sheaf on \( {\Delta }_{p} \) (as \( {\sigma }^{ * }\left( \mathcal{A}\right) \) is just the induced bundle on \( {\Delta }_{p} \) ). It follows that we can define the coboundary operator \[ d : {S}^{p}\left( {U;\mathcal{A}}\right) \rightarrow {S}^{p + 1}\left( {U;\mathcal{A}}\right) \] by \( {df}\left( \tau \right) = f\left( {\partial \tau }\right) \in \Gamma \left( {{\tau }^{ * }\left( \mathcal{A}\right) }\right) \) . Let \( {\mathcal{P}}^{p}\left( {X;\mathcal{A}}\right) = \mathcal{R} \) has \( /\left( {U \mapsto {S}^{p}\left( {U;\mathcal{A}}\right) }\right) \) with the induced differential. The presheaf \( {S}^{p}\left( {\bullet ;\mathcal{A}}\right) \) is conjunctive since if \( \left\{ {U}_{\alpha }\right\} \) is a collection of open sets with union \( U \) and if \( f\left( \sigma \right) \) is defined whenever \( \sigma \) is a singular simplex in some \( {U}_{\alpha } \) with value that is independent of the particular index \( \alpha \), then just define \( f\left( \sigma \right) = 0 \) (or anything) if \( \sigma \notin {U}_{\alpha } \) for any \( \alpha \), and this extends \( f \) to be an element of \( {S}^{p}\left( {U;\mathcal{A}}\right) \) . The classical definition of singular cohomology (with the local coefficients \( \mathcal{A} \) and supports in \( \Phi \) ) is \[ \Delta {H}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) = {H}^{p}\left( {{S}_{\Phi }^{ * }\left( {X;\mathcal{A}}\right) }\right) . \] However, it is a well-known consequence of the operation of subdivision that \[ {H}^{p}\left( {{S}_{0}^{ * }\left( {X;\mathcal{A}}\right) }\right) = 0\;\text{ for all }p. \] [We indicate the proof: Let \( \mathfrak{U} = \left\{ {U}_{\alpha }\right\} \) be a covering of \( X \) by open sets and let \( {S}^{p}\left( {\mathfrak{U};\mathcal{A}}\right) \) be the group of singular cochains based on \( \mathfrak{U} \) -small singular simplices. Then a subdivision argument shows that the surjection \[ {j}_{\mathfrak{U}} : {S}^{ * }\left( {X;\mathcal{A}}\right) \rightarrow {S}^{ * }\left( {\mathfrak{U};\mathcal{A}}\right) \] induces a cohomology isomorphism. Therefore, if we let \( {K}_{\mathfrak{U}}^{ * } = \operatorname{Ker}{j}_{\mathfrak{U}} \), then \( {H}^{ * }\left( {K}_{u}^{ * }\right) = 0 \) by the long exact cohomology sequence induced by the short exact cochain sequence \( 0 \rightarrow {K}_{\mathfrak{U}}^{ * } \rightarrow {S}^{ * }\left( {X;\mathcal{A}}\right) \rightarrow {S}^{ * }\left( {\mathfrak{U};\mathcal{A}}\right) \rightarrow 0 \) . However, \( {S}_{0}^{ * }\left( {X;\mathcal{A}}\right) = \bigcup {K}_{\mathfrak{U}}^{ * } = \mathop{\lim }\limits_{ \rightarrow }{K}_{\mathfrak{U}}^{ * } \), so that \[ \left. {{H}^{ * }\left( {{S}_{0}^{ * }\left( {X;\mathcal{A}}\right) }\right) = {H}^{ * }\left( {\underline{\lim }{K}_{\mathfrak{U}}^{ * }}\right) \approx \underline{\lim }{H}^{ * }\left( {K}_{\mathfrak{U}}^{ * }\right) = 0.}\right\rbrack \] Therefore, if \( \Phi \) is paracompactifying, then the exact sequence \[ 0 \rightarrow {S}_{0}^{ * } \rightarrow {S}_{\Phi }^{ * } \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{S}}^{ * }\right) \rightarrow 0 \] of 6.2 yields the isomorphism \[ \Delta {H}_{\Phi }^{p}\left( {X;\mathcal{A}}\right) \approx {H}^{p}\left( {{\Gamma }_{\Phi }\left( {{\mathcal{S}}^{ * }\left( {X;\mathcal{A}}\right) }\right) }\right) . \] (16) As with the case of Alexander-Spanier cohomology, the singular cup product makes \( {\mathcal{P}}^{0}\left( {X;L}\right) \) into a sheaf of rings and each \( {\mathcal{P}}^{n}\left( {X;\mathcal{A}}\right) \) into an \( {\mathcal{P}}^{0}\left( {X;L}\right) \) -module, where \( \mathcal{A} \) is a locally constant sheaf of \( L \) -modules. Remark: If \( X \) is a differentiable manifold and we let \( {S}^{ * }\left( {U;\mathcal{A}}\right) \) be the complex of singular cochains based on \( {C}^{\infty } \) singular simplices, a similar discussion applies. ## de Rham cohomology Let \( X \) be a differentiable manifold and let \( {\Omega }^{p}\left( U\right) \) be the group of differential \( p \) -forms on \( U \) with \( d : {\Omega }^{p}\left( U\right) \rightarrow {\Omega }^{p + 1}\left( U\right) \) being the exterior derivative. \( {}^{15} \) The de Rham cohomology group of \( X \) is defined to be \[ {\Omega }_{\Omega }{H}_{\Phi }^{p}\left( X\right) = {H}^{p}\left( {{\Omega }_{\Phi }^{ * }\left( X\right) }\right) . \] However, the presheaf \( U \mapsto {\Omega }^{p}\left( U\right) \) is a conjunctive monopresheaf and hence is a sheaf. Thus, trivially, we have  (17) for any family \( \Phi \) of supports. ## Čech cohomology Let \( \mathfrak{U} = \left\{ {{U}_{\alpha };\alpha \in I}\right\} \) be an open covering of a space \( X \) indexed by a set \( I \) and let \( G \) be a presheaf on \( X \) . Then an \( n \) -cochain \( c \) of \( \mathfrak{U} \) is a function defined on ordered \( \left( {n + 1}\right) \) -tuples \( \left( {{\alpha }_{0},\ldots ,{\alpha }_{n}}\right) \) of members of \( I \) such that \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n}} = {U}_{{\alpha }_{0}} \cap \cdots \cap {U}_{{\alpha }_{n}} \neq \varnothing \) with value \[ c\left( {{\alpha }_{0},\ldots ,{\alpha }_{n}}\right) \in G\left( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n}}\right) \] These form a group denoted by \( {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \) . An open set \( V \) of \( X \) is covered by \( \mathfrak{U} \cap V = \left\{ {{U}_{\alpha } \cap V;\alpha \in I}\right\} \) . Thus we have the cochain group \( {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) \) , and the assignment \( V \mapsto {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) \) gives a presheaf on \( X \), and hence a sheaf \[ {\check{\mathcal{C}}}^{n}\left( {\mathfrak{U};G}\right) = \mathcal{P}\text{ heaf }\left( {V \mapsto {\check{C}}^{n}\left( {\mathfrak{U} \cap V;G}\right) }\right) . \] Thus it makes sense to speak of the support \( \left| c\right| \) of a cochain, i.e., \( \left| c\right| = \) \( \left| {\theta \left( c\right) }\right| \), where \( \theta : {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \rightarrow \Gamma \left( {{\check{\mathcal{C}}}^{n}\left( {\mathfrak{U};G}\right) }\right) \) . This defines the cochain group \( {\check{C}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \) for a family \( \Phi \) of supports on \( X \) . The coboundary operator \( d : {\check{C}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{n + 1}\left( {\mathfrak{U};G}\right) \) is defined by \[ {dc}\left( {{\alpha }_{0},\ldots ,{\alpha }_{n + 1}}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n + 1}}{\left( -1\right) }^{i}c\left( {{\alpha }_{0},\ldots ,\widehat{{\alpha }_{i}},\ldots ,{\alpha }_{n + 1}}\right) \mid {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n + 1}}. \] --- \( {}^{15} \) See, for example, \( \left\lbrack {{19}\text{, Chapters II and V}}\right\rbrack \) . --- It is easy to see that \( {d}^{2} = 0 \) and so there are the cohomology groups \[ {\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) = {H}^{n}\left( {{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) }\right) \] A refinement of \( \mathfrak{U} \) is another open covering \( \mathfrak{V} = \left\{ {{V}_{\beta };\beta \in J}\right\} \) together with a function (called a refinement projection) \( \varphi : J \rightarrow I \) such that \( {V}_{\beta } \subset {U}_{\varphi \left( \beta \right) } \) for all \( \beta \in J \) . This yields a chain map \( {\varphi }^{ * } : {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{V};G}\right) \) by \[ {\varphi }^{ * }\left( c\right) \left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{n}\right) }\right) |{V}_{{\beta }_{0},\ldots ,{\beta }_{n}}. \] If \( \psi : J \rightarrow I \) is another refinement projection, then the functions \( D \) : \( {\check{C}}_{\Phi }^{n + 1}\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by \[ {Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldots ,\psi \left( {\beta }_{n}\right) }\right) \mid {V}_{{\beta }_{0},\ldots ,{\beta }_{n}} \] provide a chain homotopy between \( {\varphi }^{ * } \) and \( {\psi }^{ * } \) . Therefore, there is a homomorphism \[ {j}_{\mathfrak{V},\mathfrak{U}}^{n} : {\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{H}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \] induced by \( {\varphi }^{ * } \) but independent of the particular refinement projection \( \varphi \) used to define it. Thus we can define the Čech cohomology group as \[ {\check{H}}_{\Phi }^{n}\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\ri

1004_(GTM170)Sheaf Theory

ightarrow {\check{C}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \) given by \[ {Dc}\left( {{\beta }_{0},\ldots ,{\beta }_{n}}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}c\left( {\varphi \left( {\beta }_{0}\right) ,\ldots ,\varphi \left( {\beta }_{i}\right) ,\psi \left( {\beta }_{i}\right) ,\ldots ,\psi \left( {\beta }_{n}\right) }\right) \mid {V}_{{\beta }_{0},\ldots ,{\beta }_{n}} \] provide a chain homotopy between \( {\varphi }^{ * } \) and \( {\psi }^{ * } \) . Therefore, there is a homomorphism \[ {j}_{\mathfrak{V},\mathfrak{U}}^{n} : {\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \rightarrow {\check{H}}_{\Phi }^{n}\left( {\mathfrak{V};G}\right) \] induced by \( {\varphi }^{ * } \) but independent of the particular refinement projection \( \varphi \) used to define it. Thus we can define the Čech cohomology group as \[ {\check{H}}_{\Phi }^{n}\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{H}}_{\Phi }^{n}\left( {\mathfrak{U};G}\right) \] Since it does not affect the direct limit to restrict the coverings to a cofinal set of coverings, it is legitimate to restrict attention to coverings \( \mathfrak{U} = \left\{ {{U}_{x};x \in X}\right\} \) such that \( x \in {U}_{x} \) for all \( x \) . In this case there is a canonical refinement projection, the identity map \( X \rightarrow X \), for a refinement \( \mathfrak{V} = \left\{ {{V}_{x};x \in X}\right\} ,{V}_{x} \subset {U}_{x} \) . Thus there is a canonical chain map \[ {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) \rightarrow {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{V};G}\right) \] and so it is legitimate to pass to the limit and define the Čech cochain group \[ {\check{C}}_{\Phi }^{ * }\left( {X;G}\right) = {\underline{\lim }}_{\mathfrak{U}}{\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) \] Since the direct limit functor is exact, it commutes with cohomology, i.e., there is a canonical isomorphism \[ {\check{H}}_{\Phi }^{ * }\left( {X;G}\right) \approx {H}^{ * }\left( {{\check{C}}_{\Phi }^{ * }\left( {X;G}\right) }\right) \] which we shall regard as equality. We shall study this further in Chapter III. For the present, let us restrict attention to the case in which \( G \) is an abelian group regarded as a constant presheaf. We wish to define a natural homomorphism from the Alexander-Spanier groups to the Čech groups. If \( f : {X}^{n + 1} \rightarrow G \) is an Alexander-Spanier cochain and \( \mathfrak{U} = \left\{ {{U}_{x};x \in X}\right\} \) is a covering of \( X \), then \( f \) induces an element \( {f}_{\mathfrak{U}} \in {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \) by putting \( {f}_{\mathfrak{U}}\left( {{x}_{0},\ldots ,{x}_{n}}\right) = f\left( {{x}_{0},\ldots ,{x}_{n}}\right) \) when \( {U}_{{x}_{0},\ldots ,{x}_{n}} \neq \varnothing \) . Consequently, \( f \) induces \( {f}_{\infty } = \underline{\lim }{f}_{\mathfrak{U}} \in {\check{C}}^{ * }\left( {X;G}\right) \) . We claim that \( \left| {f}_{\infty }\right| = \left| f\right| \) . Indeed, \( x \notin \left| {f}_{\infty }\right| \Leftrightarrow \exists \mathfrak{U}, V, x \in V \) and \( {f}_{\mathfrak{U}}|V = 0 \) in \( \check{C}\left( {\mathfrak{U} \cap V;G}\right) \) \[ \Leftrightarrow \;\exists \mathfrak{U}, V,\;x \in V \ni {x}_{0},\ldots ,{x}_{n} \in V\; \Rightarrow \;{f}_{\mathfrak{U}}\left( {{x}_{0},\ldots ,{x}_{n}}\right) = 0 \] \[ \Leftrightarrow \;\exists W,\;x \in W \ni {x}_{0},\ldots ,{x}_{n} \in W\; \Rightarrow \;f\left( {{x}_{0},\ldots ,{x}_{n}}\right) = 0 \] \[ \Leftrightarrow \;\exists W,\;x \in W \ni f|{W}^{n + 1} = 0 \] \[ \Leftrightarrow x \notin \left| f\right| \text{. } \] Also, \( \left| {f}_{\infty }\right| = \varnothing \Leftrightarrow {f}_{\infty } = 0 \) . Now, given \( g \in {\check{C}}^{ * }\left( {X;G}\right), g \) comes from some \( {g}_{\mathfrak{U}} \in {\check{C}}^{n}\left( {\mathfrak{U};G}\right) \), and it is clear that \( {g}_{\mathfrak{U}} \) extends arbitrarily to an Alexander-Spanier cochain \( g \) . It follows that \( f \mapsto {f}_{\infty } \) induces an isomorphism \[ {A}_{\Phi }^{n}\left( {X;G}\right) /{A}_{0}^{n}\left( {X;G}\right) \overset{ \approx }{ \rightarrow }{\check{C}}_{\Phi }^{n}\left( {X;G}\right) \] whence \[ {A}_{S}{H}_{\Phi }^{n}\left( {X;G}\right) \approx {\check{H}}_{\Phi }^{n}\left( {X;G}\right) \] (18) for all spaces \( X \) and families \( \Phi \) of supports on \( X \) . Now, the Čech cohomology groups are not altered by restriction to any cofinal system of coverings. Therefore, if \( X \) is compact, we can restrict the coverings used to finite coverings. Similarly, if \( X \) is paracompact, we can restrict attention to locally finite coverings. Finally, if the covering dimension \( {}^{16}\operatorname{covdim}X = n < \infty \) then we can restrict attention to locally finite coverings \( \mathfrak{U} = \left\{ {{U}_{\alpha };\alpha \in I}\right\} \) such that \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{n + 1}} = \varnothing \) for distinct \( {\alpha }_{i} \) . Now, \( {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \) is the ordered simplicial cochain complex \( {C}^{ * }\left( {N\left( \mathfrak{U}\right) ;G}\right) \) of an \( n \) -dimensional abstract simplicial complex, namely the nerve \( N\left( \mathfrak{U}\right) \) of \( {\mathfrak{U}}^{.17} \) For \( c \in {\check{C}}^{p}\left( {\mathfrak{U};G}\right) \) we have that \( \left| c\right| = \bigcup \left\{ {{\bar{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} \mid c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) \neq 0}\right\} \), since \( \mathfrak{U} \) is locally finite. Thus \[ {\check{C}}_{\Phi }^{ * }\left( {\mathfrak{U};G}\right) = \{ c \in {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \;|\;\exists K \in \Phi \ni c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) = 0\text{ if }{\overline{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K\} \] \[ = \mathop{\lim }\limits_{{K \in \Phi }}\left\{ {c \in {\check{C}}^{ * }\left( {\mathfrak{U};G}\right) \mid c\left( {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right) = 0\text{ if }{\overline{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K}\right\} \] \[ = {\underline{\lim }}_{K \in \Phi }{C}^{ * }\left( {N\left( \mathfrak{U}\right) ,{N}_{K}\left( \mathfrak{U}\right) ;G}\right) , \] where \( {N}_{K}\left( \mathfrak{U}\right) = \left\{ {\left\{ {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right\} \in N\left( \mathfrak{U}\right) \mid {\bar{U}}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} ⊄ K}\right\} \), which is a sub-complex of \( N\left( \mathfrak{U}\right) \) . But \( {C}^{ * }\left( {N\left( \mathfrak{U}\right) ,{N}_{K}\left( \mathfrak{U}\right) ;G}\right) \) is chain equivalent to the corresponding oriented simplicial cochain complex that vanishes above degree \( n \) . Therefore \( {\check{H}}_{\Phi }^{p}\left( {\mathfrak{U};G}\right) = 0 \) for \( p > n \), whence \( {\check{H}}_{\Phi }^{p}\left( {X;G}\right) = 0 \) for \( p > n \) . Consequently, \[ {}_{AS}{H}_{\Phi }^{p}\left( {X;G}\right) = 0\text{ for }p > \operatorname{covdim}X. \] (19) \( {}^{16} \) The covering dimension of \( X \) is the least integer \( n \) (or \( \infty \) ) such that every covering of \( X \) has a refinement for which no point is contained in more than \( n + 1 \) distinct members of the covering. \( {}^{17} \) This has the members of \( I \) as vertices and the subsets \( \left\{ {{\alpha }_{0},\ldots ,{\alpha }_{p}}\right\} \subset I \), where \( {U}_{{\alpha }_{0},\ldots ,{\alpha }_{p}} \neq \varnothing \), as the \( p \) -simplices. ## Singular homology Even though the definition of singular cohomology requires a locally constant sheaf as coefficients, \( {}^{18} \) one can define singular homology with coefficients in an arbitrary sheaf \( \mathcal{A} \) . To do this, define the group of singular \( n \) -chains by \[ {S}_{n}\left( {X;\mathcal{A}}\right) = {\bigoplus }_{\sigma }\Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right) \] where the sum ranges over all singular simplices \( \sigma : {\Delta }_{n} \rightarrow X \) of \( X \) . If \( {F}_{i} \) : \( {\Delta }_{n - 1} \rightarrow {\Delta }_{n} \) is the \( i \) th face map, then we have the induced homomorphism \[ {\eta }_{i} : \Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right) \rightarrow \Gamma \left( {{F}_{i}^{ * }{\sigma }^{ * }\mathcal{A}}\right) = \Gamma \left( {{\left( \sigma \circ {F}_{i}\right) }^{ * }\mathcal{A}}\right) \] of Section 4, and so the boundary operator \[ \partial : {S}_{n}\left( {X;\mathcal{A}}\right) \rightarrow {S}_{n - 1}\left( {X;\mathcal{A}}\right) \] can be defined by \[ \partial s = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}{\eta }_{i}\left( s\right) \] for \( s \in \Gamma \left( {{\sigma }^{ * }\mathcal{A}}\right) \) . When \( \mathcal{A} \) is locally constant, then this, and the case of cohomology, is equivalent to Steenrod's definition of (co)homology with "local coefficients"; see [75] for the definition of the latter. The functor \( U \mapsto {S}_{n}\left( {U;\mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories. ## Exercises 1. (c) If \( \mathcal{A} \) is a sheaf on \( X \) and \( i : B \hookrightarrow X \), show that \( {i}^{ * }\mathcal{A} \approx \mathcal{A} \mid B \) . 2. (c) If \( \mathcal{B} \) is a sheaf on \( B \) and \( i : B \hookrightarrow X \), show that \( \left( {i\mathcal{B}}\right) \mid B \approx \mathcal{B} \) . 3. Let \( \left\{ {{B}_{\alpha },{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of presheaves [that is, for \( U \subset X \) , \( \left\{ {{B}_{\alpha }\left( U\right) ,{\pi }_{\alpha ,\beta }\left( U\right) }\right\} \) is a direct system of groups such that the \( {\pi }_{\alpha ,\beta } \) ’s commute with restrictions]. Let \( B = \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha } \) denote the

1004_(GTM170)Sheaf Theory

\mathcal{A}}\right) \) is covariant, and so it is not a presheaf. Thus it has a different nature than do the cohomology theories. See, however, Exercise 12 for a different description of singular homology that has a closer relationship to the cohomology theories. ## Exercises 1. (c) If \( \mathcal{A} \) is a sheaf on \( X \) and \( i : B \hookrightarrow X \), show that \( {i}^{ * }\mathcal{A} \approx \mathcal{A} \mid B \) . 2. (c) If \( \mathcal{B} \) is a sheaf on \( B \) and \( i : B \hookrightarrow X \), show that \( \left( {i\mathcal{B}}\right) \mid B \approx \mathcal{B} \) . 3. Let \( \left\{ {{B}_{\alpha },{\pi }_{\alpha ,\beta }}\right\} \) be a direct system of presheaves [that is, for \( U \subset X \) , \( \left\{ {{B}_{\alpha }\left( U\right) ,{\pi }_{\alpha ,\beta }\left( U\right) }\right\} \) is a direct system of groups such that the \( {\pi }_{\alpha ,\beta } \) ’s commute with restrictions]. Let \( B = \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha } \) denote the presheaf \( U \mapsto \mathop{\lim }\limits_{ \rightarrow }{B}_{\alpha }\left( U\right) \) . Let \( {\mathcal{B}}_{\alpha } = \mathcal{R} \) heaf \( \left( {B}_{\alpha }\right) \) and \( \mathcal{B} = \mathcal{R} \) heaf \( \left( B\right) \) . Show that \( \mathcal{B} \) and \( \underline{\lim }{\mathcal{B}}_{\alpha } \) are canonically isomorphic. 4. (c) A sheaf \( \mathcal{P} \) on \( X \) is called projective if the following commutative diagram, with exact row, can always be completed as indicated:  --- \( {}^{18} \) This will be generalized by another method in Chapter III. --- Show that the constant sheaf \( \mathbb{Z} \) on the unit interval is not the quotient of a projective sheaf. (Thus there are not "sufficiently many projectives" in the category of sheaves.) More generally, show that on a locally connected Hausdorff space without isolated points the only projective sheaf is 0 . 5. Show that the tensor product of two sheaves satisfies the universal property of tensor products. That is, if \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) are sheaves on \( X \) and if \( f : \mathcal{A}\Delta \mathcal{B} \rightarrow \mathcal{C} \) is a map that commutes with the projections onto \( X \) and is bilinear on each stalk, then there exists a unique homomorphism \( h : \mathcal{A} \otimes \mathcal{B} \rightarrow \mathcal{C} \) such that \( f = {hk} \), where \( k : \mathcal{A}\bigtriangleup \mathcal{B} \rightarrow \mathcal{A} \otimes \mathcal{B} \) takes \( \left( {a, b}\right) \in {\mathcal{A}}_{x} \times {\mathcal{B}}_{x} = {\left( \mathcal{A}\bigtriangleup \mathcal{B}\right) }_{x} \) into \( a \otimes b \in {\mathcal{A}}_{x} \otimes {\mathcal{B}}_{x} = {\left( \mathcal{A} \otimes \mathcal{B}\right) }_{x}. \) Treat the direct sum in a similar manner. 6. Show that the functor \( \operatorname{Hom}\left( {\bullet , \bullet }\right) \) of sheaves is left exact. 7. Let \( f : X \rightarrow Y \) and let \( \mathcal{R} \) be a sheaf of rings on \( Y \) . Show that the natural equivalence (5) of Section 4 restricts to a natural equivalence \[ \varphi : {\operatorname{Hom}}_{{f}^{ * }\mathcal{R}}\left( {{f}^{ * }\mathcal{B},\mathcal{A}}\right) \overset{ \approx }{ \rightarrow }{\operatorname{Hom}}_{\mathcal{R}}\left( {\mathcal{B}, f\mathcal{A}}\right) \] where \( \mathcal{B} \) is an \( \mathcal{R} \) -module and \( \mathcal{A} \) is an \( {f}^{ * }\mathcal{R} \) -module. [The \( \mathcal{R} \) -module structure of \( f\mathcal{A} \) is given by the composition \[ \left. {\mathcal{R}\left( U\right) \otimes \left( {f\mathcal{A}}\right) \left( U\right) \rightarrow \left( {{f}^{ * }\mathcal{R}}\right) \left( {{f}^{-1}U}\right) \otimes \mathcal{A}\left( {{f}^{-1}U}\right) \rightarrow \mathcal{A}\left( {{f}^{-1}U}\right) .}\right\rbrack \] 8. (c) Let \( f : X \rightarrow Y \) and let \( \mathcal{A} \) be a sheaf on \( X \) . Show that \( {\Gamma }_{\Phi }\left( {f\mathcal{A}}\right) = \) \( {\Gamma }_{{f}^{-1}\Phi }\left( \mathcal{A}\right) \), under the defining equality \( \left( {f\mathcal{A}}\right) \left( Y\right) = \mathcal{A}\left( X\right) \), for any family \( \Phi \) of supports on \( Y \) . [Also, see IV-3.] 9. (c) Let \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) be an exact sequence of sheaves on a locally connected Hausdorff space \( X \) . Suppose that \( {\mathcal{A}}^{\prime } \) and \( {\mathcal{A}}^{\prime \prime } \) are locally constant and that the stalks of \( {\mathcal{A}}^{\prime \prime } \) are finitely generated (over some constant base ring). Show that \( \mathcal{A} \) is also locally constant. [Hint: For \( x \in X \) find a neighborhood \( U \) such that \( \mathcal{A}\left( U\right) \rightarrow {\mathcal{A}}^{\prime \prime }\left( U\right) \) is surjective and such that \( {\mathcal{A}}^{\prime }\left( U\right) \rightarrow {\mathcal{A}}_{y}^{\prime } \) and \( {\mathcal{A}}^{\prime \prime }\left( U\right) \rightarrow {\mathcal{A}}_{y}^{\prime \prime } \) are isomorphisms for every \( y \in U \) .] 10. (c) Show by example that Exercise 9 does not hold without the condition that the stalks of \( {\mathcal{A}}^{\prime \prime } \) are finitely generated. 11. (c) If \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) is an exact sequence of constant sheaves on \( X \), show that the sequence \[ 0 \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime }\right) \rightarrow {\Gamma }_{\Phi }\left( \mathcal{A}\right) \rightarrow {\Gamma }_{\Phi }\left( {\mathcal{A}}^{\prime \prime }\right) \rightarrow 0 \] is exact for any family \( \Phi \) of supports on \( X \) . 12. (c) Let \( {\Delta }_{ * }\left( {X, A}\right) \) [respectively, \( {\Delta }_{ * }^{c}\left( {X, A}\right) \) ] be the chain complex of locally finite (respectively, finite) singular chains of \( X \) modulo those chains in \( A \) . Show that the homomorphism of generated sheaves induced by the obvious homomorphism \[ {\Delta }_{ * }^{c}\left( {X, X - U}\right) \hookrightarrow {\Delta }_{ * }\left( {X, X - U}\right) \] of presheaves is an isomorphism. Denote this generated sheaf by \( {\Delta }_{ * } \) . Show that the presheaf \( U \mapsto {\Delta }_{ * }\left( {X, X - U}\right) \) (which generates \( {\Delta }_{ * } \) ) is a mono-presheaf and that it is conjunctive for coverings of \( X \) . Deduce that \[ \theta : {\Delta }_{ * }\left( X\right) \rightarrow \Gamma \left( {\Delta }_{ * }\right) \] is an isomorphism when \( X \) is paracompact. [Note, however, that \( U \mapsto \) \( {\Delta }_{ * }\left( {X, X - U}\right) \) is not fully conjunctive and hence is not itself a sheaf.] Also show that \( \theta \) induces an isomorphism \[ {\Delta }_{ * }^{c}\left( X\right) \overset{ \approx }{ \rightarrow }{\Gamma }_{c}\left( {\Delta }_{ * }\right) \] (This provides another approach to the definition of singular homology with coefficients in a sheaf, by putting \( {\Delta }_{ * }^{c}\left( {X;\mathcal{A}}\right) = {\Gamma }_{c}\left( {{\Delta }_{ * } \otimes \mathcal{A}}\right) \) .) 13. Let \( X \) be the complex line (real 2-dimensional) and let \( C \) denote the constant sheaf of complex numbers. Let \( \mathcal{A} \) be the sheaf of germs of complex analytic functions on \( X \) . Show that \[ 0 \rightarrow C\overset{i}{ \rightarrow }\mathcal{A}\overset{d}{ \rightarrow }\mathcal{A} \rightarrow 0 \] is exact, where \( i \) is the canonical inclusion and \( d \) is differentiation. For \( U \subset X \) open show that \( d : \mathcal{A}\left( U\right) \rightarrow \mathcal{A}\left( U\right) \) need not be surjective. For which open sets \( U \) is it surjective? 14. (c) Let \( X \) be the unit circle in the plane. Let \( \mathbb{R} \) denote the constant sheaf of real numbers; \( \mathcal{D} \) the sheaf of germs of continuously differentiable real-valued functions on \( X \) ; and \( \mathcal{C} \) the sheaf of germs of continuous real-valued functions on \( X \) . Show that \( 0 \rightarrow \mathbb{R}\overset{i}{ \rightarrow }\mathcal{D}\overset{d}{ \rightarrow }\mathcal{C} \rightarrow 0 \) is exact, where \( d \) is differentiation. Show that \( \operatorname{Coker}\{ d : \mathcal{D}\left( X\right) \rightarrow \mathcal{C}\left( X\right) \} \) is isomorphic to the group of real numbers. 15. (c) Let \( X \) be the real line. Let \( \mathcal{F} \) be the sheaf of germs of all integer-valued functions on \( X \) and let \( i : \mathbb{Z} \hookrightarrow \mathcal{F} \) be the canonical inclusion. Let \( \mathcal{G} \) be the quotient sheaf of \( \mathcal{F} \) by \( \mathbb{Z} \) . Show that \( \mathcal{F}\left( X\right) \rightarrow \mathcal{G}\left( X\right) \) is surjective, while \( \operatorname{Coker}\left\{ {{\Gamma }_{c}\left( \mathcal{F}\right) \rightarrow {\Gamma }_{c}\left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \) 16. Show that there are natural isomorphisms \[ \operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \] and \[ \operatorname{Hom}\left( {\mathcal{A},\mathop{\prod }\limits_{\lambda }{\mathcal{B}}_{\lambda }}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {\mathcal{A},{\mathcal{B}}_{\lambda }}\right) \] 17. Prove or disprove that there is the following natural isomorphism of functors of sheaves \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) on \( X \) : \[ \mathcal{H}\text{on}\left( {\mathcal{A},\mathcal{H}\text{on}\left( {\mathcal{B},\mathcal{C}}\right) }\right) \approx \mathcal{H}\text{on}\left( {\mathcal{A} \otimes \mathcal{B},\mathcal{C}}\right) . \] 18. (c) If \( f : A \rightarrow X \) is a map with \( f\left( A\right) \) dense in \( X

1004_(GTM170)Sheaf Theory

\left( \mathcal{G}\right) }\right\} \approx \mathbb{Z}. \) 16. Show that there are natural isomorphisms \[ \operatorname{Hom}\left( {{\bigoplus }_{\lambda }{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {{\mathcal{A}}_{\lambda },\mathcal{B}}\right) \] and \[ \operatorname{Hom}\left( {\mathcal{A},\mathop{\prod }\limits_{\lambda }{\mathcal{B}}_{\lambda }}\right) \approx \mathop{\prod }\limits_{\lambda }\operatorname{Hom}\left( {\mathcal{A},{\mathcal{B}}_{\lambda }}\right) \] 17. Prove or disprove that there is the following natural isomorphism of functors of sheaves \( \mathcal{A},\mathcal{B} \), and \( \mathcal{C} \) on \( X \) : \[ \mathcal{H}\text{on}\left( {\mathcal{A},\mathcal{H}\text{on}\left( {\mathcal{B},\mathcal{C}}\right) }\right) \approx \mathcal{H}\text{on}\left( {\mathcal{A} \otimes \mathcal{B},\mathcal{C}}\right) . \] 18. (c) If \( f : A \rightarrow X \) is a map with \( f\left( A\right) \) dense in \( X \) and \( \mathcal{M} \) is a subsheaf of a constant sheaf on \( X \), then show that the canonical map \( \beta : \mathcal{M} \rightarrow f{f}^{ * }\mathcal{M} \) of Section 4 is a monomorphism. Also, give an example showing that this is false for arbitrary sheaves \( \mathcal{M} \) on \( X \) . 19. (c) For a given point \( x \) in the Hausdorff space \( X \), let \( x \) also denote the family \( \{ \{ x\} ,\varnothing \} \) of supports on \( X \) . Show that \( {}_{AS}{H}_{x}^{n}\left( {X;G}\right) = 0 \) for all \( n > 0 \) . (Compare II-18.) ## Chapter II Sheaf Cohomology In this chapter we shall define the sheaf-theoretic cohomology theory and shall develop many of its basic properties. The cohomology groups of a space with coefficients in a sheaf are defined in Section 2 using the canonical resolution of a sheaf due to Godement. In Section 3 it is shown that the category of sheaves contains "enough injectives," and it follows from the results of Sections 4 and 5 that the sheaf cohomology groups are just the right derived functors of the left exact functor \( \Gamma \) that assigns to a sheaf its group of sections. A sheaf \( \mathcal{A} \) is said to be acyclic if the higher cohomology groups with coefficients in \( \mathcal{A} \) are zero. Such sheaves provide a means of "computing" cohomology in particular situations. In Sections 5 and 9 some important classes of acyclic sheaves are defined and investigated. In Section 6 we prove a theorem concerning the existence and uniqueness of extensions of a natural transformation of functors (of several variables) to natural transformations of "connected systems" of functors. This result is applied in Section 7 to define, and to give axioms for, the cup product in sheaf cohomology theory. These sections are central to our treatment of many of the fundamental consequences of sheaf theory. The cohomology homomorphism induced by a map is defined in Section 8. The relationship between the cohomology of a subspace and that of its neighborhoods is investigated in Section 10, and the important notion of "tautness" of a subspace is introduced there. In Section 11 we prove the Vietoris mapping theorem and use it to prove that sheaf-theoretic cohomology, with constant coefficients, satisfies the invariance under homotopy property for general topological spaces. Relative cohomology theory is introduced into sheaf theory in Section 12, and its properties, such as invariance under excision, are developed. In Section 13 we derive some exact sequences of the Mayer-Vietoris type. Sections 14, 15, and 17 are concerned, almost exclusively, with locally compact spaces. In Section 14 we prove the "continuity" property, both for spaces and for coefficient sheaves. This property is an important feature of sheaf-theoretic cohomology that is not satisfied for such theories as singular cohomology. A general Künneth formula is derived in Section 15. Section 17 treats local connectivity in higher degrees. This section really has nothing to do with sheaf theory, but the results of this section are used repeatedly in later parts of the book. In Section 16 we study the concept of cohomological dimension, which has important applications to other parts of the book. Section 18 contains definitions of "local cohomology groups" and of cohomology groups of the "ideal boundary." If \( G \) is a finite group acting on a space \( X \) and if \( \pi : X \rightarrow X/G \) is the "orbit map," then \( \pi \) induces, as does any map, a homomorphism from the cohomology of \( X/G \) to that of \( X \) . In Section 19 the "transfer map," which takes the cohomology of \( X \) into that of \( X/G \), is defined. When \( G \) is cyclic of prime order we also obtain the exact sequences of P. A. Smith relating the cohomology of the fixed point set of \( G \) to that of \( X \) (a general Hausdorff space on which \( G \) acts). In Sections 20 and 21 we define the Steenrod cohomology operations (the squares and \( p \) th powers) in sheaf cohomology and derive several of their properties. This material is not used elsewhere in the book. All of the sections of this chapter, except for Sections 18 through 21, are used repeatedly in other parts of the book. Most of Chapter III can be read after Section 9 of the present chapter. ## 1 Differential sheaves and resolutions 1.1. Definition. A "graded sheaf" \( {\mathcal{X}}^{ * } \) is a sequence \( \left\{ {\mathcal{L}}^{p}\right\} \) of sheaves, \( p \) ranging over the integers. A "differential sheaf" is a graded sheaf together with homomorphisms \( d : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 1} \) such that \( {d}^{2} : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 2} \) is zero for all \( p \) . A "resolution" of a sheaf \( \mathcal{A} \) is a differential sheaf \( {\mathcal{L}}^{ * } \) with \( {\mathcal{L}}^{p} = 0 \) for \( p < 0 \) together with an "augmentation" homomorphism \( \varepsilon : \mathcal{A} \rightarrow {\mathcal{L}}^{0} \) such that the sequence \( 0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{L}}^{0}\overset{d}{ \rightarrow }{\mathcal{L}}^{1}\overset{d}{ \rightarrow }{\mathcal{L}}^{2} \rightarrow \cdots \) is exact. Similarly, one can define graded and differential presheaves. Since exact sequences and direct limits commute, it follows that the functor Meal, assigning to a presheaf its associated sheaf, is an exact functor. Thus if \( A\overset{f}{ \rightarrow }B\overset{g}{ \rightarrow }C \) is a sequence of presheaves of order two [i.e., \( g\left( U\right) \circ f\left( U\right) = 0 \) for all \( U \) ] and if \( \mathcal{A}\overset{{f}^{\prime }}{ \rightarrow }\mathcal{B}\overset{{g}^{\prime }}{ \rightarrow }\mathcal{C} \) is the induced sequence of generated sheaves, then \( \operatorname{Im}{f}^{\prime } \) and \( \operatorname{Ker}{g}^{\prime } \) are generated respectively by the presheaves \( \operatorname{Im}f \) and \( \operatorname{Ker}g \) . Similarly, the sheaf \( \operatorname{Ker}{g}^{\prime }/\operatorname{Im}{f}^{\prime } \) is (naturally isomorphic to) the sheaf generated by the pre-sheaf \( \operatorname{Ker}g/\operatorname{Im}f : U \mapsto \operatorname{Ker}g\left( U\right) /\operatorname{Im}f\left( U\right) \) . If \( {\mathcal{L}}^{ * } \) is a differential sheaf then we define its homology sheaf (or "derived sheaf") to be the graded sheaf \( {\mathcal{H}}^{ * }\left( {\mathcal{L}}^{ * }\right) \), where as usual, \[ {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \operatorname{Ker}\left( {d : {\mathcal{L}}^{p} \rightarrow {\mathcal{L}}^{p + 1}}\right) /\operatorname{Im}\left( {d : {\mathcal{L}}^{p - 1} \rightarrow {\mathcal{L}}^{p}}\right) . \] The preceding remarks show that \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \mathcal{R} \) eq. \( \left( {U \mapsto {H}^{p}\left( {{\mathcal{L}}^{ * }\left( U\right) }\right) }\right) \) , and in general, if \( {\mathcal{L}}^{ * } \) is generated by the differential presheaf \( {L}^{ * } \), then \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) = \mathcal{P} \) heaf \( \left( {U \mapsto {H}^{p}\left( {{L}^{ * }\left( U\right) }\right) }\right) \) . Note that in general, \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) \left( U\right) ≉ {H}^{p}\left( {{\mathcal{L}}^{ * }\left( U\right) }\right) \) . [For example, if we let \( {\mathcal{L}}^{0} = {\mathcal{L}}^{1} \) be the "twisted" sheaf with stalks \( \mathbb{Z} \) on \( X = {\mathbb{S}}^{1} \) and let \( {\mathcal{L}}^{2} = {\mathbb{Z}}_{2} \), the constant sheaf, then \( 0 \rightarrow {\mathcal{L}}^{0}\overset{2}{ \rightarrow }{\mathcal{L}}^{1} \rightarrow {\mathcal{L}}^{2} \rightarrow 0 \) is exact, so that \( {\mathcal{H}}^{p}\left( {\mathcal{L}}^{ * }\right) \left( X\right) = 0 \) for all \( p \) . However, \( {\mathcal{L}}^{0}\left( X\right) = 0 = {\mathcal{L}}^{1}\left( X\right) \) and \( \left. {{\mathcal{L}}^{2}\left( X\right) \approx {\mathbb{Z}}_{2}\text{, so that }{H}^{2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \) 1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters). We have the differential presheaf \[ 0 \rightarrow G \rightarrow {S}^{0}\left( {U;G}\right) \rightarrow {S}^{1}\left( {U;G}\right) \rightarrow \cdots , \] (1) where \( G \rightarrow {S}^{0}\left( {U;G}\right) \) is the usual augmentation. Here we regard \( G \) as the constant presheaf \( U \mapsto G\left( U\right) = G \) . This generates the differential sheaf \[ 0 \rightarrow G \rightarrow {\mathcal{S}}^{0}\left( {X;G}\right) \rightarrow {\mathcal{S}}^{1}\left( {X;G}\right) \rightarrow \cdots \] (2) When is this exact? That is, when is \( {\mathcal{P}}^{ * }\left( {X;G}\right) \) a resolution of \( G \) ? Clearly this sequence is exact at \( G \) since (1) is. The homology

1004_(GTM170)Sheaf Theory

{2}\left( {{\mathcal{L}}^{ * }\left( X\right) }\right) \approx {\mathbb{Z}}_{2}\text{. }}\right\rbrack \) 1.2. Example. In singular cohomology let \( G \) be the coefficient group (that is, the constant sheaf with stalk \( G \) ; this is no loss of generality since we are interested here in local matters). We have the differential presheaf \[ 0 \rightarrow G \rightarrow {S}^{0}\left( {U;G}\right) \rightarrow {S}^{1}\left( {U;G}\right) \rightarrow \cdots , \] (1) where \( G \rightarrow {S}^{0}\left( {U;G}\right) \) is the usual augmentation. Here we regard \( G \) as the constant presheaf \( U \mapsto G\left( U\right) = G \) . This generates the differential sheaf \[ 0 \rightarrow G \rightarrow {\mathcal{S}}^{0}\left( {X;G}\right) \rightarrow {\mathcal{S}}^{1}\left( {X;G}\right) \rightarrow \cdots \] (2) When is this exact? That is, when is \( {\mathcal{P}}^{ * }\left( {X;G}\right) \) a resolution of \( G \) ? Clearly this sequence is exact at \( G \) since (1) is. The homology of (1) is just the reduced singular cohomology group \( {}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) \) . Thus (2) is exact \( \Leftrightarrow \) the sheaf \( {}_{\Delta }{\widetilde{\mathcal{H}}}^{ * }\left( {X;G}\right) = \mathcal{R} \) eaf \( \left( {U \mapsto {}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) }\right) \) is trivial. This is the case \( \Leftrightarrow \) \[ \mathop{\lim }\limits_{ \rightarrow }{}_{\Delta }{\widetilde{H}}^{ * }\left( {U;G}\right) = 0 \] (3) for all \( x \in X \), where \( U \) ranges over the neighborhoods of \( x \) . In the terminology of Spanier,(3) is the condition that the point \( x \) be "taut" with respect to singular cohomology with coefficients in \( G \) . Note that this condition is implied by the condition \( {HL}{C}_{\mathbb{Z}}^{\infty } \) . [A space \( X \) is said to be \( {HL}{C}_{L}^{n} \) (homologically locally connected) if for each \( x \in X \) and neighborhood \( \bar{U} \) of \( x \), there is a neighborhood \( V \subset U \) of \( x \), depending on \( p \), such that the homomorphism \( {}_{\Delta }{\widetilde{H}}_{p}\left( {V;L}\right) \rightarrow {}_{\Delta }{\widetilde{H}}_{p}\left( {U;L}\right) \) is trivial for \( p \leq n \) . Obviously any locally contractible space, and hence any manifold or CW-complex, is HLC.] An example of a space that does not satisfy this condition is the union \( X \) of circles of radius \( 1/n \) all tangent to the \( x \) -axis at the origin. It is clear that this is not \( {HL}{C}^{1} \), and at least in the case of rational coefficients, the sheaf \( {}_{\Delta }{\mathcal{H}}^{1}\left( {X;\mathbb{Q}}\right) \neq 0 \) for this space. 1.3. Example. The Alexander-Spanier presheaf \( {A}^{ * }\left( {\bullet ;G}\right) \) provides a differential presheaf \[ 0 \rightarrow G \rightarrow {A}^{0}\left( {U;G}\right) \rightarrow {A}^{1}\left( {U;G}\right) \rightarrow \cdots \] (4) and hence a differential sheaf. However, in this case (4) is already exact. [For if \( f : {U}^{p + 1} \rightarrow G \) and \( {df} = 0 \), define \( g : {U}^{p} \rightarrow G \) by \( g\left( {{x}_{0},\ldots ,{x}_{p - 1}}\right) = \) \( f\left( {x,{x}_{0},\ldots ,{x}_{p - 1}}\right) \), where \( x \) is an arbitrary element of \( U \) . Then \[ {dg}\left( {{x}_{0},\ldots ,{x}_{p}}\right) = \sum {\left( -1\right) }^{i}g\left( {{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right) \] \[ = \sum {\left( -1\right) }^{i}f\left( {x,{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right) \] \[ = \overline{f}\left( {{x}_{0},\ldots ,{x}_{p}}\right) - {df}\left( {x,{x}_{0},\ldots ,\widehat{{x}_{i}},\ldots ,{x}_{p}}\right) \] \[ = f\left( {{x}_{0},\ldots ,{x}_{p}}\right) \] so that \( f = {dg} \) .] Thus the Alexander-Spanier sheaf \( {\mathcal{A}}^{ * }\left( {X;G}\right) \) is always a resolution of \( G \) . 1.4. Example. The de Rham sheaf \( {\Omega }^{ * } \) on any differentiable manifold is a differential sheaf and has an augmentation \( \mathbb{R} \rightarrow {\Omega }^{0} \) defined by taking a real number \( r \) into the constant function on \( X \) with value \( r \) . Moreover, \( 0 \rightarrow \mathbb{R} \rightarrow {\Omega }^{0} \rightarrow {\Omega }^{1} \rightarrow \cdots \) is an exact sequence of sheaves, as follows from the Poincaré Lemma, which states that every closed differential form on euclidean space is exact; see [19, V-9.2]. Therefore, \( {\Omega }^{ * } \) is a resolution of \( \mathbb{R} \) . ## 2 The canonical resolution and sheaf cohomology For any sheaf \( \mathcal{A} \) on \( X \) and open set \( U \subset X \) we let \( {C}^{0}\left( {U;\mathcal{A}}\right) \) be the collection of all functions (not necessarily continuous) \( f : U \rightarrow \mathcal{A} \) such that \( \pi \circ f \) is the identity on \( U,\pi : \mathcal{A} \rightarrow X \) being the canonical projection. Such possibly discontinuous sections are called serrations, a terminology introduced by Bourgin [10]. That is, \[ {C}^{0}\left( {U;\mathcal{A}}\right) = \mathop{\prod }\limits_{{x \in U}}{\mathcal{A}}_{x} \] Under pointwise operations, this is a group, and the functor \( U \mapsto {C}^{0}\left( {U;\mathcal{A}}\right) \) is a conjunctive monopresheaf on \( X \) . Hence this presheaf is a sheaf, which will be denoted by \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \) . Note that if \( {X}_{d} \) denotes the point set of \( X \) with the discrete topology and if \( f : {X}_{d} \rightarrow X \) is the canonical map, then \( {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \approx f{f}^{ * }\mathcal{A} \), as already mentioned in I-4. Inclusion of the collection of sections of \( \mathcal{A} \) in the collection of all ser-rations gives an inclusion \( \mathcal{A}\left( U\right) \hookrightarrow {C}^{0}\left( {U;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \left( U\right) \) and hence provides a natural monomorphism \[ \varepsilon : \mathcal{A} \rightarrowtail {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) . \] (For \( f : {X}_{d} \rightarrow X \) as above, this inclusion coincides with the monomorphism \( \beta : \mathcal{A} \rightarrowtail f{f}^{ * }\mathcal{A} \) of (6) on page 15.) If \( \Phi \) is a family of supports on \( X \), we put \[ {C}_{\Phi }^{0}\left( {X;\mathcal{A}}\right) = {\Gamma }_{\Phi }\left( {{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) }\right) \] Then for any exact sequence \( 0 \rightarrow {\mathcal{A}}^{\prime } \rightarrow \mathcal{A} \rightarrow {\mathcal{A}}^{\prime \prime } \rightarrow 0 \) of sheaves, the corresponding sequence of presheaves \[ 0 \rightarrow {C}^{0}\left( {\bullet ;{\mathcal{A}}^{\prime }}\right) \rightarrow {C}^{0}\left( {\bullet ;\mathcal{A}}\right) \rightarrow {C}^{0}\left( {\bullet ;{\mathcal{A}}^{\prime \prime }}\right) \rightarrow 0 \] is obviously exact. Moreover, for any family \( \Phi \) of supports, the sequence \[ 0 \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;{\mathcal{A}}^{\prime }}\right) \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;\mathcal{A}}\right) \rightarrow {C}_{\Phi }^{0}\left( {\bullet ;{\mathcal{A}}^{\prime \prime }}\right) \rightarrow 0 \] is exact. [To see that the last map is onto, we recall that \( f \in {C}_{\Phi }^{0}\left( {X;{\mathcal{A}}^{\prime \prime }}\right) \) can be regarded as a serration \( \widehat{f} : X \rightarrow {\mathcal{A}}^{\prime \prime } \) . Then \( \left| f\right| \) is the closure of \( \{ x \mid \widehat{f}\left( x\right) = 0\} \) . Clearly \( \widehat{f} \) is the image of a serration \( \widehat{g} \) of \( \mathcal{A} \) that vanishes wherever \( \widehat{f} \) vanishes. Thus \( \left. {\left| g\right| = \left| f\right| \in \Phi \text{.}}\right\rbrack \) Let \( {\mathcal{Z}}^{1}\left( {X;\mathcal{A}}\right) = \operatorname{Coker}\left\{ {\varepsilon : \mathcal{A} \rightarrow {\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) }\right\} \), so that the sequence \[ 0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{1}\left( {X;\mathcal{A}}\right) \rightarrow 0 \] is exact. We also define, inductively, \[ {\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) = {\mathcal{C}}^{0}\left( {X;{\mathcal{J}}^{n}\left( {X;\mathcal{A}}\right) }\right) \] \[ {\mathcal{Z}}^{n + 1}\left( {X;\mathcal{A}}\right) = {\mathcal{Z}}^{1}\left( {X;{\mathcal{Z}}^{n}\left( {X;\mathcal{A}}\right) }\right) \] so that \[ 0 \rightarrow {\mathcal{J}}^{n}\left( {X;\mathcal{A}}\right) \overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{n + 1}\left( {X;\mathcal{A}}\right) \rightarrow 0 \] is exact. Let \( d = \varepsilon \circ \partial \) be the composition \[ {\mathcal{C}}^{n}\left( {X;\mathcal{A}}\right) \overset{\partial }{ \rightarrow }{\mathcal{J}}^{n + 1}\left( {X;\mathcal{A}}\right) \overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{n + 1}\left( {X;\mathcal{A}}\right) \] Then the sequence \[ 0 \rightarrow \mathcal{A}\overset{\varepsilon }{ \rightarrow }{\mathcal{C}}^{0}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{1}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }{\mathcal{C}}^{2}\left( {X;\mathcal{A}}\right) \overset{d}{ \rightarrow }\ldots \] is exact. That is, \( {\mathcal{C}}^{ * }\left( {X;\mathcal{A}}\right) \) is a resolution of \( \mathcal{A} \) . It is called the canonical resolution of \( \mathcal{A} \) and is due to Godement [40]. This resolution satisfies the stronger property of being naturally "pointwise homotopically trivial." In fact, for \( x \in U \subset X \) consider the homomorphism \( {C}^{0}\left( {U;\mathcal{A}}\right) \rightarrow {\mathcal{A}}_{x} \) that assigns to a serration \( U \rightarrow \mathcal{A} \) its value at \( x \) . Passing to the limit over neighborhoods of \( x \), this induces a homomorphism \( {\eta }_{x} : {\mathcal{C}}^{0}{\left( X;\mathcal{A}\right) }_{x} \rightarr