Coherent sheaf cohomology

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.
Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.
Coherent sheaves.
Coherent sheaves can be seen as a generalization of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on a scheme. In both cases, the given space formula_1 comes with a sheaf of rings formula_2, the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a full subcategory of the category of formula_2-modules (that is, sheaves of formula_2-modules).
Vector bundles such as the tangent bundle play a fundamental role in geometry. More generally, for a closed subvariety formula_5 of formula_1 with inclusion formula_7, a vector bundle formula_8 on formula_5 determines a coherent sheaf on formula_1, the direct image sheaf formula_11, which is zero outside formula_5. In this way, many questions about subvarieties of formula_1 can be expressed in terms of coherent sheaves on formula_1.
Unlike vector bundles, coherent sheaves (in the analytic or algebraic case) form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. On a scheme, the quasi-coherent sheaves are a generalization of coherent sheaves, including the locally free sheaves of infinite rank.
Sheaf cohomology.
For a sheaf formula_15 of abelian groups on a topological space formula_1, the sheaf cohomology groups formula_17 for integers formula_18 are defined as the right derived functors of the functor of global sections, formula_19. As a result, formula_17 is zero for formula_21, and formula_22 can be identified with formula_23. For any short exact sequence of sheaves formula_24, there is a long exact sequence of cohomology groups:
If formula_15 is a sheaf of formula_2-modules on a scheme formula_1, then the cohomology groups formula_17 (defined using the underlying topological space of formula_1) are modules over the ring formula_31 of regular functions. For example, if formula_1 is a scheme over a field formula_33, then the cohomology groups formula_17 are formula_33-vector spaces. The theory becomes powerful when formula_15 is a coherent or quasi-coherent sheaf, because of the following sequence of results.
Vanishing theorems in the affine case.
Complex analysis was revolutionized by Cartan's theorems A and B in 1953. These results say that if formula_15 is a coherent analytic sheaf on a Stein space formula_1, then formula_15 is spanned by its global sections, and formula_40 for all formula_41. (A complex space formula_1 is Stein if and only if it is isomorphic to a closed analytic subspace of formula_43 for some formula_44.) These results generalize a large body of older work about the construction of complex analytic functions with given singularities or other properties.
In 1955, Serre introduced coherent sheaves into algebraic geometry (at first over an algebraically closed field, but that restriction was removed by Grothendieck). The analogs of Cartan's theorems hold in great generality: if formula_15 is a quasi-coherent sheaf on an affine scheme formula_1, then formula_15 is spanned by its global sections, and formula_40 for formula_49. This is related to the fact that the category of quasi-coherent sheaves on an affine scheme formula_1 is equivalent to the category of formula_31-modules, with the equivalence taking a sheaf formula_15 to the formula_31-module formula_22. In fact, affine schemes are characterized among all quasi-compact schemes by the vanishing of higher cohomology for quasi-coherent sheaves.
Čech cohomology and the cohomology of projective space.
As a consequence of the vanishing of cohomology for affine schemes: for a separated scheme formula_1, an affine open covering formula_56 of formula_1, and a quasi-coherent sheaf formula_15 on formula_1, the cohomology groups formula_60 are isomorphic to the Čech cohomology groups with respect to the open covering formula_56. In other words, knowing the sections of formula_15 on all finite intersections of the affine open subschemes formula_63 determines the cohomology of formula_1 with coefficients in formula_15.
Using Čech cohomology, one can compute the cohomology of projective space with coefficients in any line bundle. Namely, for a field formula_33, a positive integer formula_44, and any integer formula_68, the cohomology of projective space formula_69 over formula_33 with coefficients in the line bundle formula_71 is given by:
In particular, this calculation shows that the cohomology of projective space over formula_33 with coefficients in any line bundle has finite dimension as a formula_33-vector space.
The vanishing of these cohomology groups above dimension formula_44 is a very special case of Grothendieck's vanishing theorem: for any sheaf of abelian groups formula_15 on a Noetherian topological space formula_1 of dimension formula_78, formula_79 for all formula_80. This is especially useful for formula_1 a Noetherian scheme (for example, a variety over a field) and formula_15 a quasi-coherent sheaf.
Sheaf cohomology of plane-curves.
Given a smooth projective plane curve formula_83 of degree formula_84, the sheaf cohomology formula_85 can be readily computed using a long exact sequence in cohomology. First note that for the embedding formula_86 there is the isomorphism of cohomology groups
since formula_88 is exact. This means that the short exact sequence of coherent sheaves
on formula_90, called the ideal sequence, can be used to compute cohomology via the long exact sequence in cohomology. The sequence reads as
which can be simplified using the previous computations on projective space. For simplicity, assume the base ring is formula_92 (or any algebraically closed field). Then there are the isomorphisms
which shows that formula_94 of the curve is a finite dimensional vector space of rank
Kunneth Theorem.
There is an analogue of the Kunneth formula in coherent sheaf cohomology for products of varieties. Given quasi-compact schemes formula_96 with affine-diagonals over a field formula_33, (e.g. separated schemes), and let formula_98 and formula_99, then there is an isomorphism formula_100 where formula_101 are the canonical projections of formula_102 to formula_96.
Computing sheaf cohomology of curves.
In formula_104, a generic section of formula_105 defines a curve formula_83, giving the ideal sequenceformula_107Then, the long exact sequence reads asformula_108giving formula_109Since formula_94is the genus of the curve, we can use the Kunneth formula to compute its Betti numbers. This isformula_111which is of rankformula_112for formula_113. In particular, if formula_83 is defined by the vanishing locus of a generic section of formula_115, it is of genusformula_116hence a curve of any genus can be found inside of formula_117.
Finite-dimensionality.
For a proper scheme formula_1 over a field formula_33 and any coherent sheaf formula_15 on formula_1, the cohomology groups formula_122 have finite dimension as formula_33-vector spaces. In the special case where formula_1 is projective over formula_33, this is proved by reducing to the case of line bundles on projective space, discussed above. In the general case of a proper scheme over a field, Grothendieck proved the finiteness of cohomology by reducing to the projective case, using Chow's lemma.
The finite-dimensionality of cohomology also holds in the analogous situation of coherent analytic sheaves on any compact complex space, by a very different argument. Cartan and Serre proved finite-dimensionality in this analytic situation using a theorem of Schwartz on compact operators in Fréchet spaces. Relative versions of this result for a proper morphism were proved by Grothendieck (for locally Noetherian schemes) and by Grauert (for complex analytic spaces). Namely, for a proper morphism formula_126 (in the algebraic or analytic setting) and a coherent sheaf formula_15 on formula_1, the higher direct image sheaves formula_129 are coherent. When formula_5 is a point, this theorem gives the finite-dimensionality of cohomology.
The finite-dimensionality of cohomology leads to many numerical invariants for projective varieties. For example, if formula_1 is a smooth projective curve over an algebraically closed field formula_33, the genus of formula_1 is defined to be the dimension of the formula_33-vector space formula_135. When formula_33 is the field of complex numbers, this agrees with the genus of the space formula_137 of complex points in its classical (Euclidean) topology. (In that case, formula_138 is a closed oriented surface.) Among many possible higher-dimensional generalizations, the geometric genus of a smooth projective variety formula_1 of dimension formula_44 is the dimension of formula_141, and the arithmetic genus (according to one convention) is the alternating sum
Serre duality.
Serre duality is an analog of Poincaré duality for coherent sheaf cohomology. In this analogy, the canonical bundle formula_143 plays the role of the orientation sheaf. Namely, for a smooth proper scheme formula_1 of dimension formula_44 over a field formula_33, there is a natural trace map formula_147, which is an isomorphism if formula_1 is geometrically connected, meaning that the base change of formula_1 to an algebraic closure of formula_33 is connected. Serre duality for a vector bundle formula_8 on formula_1 says that the product
is a perfect pairing for every integer formula_18. In particular, the formula_33-vector spaces formula_156 and formula_157 have the same (finite) dimension. (Serre also proved Serre duality for holomorphic vector bundles on any compact complex manifold.) Grothendieck duality theory includes generalizations to any coherent sheaf and any proper morphism of schemes, although the statements become less elementary.
For example, for a smooth projective curve formula_1 over an algebraically closed field formula_33, Serre duality implies that the dimension of the space formula_160 of 1-forms on formula_1 is equal to the genus of formula_1 (the dimension of formula_135).
GAGA theorems.
GAGA theorems relate algebraic varieties over the complex numbers to the corresponding analytic spaces. For a scheme "X" of finite type over C, there is a functor from coherent algebraic sheaves on "X" to coherent analytic sheaves on the associated analytic space "X"an. The key GAGA theorem (by Grothendieck, generalizing Serre's theorem on the projective case) is that if "X" is proper over C, then this functor is an equivalence of categories. Moreover, for every coherent algebraic sheaf "E" on a proper scheme "X" over C, the natural map
of (finite-dimensional) complex vector spaces is an isomorphism for all "i". (The first group here is defined using the Zariski topology, and the second using the classical (Euclidean) topology.) For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CP"n" is algebraic.
Vanishing theorems.
Serre's vanishing theorem says that for any ample line bundle formula_165 on a proper scheme formula_1 over a Noetherian ring, and any coherent sheaf formula_15 on formula_1, there is an integer formula_169 such that for all formula_170, the sheaf formula_171 is spanned by its global sections and has no cohomology in positive degrees.
Although Serre's vanishing theorem is useful, the inexplicitness of the number formula_169 can be a problem. The Kodaira vanishing theorem is an important explicit result. Namely, if formula_1 is a smooth projective variety over a field of characteristic zero, formula_165 is an ample line bundle on formula_1, and formula_143 a canonical bundle, then
for all formula_178. Note that Serre's theorem guarantees the same vanishing for large powers of formula_165. Kodaira vanishing and its generalizations are fundamental to the classification of algebraic varieties and the minimal model program. Kodaira vanishing fails over fields of positive characteristic.
Hodge theory.
The Hodge theorem relates coherent sheaf cohomology to singular cohomology (or de Rham cohomology). Namely, if formula_1 is a smooth complex projective variety, then there is a canonical direct-sum decomposition of complex vector spaces:
for every formula_182. The group on the left means the singular cohomology of formula_183 in its classical (Euclidean) topology, whereas the groups on the right are cohomology groups of coherent sheaves, which (by GAGA) can be taken either in the Zariski or in the classical topology. The same conclusion holds for any smooth proper scheme formula_1 over formula_185, or for any compact Kähler manifold.
For example, the Hodge theorem implies that the definition of the genus of a smooth projective curve formula_1 as the dimension of formula_187, which makes sense over any field formula_33, agrees with the topological definition (as half the first Betti number) when formula_33 is the complex numbers. Hodge theory has inspired a large body of work on the topological properties of complex algebraic varieties.
Riemann–Roch theorems.
For a proper scheme "X" over a field "k", the Euler characteristic of a coherent sheaf "E" on "X" is the integer
The Euler characteristic of a coherent sheaf "E" can be computed from the Chern classes of "E", according to the Riemann–Roch theorem and its generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem. For example, if "L" is a line bundle on a smooth proper geometrically connected curve "X" over a field "k", then
where deg("L") denotes the degree of "L".
When combined with a vanishing theorem, the Riemann–Roch theorem can often be used to determine the dimension of the vector space of sections of a line bundle. Knowing that a line bundle on "X" has enough sections, in turn, can be used to define a map from "X" to projective space, perhaps a closed immersion. This approach is essential for classifying algebraic varieties.
The Riemann–Roch theorem also holds for holomorphic vector bundles on a compact complex manifold, by the Atiyah–Singer index theorem.
Growth.
Dimensions of cohomology groups on a scheme of dimension "n" can grow up at most like a polynomial of degree "n".
Let "X" be a projective scheme of dimension "n" and "D" a divisor on "X". If formula_15 is any coherent sheaf on "X" then
formula_193 for every "i".
For a higher cohomology of nef divisor "D" on "X";
formula_194
Applications.
Given a scheme "X" over a field "k", deformation theory studies the deformations of "X" to infinitesimal neighborhoods. The simplest case, concerning deformations over the ring formula_195 of dual numbers, examines whether there is a scheme "X""R" over Spec "R" such that the special fiber
is isomorphic to the given "X". Coherent sheaf cohomology with coefficients in the tangent sheaf formula_197 controls this class of deformations of "X", provided "X" is smooth. Namely,