Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.
Definitions.
A quasi-coherent sheaf on a ringed space formula_1 is a sheaf formula_2 of formula_3-modules which has a local presentation, that is, every point in formula_4 has an open neighborhood formula_5 in which there is an exact sequence
for some (possibly infinite) sets formula_7 and formula_8.
A coherent sheaf on a ringed space formula_1 is a sheaf formula_2 satisfying the following two properties:
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of formula_3-modules.
The case of schemes.
When formula_4 is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf formula_2 of formula_3-modules is quasi-coherent if and only if over each open affine subscheme formula_27 the restriction formula_28 is isomorphic to the sheaf formula_29 associated to the module formula_30 over formula_31. When formula_4 is a locally Noetherian scheme, formula_2 is coherent if and only if it is quasi-coherent and the modules formula_34 above can be taken to be finitely generated.
On an affine scheme formula_35, there is an equivalence of categories from formula_31-modules to quasi-coherent sheaves, taking a module formula_34 to the associated sheaf formula_29. The inverse equivalence takes a quasi-coherent sheaf formula_2 on formula_5 to the formula_31-module formula_42 of global sections of formula_2.
Here are several further characterizations of quasi-coherent sheaves on a scheme.
Properties.
On an arbitrary ringed space quasi-coherent sheaves do not necessarily form an abelian category. On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.
On any ringed space formula_4, the coherent sheaves form an abelian category, a full subcategory of the category of formula_3-modules. (Analogously, the category of coherent modules over any ring formula_31 is a full abelian subcategory of the category of all formula_31-modules.) So the kernel, image, and cokernel of any map of coherent sheaves are coherent. The direct sum of two coherent sheaves is coherent; more generally, an formula_3-module that is an extension of two coherent sheaves is coherent.
A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always an formula_3-module of "finite presentation", meaning that each point formula_50 in formula_4 has an open neighborhood formula_5 such that the restriction formula_28 of formula_2 to formula_5 is isomorphic to the cokernel of a morphism formula_56 for some natural numbers formula_17 and formula_58. If formula_3 is coherent, then, conversely, every sheaf of finite presentation over formula_3 is coherent.
The sheaf of rings formula_3 is called coherent if it is coherent considered as a sheaf of modules over itself. In particular, the Oka coherence theorem states that the sheaf of holomorphic functions on a complex analytic space formula_4 is a coherent sheaf of rings. The main part of the proof is the case formula_63. Likewise, on a locally Noetherian scheme formula_4, the structure sheaf formula_3 is a coherent sheaf of rings.
Functoriality.
Let formula_160 be a morphism of ringed spaces (for example, a morphism of schemes). If formula_2 is a quasi-coherent sheaf on formula_162, then the inverse image formula_3-module (or pullback) formula_164 is quasi-coherent on formula_4. For a morphism of schemes formula_160 and a coherent sheaf formula_2 on formula_162, the pullback formula_164 is not coherent in full generality (for example, formula_170, which might not be coherent), but pullbacks of coherent sheaves are coherent if formula_4 is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle.
If formula_160 is a quasi-compact quasi-separated morphism of schemes and formula_2 is a quasi-coherent sheaf on formula_4, then the direct image sheaf (or pushforward) formula_175 is quasi-coherent on formula_162.
The direct image of a coherent sheaf is often not coherent. For example, for a field formula_177, let formula_4 be the affine line over formula_177, and consider the morphism formula_180; then the direct image formula_181 is the sheaf on formula_182 associated to the polynomial ring formula_183, which is not coherent because formula_183 has infinite dimension as a formula_177-vector space. On the other hand, the direct image of a coherent sheaf under a proper morphism is coherent, by results of Grauert and Grothendieck.
Local behavior of coherent sheaves.
An important feature of coherent sheaves formula_2 is that the properties of formula_2 at a point formula_50 control the behavior of formula_2 in a neighborhood of formula_50, more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if formula_2 is a coherent sheaf on a scheme formula_4, then the fiber formula_193 of formula_2 at a point formula_50 (a vector space over the residue field formula_196) is zero if and only if the sheaf formula_197 is zero on some open neighborhood of formula_50. A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous. Thus a coherent sheaf has constant rank on an open set, while the rank can jump up on a lower-dimensional closed subset.
In the same spirit: a coherent sheaf formula_2 on a scheme formula_4 is a vector bundle if and only if its stalk formula_201 is a free module over the local ring formula_202 for every point formula_50 in formula_4.
On a general scheme, one cannot determine whether a coherent sheaf is a vector bundle just from its fibers (as opposed to its stalks). On a reduced locally Noetherian scheme, however, a coherent sheaf is a vector bundle if and only if its rank is locally constant.
Examples of vector bundles.
For a morphism of schemes formula_205, let formula_206 be the diagonal morphism, which is a closed immersion if formula_4 is separated over formula_162. Let formula_209 be the ideal sheaf of formula_4 in formula_211. Then the sheaf of differentials formula_212 can be defined as the pullback formula_213 of formula_209 to formula_4. Sections of this sheaf are called 1-forms on formula_4 over formula_162, and they can be written locally on formula_4 as finite sums formula_219 for regular functions formula_220 and formula_221. If formula_4 is locally of finite type over a field formula_177, then formula_224 is a coherent sheaf on formula_4.
If formula_4 is smooth over formula_177, then formula_228 (meaning formula_224) is a vector bundle over formula_4, called the cotangent bundle of formula_4. Then the tangent bundle formula_232 is defined to be the dual bundle formula_233. For formula_4 smooth over formula_177 of dimension formula_17 everywhere, the tangent bundle has rank formula_17.
If formula_162 is a smooth closed subscheme of a smooth scheme formula_4 over formula_177, then there is a short exact sequence of vector bundles on formula_162:
which can be used as a definition of the normal bundle formula_243 to formula_162 in formula_4.
For a smooth scheme formula_4 over a field formula_177 and a natural number formula_248, the vector bundle formula_249 of "i"-forms on formula_4 is defined as the formula_248-th exterior power of the cotangent bundle, formula_252. For a smooth variety formula_4 of dimension formula_17 over formula_177, the canonical bundle formula_256 means the line bundle formula_257. Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on formula_4. For example, a section of the canonical bundle of affine space formula_259 over formula_177
can be written as
where formula_111 is a polynomial with coefficients in formula_177.
Let formula_95 be a commutative ring and formula_17 a natural number. For each integer formula_266, there is an important example of a line bundle on projective space formula_267 over formula_95, called formula_269. To define this, consider the morphism of formula_95-schemes
given in coordinates by formula_272. (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of formula_269 over an open subset formula_5 of formula_267 is defined to be a regular function formula_111 on formula_277 that is homogeneous of degree formula_266, meaning that
as regular functions on (formula_280. For all integers formula_248 and formula_266, there is an isomorphism formula_283 of line bundles on formula_267.
In particular, every homogeneous polynomial in formula_285 of degree formula_266 over formula_95 can be viewed as a global section of formula_269 over formula_267. Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles formula_269. This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space formula_267 over formula_95 are just the "constants" (the ring formula_95), and so it is essential to work with the line bundles formula_269.
Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let formula_95 be a Noetherian ring (for example, a field), and consider the polynomial ring formula_296 as a graded ring with each formula_297 having degree 1. Then every finitely generated graded formula_298-module formula_34 has an associated coherent sheaf formula_300 on formula_267 over formula_95. Every coherent sheaf on formula_267 arises in this way from a finitely generated graded formula_298-module formula_34. (For example, the line bundle formula_269 is the sheaf associated to the formula_298-module formula_298 with its grading lowered by formula_266.) But the formula_298-module formula_34 that yields a given coherent sheaf on formula_267 is not unique; it is only unique up to changing formula_34 by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on formula_267 is the quotient of the category of finitely generated graded formula_298-modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.
The tangent bundle of projective space formula_267 over a field formula_177 can be described in terms of the line bundle formula_318. Namely, there is a short exact sequence, the Euler sequence:
It follows that the canonical bundle formula_320 (the dual of the determinant line bundle of the tangent bundle) is isomorphic to formula_321. This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle formula_318 means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.
Vector bundles on a hypersurface.
Consider a smooth degree-formula_323 hypersurface formula_324 defined by the homogeneous polynomial formula_111 of degree formula_323. Then, there is an exact sequence
where the second map is the pullback of differential forms, and the first map sends
Note that this sequence tells us that formula_329 is the conormal sheaf of formula_4 in formula_267. Dualizing this yields the exact sequence
hence formula_333 is the normal bundle of formula_4 in formula_267. If we use the fact that given an exact sequence
of vector bundles with ranks formula_337,formula_338,formula_339, there is an isomorphism
of line bundles, then we see that there is the isomorphism
showing that
Chern classes and algebraic "K"-theory.
A vector bundle formula_79 on a smooth variety formula_4 over a field has Chern classes in the Chow ring of formula_4, formula_346 in formula_347 for formula_348. These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence
of vector bundles on formula_4, the Chern classes of formula_351 are given by
It follows that the Chern classes of a vector bundle formula_79 depend only on the class of formula_79 in the Grothendieck group formula_355. By definition, for a scheme formula_4, formula_355 is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on formula_4 by the relation that formula_359 for any short exact sequence as above. Although formula_355 is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups formula_361 for integers formula_362.
A variant is the group formula_363 (or formula_364), the Grothendieck group of coherent sheaves on formula_4. (In topological terms, "G"-theory has the formal properties of a Borel–Moore homology theory for schemes, while "K"-theory is the corresponding cohomology theory.) The natural homomorphism formula_366 is an isomorphism if formula_4 is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case. For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.
More generally, a Noetherian scheme formula_4 is said to have the resolution property if every coherent sheaf on formula_4 has a surjection from some vector bundle on formula_4. For example, every quasi-projective scheme over a Noetherian ring has the resolution property.
Applications of resolution property.
Since the resolution property states that a coherent sheaf formula_371 on a Noetherian scheme is quasi-isomorphic in the derived category to the complex of vector bundles :formula_372
we can compute the total Chern class of formula_371 with
For example, this formula is useful for finding the Chern classes of the sheaf representing a subscheme of formula_4. If we take the projective scheme formula_135 associated to the ideal formula_377, then
since there is the resolution
over formula_380.
Bundle homomorphism vs. sheaf homomorphism.
When vector bundles and locally free sheaves of finite constant rank are used interchangeably,
care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms. Specifically, given vector bundles formula_381, by definition, a bundle homomorphism formula_382 is a scheme morphism over formula_4 (i.e., formula_384) such that, for each geometric point formula_50 in formula_4, formula_387 is a linear map of rank independent of formula_50. Thus, it induces the sheaf homomorphism formula_389 of constant rank between the corresponding locally free formula_3-modules (sheaves of dual sections). But there may be an formula_3-module homomorphism that does not arise this way; namely, those not having constant rank.
In particular, a subbundle formula_392 is a subsheaf (i.e., formula_371 is a subsheaf of formula_2). But the converse can fail; for example, for an effective Cartier divisor formula_395 on formula_4, formula_397 is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).
The category of quasi-coherent sheaves.
Quasi-coherent sheaves on any scheme form an abelian category. Gabber showed that, in fact, the quasi-coherent sheaves on any scheme form a particularly well-behaved abelian category, a Grothendieck category. A quasi-compact quasi-separated scheme formula_4 (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on formula_4, by Rosenberg, generalizing a result of Gabriel.
Coherent cohomology.
The fundamental technical tool in algebraic geometry is the cohomology theory of coherent sheaves. Although it was introduced only in the 1950s, many earlier techniques of algebraic geometry are clarified by the language of sheaf cohomology applied to coherent sheaves. Broadly speaking, coherent sheaf cohomology can be viewed as a tool for producing functions with specified properties; sections of line bundles or of more general sheaves can be viewed as generalized functions. In complex analytic geometry, coherent sheaf cohomology also plays a foundational role.
Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann–Roch theorem.