Generalization of vector bundles
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 that has a local presentation, that is, every point in formula_4 has an open neighborhood formula_5 in which there is an exact sequence
formula_6
for some (possibly infinite) sets formula_7 and formula_8.
A coherent sheaf on a ringed space formula_1 is a sheaf formula_2 of formula_3-modules 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_28 the restriction formula_29 is isomorphic to the sheaf formula_30 associated to the module formula_31 over formula_32. When formula_4 is a locally Noetherian scheme, formula_2 is coherent if and only if it is quasi-coherent and the modules formula_35 above can be taken to be finitely generated.
On an affine scheme formula_36, there is an equivalence of categories from formula_32-modules to quasi-coherent sheaves, taking a module formula_35 to the associated sheaf formula_30. The inverse equivalence takes a quasi-coherent sheaf formula_2 on formula_5 to the formula_32-module formula_43 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_32 is a full abelian subcategory of the category of all formula_32-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_51 in formula_4 has an open neighborhood formula_5 such that the restriction formula_29 of formula_2 to formula_5 is isomorphic to the cokernel of a morphism formula_57 for some natural numbers formula_18 and formula_59. 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_64. Likewise, on a locally Noetherian scheme formula_4, the structure sheaf formula_3 is a coherent sheaf of rings.
Vector bundles in this sheaf-theoretic sense over a scheme formula_4 are equivalent to vector bundles defined in a more geometric way, as a scheme formula_80 with a morphism formula_81 and with a covering of formula_4 by open sets formula_83 with given isomorphisms formula_84 over formula_83 such that the two isomorphisms over an intersection formula_86 differ by a linear automorphism. (The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle formula_80 in this geometric sense, the corresponding sheaf formula_2 is defined by: over an open set formula_5 of formula_4, the formula_91-module formula_43 is the set of sections of the morphism formula_93. The sheaf-theoretic interpretation of vector bundles has the advantage that vector bundles (on a locally Noetherian scheme) are included in the abelian category of coherent sheaves.
formula_134
this is because formula_135 restricted to the vanishing locus of the two polynomials has two-dimensional fibers, and has one-dimensional fibers elsewhere.
formula_153
formula_160
Since this sheaf has non-trivial stalks, but zero global sections, this cannot be a quasi-coherent sheaf. This is because quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global sections.
Functoriality.
Let formula_161 be a morphism of ringed spaces (for example, a morphism of schemes). If formula_2 is a quasi-coherent sheaf on formula_163, then the inverse image formula_3-module (or pullback) formula_165 is quasi-coherent on formula_4. For a morphism of schemes formula_161 and a coherent sheaf formula_2 on formula_163, the pullback formula_165 is not coherent in full generality (for example, formula_171, 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_161 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_176 is quasi-coherent on formula_163.
The direct image of a coherent sheaf is often not coherent. For example, for a field formula_178, let formula_4 be the affine line over formula_178, and consider the morphism formula_181; then the direct image formula_182 is the sheaf on formula_183 associated to the polynomial ring formula_184, which is not coherent because formula_184 has infinite dimension as a formula_178-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_51 control the behavior of formula_2 in a neighborhood of formula_51, 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_194 of formula_195 at a point formula_51 (a vector space over the residue field formula_197) is zero if and only if the sheaf formula_2 is zero on some open neighborhood of formula_51. 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_202 is a free module over the local ring formula_203 for every point formula_51 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_206, let formula_207 be the diagonal morphism, which is a closed immersion if formula_4 is separated over formula_163. Let formula_210 be the ideal sheaf of formula_4 in formula_212. Then the sheaf of differentials formula_213 can be defined as the pullback formula_214 of formula_210 to formula_4. Sections of this sheaf are called 1-forms on formula_4 over formula_163, and they can be written locally on formula_4 as finite sums formula_220 for regular functions formula_221 and formula_222. If formula_4 is locally of finite type over a field formula_178, then formula_225 is a coherent sheaf on formula_4.
If formula_4 is smooth over formula_178, then formula_229 (meaning formula_225) is a vector bundle over formula_4, called the cotangent bundle of formula_4. Then the tangent bundle formula_233 is defined to be the dual bundle formula_234. For formula_4 smooth over formula_178 of dimension formula_18 everywhere, the tangent bundle has rank formula_18.
If formula_163 is a smooth closed subscheme of a smooth scheme formula_4 over formula_178, then there is a short exact sequence of vector bundles on formula_163:
formula_243
which can be used as a definition of the normal bundle formula_244 to formula_163 in formula_4.
For a smooth scheme formula_4 over a field formula_178 and a natural number formula_249, the vector bundle formula_250 of "i"-forms on formula_4 is defined as the formula_249-th exterior power of the cotangent bundle, formula_253. For a smooth variety formula_4 of dimension formula_18 over formula_178, the canonical bundle formula_257 means the line bundle formula_258. 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_260 over formula_178
can be written as
formula_262
where formula_112 is a polynomial with coefficients in formula_178.
Let formula_96 be a commutative ring and formula_18 a natural number. For each integer formula_267, there is an important example of a line bundle on projective space formula_268 over formula_96, called formula_270. To define this, consider the morphism of formula_96-schemes
formula_272
given in coordinates by formula_273. (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_270 over an open subset formula_5 of formula_268 is defined to be a regular function formula_112 on formula_278 that is homogeneous of degree formula_267, meaning that
formula_280
as regular functions on (formula_281. For all integers formula_249 and formula_267, there is an isomorphism formula_284 of line bundles on formula_268.
In particular, every homogeneous polynomial in formula_286 of degree formula_267 over formula_96 can be viewed as a global section of formula_270 over formula_268. 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_270. 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_268 over formula_96 are just the "constants" (the ring formula_96), and so it is essential to work with the line bundles formula_270.
Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let formula_96 be a Noetherian ring (for example, a field), and consider the polynomial ring formula_297 as a graded ring with each formula_298 having degree 1. Then every finitely generated graded formula_299-module formula_35 has an associated coherent sheaf formula_301 on formula_268 over formula_96. Every coherent sheaf on formula_268 arises in this way from a finitely generated graded formula_299-module formula_35. (For example, the line bundle formula_270 is the sheaf associated to the formula_299-module formula_299 with its grading lowered by formula_267.) But the formula_299-module formula_35 that yields a given coherent sheaf on formula_268 is not unique; it is only unique up to changing formula_35 by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on formula_268 is the quotient of the category of finitely generated graded formula_299-modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.
The tangent bundle of projective space formula_268 over a field formula_178 can be described in terms of the line bundle formula_319. Namely, there is a short exact sequence, the Euler sequence:
formula_320
It follows that the canonical bundle formula_321 (the dual of the determinant line bundle of the tangent bundle) is isomorphic to formula_322. 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_319 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_324 hypersurface formula_325 defined by the homogeneous polynomial formula_112 of degree formula_324. Then, there is an exact sequence
formula_328
where the second map is the pullback of differential forms, and the first map sends
formula_329
Note that this sequence tells us that formula_330 is the conormal sheaf of formula_4 in formula_268. Dualizing this yields the exact sequence
formula_333
hence formula_334 is the normal bundle of formula_4 in formula_268. If we use the fact that given an exact sequence
formula_337
of vector bundles