Векторлық шоштардың жалпыламасы

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.

When is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf of modules is quasi coherent if and only if over each open affine subscheme the restriction is isomorphic to the sheaf associated to the module over When is a locally Noetherian scheme, is coherent if and only if it is quasi coherent and the modules above can be taken to be finitely generated. On an affine scheme , there is an equivalence of categories from modules to quasi coherent sheaves, taking a module to the associated sheaf The inverse equivalence takes a quasi coherent sheaf on to the module of global sections of
Here are several further characterizations of quasi coherent sheaves on a scheme.

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 , the coherent sheaves form an abelian category, a full subcategory of the category of modules. (Analogously, the category of coherent modules over any ring is a full abelian subcategory of the category of all 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 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 module of finite presentation, meaning that each point in has an open neighborhood such that the restriction of to is isomorphic to the cokernel of a morphism for some natural numbers and If is coherent, then, conversely, every sheaf of finite presentation over is coherent. The sheaf of rings 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 is a coherent sheaf of rings. The main part of the proof is the case Likewise, on a locally Noetherian scheme , the structure sheaf is a coherent sheaf of rings.

Let be a morphism of ringed spaces (for example, a morphism of schemes). If is a quasi coherent sheaf on , then the inverse image module (or pullback) is quasi coherent on For a morphism of schemes and a coherent sheaf on , the pullback is not coherent in full generality (for example, , which might not be coherent), but pullbacks of coherent sheaves are coherent if is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle. If is a quasi compact quasi separated morphism of schemes and is a quasi coherent sheaf on , then the direct image sheaf (or pushforward) is quasi coherent on 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 on a scheme is a vector bundle if and only if its stalk is a free module over the local ring for every point in
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.

The quasi coherent sheaves on any fixed 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 (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi coherent sheaves on , by Rosenberg, generalizing a result of Gabriel.

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.