In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of O_X-modules Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

O_X^m → O_Xⁿ.

Here O_X is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented module; given a ring R such modules are cokernels of homomorphisms In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

R^m → Rⁿ.

Under some noetherian conditions, the condition of being finitely-presented can be replaced by that of being finitely generated (see finitely generated module), which is in general, though, a weaker condition. (In the module case this says that the submodule of relations, in Rⁿ, can in the noetherian case be taken to be finitely generated.) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ...

For a sheaf of rings R, a sheaf F of R-modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by U_i of the topological space and an exact sequence In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

If F is finitely presented, i.e. I_i and J_i both finite, then F is said to be coherent.

For an affine variety X with affine coordinate ring R, there exists a covariant equivalence of categories between that of quasi-coherent sheaves and sheaf morphisms on the one hand, and R-modules and module homomorphisms on the other hand. This article is about algebraic varieties. ... In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ... This word should not be confused with homeomorphism. ...

Coherence in sheaves makes some lemmata from commutative algebra work, e.g. Nakayama's lemma, which states that if F is a coherent sheaf, then the stalk F_x = 0 if and only if there is a neighborhood U of x so that F | _U = 0 In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, Nakayamas lemma is an important technical lemma in commutative algebra and algebraic geometry. ... The word stalk has several basic meanings. ...

The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf, or sheaf of sections of a (holomorphic) vector bundle — but still with manageable properties. The generality is desirable, to be able to take kernels and cokernels of morphisms, for example, without moving outside the given class of sheaves. To put that more formally, suppose one wants, given a short exact sequence of sheaves, to be able to infer that if any two are in a class of sheaves, then the third should be. Then the coherent sheaves are the smallest such class containing O_X. This makes consideration of them natural, from the perspective of homological algebra. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... The word kernel has several meanings in mathematics, some related to each other and some not. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

Coherent cohomology

The sheaf cohomology theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories. In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...

In the basic work of Serre, it was shown first that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension. This result was then carried over to any projective variety; the dimensions of such spaces had in many cases (and under other names) been studied by geometers, and this was a very general finiteness result backing up the theory. Versions of this result for a proper morphism were proved, by Grothendieck, Grauert and Remmert. For example Grothendieck's result concerns the functor Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... This article is about algebraic varieties. ... This is a glossary of scheme theory. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... Hans Grauert (b. ... Reinhold Remmert (June 22, 1930) is a German mathematician born in Osnabrück, Germany. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Rf_*

or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The Serre result is the case of a morphism to a point (which is therefore already a deep result). In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

The duality theory in scheme theory that extends Serre duality is called coherent duality (sometimes Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kähler differentials on an algebraic variety is a coherent sheaf Ω¹. When the variety is non-singular its 'top' exterior power acts as the dualising object; and it is locally free (effectively it is the sheaf of sections of the cotangent bundle, when working over the complex numbers, but that is a statement that requires more precision since only holomorphic 1-forms count as sections). The successful extension of the theory beyond this case was a major step. In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ... Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory. ... In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...