Coherent sheaf

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

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

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.)

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 <math>U_i<math> of the topological space and an exact sequence

<math>R|_{U_i}^{I_i} \to R|_{U_i}^{J_i} \to F|_{U_i} \to 0<math>

If F is finitely presented, i.e. <math>I_i<math> and <math>J_i<math> 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.

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 <math>F_x=0<math> if and only if there is a neighborhood U of x so that <math>F|_U=0<math>

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.

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 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

Rf_*

or push-forward, in sheaf cohomology. 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).

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.