Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometrical information. In addition, there is a related concept of quasi-coherent sheaves. Many results and properties in algebraic geometry and complex analytic geometry are both formulated in terms of coherent sheaves and their cohomology.

Coherent sheaves can be seen as a generalization of (sheaves of sections of) vector bundles. They form a category closed under usual operations such as taking kernels, cokernels and finite direct sums. In addition, under suitable compactness conditions they are preserved under maps of the underlying spaces and have finite dimensional cohomology spaces.

Definition

A coherent sheaf on a ringed space $(X,{\mathcal {O}}_{X})$ is a sheaf ${\mathcal {F}}$ of ${\mathcal {O}}_{X}$-modules with the following two properties:

1. ${\mathcal {F}}$ is of finite type over ${\mathcal {O}}_{X}$, i.e., for any point $x\in X$ there is an open neighbourhood $U\subset X$ such that the restriction ${\mathcal {F}}|_{U}$ of ${\mathcal {F}}$ to $U$ is generated by a finite number of sections (in other words, there is a surjective morphism ${\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}$ for some $n\in \mathbb {N}$); and
2. for any open set $U\subset X$, any $n\in \mathbb {N}$ and any morphism $\phi \colon {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}$ of ${\mathcal {O}}_{X}$-modules, the kernel of $\phi$ is of finite type.

The sheaf of rings ${\mathcal {O}}_{X}$ is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifolds and the structure sheaf of a Noetherian scheme from algebraic geometry.

A coherent sheaf is always a sheaf of finite presentation, or in other words each point $x\in X$ has an open neighbourhood $U$ such that the restriction ${\mathcal {F}}|_{U}$ of ${\mathcal {F}}$ to $U$ is isomorphic to the cokernel of a morphism ${\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {O}}_{X}^{m}|_{U}$ for some integers $n$ and $m$. If ${\mathcal {O}}_{X}$ is coherent, then the converse is true and each sheaf of finite presentation over ${\mathcal {O}}_{X}$ is coherent.

For a sheaf of rings ${\mathcal {O}}_{}$ , a sheaf ${\mathcal {F}}$ of ${\mathcal {O}}_{}$-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

${\mathcal {O}}^{(I_{i})}|_{U_{i}}\to {\mathcal {O}}^{(J_{i})}|_{U_{i}}\to {\mathcal {F}}|_{U_{i}}\to 0$

where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf.

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. In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules.

Coherence in sheaves makes some lemmata from commutative algebra work, e.g. Nakayama's lemma, which states that if ${\mathcal {F}}$ is a coherent sheaf, then the fiber $k(x)\otimes _{{\mathcal {O}}_{X,x}}{\mathcal {F}}_{x}=0$ if and only if there is a neighborhood $U$ of $x$ so that ${\mathcal {F}}|_{U}=0$.

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 ${\mathcal {O}}_{X}$. This makes consideration of them natural from the perspective of homological algebra.

Examples of coherent sheaves

• On noetherian schemes, the structure sheaf O_X itself.
• Sheaves of sections in vector bundles.
• Ideal sheaves: If Z is a closed complex subspace of a complex analytic space X, the sheaf I_Z of all holomorphic functions vanishing on Z is coherent.
• Structure sheaves of subspaces.

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.

Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension. This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds. It plays a major role in the proof of the "GAGA" equivalence analytic <-> algebraic. An algebraic (and much easier) version of this theorem was proved by Serre. Relative versions of this result for a proper morphism were proved by Grothendieck in the algebraic case and by Grauert and Remmert in the analytic case. For example Grothendieck's result concerns the functor 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 result of Serre is the case of a morphism to a point.

The duality theory in scheme theory that extends Serre duality is called coherent duality (or 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.

References

• "Coherent sheaf", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Section 0.5.3 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977, ISBN 0-387-90244-9