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.

Coherent sheaves can be seen as a generalization of vector bundles, or of locally free sheaves of finite rank. Unlike vector bundles, they form a "nice" category closed under usual operations such as taking kernels, cokernels and finite direct sums. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.

Many results and properties in algebraic geometry and complex analytic geometry are formulated in terms of coherent or quasi-coherent sheaves and their cohomology.

Definitions

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 $\varphi\colon \mathcal{O}_X^n|_U \to \mathcal{F}|_U$ of $\mathcal{O}_X$-modules, the kernel of $\varphi$ is finitely generated.

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 manifold (Oka coherence theorem) and the structure sheaf of a Noetherian scheme from algebraic geometry.

A submodule of a coherent sheaf is coherent if it is of finite type. 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.

A sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-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 $X$ and an exact sequence

$\mathcal{O}_X^{(I_i)}|_{U_i} \to \mathcal{O}_X^{(J_i)}|_{U_i} \to \mathcal{F}|_{U_i} \to 0$

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

Note: Some authors, notably Hartshorne, use a different but essentially equivalent definition of coherent and quasi-coherent sheaves on a scheme (cf. #Properties). Let X be a scheme and F an $\mathcal{O}_X$-module. Then:

• F is quasi-coherent if there are open affine cover $U_i = \operatorname{Spec} A_i$ of X and A_i-modules M_i such that $F|_{U_i} \simeq \widetilde{M}_i$ as $O_{U_i}$-modules, where $\widetilde{M}_i$ are sheaves associated to $M_i$.
• When X is a Noetherian scheme, F is coherent if it is quasi-coherent and the $M_i$ above can be taken to be finitely generated.

Examples of coherent sheaves

• On a Noetherian scheme X, the structure sheaf $\mathcal{O}_X$ is a coherent sheaf of rings.
• A sheaf of $\mathcal{O} _X$-modules $\mathcal{F}$ on a ringed space $X$ is called locally free if for each point $p\in X$, there is an open neighborhood $U$ of $p$ such that $\mathcal{F}| _U$ is free as an $\mathcal{O} _X| _U$-module. This implies that $\mathcal{F}_p$, the stalk of $\mathcal{F}$ at $p$, is free as a $(\mathcal{O} _X)_p$-module for all $p$. The converse is true if $\mathcal{F}$ is moreover coherent. If $\mathcal{F}_p$ is of finite rank $n$ for every $p\in X$, then $\mathcal{F}$ is said to be of rank $n.$
• Let $X = \operatorname{Spec}(R)$, R a Noetherian ring. Then any finitely generated projective module over R can be viewed as a locally free $\mathcal{O}_X$-module. (see also Proj construction for the case when R is a graded ring.)
• The Oka coherence theorem states that the sheaf of holomorphic functions on a complex manifold is a coherent sheaf of rings.
• The sheaf of sections of a vector bundle (on a scheme, or a complex analytic space) is coherent.
• Ideal sheaves: If Z is a closed complex subspace of a complex analytic space X, the sheaf I_Z/X of all holomorphic functions vanishing on Z is coherent. Likewise, the ideal sheaf of regular functions vanishing on a closed subscheme is coherent.
• The structure sheaf O_Z of a closed subscheme Z of X, or of a closed analytic subspace, is a coherent sheaf on X. The sheaf O_Z has fiber dimension (defined below) equal to zero at points in the open set X−Z, and fiber dimension one at points in Z.

Properties

• A finite direct sum of quasi-coherent sheaves is quasi-coherent.
• Let ƒ:X→Y be a morphism of ringed spaces (e.g., a morphism of schemes). If F is a quasi-coherent sheaf on Y, then the inverse image $f^* F$ is quasi-coherent on X.
• Given an exact sequence of coherent sheaves $0 \to F \to G \to H \to 0$, if two of them are coherent, then the third is also coherent.

For schemes, there is the following basic result:

Theorem — Let X be a scheme and F an $\mathcal{O}_X$-module on it. Then the following are equivalent.

• F is quasi-coherent (i.e., it locally admits a free presentation).
• For each open affine subset U of X, F|_U is isomorphic as O_U-module to the sheaf associated to some $\Gamma(U, \mathcal{O}_U)$-module.
• There exists an open affine cover U_i of X such that for each i, F|_Ui is isomorphic to the sheaf associated to some $\Gamma(U_i, \mathcal{O}_{U_i})$-module.
• For each pair of open affine subsets V ⊂ U of X, the natural map

  $\Gamma(U, F) \otimes_{\Gamma(U, \mathcal{O}_U)} \Gamma(V, \mathcal{O}_V) \to \Gamma(V, F), \, s \otimes f \mapsto s|_V f$

is an isomorphism.

• For each open affine subset U = Spec A of X and $D(f) = \operatorname{Spec}A[f^{-1}] \subset U$, the natural map

  $\Gamma(U, F)[f^{-1}] \to \Gamma(D(f), F)$

is an isomorphism, where the natural map is obtained by the universal property of localization.

If X is a scheme, then the following holds.

• Given an exact sequence of coherent sheaves $0 \to F \to G \to H \to 0$ on X, if two of them are quasi-coherent, then the third is also quasi-coherent.

The category of quasi-coherent sheaves

The category of coherent sheaves on $(X,\mathcal{O}_X)$ is an abelian category, a full subcategory of the (much more unwieldy) abelian category of all sheaves on $(X,\mathcal{O}_X)$. (Analogously, the category of coherent modules over any ring R is a full abelian subcategory of the category of all R-modules.)

If R denotes the ring of regular functions $\Gamma(X,\mathcal{O}_X)$, then every R-module gives rise to a quasi-coherent sheaf of $\mathcal{O}_X$-modules in a natural fashion, yielding a functor from R-modules to quasi-coherent sheaves. In general, not every quasi-coherent sheaf arises from an R-module in this fashion. However, for an affine scheme X with coordinate ring R, this construction gives an equivalence of categories between R-modules and quasi-coherent sheaves on X. In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules.

Some results in commutative algebra are naturally interpreted using coherent sheaves. For example, Nakayama's lemma says that if F is a coherent sheaf, then the fiber F_x⊗_OX,xk(x) of F at a point x (a vector space over the residue field k(x)) is zero if and only if the sheaf F is zero on some open neighborhood of x. 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 (where it is a vector bundle), while the rank can jump up on a lower-dimensional closed subset.

Caution: Depending on X, the category of quasi-coherent sheaves on X need not be abelian.

Given an algebraic variety X (or more generally: a quasi-compact quasi-separated scheme), the category of quasi-coherent sheaves on X is a very well-behaved abelian category, a Grothendieck category. It follows that the category of quasi-coherent sheaves (unlike the category of coherent sheaves) has enough injectives, which makes it a convenient setting for sheaf cohomology. The scheme X is determined up to isomorphism by the abelian category of quasi-coherent sheaves on X.

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 Fréchet 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. 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, 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 smooth, Ω¹ is a vector bundle, the cotangent bundle of X. For a smooth projective variety X of dimension n, Serre duality says that the top exterior power Ωⁿ = ΛⁿΩ¹ acts as the dualizing object for coherent sheaf cohomology.

See also

• Picard group
• Swan's theorem
• Algebraic vector bundle
• Quot scheme
• Reflexive sheaf

References

• Sections 0.5.3 and 0.5.4 of
• Vakil, Math 216: Foundations of algebraic geometry, Part V.

External links

• This article incorporates material from Locally free on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
• Sheaves of Modules, from the Stacks Project