Coherent sheaf

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.

Definitions

A quasi-coherent sheaf on a ringed space (X,OX) is a sheaf F of OX-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exact sequence

${\displaystyle {\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0}$

for some sets I and J (possibly infinite).

A coherent sheaf on a ringed space (X,OX) is a quasi-coherent sheaf F satisfying the following two properties:

1. F is of finite type over OX, that is, every point in X has an open neighborhood U in X such that there is a surjective morphism On
X
|UF|U for some natural number n;
2. for any open set UX, any natural number n, and any morphism φ: On
X
|UF|U of OX-modules, the kernel of φ is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of OX-modules.

The case of schemes

When X is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf F of OX-modules is quasi-coherent if and only if over each open affine subscheme U=Spec(R) the restriction F|U is isomorphic to a sheaf ${\displaystyle {\tilde {M}}}$ associated to the module M=Γ(U, F) over R. When X is a locally Noetherian scheme, F is coherent if and only if it is quasi-coherent and the modules M above can be taken to be finitely generated.

On an affine scheme U = Spec A, there is an equivalence of categories from A-modules to quasi-coherent sheaves, taking a module M to the associated sheaf . The inverse equivalence takes a quasi-coherent sheaf F on U to the A-module F(U) of global sections of F.

Here are several further characterizations of quasi-coherent sheaves on a scheme.[1]

Theorem  Let X be a scheme and F an OX-module on it. Then the following are equivalent.

• F is quasi-coherent.
• For each open affine subscheme U of X, F|U is isomorphic as an OU-module to the sheaf associated to some O(U)-module M.
• There is an open affine cover ${\displaystyle \{U_{\alpha }\}}$ of X such that for any Uα from the cover F|Uα is the sheaf associated to some O(Uα)-module.
• For each pair of open affine subschemes VU of X, the natural homomorphism
${\displaystyle O(V)\otimes _{O(U)}F(U)\to F(V),\,f\otimes s\mapsto f\cdot s|_{V}}$
is an isomorphism.
• For each open affine subscheme U = Spec A of X and each f in A, writing {f ≠ 0} for the open subscheme of U where f is not zero, the natural homomorphism
${\displaystyle F(U){\bigg [}{\frac {1}{f}}{\bigg ]}\to F(\{f\neq 0\})}$
is an isomorphism. The homomorphism comes from the universal property of localization.

Properties

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.[2]

On any ringed space X, the coherent sheaves form an abelian category, a full subcategory of the category of OX-modules.[3] (Analogously, the category of coherent modules over any ring R is a full abelian subcategory of the category of all R-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 OX-module that is an extension of two coherent sheaves is coherent.[4]

A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always an OX-module of finite presentation, meaning that each point x in X has an open neighborhood U such that the restriction F|U of F to U is isomorphic to the cokernel of a morphism OXn|UOXm|U for some natural numbers n and m. If OX is coherent, then, conversely, every sheaf of finite presentation over OX is coherent.

The sheaf of rings OX 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 X is a coherent sheaf of rings. The main part of the proof is the case X = Cn. Likewise, on a locally Noetherian scheme X, the structure sheaf OX is a coherent sheaf of rings.[5]

Basic constructions of coherent sheaves

• An OX-module F on a ringed space X is called locally free of finite rank, or a vector bundle, if every point in X has an open neighborhood U such that the restriction F|U is isomorphic to a finite direct sum of copies of OX|U. If F is free of the same rank n near every point of X, then the vector bundle F is said to be of rank n.
Vector bundles in this sheaf-theoretic sense over a scheme X are equivalent to vector bundles defined in a more geometric way, as a scheme E with a morphism f: EX and with a covering of X by open sets Uα with given isomorphisms f−1(Uα) ≅ An × Uα over Uα such that the two isomorphisms over an intersection UαUβ differ by a linear automorphism.[6] (The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle E in this geometric sense, the corresponding sheaf is defined by: the group of sections E(U) over an open set U in X is the set of sections of the morphism f−1(U) → U. The sheaf-theoretic interpretation of vector bundles has the advantage that vector bundles (on a locally Noetherian scheme) are included in the abelian category of coherent sheaves.
• Locally free sheaves come equipped with the standard ${\displaystyle {\mathcal {O}}_{X}}$-module operations, but these give back locally free sheaves.
• Let X = Spec(R), R a Noetherian ring. Then vector bundles on X are exactly the sheaves associated to finitely generated projective modules over R, or (equivalently) to finitely generated flat modules over R.[7]
• Let X = Proj(R), R a Noetherian ${\displaystyle \mathbb {N} }$-graded ring, be a projective scheme over a Noetherian ring R0. Then each ${\displaystyle \mathbb {Z} }$-graded R-module M determines a quasi-coherent sheaf F on X such that ${\displaystyle F|_{\{f\neq 0\}}}$ is the sheaf associated to the ${\displaystyle R[f^{-1}]_{0}}$-module ${\displaystyle M[f^{-1}]_{0}}$, where f is a homogeneous element of R of positive degree and ${\displaystyle \{f\neq 0\}=\operatorname {Spec} R[f^{-1}]_{0}}$ is the locus where f doesn't vanish.
• For example, for each integer n, let R(n) denote the graded R-module given by ${\displaystyle R(n)_{l}=R_{n+l}}$. Then each ${\displaystyle R(n)}$ determines the quasi-coherent sheaf ${\displaystyle {\mathcal {O}}_{X}(n)}$ on X. If R is generated as R0-algebra by R1, then ${\displaystyle {\mathcal {O}}_{X}(n)}$ are line bundles (invertible sheaves) on X and ${\displaystyle {\mathcal {O}}_{X}(n)}$ is the n-th tensor power of ${\displaystyle {\mathcal {O}}_{X}(1)}$. In particular, ${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)}$ is called the tautological line bundle on the projective n-space.
• A simple example of a coherent sheaf on ${\displaystyle \mathbb {P} ^{2}}$ which is not a vector bundle is given by the cokernel in the following sequence
${\displaystyle {\mathcal {O}}(1){\xrightarrow {\cdot (x^{2}-yz,y^{3}+xy^{2}-xyz)}}{\mathcal {O}}(3)\oplus {\mathcal {O}}(4)\to {\mathcal {E}}\to 0}$
this is because ${\displaystyle {\mathcal {E}}}$ restricted to the vanishing locus of the two polynomials is the zero object.
• Ideal sheaves: If Z is a closed subscheme of a locally Noetherian scheme X, the sheaf IZ/X of all regular functions vanishing on Z is coherent. Likewise, if Z is a closed analytic subspace of a complex analytic space X, the ideal sheaf IZ/X is coherent.
• The structure sheaf OZ of a closed subscheme Z of a locally Noetherian scheme X can be viewed as a coherent sheaf on X. To be precise, this is the direct image sheaf i*OZ, where i: ZX is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf i*OZ has fiber (defined below) of dimension zero at points in the open set XZ, and fiber of dimension 1 at points in Z. There is a short exact sequence of coherent sheaves on X:
${\displaystyle 0\to I_{Z/X}\to O_{X}\to i_{*}O_{Z}\to 0.}$
• A simple non-example of a quasi-coherent sheaf is given by the extension by zero functor. For example, consider ${\displaystyle i_{!}{\mathcal {O}}_{X}}$ for
${\displaystyle X={\text{Spec}}(\mathbb {C} [x,x^{-1}]){\xrightarrow {i}}{\text{Spec}}(\mathbb {C} [x])=Y}$[9]
Since this sheaf has non-trivial stalks, but zero global sections, this cannot be a quasi-coherent sheaf. This is because quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global sections.

Functoriality

Let ƒ: XY be a morphism of ringed spaces (for example, a morphism of schemes). If F is a quasi-coherent sheaf on Y, then the inverse image OX-module (or pullback) f*F is quasi-coherent on X.[10] For a morphism of schemes f: XY and a coherent sheaf F on Y, the pullback f*F is not coherent in full generality (for example, f*OY = OX, which might not be coherent), but pullbacks of coherent sheaves are coherent if X is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle.

If f: XY is a quasi-compact quasi-separated morphism of schemes and E is a quasi-coherent sheaf on X, then the direct image sheaf (or pushforward) f*E is quasi-coherent on Y.[2]

The direct image of a coherent sheaf is often not coherent. For example, for a field k, let X be the affine line over k, and consider the morphism f: X → Spec(k); then the direct image f*OX is the sheaf on Spec(k) associated to the polynomial ring k[x], which is not coherent because k[x] has infinite dimension as a k-vector space. On the other hand, the direct image of a coherent sheaf under a proper morphism is coherent, by results of Grauert and Grothendieck.

Local behavior of coherent sheaves

An important feature of coherent sheaves F is that the properties of F at a point p control the behavior of F in a neighborhood of p, more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if F is a coherent sheaf on a scheme X, then the fiber FpOX,pk(p) of F at a point p (a vector space over the residue field k(p)) is zero if and only if the sheaf F is zero on some open neighborhood of p. A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous.[11] 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 F on a scheme X is a vector bundle if and only if its stalk Fp is a free module over the local ring OX,p for every point p in X.[12]

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.[13]

Examples of vector bundles

For a morphism of schemes XY, let Δ: XX ×Y X be the diagonal morphism, which is a closed immersion if X is separated over Y. Let I be the ideal sheaf of X in X ×Y X. Then the sheaf of differentials Ω1X/Y can be defined as the pullback Δ*(I) of I to X. Sections of this sheaf are called 1-forms on X over Y, and they can be written locally on X as finite sums ∑ fj dgj for regular functions fj and gj. If X is locally of finite type over a field k, then Ω1X/k is a coherent sheaf on X.

If X is smooth over k, then Ω1 (meaning Ω1X/k) is a vector bundle over X, called the cotangent bundle of X. Then the tangent bundle TX is defined to be the dual bundle (Ω1)*. For X smooth over k of dimension n everywhere, the tangent bundle has rank n.

If Y is a smooth closed subscheme of a smooth scheme X over k, then there is a short exact sequence of vector bundles on Y:

${\displaystyle 0\to TY\to TX|_{Y}\to N_{Y/X}\to 0,}$

which can be used as a definition of the normal bundle NY/X to Y in X.

For a smooth scheme X over a field k and a natural number a, the vector bundle Ωa of a-forms on X is defined as the ath exterior power of the cotangent bundle, Ωa = ΛaΩ1. For a smooth variety X of dimension n over k, the canonical bundle KX means the line bundle Ωn. Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on X. For example, a section of the canonical bundle of affine space An over k can be written as

${\displaystyle f(x_{1},\ldots ,x_{n})\;dx_{1}\wedge \cdots \wedge dx_{n},}$

where f is a polynomial with coefficients in k.

Let R be a commutative ring and n a natural number. For each integer j, there is an important example of a line bundle on projective space Pn over R, called O(j). To define this, consider the morphism of R-schemes

${\displaystyle \pi \colon A^{n+1}-0\to \mathbf {P} ^{n}}$

given in coordinates by (x0,...,xn) ↦ [x0,...,xn]. (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of O(j) over an open subset U of Pn is defined to be a regular function f on π−1(U) that is homogeneous of degree j, meaning that

${\displaystyle f(ax)=a^{j}f(x)}$

as regular functions on (A1 − 0) × π−1(U). For all integers i and j, there is an isomorphism O(i) ⊗ O(j) ≅ O(i+j) of line bundles on Pn.

In particular, every homogeneous polynomial in x0,...,xn of degree j over R can be viewed as a global section of O(j) over Pn. Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles O(j).[14] This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space Pn over R are just the "constants" (the ring R), and so it is essential to work with the line bundles O(j).

Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let R be a Noetherian ring (for example, a field), and consider the polynomial ring S = R[x0,...,xn] as a graded ring with each xi having degree 1. Then every finitely generated graded S-module M has an associated coherent sheaf on Pn over R. Every coherent sheaf on Pn arises in this way from a finitely generated graded S-module M. (For example, the line bundle O(j) is the sheaf associated to the S-module S with its grading lowered by j.) But the S-module M that yields a given coherent sheaf on Pn is not unique; it is only unique up to changing M by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on Pn is the quotient of the category of finitely generated graded S-modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.[15]

The tangent bundle of projective space Pn over a field k can be described in terms of the line bundle O(1). Namely, there is a short exact sequence, the Euler sequence:

${\displaystyle 0\to O_{\mathbf {P} ^{n}}\to O(1)^{\oplus \;n+1}\to T\mathbf {P} ^{n}\to 0.}$

It follows that the canonical bundle KPn (the dual of the determinant line bundle of the tangent bundle) is isomorphic to O(−n−1). This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle O(1) means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.

Vector bundles on a hypersurface

Consider a smooth degree ${\displaystyle d}$ hypersurface ${\displaystyle X\subset \mathbb {P} ^{n}}$ defined by the homogeneous polynomial ${\displaystyle f}$. Then, there is an exact sequence

${\displaystyle 0\to {\mathcal {O}}_{X}(-d)\to i^{*}\Omega _{\mathbb {P} ^{n}}\to \Omega _{X}\to 0}$

where the second map is the pullback of differential forms, and the first map sends

${\displaystyle \phi \mapsto d(f\cdot \phi )}$

Note that this sequence tells us that ${\displaystyle {\mathcal {O}}(-d)}$ is the conormal sheaf of ${\displaystyle X}$ in ${\displaystyle \mathbb {P} ^{n}}$. Dualizing this yields the exact sequence

${\displaystyle 0\to T_{X}\to i^{*}T_{\mathbb {P} ^{n}}\to {\mathcal {O}}(d)\to 0}$

hence ${\displaystyle {\mathcal {O}}(d)}$ is the normal bundle of ${\displaystyle X}$ in ${\displaystyle \mathbb {P} ^{n}}$. If we use the fact that given an exact sequence

${\displaystyle 0\to {\mathcal {E}}_{1}\to {\mathcal {E}}_{2}\to {\mathcal {E}}_{3}\to 0}$

of vector bundles with ranks ${\displaystyle e_{1},e_{2},e_{3}}$, there is an isomorphism

${\displaystyle \Lambda ^{e_{2}}{\mathcal {E}}_{2}\cong \Lambda ^{e_{1}}{\mathcal {E}}_{1}\otimes \Lambda ^{e_{3}}{\mathcal {E}}_{3}}$

of line bundles, then we see that there is the isomorphism

${\displaystyle i^{*}\omega _{\mathbb {P} ^{n}}\cong \omega _{X}\otimes {\mathcal {O}}_{X}(-d)}$

showing that

${\displaystyle \omega _{X}\cong {\mathcal {O}}_{X}(d-n-1)}$

Chern classes and algebraic K-theory

A vector bundle E on a smooth variety X over a field has Chern classes in the Chow ring of X, ci(E) in CHi(X) for i ≥ 0.[16] These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence

${\displaystyle 0\to A\to B\to C\to 0}$

of vector bundles on X, the Chern classes of B are given by

${\displaystyle c_{i}(B)=c_{i}(A)+c_{1}(A)c_{i-1}(C)+\cdots +c_{i-1}(A)c_{1}(C)+c_{i}(C).}$

It follows that the Chern classes of a vector bundle E depend only on the class of E in the Grothendieck group K0(X). By definition, for a scheme X, K0(X) is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on X by the relation that [B] = [A] + [C] for any short exact sequence as above. Although K0(X) is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups Ki(X) for integers i.

A variant is the group G0(X) (or K0'(X)), the Grothendieck group of coherent sheaves on X. (In topological terms, G-theory has the formal properties of a Borel–Moore homology theory for schemes, while K-theory is the corresponding cohomology theory.) The natural homomorphism K0(X) → G0(X) is an isomorphism if X is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case.[17] For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.

More generally, a Noetherian scheme X is said to have the resolution property if every coherent sheaf on X has a surjection from some vector bundle on X. For example, every quasi-projective scheme over a Noetherian ring has the resolution property.

Bundle homomorphism vs. sheaf homomorphism

When vector bundles and locally free sheaves of finite constant rank are used interchangeably, care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms. Specifically, given vector bundles ${\displaystyle p:E\to X,\,q:F\to X}$, by definition, a bundle homomorphism ${\displaystyle \varphi :E\to F}$ is a scheme morphism over X (i.e., ${\displaystyle p=q\circ \varphi }$) such that, for each geometric point x in X, ${\displaystyle \varphi :p^{-1}(x)\to q^{-1}(x)}$ is a linear map of rank independent of x. Thus, it induces the sheaf homomorphism ${\displaystyle {\widetilde {\varphi }}:{\widetilde {E}}\to {\widetilde {F}}}$ of constant rank between the corresponding locally free ${\displaystyle {\mathcal {O}}_{X}}$-modules (sheaves of dual sections). But there may be an ${\displaystyle {\mathcal {O}}_{X}}$-module homomorphism that does not arise this way; namely, those not having constant rank.

In particular, a subbundle ${\displaystyle E\subset F}$ is a subsheaf (${\displaystyle E,F}$ viewed as sheaves). But the converse can fail; for example, for an effective Cartier divisor D on X, ${\displaystyle {\mathcal {O}}_{X}(-D)\subset {\mathcal {O}}_{X}}$ is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).

The category of quasi-coherent sheaves

Quasi-coherent sheaves on any 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.[18] A quasi-compact quasi-separated scheme X (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on X, by Rosenberg, generalizing a result of Gabriel.[19]

Coherent cohomology

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.

Notes

1. Mumford, Ch. III, § 1, Theorem-Definition 3.
2. Serre (1955), section 13.
3. Grothendieck, EGA I, Corollaire 1.5.2.
4. Hartshorne (1977), Exercise II.5.18.
5. Serre (1955), section 14.
6. Hartshorne, Robin. Algebraic Geometry.
7. Hartshorne (1977), Example III.12.7.2.
8. Grothendieck, EGA I, Ch. 0, 5.2.7.
9. Eisenbud (1995), Exercise 20.13.
10. Hartshorne (1977), Corollary II.5.16.
11. Fulton (1998), section 3.2 and Example 8.3.3.
12. Fulton (1998), B.8.3.
13. Antieau (2016), Corollary 4.2.