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.
- F is of finite type over OX, that is, every point in X has an open neighborhood U in X such that the restriction F|U of F to U is generated by a finite number of sections (in other words, there is a surjective morphism OXn|U → F|U for some natural number n); and
- for any open set U ⊂ X, any natural number n, and any morphism φ: OXn|U → F|U of OX-modules, the kernel of φ is of finite type.
Note that an "OX-module" is shorthand for a "sheaf of OX-modules".
On any ringed space X, the coherent sheaves form an abelian category, a full subcategory of the category of OX-modules. (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.
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.
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 neighbourhood U such that the restriction F|U of F to U is isomorphic to the cokernel of a morphism OXn|U → OXm|U for some natural numbers n and m. If OX is coherent, then, conversely, every sheaf of finite presentation over OX is coherent.
For a ringed space X, an OX-module F is said to be quasi-coherent if it has a local presentation, that is, if every point in X has an open neighborhood U in which there is an exact sequence
for some sets I and J (possibly infinite). This definition is not of much use on an arbitrary ringed space, where quasi-coherent sheaves need not 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.
Note: Some authors, notably Hartshorne, use a different but essentially equivalent definition of coherent and quasi-coherent sheaves on a scheme. Let X be a scheme and F an OX-module. Then:
- F is quasi-coherent if every point in X has an affine open neighborhood U = Spec A such that the restriction of F to U is the sheaf associated to an A-module M.
- 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.
- 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.
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: E → X 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 Uα ∩ Uβ differ by a linear automorphism. (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.
- 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.
- 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: Z → X 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 X−Z, and fiber of dimension 1 at points in Z. There is a short exact sequence of coherent sheaves on X:
- Most operations of linear algebra preserve coherent sheaves. In particular, for coherent sheaves F and G on a ringed space X, the tensor product sheaf F⊗ OXG and the sheaf of homomorphisms HomOX(F,G) are coherent.
Let ƒ: X → Y 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. For a morphism of schemes f: X → Y 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.
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 Fp⊗OX,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. Thus a coherent sheaf has constant rank on an open set, while the rank can jump up on a lower-dimensional closed subset.
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.
Examples of vector bundles
For a morphism of schemes X → Y, let Δ: X → X ×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:
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
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
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
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). 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.
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:
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.
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. These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence
of vector bundles on X, the Chern classes of B are given by
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. 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.
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. 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.
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.
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