Coherent sheaf cohomology

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.

Coherent sheaves

Main article: Coherent sheaf

Coherent sheaves can be seen as a generalization of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on a scheme. In both cases, the given space X comes with a sheaf of rings O_X, the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a full subcategory of the category of O_X-modules (that is, sheaves of O_X-modules).

Vector bundles such as the tangent bundle play a fundamental role in geometry. More generally, for a closed subvariety Y of X with inclusion i: Y → X, a vector bundle E on Y determines a coherent sheaf on X, the direct image sheaf i_*E, which is zero outside Y. In this way, many questions about subvarieties of X can be expressed in terms of coherent sheaves on X.

Unlike vector bundles, coherent sheaves (in the analytic or algebraic case) form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. On a scheme, the quasi-coherent sheaves are a generalization of coherent sheaves, including the locally free sheaves of infinite rank.

Sheaf cohomology

For a sheaf E of abelian groups on a topological space X, the sheaf cohomology groups Hⁱ(X,E) for integers i are defined as the right derived functors of the functor of global sections, E ↦ E(X). As a result, Hⁱ(X,E) is zero for i < 0, and H⁰(X,E) can be identified with E(X). For any short exact sequence of sheaves 0 → A → B → C → 0, there is a long exact sequence of cohomology groups:^[1]

0\to H^{0}(X,A)\to H^{0}(X,B)\to H^{0}(X,C)\to H^{1}(X,A)\to \cdots .

If E is a sheaf of O_X-modules on a scheme X, then the cohomology groups Hⁱ(X,E) (defined using the underlying topological space of X) are modules over the ring O(X) of regular functions. For example, if X is a scheme over a field k, then the cohomology groups Hⁱ(X,E) are k-vector spaces. The theory becomes powerful when E is a coherent or quasi-coherent sheaf, because of the following sequence of results.

Vanishing theorems in the affine case

Complex analysis was revolutionized by Cartan's theorems A and B in 1953. These results say that if E is a coherent analytic sheaf on a Stein space X, then E is spanned by its global sections, and Hⁱ(X,E) = 0 for all i > 0. (A complex space X is Stein if and only if it is isomorphic to a closed analytic subspace of Cⁿ for some n.) These results generalize a large body of older work about the construction of complex analytic functions with given singularities or other properties.

In 1955, Serre introduced coherent sheaves into algebraic geometry (at first over an algebraically closed field, but that restriction was removed by Grothendieck). The analogs of Cartan's theorems hold in great generality: if E is a quasi-coherent sheaf on an affine scheme X, then E is spanned by its global sections, and Hⁱ(X,E) = 0 for i > 0.^[2] This is related to the fact that the category of quasi-coherent sheaves on an affine scheme X is equivalent to the category of O(X)-modules, with the equivalence taking a sheaf E to the O(X)-module H⁰(X,E). In fact, affine schemes are characterized among all quasi-compact schemes by the vanishing of higher cohomology for quasi-coherent sheaves.^[3]

Cech cohomology, and the cohomology of projective space

As a consequence of the vanishing of cohomology for affine schemes: for a separated scheme X, an affine open covering {U_i} of X, and a quasi-coherent sheaf E on X, the cohomology groups H*(X,E) are isomorphic to the Čech cohomology groups with respect to the open covering {U_i}.^[2] In other words, knowing the sections of E on all finite intersections of the affine open subschemes U_i determines the cohomology of X with coefficients in E.

Using Čech cohomology, one can compute the cohomology of projective space with coefficients in any line bundle. Namely, for a field k, a positive integer n, and any integer j, the cohomology of projective space Pⁿ over k with coefficients in the line bundle O(j) is given by:^[4]

H^{i}(\mathbf {P} ^{n},O(j))\cong {\begin{cases}\bigoplus _{a_{0}\geq 0,\ldots ,a_{n}\geq 0,\;a_{0}+\cdots +a_{n}=j}\;k\cdot x_{0}^{a_{0}}\cdots x_{n}^{a_{n}}&{\text{if }}i=0\\0&{\text{if }}i\neq 0{\text{ and }}i\neq n\\\bigoplus _{a_{0}<0,\ldots ,a_{n}<0,\;a_{0}+\cdots +a_{n}=j}\;k\cdot x_{0}^{a_{0}}\cdots x_{n}^{a_{n}}&{\text{if }}i=n.\end{cases}}

In particular, this calculation shows that the cohomology of projective space over k with coefficients in any line bundle has finite dimension as a k-vector space.

The vanishing of these cohomology groups above dimension n is a very special case of Grothendieck's vanishing theorem: for any sheaf of abelian groups E on a Noetherian topological space X of dimension n < ∞, Hⁱ(X,E) = 0 for all i > n.^[5] This is especially useful for X a Noetherian scheme (for example, a variety over a field) and E a quasi-coherent sheaf.

Finite-dimensionality of cohomology

For a proper scheme X over a field k and any coherent sheaf E on X, the cohomology groups Hⁱ(X,E) have finite dimension as k-vector spaces.^[6] In the special case where X is projective over k, this is proved by reducing to the case of line bundles on projective space, discussed above. In the general case of a proper scheme over a field, Grothendieck proved the finiteness of cohomology by reducing to the projective case, using Chow's lemma.

The finite-dimensionality of cohomology also holds in the analogous situation of coherent analytic sheaves on any compact complex space, by a very different argument. Cartan and Serre proved finite-dimensionality in this analytic situation using a theorem of Schwartz on compact operators in Fréchet spaces. Relative versions of this result for a proper morphism were proved by Grothendieck for locally Noetherian schemes) and by Grauert (for complex analytic spaces). Namely, for a proper morphism f: X → Y (in the algebraic or analytic setting) and a coherent sheaf E on X, the higher direct image sheaves Rⁱf_*E are coherent.^[7] When Y is a point, this theorem gives the finite-dimensionality of cohomology.

The finite-dimensionality of cohomology leads to many numerical invariants for projective varieties. For example, if X is a smooth projective curve over an algebraically closed field k, the genus of X is defined to be the dimension of the k-vector space H¹(X,O). When k is the field of complex numbers, this agrees with the genus of the space X(C) of complex points in its classical (Euclidean) topology. (In that case, X(C) = X^an is a closed oriented surface.) Among many possible higher-dimensional generalizations, the geometric genus of a smooth projective variety X of dimension n is the dimension of Hⁿ(X,O), and the arithmetic genus (according to one convention^[8]) is the alternating sum

\chi (X,O)=\sum _{j}(-1)^{j}\dim _{k}(H^{j}(X,O)).

Serre duality

Serre duality is an analog of Poincaré duality for coherent sheaf cohomology. In this analogy, the canonical bundle K_X plays the role of the orientation sheaf. Namely, for a smooth proper scheme X of dimension n over a field k, there is a natural trace map Hⁿ(X,K_X) → k, which is an isomorphism if X is geometrically connected, meaning that the base change of X to an algebraic closure of k is connected. Serre duality for a vector bundle E on X says that the product

H^{i}(X,E)\times H^{n-i}(X,K_{X}\otimes E^{*})\to H^{n}(X,K_{X})\to k

is a perfect pairing for every integer i.^[9] In particular, the k-vector spaces Hⁱ(X,E) and Hⁿ⁻ⁱ(X,K_X ⊗ E*) have the same (finite) dimension. (Serre also proved Serre duality for holomorphic vector bundles on any compact complex manifold.) Grothendieck duality theory includes generalizations to any coherent sheaf and any proper morphism of schemes, although the statements become less elementary.

For example, for a smooth projective curve X over an algebraically closed field k, Serre duality implies that the dimension of the space H⁰(X,Ω¹) = H⁰(X,K_X) of 1-forms on X is equal to the genus of X (the dimension of H¹(X,O)).

GAGA theorems

GAGA theorems relate algebraic varieties over the complex numbers to the corresponding analytic spaces. For a scheme X of finite type over C, there is a functor from coherent algebraic sheaves on X to coherent analytic sheaves on the associated analytic space X^an. The key GAGA theorem (by Grothendieck, generalizing Serre's theorem on the projective case) is that if X is proper over C, then this functor is an equivalence of categories. Moreover, for every coherent algebraic sheaf E on a proper scheme X over C, the natural map

H^{i}(X,E)\to H^{i}(X^{\text{an}},E)

of (finite-dimensional) complex vector spaces is an isomorphism for all i.^[10] (The first group here is defined using the Zariski topology, and the second using the classical (Euclidean) topology.) For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CPⁿ is algebraic.

Vanishing theorems

Serre's vanishing theorem says that for any ample line bundle L on a proper scheme X over a Noetherian ring, and any coherent sheaf F on X, there is an integer m₀ such that for all m ≥ m₀, the sheaf F ⊗ L^⊗m is spanned by its global sections and has no cohomology in positive degrees.^[11]

Although Serre's vanishing theorem is useful, the inexplicitness of the number m₀ can be a problem. The Kodaira vanishing theorem is an important explicit result. Namely, if X is a smooth projective variety over a field of characteristic zero and L is an ample line bundle on X, then

H^{j}(X,K_{X}\otimes L)=0

for all j > 0. Kodaira vanishing and its generalizations are fundamental to the classification of algebraic varieties and the minimal model program.

Hodge theory

The Hodge theorem relates coherent sheaf cohomology to singular cohomology (or de Rham cohomology). Namely, if X is a smooth complex projective variety, then there is a canonical direct-sum decomposition of complex vector spaces:

H^{a}(X,\mathbf {C} )\cong \bigoplus _{b=0}^{a}H^{a-b}(X,\Omega ^{b}),

for every a. The group on the left means the singular cohomology of X(C) in its classical (Euclidean) topology, whereas the groups on the right are cohomology groups of coherent sheaves, which (by GAGA) can be taken either in the Zariski or in the classical topology. The same conclusion holds for any smooth proper scheme X over C, or for any compact Kähler manifold.

For example, the Hodge theorem implies that the definition of the genus of a smooth projective curve X as the dimension of H¹(X,O), which makes sense over any field k, agrees with the topological definition (as half the first Betti number) when k is the complex numbers. Hodge theory has inspired a large body of work on the topological properties of complex algebraic varieties.

Riemann–Roch theorems

For a proper scheme X over a field k, the Euler characteristic of a coherent sheaf E on X is the integer

\chi (X,E)=\sum _{j}(-1)^{j}\dim _{k}(H^{j}(X,E)).

The Euler characteristic of a coherent sheaf E can be computed from the Chern classes of E, according to the Riemann–Roch theorem and its generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem. For example, if L is a line bundle on a smooth proper geometrically connected curve X over a field k, then

\chi (X,L)={\text{deg}}(L)-{\text{genus}}(X)+1,

where deg(L) denotes the degree of L.

When combined with a vanishing theorem, the Riemann–Roch theorem can often be used to determine the dimension of the vector space of sections of a line bundle. Knowing that a line bundle on X has enough sections, in turn, can be used to define a map from X to projective space, perhaps a closed immersion. This approach is essential for classifying algebraic varieties.

The Riemann–Roch theorem also holds for holomorphic vector bundles on a compact complex manifold, by the Atiyah–Singer index theorem.

Notes

↑ Hartshorne (1977), (III.1.1A) and section III.2.
1 2 Stacks Project, Tag 01X8 .
↑ Stacks Project, Tag 01XE .
↑ Hartshorne (1977), Theorem III.5.1.
↑ Hartshorne (1977), Theorem III.2.7.
↑ Stacks Project, Tag 02O3 .
↑ EGA III, 3.2.1; Grauert & Remmert (1984), Theorem 10.4.6.
↑ Serre (1955), section 80.
↑ Hartshorne (1977), Theorem III.7.6.
↑ Grothendieck & Raynaud, SGA 1, Exposé XII.
↑ Hartshorne (1977), Theorem II.5.17 and Proposition III.5.3.

References

Grauert, Hans; Remmert, Reinhold (1984), Coherent Analytic Sheaves, Springer-Verlag, ISBN 3-540-13178-7, MR 0755331
Grothendieck, Alexandre; Raynaud, Michèle (2003) [1971], Séminaire de Géométrie Algébrique du Bois Marie – 1960–61 – Revêtements étales et groupe fondamental (SGA 1) (Documents Mathématiques 3), Paris: Société Mathématique de France, arXiv:math.AG/0206203, ISBN 978-2-85629-141-2, MR 2017446
Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents", Annals of Mathematics, 61: 197–278, doi:10.2307/1969915, MR 0068874

External links

The Stacks Project Authors, The Stacks Project

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