Coherent sheaf 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 \mathcal O_X, the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a full subcategory of the category of \mathcal O_X-modules (that is, sheaves of \mathcal 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 \to 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 \mathcal F of abelian groups on a topological space X, the sheaf cohomology groups H^i(X, \mathcal F) for integers i are defined as the right derived functors of the functor of global sections, \mathcal F \mapsto \mathcal F(X). As a result, H^i(X, \mathcal F) is zero for i , and H^0(X, \mathcal F) can be identified with \mathcal F(X). For any short exact sequence of sheaves 0\to \mathcal A \to \mathcal B \to \mathcal C\to 0, there is a long exact sequence of cohomology groups: : 0\to H^0(X,\mathcal A) \to H^0(X,\mathcal B) \to H^0(X,\mathcal C) \to H^1(X,\mathcal A) \to \cdots.

If \mathcal F is a sheaf of \mathcal O_X-modules on a scheme X, then the cohomology groups H^i(X, \mathcal F) (defined using the underlying topological space of X) are modules over the ring \mathcal O(X) of regular functions. For example, if X is a scheme over a field k, then the cohomology groups H^i(X, \mathcal F) are k-vector spaces. The theory becomes powerful when \mathcal F 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 \mathcal F is a coherent analytic sheaf on a Stein space X, then \mathcal F is spanned by its global sections, and H^i(X, \mathcal F) = 0 for all i 0. (A complex space X is Stein if and only if it is isomorphic to a closed analytic subspace of \Complex^n 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 \mathcal F is a quasi-coherent sheaf on an affine scheme X, then \mathcal F is spanned by its global sections, and H^i(X, \mathcal F) = 0 for i0. This is related to the fact that the category of quasi-coherent sheaves on an affine scheme X is equivalent to the category of \mathcal O(X)-modules, with the equivalence taking a sheaf \mathcal F to the \mathcal O(X)-module H^0(X, \mathcal F). In fact, affine schemes are characterized among all quasi-compact schemes by the vanishing of higher cohomology for quasi-coherent sheaves.

Čech 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 \mathcal F on X, the cohomology groups H^*(X,\mathcal F) are isomorphic to the Čech cohomology groups with respect to the open covering {U_i}. In other words, knowing the sections of \mathcal F on all finite intersections of the affine open subschemes U_i determines the cohomology of X with coefficients in \mathcal F.

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 \mathbb{P}^n over k with coefficients in the line bundle \mathcal O(j) is given by:

: H^i(\mathbb{P}^n,\mathcal O(j)) \cong \begin{cases} \bigoplus_{a_0,\ldots,a_n\geq 0,; a_0+\cdots+a_n=j}; k\cdot x_0^{a_0}\cdots x_n^{a_n} & i=0\[6pt] 0 & i \neq 0, n\[6pt] \bigoplus_{a_0, \ldots,a_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 \mathcal F on a Noetherian topological space X of dimension n, H^i(X,\mathcal F) = 0 for all in. This is especially useful for X a Noetherian scheme (for example, a variety over a field) and \mathcal F a quasi-coherent sheaf.

Sheaf cohomology of plane-curves

Given a smooth projective plane curve C of degree d, the sheaf cohomology H^*(C,\mathcal{O}_C) can be readily computed using a long exact sequence in cohomology. First note that for the embedding i:C \to \mathbb{P}^2 there is the isomorphism of cohomology groups

:H^(\mathbb{P}^2, i_\mathcal{O}_C) \cong H^*(C, \mathcal{O}_C)

since i_* is exact. This means that the short exact sequence of coherent sheaves

:0 \to \mathcal{O}(-d) \to \mathcal{O} \to i_*\mathcal{O}_C \to 0

on \mathbb{P}^2, called the ideal sequence**,** can be used to compute cohomology via the long exact sequence in cohomology. The sequence reads as

:\begin{align} 0&\to H^0(\mathbb{P}^2, \mathcal{O}(-d)) \to H^0(\mathbb{P}^2, \mathcal{O}) \to H^0(\mathbb{P}^2, \mathcal{O}_C)\ &\to H^1(\mathbb{P}^2, \mathcal{O}(-d)) \to H^1(\mathbb{P}^2, \mathcal{O}) \to H^1(\mathbb{P}^2, \mathcal{O}_C)\ &\to H^2(\mathbb{P}^2, \mathcal{O}(-d)) \to H^2(\mathbb{P}^2, \mathcal{O}) \to H^2(\mathbb{P}^2, \mathcal{O}_C) \end{align}

which can be simplified using the previous computations on projective space. For simplicity, assume the base ring is \C (or any algebraically closed field). Then there are the isomorphisms

:\begin{align} H^0(C,\mathcal{O}_C) &\cong H^0(\mathbb{P}^2,\mathcal{O}) \ H^1(C,\mathcal{O}_C) &\cong H^2(\mathbb{P}^2,\mathcal{O}(-d)) \end{align}

which shows that H^1 of the curve is a finite dimensional vector space of rank

:{d-1 \choose d-3 } = \frac{(d-1)(d-2)}{2}.

Künneth Theorem

There is an analogue of the Künneth formula in coherent sheaf cohomology for products of varieties. Given quasi-compact schemes X,Y with affine-diagonals over a field k, (e.g. separated schemes), and let \mathcal{F} \in \text{Coh}(X) and \mathcal{G} \in \text{Coh}(Y), then there is an isomorphism H^k(X\times_{\text{Spec}(k)}Y, \pi_1^\mathcal{F}\otimes_{\mathcal{O}{X\times{\text{Spec}(k)} Y}}\pi_2^\mathcal{G}) \cong \bigoplus_{i+j = k} H^i(X,\mathcal{F})\otimes_k H^j(Y,\mathcal{G}) where \pi_1,\pi_2 are the canonical projections of X\times_{\text{Spec}(k)} Y to X,Y.

Computing sheaf cohomology of curves

In X = \mathbb{P}^1 \times \mathbb{P}^1, a generic section of \mathcal{O}X(a,b) = \pi_1^*\mathcal{O}{\mathbb{P}^1}(a) \otimes_{\mathcal{O}X} \pi_2^*\mathcal{O}{\mathbb{P}^1}(b) defines a curve C, giving the ideal sequence0 \to \mathcal{O}_X(-a,-b) \to \mathcal{O}_X \to \mathcal{O}_C \to 0Then, the long exact sequence reads as\begin{align} 0&\to H^0(X, \mathcal{O}(-a,-b)) \to H^0(X, \mathcal{O}) \to H^0(X, \mathcal{O}_C)\ &\to H^1(X, \mathcal{O}(-a,-b)) \to H^1(X, \mathcal{O}) \to H^1(X, \mathcal{O}_C)\ &\to H^2(X, \mathcal{O}(-a,-b)) \to H^2(X, \mathcal{O}) \to H^2(X, \mathcal{O}_C) \end{align}giving \begin{align} H^0(C,\mathcal{O}_C) &\cong H^0(X,\mathcal{O}) \ H^1(C,\mathcal{O}_C) &\cong H^2(X,\mathcal{O}(-a,-b)) \end{align}Since H^1 is the genus of the curve, we can use the Künneth formula to compute its Betti numbers. This isH^2(X, \mathcal{O}_X(-a,-b)) \cong H^1(\mathbb{P}^1,\mathcal{O}(-a))\otimes_kH^1(\mathbb{P}^1,\mathcal{O}(-b))which is of rank\binom{a-1}{a-2}\binom{b-1}{b-2} = (a-1)(b-1) = ab - a - b +1for -a,-b \leq -2. In particular, if C is defined by the vanishing locus of a generic section of \mathcal{O}(2,k), it is of genus2k-2-k+1 = k-1hence a curve of any genus can be found inside of \mathbb{P}^1\times\mathbb{P}^1.

Finite-dimensionality

For a proper scheme X over a field k and any coherent sheaf \mathcal F on X, the cohomology groups H^i(X,\mathcal F) have finite dimension as k-vector spaces. 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\to Y (in the algebraic or analytic setting) and a coherent sheaf \mathcal F on X, the higher direct image sheaves R^i f_*\mathcal F are coherent. 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 H^1(X,\mathcal O_X). When k is the field of complex numbers, this agrees with the genus of the space X(\Complex) of complex points in its classical (Euclidean) topology. (In that case, X(\Complex) = 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 H^n(X, \mathcal O_X), and the arithmetic genus (according to one convention) is the alternating sum ::\chi(X, \mathcal{O}_X)=\sum_j (-1)^j\dim_k(H^j(X, \mathcal O_X)).

Serre duality

Main article: 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 H^n(X, K_X)\to 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