Coherent sheaf cohomology

Sheaf cohomology

For a sheaf ${\displaystyle \mathcal{F}}$ of abelian groups on a topological space ${\displaystyle X}$, the sheaf cohomology groups ${\displaystyle H^i(X,\mathcal{F})}$ for integers ${\displaystyle i}$ are defined as the right derived functors of the functor of global sections, ${\displaystyle \mathcal{F}\mapsto \mathcal{F}(X)}$. As a result, ${\displaystyle H^i(X,\mathcal{F})}$ is zero for ${\displaystyle i<0}$, and ${\displaystyle H^0(X,\mathcal{F})}$ can be identified with ${\displaystyle \mathcal{F}(X)}$. For any short exact sequence of sheaves ${\displaystyle 0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0}$, there is a long exact sequence of cohomology groups:

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

If ${\displaystyle \mathcal{F}}$ is a sheaf of ${\displaystyle {\mathcal{O}}_X}$-modules on a scheme ${\displaystyle X}$, then the cohomology groups ${\displaystyle H^i(X,\mathcal{F})}$ (defined using the underlying topological space of ${\displaystyle X}$) are modules over the ring ${\displaystyle \mathcal{O}(X)}$ of regular functions. For example, if ${\displaystyle X}$ is a scheme over a field ${\displaystyle k}$, then the cohomology groups ${\displaystyle H^i(X,\mathcal{F})}$ are ${\displaystyle k}$-vector spaces. The theory becomes powerful when ${\displaystyle \mathcal{F}}$ 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 ${\displaystyle \mathcal{F}}$ is a coherent analytic sheaf on a Stein space ${\displaystyle X}$, then ${\displaystyle \mathcal{F}}$ is spanned by its global sections, and ${\displaystyle H^i(X,\mathcal{F})=0}$ for all ${\displaystyle i>0}$. (A complex space ${\displaystyle X}$ is Stein if and only if it is isomorphic to a closed analytic subspace of ${\displaystyle {\mathbb{C}}^n}$ for some ${\displaystyle n}$.) 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 ${\displaystyle \mathcal{F}}$ is a quasi-coherent sheaf on an affine scheme ${\displaystyle X}$, then ${\displaystyle \mathcal{F}}$ is spanned by its global sections, and ${\displaystyle H^i(X,\mathcal{F})=0}$ for ${\displaystyle i>0}$. This is related to the fact that the category of quasi-coherent sheaves on an affine scheme ${\displaystyle X}$ is equivalent to the category of ${\displaystyle \mathcal{O}(X)}$-modules, with the equivalence taking a sheaf ${\displaystyle \mathcal{F}}$ to the ${\displaystyle \mathcal{O}(X)}$-module ${\displaystyle H^0(X,\mathcal{F})}$. 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 ${\displaystyle X}$, an affine open covering ${\displaystyle \{U_i\}}$ of ${\displaystyle X}$, and a quasi-coherent sheaf ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$, the cohomology groups ${\displaystyle H^{\ast }(X,\mathcal{F})}$ are isomorphic to the Čech cohomology groups with respect to the open covering ${\displaystyle \{U_i\}}$. In other words, knowing the sections of ${\displaystyle \mathcal{F}}$ on all finite intersections of the affine open subschemes ${\displaystyle U_i}$ determines the cohomology of ${\displaystyle X}$ with coefficients in ${\displaystyle \mathcal{F}}$.

Using Čech cohomology, one can compute the cohomology of projective space with coefficients in any line bundle. Namely, for a field ${\displaystyle k}$, a positive integer ${\displaystyle n}$, and any integer ${\displaystyle j}$, the cohomology of projective space ${\displaystyle {\mathbb{P}}^n}$ over ${\displaystyle k}$ with coefficients in the line bundle ${\displaystyle \mathcal{O}(j)}$ is given by:

${\displaystyle H^i({\mathbb{P}}^n,\mathcal{O}(j))\cong \{\begin{array}{ll}\underset{a_0,\dots ,a_n\ge 0,a_0+\cdots +a_n=j}{⨁}k\cdot x_0^{a_0}\cdots x_n^{a_n}& i=0\\ 0& i\ne 0,n\\ \underset{a_0,\dots ,a_n<0,a_0+\cdots +a_n=j}{⨁}k\cdot x_0^{a_0}\cdots x_n^{a_n}& i=n\end{array}}$

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

The vanishing of these cohomology groups above dimension ${\displaystyle n}$ is a very special case of Grothendieck's vanishing theorem: for any sheaf of abelian groups ${\displaystyle \mathcal{F}}$ on a Noetherian topological space ${\displaystyle X}$ of dimension ${\displaystyle n<\mathrm{\infty }}$, ${\displaystyle H^i(X,\mathcal{F})=0}$ for all ${\displaystyle i>n}$. This is especially useful for ${\displaystyle X}$ a Noetherian scheme (for example, a variety over a field) and ${\displaystyle \mathcal{F}}$ a quasi-coherent sheaf.

Sheaf cohomology of plane-curves

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

${\displaystyle H^{\ast }({\mathbb{P}}^2,i_{\ast }{\mathcal{O}}_C)\cong H^{\ast }(C,{\mathcal{O}}_C)}$

since ${\displaystyle i_{\ast }}$ is exact. This means that the short exact sequence of coherent sheaves

${\displaystyle 0\to \mathcal{O}(-d)\to \mathcal{O}\to i_{\ast }{\mathcal{O}}_C\to 0}$

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

${\displaystyle \begin{array}{cc}0& \to H^0({\mathbb{P}}^2,\mathcal{O}(-d))\to H^0({\mathbb{P}}^2,\mathcal{O})\to H^0({\mathbb{P}}^2,{\mathcal{O}}_C)\\ & \to H^1({\mathbb{P}}^2,\mathcal{O}(-d))\to H^1({\mathbb{P}}^2,\mathcal{O})\to H^1({\mathbb{P}}^2,{\mathcal{O}}_C)\\ & \to H^2({\mathbb{P}}^2,\mathcal{O}(-d))\to H^2({\mathbb{P}}^2,\mathcal{O})\to H^2({\mathbb{P}}^2,{\mathcal{O}}_C)\end{array}}$

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

${\displaystyle \begin{array}{cc}{H}^0(C,{\mathcal{O}}_C)& \cong H^0({\mathbb{P}}^2,\mathcal{O})\\ H^1(C,{\mathcal{O}}_C)& \cong H^2({\mathbb{P}}^2,\mathcal{O}(-d))\end{array}}$

which shows that ${\displaystyle H^1}$ of the curve is a finite dimensional vector space of rank

${\displaystyle (\genfrac{}{}{0ex}{}{d-1}{d-3})=\frac{(d-1)(d-2)}{2}}$.

Künneth Theorem

There is an analogue of the Künneth formula in coherent sheaf cohomology for products of varieties. Given quasi-compact schemes ${\displaystyle X,Y}$ with affine-diagonals over a field ${\displaystyle k}$, (e.g. separated schemes), and let ${\displaystyle \mathcal{F}\in \text{Coh}(X)}$ and ${\displaystyle \mathcal{G}\in \text{Coh}(Y)}$, then there is an isomorphism

${\displaystyle H^k(X{\times }_{\text{Spec}(k)}Y,{\pi }_1^{\ast }\mathcal{F}{\otimes }_{{\mathcal{O}}_{X{\times }_{\text{Spec}(k)}Y}}{\pi }_2^{\ast }\mathcal{G})\cong \underset{i+j=k}{⨁}{H}^i(X,\mathcal{F}){\otimes }_k{H}^j(Y,\mathcal{G})}$

where ${\displaystyle {\pi }_1,{\pi }_2}$ are the canonical projections of ${\displaystyle X{\times }_{\text{Spec}(k)}Y}$ to ${\displaystyle X,Y}$.

Computing sheaf cohomology of curves

In ${\displaystyle X={\mathbb{P}}^1\times {\mathbb{P}}^1}$, a generic section of ${\displaystyle {\mathcal{O}}_X(a,b)={\pi }_1^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(a){\otimes }_{{\mathcal{O}}_X}{\pi }_2^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(b)}$ defines a curve ${\displaystyle C}$, giving the ideal sequence

${\displaystyle 0\to {\mathcal{O}}_X(-a,-b)\to {\mathcal{O}}_X\to {\mathcal{O}}_C\to 0}$

Then, the long exact sequence reads as

${\displaystyle \begin{array}{cc}0& \to H^0(X,\mathcal{O}(-a,-b))\to H^0(X,\mathcal{O})\to H^0(X,{\mathcal{O}}_C)\\ & \to H^1(X,\mathcal{O}(-a,-b))\to H^1(X,\mathcal{O})\to H^1(X,{\mathcal{O}}_C)\\ & \to H^2(X,\mathcal{O}(-a,-b))\to H^2(X,\mathcal{O})\to H^2(X,{\mathcal{O}}_C)\end{array}}$

giving

${\displaystyle \begin{array}{cc}{H}^0(C,{\mathcal{O}}_C)& \cong H^0(X,\mathcal{O})\\ H^1(C,{\mathcal{O}}_C)& \cong H^2(X,\mathcal{O}(-a,-b))\end{array}}$

Since ${\displaystyle H^1}$ is the genus of the curve, we can use the Künneth formula to compute its Betti numbers. This is

${\displaystyle H^2(X,{\mathcal{O}}_X(-a,-b))\cong H^1({\mathbb{P}}^1,\mathcal{O}(-a)){\otimes }_k{H}^1({\mathbb{P}}^1,\mathcal{O}(-b))}$

which is of rank

${\displaystyle (\genfrac{}{}{0ex}{}{a-1}{a-2})(\genfrac{}{}{0ex}{}{b-1}{b-2})=(a-1)(b-1)=ab-a-b+1}$

for ${\displaystyle -a,-b\le -2}$. In particular, if ${\displaystyle C}$ is defined by the vanishing locus of a generic section of ${\displaystyle \mathcal{O}(2,k)}$, it is of genus

${\displaystyle 2k-2-k+1=k-1}$

hence a curve of any genus can be found inside of ${\displaystyle {\mathbb{P}}^1\times {\mathbb{P}}^1}$.

Finite-dimensionality

For a proper scheme ${\displaystyle X}$ over a field ${\displaystyle k}$ and any coherent sheaf ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$, the cohomology groups ${\displaystyle H^i(X,\mathcal{F})}$ have finite dimension as ${\displaystyle k}$-vector spaces. In the special case where ${\displaystyle X}$ is projective over ${\displaystyle k}$, 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 ${\displaystyle f:X\to Y}$ (in the algebraic or analytic setting) and a coherent sheaf ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$, the higher direct image sheaves ${\displaystyle R^i{f}_{\ast }\mathcal{F}}$ are coherent. When ${\displaystyle Y}$ 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 ${\displaystyle X}$ is a smooth projective curve over an algebraically closed field ${\displaystyle k}$, the genus of ${\displaystyle X}$ is defined to be the dimension of the ${\displaystyle k}$-vector space ${\displaystyle H^1(X,{\mathcal{O}}_X)}$. When ${\displaystyle k}$ is the field of complex numbers, this agrees with the genus of the space ${\displaystyle X(\mathbb{C})}$ of complex points in its classical (Euclidean) topology. (In that case, ${\displaystyle X(\mathbb{C})=X^{an}}$ is a closed oriented surface.) Among many possible higher-dimensional generalizations, the geometric genus of a smooth projective variety ${\displaystyle X}$ of dimension ${\displaystyle n}$ is the dimension of ${\displaystyle H^n(X,{\mathcal{O}}_X)}$, and the arithmetic genus (according to one convention) is the alternating sum

${\displaystyle \chi (X,{\mathcal{O}}_X)=\sum _j(-1{)}^j{\dim }_k(H^j(X,{\mathcal{O}}_X)).}$

Serre duality

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

${\displaystyle H^i(X,E)\times H^{n-i}(X,K_X\otimes E^{\ast })\to H^n(X,K_X)\to k}$

is a perfect pairing for every integer ${\displaystyle i}$. In particular, the ${\displaystyle k}$-vector spaces ${\displaystyle H^i(X,E)}$ and ${\displaystyle H^{n-i}(X,K_X\otimes E^{\ast })}$ 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 ${\displaystyle X}$ over an algebraically closed field ${\displaystyle k}$, Serre duality implies that the dimension of the space ${\displaystyle H^0(X,{\mathrm{\Omega }}^1)=H^0(X,K_X)}$ of 1-forms on ${\displaystyle X}$ is equal to the genus of ${\displaystyle X}$ (the dimension of ${\displaystyle H^1(X,{\mathcal{O}}_X)}$).

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

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

of (finite-dimensional) complex vector spaces is an isomorphism for all i. (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 ${\displaystyle L}$ on a proper scheme ${\displaystyle X}$ over a Noetherian ring, and any coherent sheaf ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$, there is an integer ${\displaystyle m_0}$ such that for all ${\displaystyle m\ge m_0}$, the sheaf ${\displaystyle \mathcal{F}\otimes L^{\otimes m}}$ is spanned by its global sections and has no cohomology in positive degrees.

Although Serre's vanishing theorem is useful, the inexplicitness of the number ${\displaystyle m_0}$ can be a problem. The Kodaira vanishing theorem is an important explicit result. Namely, if ${\displaystyle X}$ is a smooth projective variety over a field of characteristic zero, ${\displaystyle L}$ is an ample line bundle on ${\displaystyle X}$, and ${\displaystyle K_X}$ a canonical bundle, then

${\displaystyle H^j(X,K_X\otimes L)=0}$

for all ${\displaystyle j>0}$. Note that Serre's theorem guarantees the same vanishing for large powers of ${\displaystyle L}$. Kodaira vanishing and its generalizations are fundamental to the classification of algebraic varieties and the minimal model program. Kodaira vanishing fails over fields of positive characteristic.

Hodge theory

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

${\displaystyle H^a(X,\mathbf{C})\cong \underset{b=0}{\overset{a}{⨁}}{H}^{a-b}(X,{\mathrm{\Omega }}^b),}$

for every ${\displaystyle a}$. The group on the left means the singular cohomology of ${\displaystyle X(\mathbf{C})}$ 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 ${\displaystyle X}$ over ${\displaystyle \mathbf{C}}$, or for any compact Kähler manifold.

For example, the Hodge theorem implies that the definition of the genus of a smooth projective curve ${\displaystyle X}$ as the dimension of ${\displaystyle H^1(X,\mathcal{O})}$, which makes sense over any field ${\displaystyle k}$, agrees with the topological definition (as half the first Betti number) when ${\displaystyle k}$ 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

${\displaystyle \chi (X,E)=\sum _j(-1{)}^j{\dim }_k(H^j(X,E)).}$

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

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

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.

Growth

Dimensions of cohomology groups on a scheme of dimension n can grow up at most like a polynomial of degree n.

Let X be a projective scheme of dimension n and D a divisor on X. If ${\displaystyle \mathcal{F}}$ is any coherent sheaf on X then

${\displaystyle h^i(X,\mathcal{F}(mD))=O(m^n)}$ for every i.

For a higher cohomology of nef divisor D on X;

${\displaystyle h^i(X,{\mathcal{O}}_X(mD))=O(m^{n-1})}$

Applications

Given a scheme X over a field k, deformation theory studies the deformations of X to infinitesimal neighborhoods. The simplest case, concerning deformations over the ring ${\displaystyle R:=k[ϵ]/{ϵ}^2}$ of dual numbers, examines whether there is a scheme X_R over Spec R such that the special fiber

${\displaystyle X_R{\times }_{\mathrm{Spec}R}\mathrm{Spec}k}$

is isomorphic to the given X. Coherent sheaf cohomology with coefficients in the tangent sheaf ${\displaystyle T_X}$ controls this class of deformations of X, provided X is smooth. Namely,

• isomorphism classes of deformations of the above type are parametrized by the first coherent cohomology ${\displaystyle H^1(X,T_X)}$,
• there is an element (called the obstruction class) in ${\displaystyle H^2(X,T_X)}$ which vanishes if and only if a deformation of X over Spec R as above exists.