Sheaf Cohomology as a Cohomology Theory on Schemes

This post isn’t going to be a complete recount of sheaf cohomology. There is a succinct article fulfilling that purpose provided by the Stacks Project.

Instead, I want this post to outline some of the aspects which I felt like I missed when I first read through Hartshorne’s chapter on sheaf cohomology. The reason being: I’ve been attempting to write up some notes on certain categories of motives, and how one characterization of motives could be as a “universal Weil cohomology theory”; in looking at examples of Weil cohomology theories, I realized I should probably just refresh my memory  of (read: learn more properly) some of the existing cohomology theories.

For the above reason, I don’t know where this post will go. Sheaf cohomology (for the Zariski site on smooth projective schemes) is not a Weil cohomology theory. It is, relative to other theories, somewhat computable by means of Čech cohomology of alternating cochains. Although, proving that one can use alternating cochains isn’t fun (thanks, Serre, for doing it for me).

Let me just start and see what happens.

An Introduction of Sorts

Cohomology is one of those subjects which is still elusive to some extent. One could focus on extreme concreteness with constructions like simplicial cohomology or de Rham cohomology, where to a space $X$ one assigns a complex $C_\bullet(X)$, and cohomology is calculated as the homology of this complex. Such a theory would deserve to be called a cohomology theory if it satisfied usual cohomological properties: pullbacks, cup products, etc.

One could easily argue cohomological ideas began with Euler’s definition of the Euler characteristic for a polyhedron or, the sum vertices-edges+faces. This is generalized when one studies simplicial homology as the alternating sum of the dimensions of the homology of the simplicial chain complex associated to, say, a polyhedron. Poincaré used these dimensions implictly, calling them betti numbers, and not knowing they were in fact the dimensions of some homology.

At some point, the focus turned to computing homology of chain complexes as it was realized this idea captured all of the historical ones. Here the focus would be on the homology associated to a complex, discarding the complex afterwards. Today, I’d say this idea is outdated as well. A more modern approach would be to focus on the chain complexes but, to mod out by “homological equivalence” of some sort. This idea is captured by the derived category.

The unbounded derived category of an additive category $X$ is defined then like so: take the category of complexes of objects of $X$, call it $Kom(X)$. Mod out the hom sets by those morphisms homotopy equivalent to 0, which define equal on-the-nose maps of homology between chain complexes, and call this category $K(X)$. Then, if a morphism $f$ from the category $K(X)$ induces an isomorphism on all homology of the chain complex, give it a formal inverse, and call this new category $D(X)$.

This is the setting for which we’ll define a “cohomology theory”.

Construction of Cohomology

Let $X$ be a scheme. I’ll always work with schemes. However, I will remark quite a number of the below results hold for arbitrary ringed spaces.

(1.1) Definition: the functor $\Gamma(X,-): \mathsf{AbSh}_X\rightarrow \mathsf{Ab}$, from the category of abelian sheaves over $X$ (sheaves of abelian groups on the Zariski topology of $X$) to the category of abelian groups, defined by

$\Gamma(X,\mathscr{F})=\mathscr{F}(X)$

is called the global sections functor for $X$.

$\Gamma(X,-)$ is a left-exact functor but, it is not in general right exact (this can be checked very easily on short exact sequences of sheaves). Since the category $\mathsf{AbSh}_X$ has enough injectives, we can construct a right derived functor for $\Gamma(X,-)$.

(1.2) Proposition: the right derived functor $R\Gamma(X,-):D^+(\mathsf{AbSh}_X)\rightarrow D^+(\mathsf{Ab})$ exists.

Proof. This follows from the fact that $\mathsf{AbSh}_X$ has enough injectives. We can construct, for any complex of sheaves $A^\bullet$ in $D^+(\mathsf{AbSh}_X)$ a quasi-isomorphism $A^\bullet\rightarrow I^\bullet$ with $I^\bullet$ consisting of injectives of $\mathsf{AbSh}_X$. Then $R\Gamma(X,A^\bullet)=\Gamma(X,I^\bullet)$.$\square$

(1.3) Definition: let $F$ be an abelian sheaf on $X$. We define the $i$th cohomology of $X$ with values in the sheaf $\mathscr{F}$ to be

$H^i(X,\mathscr{F})=H^i(R\Gamma(X,\mathscr{F})).$

Another way to write this is

$H^i(X,\mathscr{F}):=R^i\Gamma(X,\mathscr{F}).$

(1.4) Corollary: For any abelian sheaf $\mathscr{F}$ we have $H^0(X,\mathscr{F})=\Gamma(X,\mathscr{F})=\mathscr{F}(X)$.

Proof. Considering $\mathscr{F}$ as a complex in degree 0, we can find a resolution of $\mathscr{F}$ by injective abelian sheaves $\mathscr{F}\rightarrow I^\bullet$. Following the construction,

$H^0(X,\mathscr{F})=H^0(R\Gamma(X,\mathscr{F})=H^0(\Gamma(X,I^\bullet))=\ker(\Gamma(X,I^0)\rightarrow \Gamma(X,I^1))=\text{Im}(\Gamma(X,\mathscr{F})\rightarrow \Gamma(X,I^0))=\mathscr{F}(X)$

since $\Gamma(X,-)$ is left-exact.$\square$

While the above notion is fine for most-intents-and-purposes, it sometimes falls short in using all of the information we really have. For example, any $\mathscr{O}_X$-module, say $\mathscr{M}$, is an abelian sheaf. Then the above says we can compute the cohomology of $\mathscr{M}$ regarded as an abelian sheaf; this corresponds to first forgetting the module structure of $\mathscr{M}$ and then taking the right derived functor $R\Gamma(X,\mathscr{M})$ to obtain a chain complex of abelian groups. Alternatively, knowing $\mathsf{ModSh}_X$ has enough injectives, we could mimic the proof of Lemma (1.3) to show

(1.5) Proposition: there is a right derived functor $R\widehat{\Gamma}(X,-): D^+(\mathsf{ModSh}_X)\rightarrow D^+(\mathscr{O}_X-\mathsf{mod})$.

Proof. Similar to (1.3) but in different categories. $\square$

Computing $R\widehat{\Gamma}(X,\mathscr{M})$ then forgetting the module structure of the complex gives us another way to obtain a complex of abelian groups from $\mathscr{M}$, which just as reasonably could be used to define the cohomology of $X$ with values in $\mathscr{M}$. Ideally, it won’t matter which way we decide to calculate the cohomology. Indeed, this will be the case. But, in order to show this, we’ll need the notion of a flasque sheaf.

(1.6) Definition: A sheaf, $\mathscr{F}$, is called flasque if the restrictions

$\mathscr{F}(U)\rightarrow \mathscr{F}(V)$

are surjective for any inclusion $V\hookrightarrow U$ of open sets of $X$.

To reiterate, we want to check the commutativity of the diagram

$\begin{matrix} D^+(\mathsf{ModSh}_X) & \xrightarrow{R\widehat{\Gamma}(X,-)} & D^+(\mathscr{O}_X(X)-\mathsf{mod})\\ \downarrow & & \downarrow\\ D^+(\mathsf{AbSh}_X)& \xrightarrow{R\Gamma(X,-)} & D^+(\mathsf{Ab})\end{matrix}$

where the vertical arrows are forgetful functors.

The right derived functor $R\widehat{\Gamma}(X,-):D^+(\mathsf{ModSh}_X)\rightarrow D^+(\mathscr{O}_X(X)-\mathsf{mod})$ is defined similar to before: for any complex of $\mathscr{O}_X$-modules $A^\bullet$ we have $R\widehat{\Gamma}(X,A^\bullet)=\Gamma(X,\widehat{I}^\bullet)$ for a quasi-isomorphism $A^\bullet\rightarrow \widehat{I}^\bullet$ with $\widehat{I}^\bullet$ a complex of injectives of $\mathsf{ModSh}_X$.

As a first approach to showing commutativity of the above diagram, it would be tempting to say the complex $\widehat{I}^\bullet$ is also a complex of injectives of $\mathsf{Ab}$. If this were true then all of our work would be for naught! So of course this is not the case; the complex $\widehat{I}^\bullet$ considered as a complex of abelian sheaves is, in general, not a complex of injectives in the category $\mathsf{Ab}$.

However, the next proposition will show an $\mathscr{O}_X$-module is flasque. Flasque sheaves are very nice in the sense they are flasque in any category you consider them. That is, a flasque $\mathscr{O}_X$-module is also a flasque abelian sheaf after you forget the module structure. Combining this with proposition (1.7) below, that flasque abelian sheaves are acyclic (or right-adapted, depending on your terminology) for $\Gamma(X,-)$, we get the desired commutativity.

(1.7) Proposition: Injective $\mathscr{O}_X$-modules are flasque.

Proof. Proposition 6.2 (1) in [Ill].

(1.8) Proposition: Flasque sheaves are acyclic for $\Gamma(X,-)$.

Proof. Proposition 6.2 (3) in [Ill].

(1.9) Corollay: If an abelian sheaf $\mathscr{F}$ has the structure of an $\mathscr{O}_X$-module, then its cohomology groups, $H^i(X,\mathscr{F})$, have the structure of $\mathscr{O}_X(X)$-modules.

Proof. Indeed, by our discussion above we can compute cohomology by applying the forgetful functor to $R\widehat{\Gamma}(X,-):D^+(\mathsf{ModSh}_X)\rightarrow D^+(\mathscr{O}_X(X)-\mathsf{mod})$. So don’t apply the forgetful functor, and then we have a module structure on the cohomology.$\square$

A second, arguably as important, purpose for introducing flasque sheaves is for the study of constant sheaves on irreducible topological spaces and of skyscraper sheaves. In particular, we have a sheaf $K_X$ for any variety (separated scheme of finite type over a field $k$) $X$, which will turn out to be constant on any integral variety.

Recall it suffices to define a presheaf on $X$ by defining what happens only for affine opens. We obtain a sheaf by taking the Zariski sheafification of this presheaf. So, let $U$ be an open affine subset of $X$.

(1.10) Definition: In the setting described above we define the sheaf of rational functions on $X$, denoted $K_X$, to be the Zariski sheafification of the presheaf defined by

$U\rightsquigarrow S_U^{-1}\mathscr{O}_X(U)$

$S_U=\{ s\in \mathscr{O}_X(U): s \text{ is not a zero divisor in } \mathscr{O}_{X,x}\text{ for any } x\in U \}.$

Note this is different from the sheaf defined in Hartshorne’s book. The definition for the sheaf of rational functions which is given in Hartshorne’s book is, unfortunately, incorrect in it’s full generality. (See here for a related question, and here for a discussion). For this reason we provide proof our definition of a presheaf is a presheaf.

(1.11) Proposition: The above definition is actually a definition. That is to say, $K_X$ as defined above is, in fact, a sheaf.

Proof. We need only check that for any inclusion $V\hookrightarrow U$ of open affine sets $V,U$ of $X$, we have a well-defined restriction $S_U^{-1}\mathscr{O}_X(U)\rightarrow S_V^{-1}\mathscr{O}_X(V)$. This gives the necessary presheaf condition — which is the only thing we have to check since sheafification always exists.

The map $S^{-1}_U\mathscr{O}_X(U)\rightarrow S_V^{-1}\mathscr{O}_X(V)$ is actually the one which descends via the diagram

$\begin{matrix} \mathscr{O}_X(U) & \rightarrow & \mathscr{O}_X(V) \\ \downarrow & & \downarrow \\ S_U^{-1}\mathscr{O}_X(U) & \dashrightarrow & S_V^{-1}\mathscr{O}_X(V) \end{matrix}$

with all solid arrows the natural restriction and localization. To see this, note that we have defined maps

$\mathscr{O}_X(U)\rightarrow \mathscr{O}_X(V)\rightarrow S_V^{-1}\mathscr{O}_X(V)$

which take $S_U\subset \mathscr{O}_X(U)$ to invertible elements in $S_V^{-1}\mathscr{O}_X(V)$ (by our definition of $S_U$!). Hence, by the universal property of localization, the map factors

$\mathscr{O}_X(U)\rightarrow S_U^{-1}\mathscr{O}_X(U)\rightarrow S_V^{-1}\mathscr{O}_X(V)$

completing the above square as desired. By the functorality of our construction of the restriction maps, the morphisms are compatible in the required sense. $\square$

As alluded to above, $K_X$ admits the following characterization.

(1.12) Proposition: If $X$ is integral, then $K_X$ is constant. In particular, as $K_X$ is a constant sheaf on an irreducible topological space, it is flasque.

Proof. Since $X$ is integral, $\mathscr{O}_{X,x}$ has no zero divisors for any point $x\in X$. Then $K_X$ is the sheafification of the presheaf $U\rightsquigarrow \text{Frac}(\mathscr{O}_X(U))$. But this presheaf is the constant presheaf. Indeed, $\text{Frac}(\mathscr{O_X}(U))$ is just localization at the generic point of $U$, which is equal to the localization at the generic point of any other affine open $V$ by the assumption $X$ is integral. Hence, $K_X$ is a constant sheaf. The second claim follows by noting the constant presheaf on an irreducible topological space is already a sheaf. To see this one can check the sheaf exact sequence. But the constant presheaf on an irreducible topological space is flasque, as all the restriction maps are the identity.$\square$

To end this section, we make slight remarks on the relation between the pushforward functor of sheaves and cohomology. Although I used this functor in the last proof, I recall the definition purely for convenience.

(1.12) Definition: let $f:X\rightarrow Y$ be a morphism of schemes. We define the functor $f_\ast:\mathsf{ModSh}_X\rightarrow \mathsf{ModSh}_Y$, called the direct image functor, by the rule

$f_\ast\mathscr{F}(V):=\mathscr{F}(f^{-1}(V))$

for any $V$ open in $Y$. For completeness, $f_\ast$ acts on morphisms of sheaves on $X$, $s:\mathscr{F}\rightarrow \mathscr{G}$, by the rule

$f_\ast s(V):=s(f^{-1}(V)).$

The functor $f_\ast$ is left-exact also (and this can also be checked very easily on short exact sequences of sheaves).

(1.13) Proposition$\Gamma(Y,-)\circ \pi_{X \ast}=\Gamma(X,-)$

We have, on objects $\mathscr{F}$,

$\Gamma(Y,\pi_{X\ast}\mathscr{F})=\pi_{X\ast}\mathscr{F}(Y)=\mathscr{F}(X)=\Gamma(X,\mathscr{F})$

and, on morphisms $\mathscr{F}\xrightarrow{s}\mathscr{G}$,

$\Gamma(Y,\pi_{X\ast}s)=\pi_{X\ast}s(Y)=s(X)=\Gamma(X,\mathscr{F}))$.$\square$

(1.14) Corollary: There is a convergent spectral sequence

$E_2^{pq}= H^p(Y,R^qf_\ast\mathscr{F})\implies H^{p+q}(X,\mathscr{F})$

Proof. This is the Grothendieck spectral sequence applied to composition on the left hand side of proposition (1.13). The convergence claim follows from proposition (1.13) as the functors are equal on-the-nose so are their derived functors. This spectral sequence is actually called the Leray spectral sequence.$\square$

Cohomological Properties

Here we collect some of the cohomological properties of sheaf cohomology for more general schemes.

(2.1) Characterization of Affines: A quasi-compact quasi-separated scheme $X$ is affine if, and only if, $H^i(X,\mathscr{F})=0$ for every quasi-coherent sheaf $\mathscr{F}$ on $X$ and all $i>0$.

(2.2) Vanishing: If $X$ has a underlying Noetherian topological space and $\dim(X)\leq d$, then $H^i(X,\mathscr{F})=0$ for all $i>d$ and any abelian sheaf $\mathscr{F}$ on $X$.

(2.3) Finite Generation: If $X$ is projective, and $\mathscr{F}$ a coherent sheaf on $X$, then $H^i(X,\mathscr{F})$ is a finitely generated $\mathscr{O}_X(X)$-module for all $i$.

(2.4) Sort of Contravariance in the First Variable; Covariance in the Second Variable: For a morphism of schemes $X\xrightarrow{f}Y$ and $\mathscr{F}$ an $\mathscr{O}_X$-module there is a natural isomorphism $H^i(Y,f_\ast\mathscr{F})\xrightarrow{\sim} H^i(X,\mathscr{F})$. For two $\mathscr{O}_X$-modules, $\mathscr{F},\mathscr{G}$ and a morphism $\mathscr{F}\rightarrow \mathscr{G}$ there is a natural morphism $H^i(X,\mathscr{F})\rightarrow H^i(X,\mathscr{G})$. In particular, the map $\mathscr{O}_Y\rightarrow f_*\mathscr{O}_X$ induces a map $H^i(Y,\mathscr{O}_Y)\rightarrow H^i(Y,f_*\mathscr{O}_X)=H^i(X,\mathscr{O}_X)$.

(2.5) Long Exact Sequence: Associated to any exact sequence of abelian sheaves

$0\rightarrow \mathscr{F}\rightarrow \mathscr{G}\rightarrow \mathscr{H}\rightarrow 0$

is a long exact sequence of cohomology

$0\rightarrow H^0(X,\mathscr{F})\rightarrow H^0(X,\mathscr{G})\rightarrow H^0(X,\mathscr{H})\rightarrow H^1(X,\mathscr{F})\rightarrow \cdots.$

(2.6) Cup Product: For $\mathscr{F},\mathscr{G}$ two $\mathscr{O}_X$-modules, there is a product

$\cup:H^i(X,\mathscr{F})\times H^j(X,\mathscr{G})\rightarrow H^{i+j}(X,\mathscr{F}\otimes_{\mathscr{O}_X} \mathscr{G}).$

In particular, if $\mathscr{F}=\mathscr{G}=\mathscr{O}_X$ then the direct sum $\bigoplus_i H^i(X,\mathscr{O}_X)$ inherits a ring structure which is skew-commutative in the sense $\alpha\cup \beta = (-1)^{\deg(\alpha)\deg(\beta)}\beta\cup \alpha$ for homogeneous elements $\alpha,\beta$.

(2.7) Künneth Formula: Let $X$, $Y$ be separated schemes over a field $k$ and $\mathscr{F},\mathscr{G}$ quasi-coherent sheaves on $X,Y$ respectively. There is an isomrphism

$H^n(X\times Y,\mathscr{F}\boxtimes \mathscr{G})=\bigoplus\limits_{p+q=n}H^p(X,\mathscr{F})\otimes_k H^q(Y,\mathscr{G})$

where $\mathscr{F}\boxtimes \mathscr{G}=p^\ast_1\mathscr{F}\otimes_{\mathscr{O}_{X\times Y}} p^\ast_2\mathscr{G}$ for the projections $p_1:X\times Y\rightarrow X$, $p_2:X\times Y\rightarrow Y$.

(2.8) Serre Duality: Let $X$ be a smooth projective $k$-scheme of pure dimension $n$. Then, with $\omega_X=\wedge^n\Omega_X$, there are perfect pairings

i) $H^i(X,\mathscr{F})\times H^{n-i}(X,\mathscr{H}om(\mathscr{F},\mathscr{O}_X)\otimes \omega_X)\rightarrow H^n(X,\omega_X)\cong k$ for any locally free coherent sheaf $\mathscr{F}$

ii) $\text{Ext}^i(\mathscr{F},\omega_X)\times H^{n-i}(X,\mathscr{F})\rightarrow H^n(X,\omega_X)\cong k$ for any coherent sheaf $\mathscr{F}$

A Conclusion of Sorts

I’m deciding to stop here. Mostly because there are various directions one can take sheaf cohomology from this point and none of them are currently what I want to look at.

Some ideas, for those curious where to go:

• Take the Čech approach to sheaf cohomology, and find when they agree. In particular, on a separated scheme using a cover by affine opens.
• The classification of line bundles via the group $H^1(X,\mathscr{O}_X^\times)$. More generally, how $H^1(X,\mathscr{A}ut(S))$ classifies $S$-torsors.
• The description of Cartier divisors as the group $H^0(X,K_X^\times/\mathscr{O}_X^\times)$. Their relation to Weil divisors. More generally, the construction of the Chow ring via Bloch’s formula: $\text{CH}^p(X)\simeq H^p(X,\mathscr{K}_p(\mathscr{O}_X))$.
• The use of sheaf cohomology to define numerical invariants and their use as discrete invariants in the classification problem. Examples: the arithmetic and geometric genus (when they agree and when they differ), Hilbert polynomials, Euler characteristic.

References:

[Ill] Topics in Algebraic Geometry – Illusie. Link to text.