nLab abelian sheaf cohomology

Idea

General

The cohomology $H^n(X,F)$ of a topological space $X$ with values in a sheaf of abelian groups / abelian sheaf $F$ was originally defined as the value of the right derived functor of the global section functor, the derived direct image functor.

But by embedding sheaves with values in abelian groups as special cases of simplicial sheaves into the more general context of ∞-groupoid-valued sheaves via the Dold-Kan correspondence and thus the abelian sheaf cohomology into the more general context of the intrinsic nonabelian cohomology of an (∞,1)-topos $\mathbf{H} = Sh_{(\infty,1)}(C)$, this definition becomes equivalent to a special case of the general notion of nonabelian cohomology defined simply as the set of homotopy classes of maps

$$H^n(X,F) = \pi_0 \mathbf{H}(X,\mathbf{B}^n F)$$

from the space $X$ regarded a (“nonabelian”!) sheaf, to the Eilenberg-MacLane object in degree $n$, defined by $F$.

The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in

• Kenneth Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

(whose proof is reproduced below).

Brown constructed effectively the homotopy category of $\mathbf{H}$ using a model of a category of fibrant objects paralleling the model structure on simplicial presheaves as a presentation of the (∞,1)-category of (∞,1)-sheaves. This says that ordinary abelian sheaf cohomology in fact computes the equivalence classes of the ∞-stackification of a sheaf with values in chain complexes of abelian groups.

The general (∞,1)-topos-theoreric perspective on cohomology is described in more detail at cohomology. The details on how to realize abelian sheaf cohomology as an example of this are discussed below.

More details on this idea

Using the Dold-Kan correspondence in higher topos theory, complexes of abelian sheaves can be understood as a generalization of topological spaces, in a precise sense recalled below. This induces a notion of cohomology of complexes of abelian sheaves from the familiar notion of cohomology of spaces.

Which is a simple one: recall that the cohomology of one topological space $X$ with coefficients in another space $A$ is nothing but the space of morphisms (continuous maps) $H(X,A) := [X,A]$ from $X$ to $A$ – or, in a more restrictive sense traditionally adopted, the set $\Pi_0[X,A]$ of connected path components of this space.

Similarly, when considering chain complexes of abelian sheaves in their natural higher topos theoretic home, the cohomology of a complex of sheaves $A$ on a space $X$ is nothing but the hom-space $H(X,A) = [X,A]$ – where the space $X$ itself is regarded as a special case of a sheaf.

One reason this conceptually simple picture is not usually presented is that the space $X$ is typically not represented by an abelian complex of sheaves, so that the full simplicity of the story becomes manifest only in general nonabelian cohomology.

More precisely, via the Dold-Kan correspondence (non-negatively graded) complexes of abelian sheaves – which are equivalently sheaves with values in (non-negatively graded) categories of chain complexes – can be regarded as special cases of simplicial sheaves. But thanks to the model category structure on the category of simplicial sheaves, this in turn is a model for the (infinity,1)-topos of generalized spaces called infinity-stacks. The very point of $(\infty,1)$-topoi is that they are (infintiy,1)-categories which behave in all structural aspects relevant for homotopy theory as the archetypical example Top does. In particular, as in Top, the notion of cohomology in any (infinity,1)-topos simply coincides with that of hom-spaces.

In the 1-categorical model theoretic models these hom-spaces are computed technically by derived functors. More precisely, the Hom-space $[X,A]$ for $X$ an ordinary space computes the global sections $\Gamma(X,A)$ of the complex of abelian sheaves $A$ which is computed by the right derived functor of the global section $R \Gamma(X,-)$ of the global section functor $\Gamma(X,-)$, which does exist entirely within the abelian context.

This, then, is the definition of sheaf cohomology as usually presented: the cohomology of the complex $R \Gamma(X,A)$.

Properties

As intrinsic (∞,1)-topos cohomology

Under the (stable) Dold-Kan correspondence we have the following identification of sheaves taking values in chain complexes with sheaves taking values in infinity-groupoids and spectra, crucial for a conceptual understanding of abelian sheaf cohomology.

Let $X$ be a site.

• The category $Sh(X, Ch_+(Ab))$ of non-negatively graded chain complexes of abelian sheaves is homotopically equivalent to the category $Sh(X, sAb)$ of sheaves with values in simplicial abelian groups (i.e. Kan complexes with strict abelian group structure);

• The category $Sh(X, Ch(Ab))$ of unbounded chain complexes of abelian sheaves is equivalent to $Sh(X, Sp(Ab))$, the category of sheaves with values in combinatorial spectra internal to abelian groups.

Let how $F \in Sh(X,Ab)$ be a sheaf on a site $X$ with values in the category Ab of abelian groups.

For $n \in \mathbb{N}$ write $B^n F \in Sh(X, Ch_+(Ab))$ for the complex of sheaves with values in abelian groups which is trivial everywhere except in degree $n$, where it is given by $F$.

By the Dold-Kan correspondence we can regard $B^n F$ equivalently as a complex of sheaves of abelian groups as well as sheaf with values in infinity-groupoids.

Write $H$ for the (infinity,1)-category of simplicial sheaves on $X$ and $H_{ab}$ for the (infinity,1)-category of complexes of abelian sheaves on $X$.

Write $X$ for the terminal sheaf of $X$, i.e. for the sheaf that corresponds to the space $X$ itself.

Then

$$H^n(X,A) := \pi_0 H(X,\mathbf{B}^n F)$$

is the degree $n$ cohomology class of $X$ with values in $F$, regarded as computed in nonabelian cohomology.

Now write $\mathbb{Z}[X]$ for the free abelianization of the sheaf $X$. This is the sheaf constant on the abelian group $\mathbb{Z}$ of integers. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of $F$, is canonically isomorphic to the cohomology set

$$\cdots \simeq \pi_0 H_{ab}(\mathbb{Z}[X], \mathbf{B}^n F)$$

which can be regarded as the hom-set in the derived category of complexes of abelian sheaves. This, in turn, is the same as the traditional expression

$$\cdots \simeq R^n \Gamma(X,F)$$

giving the $n$th derived functor of the global section functor of the abelian sheaf $F$.

This, finally, is the same group as obtained by choosing any complex $I_F$ of abelian sheaves that is injective? and quasi-isomorphic to $F$ regarded as a complex concentrated in degree 0 and then computing the $n$ homology group of the complex $\Gamma(X,I_F)$ of global sections of $F$:

$$\cdots \simeq H_n(\Gamma(X,I_F)) \,.$$

Historically the development of abelian sheaf cohomology was precisely in reverse order to this derivation from the general $(\infty,1)$-categorical cohomology.

Theorem (K. Brown, 1973)

Let $X$ be a topological space, $F$ a sheaf on (the category of open subsets of) $X$ with values in abelian groups, and $\mathbf{B}^n F = K(F,n)$ the image of the complex of abelian sheaves $F[n]$ ($F$ in degree $n$, trivial elsewhere) under the Dold-Kan correspondence in sheaves with values in Kan complexes

$$\Gamma \circ (-) : Sh(X,Ch_+(Ab)) \to Sh(X, AbSimpGrp)$$

$$F[n] \mapsto K(F,n) =: \mathbf{B}^n F \,.$$

Then we have the following natural isomorphism of cohomologies:

$$H^n(X,F) \simeq H(X, \mathbf{B}^n F)$$

where

• on the left we have ordinary abelian sheaf cohomology defined as the right derived functor of the global sections functor

  $$H^n(X,F) := (R \Gamma_X)(F) \,;$$

• on the right we have nonabelian cohomology, namely the hom-set in the homotopy category of Kan complex valued simplicial sheaves

  $$H(X, \mathbf{B}^n F) := Ho_{Sh(X,\infty Grpd)}(X,\mathbf{B}^n X) \,.$$

Proof

This is the first four steps in the proof of theorem 2 in BrownAHT.

The proof proceeds along the following four steps, which we describe in more detail below:

$$\begin{aligned} H^n(X,F) & \simeq Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n]) \\ & \simeq Ho_{Sh(X,Ch_+(Ab))}(\mathbb{Z}, F[n]) \\ & \simeq Ho_{Sh(X,AbSimpGrp)}(\mathbb{Z}X, K(F,n)) \\ & \simeq Ho_{Sh(X,\infty Grpd)}(X, K(F,n)) \end{aligned}$$

1. By the derived functor definition of sheaf cohomology, $H^n(X,F)$ is the cohomology of any complex of sheaves $I^\bullet \in Sh(X,Ch(Ab))$ that is injective and weakly equivalent to $F[n]$, $F[n] \stackrel{\simeq}{\to} I^\bullet$:

   $$H^n(X,F) \simeq H^0(I^\bullet(X)) \,.$$

   On the other hand, by the general formula for hom-sets in homtotopy categories obtained by localizing at the multiplicative system given by quasi-isomorphisms of complexes (e.g. def. 13.1.2 in CaS) we have

   $$Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n]) \simeq colim_{Y^\bullet \stackrel{\simeq}{\to} \mathbb{Z}} Hom_{K(Sh(X,Ab))}(Y, I^\bullet) \,.$$

   But due to the injectiveness of $I^\bullet$, the integrand on the right is constant (lemma 14.1.5 in CaS) and hence the colimit is isomorphic to $\cdots \simeq Hom_{K(Sh(X,Ab))}(\mathbb{Z}, I^\bullet) \simeq H^0(I^\bullet(X))$, as desired.

2. The second step uses that the inclusion functor

   $$Ho_{Sh(X,Ch_+(Ab))} \hookrightarrow Ho_{Sh(X,Ch(Ab))}$$

   is full and faithful. This in turn follows from

   • first observing that the inclusion $S : Sh(X,Ch_+(Ab)) \hookrightarrow Sh(X, Ch(Ab))$ of chain complexes concentrated in non-negative degree into all complexes of sheaves is full and faithful and has the obvious right adjoint $T : Sh(X,Ch(Ab)) \to Sh(X, Ch_+(Ab))$ obtained by truncating a complex.

   • By inspection, or else by the general properties of adjoint functors (see the list of properties given there) this implies that $Id \to T \circ S$ is an isomorphism. This implies that also $Id \to Ho T \circ Ho S$ is an isomorphism.

   • But by the adjoint functor lemma for homotopical categories, $Ho S$ is also left adjoint to $Ho T$ (since both preserve weak equivalences). So that once again with the general properties of

   adjoint functors it follows that

   $Ho S$ is full and faithful.

3. The third step uses that the normalized chain complex functor $Sh(X,AbSimpGrp) \to Sh(X, Ch_+(Ab))$ is an equivalence of categories that preserves the respective weak equivalences and homotopies.

4. The fourth step finally uses that the forgetful functor $Sh(X, SimpAbGrp) \to Sh(X, \infty Grpd)$ that only remembers the Kan complex underlying a simplicial group has a left adjoint, the free abelian group functor $\mathbb{Z} : Sh(X,\infty Grpd) \to Sh(X, AbSimpGrp)$ (see Dold-Kan correspondence for details), and that preserves weak equivalences (see the discussion at simplicial group for more on that).

Relation to derived direct images

Proposition

Let $f^{-1} \colon Y \to X$ be a morphism of sites. Then the $q$th derived functor $R^q f_\ast$ of the induced direct image functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the sheafification of the presheaf

$$U_Y \mapsto H^q(f^{-1}(U_Y), \mathcal{F}) \,,$$

where on the right we have the degree $q$ abelian sheaf cohomology group with coefficients in the given $\mathcal{F}$.

(e.g. Tamme, I (3.7.1), II (1.3.4), Milne, 12.1).

Proof

We have a commuting diagram

$$\array{ Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y)) \\ \uparrow^{\mathrlap{inc}} && \downarrow^{L} \\ Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y)) } \,,$$

where the right vertical morphism is sheafification. Because $(-) \circ f^{-1}$ and $L$ are both exact functors it follows that for $I^\bullet \to \mathcal{F}$ an injective resolution that

$$\begin{aligned} R^p f_\ast(\mathcal{F}) & :\simeq H^p( f_\ast I) \\ & = H^p(L I^\bullet(f^{-1}(-))) \\ & = L (H^p(I^\bullet)(f^{-1}(-))) \end{aligned}$$

Examples

• de Rham cohomology

• Dolbeault cohomology

• Spencer cohomology

• Deligne cohomology

• etale cohomology

• crystalline cohomology

• syntomic cohomology

Related concepts

• sheaf, cohomology

• abelian sheaf cohomology

  • model structure on chain complexes

  • resolutions: soft sheaf, fine sheaf, flabby sheaf

  • triangulated category of sheaves

  • Verdier duality

References

Textbook account:

• Jean Gallier, Jocelyn Quaintance, Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, World Scientific (2022) [doi:10.1142/12495, webpage]

Original texts:

• Roger Godement, Topologie algébrique et theorie des faisceaux, Hermann, Paris (1958) [webpage, pdf]

The traditional definition of sheaf cohomology in terms of the right derived functor of the global sections functor:

• Ugo Bruzzo, Derived Functors and Sheaf Cohomology, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2 (doi:10.1142/11473)

• Chênevert, Kassaei, Sheaf Cohomology (pdf)

• Cibotaru, Sheaf cohomology (pdf)

• Patrick Morandi, Sheaf cohomology (pdf)

Its discussion in the more general nonabelian cohomology and infinity-stack context emphasized above is due to

• Kenneth Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

This uses homotopical structures of a category of fibrant objects on complexes of abelian sheaves. Discussion of actual model structure on chain complexes of abelian sheaves is in

• Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353, 6 (pdf)

A discussion of the Čech cohomology description of sheaf cohomology along the above lines is in

• Tibor Beke, Higher Čech Theory (web, pdf)

See also:

• John Duskin, Higher-dimensional torsors and the cohomology of topoi: the abelian theory, p. 255-279 in: Applications of sheaves, Lecture Notes in Mathematics 753, Springer (1979) [doi:10.1007/BFb0061822]

• Günter Tamme, section II 1 of Introduction to Étale Cohomology

• James Milne, section 7 of Lectures on Étale Cohomology