Holmstrom Sheaf cohomology

Sheaf cohomology

Here is an interesting paper: Topological representation of sheaf cohomology of sites, by Carsten Butz and Ieke Moerdijk

---

Sheaf cohomology

We consider a Grothendieck topology $T$.

The category $P$ of presheaves of abelian groups on $T$ is abelian and has enough injectives. Any section functor $P \to Ab$ is representable. For a functor $F: I \to P$, the inductive limit exists, and is computed “pointwise”. The functor $\lim_{\to}: Hom(I, P) \to P$ is additive and left exact. If $I$ is pseudofiltered, it is exact.

By standard results of homological algebra, th right derived functors exist for each left exact functor from $P$ into an abelian category. Any section functor on $P$ is exact, so the derived functors vanish.

Cech cohomology associated to a covering

Let $\{ U_i \to U \}$ be a covering. Consider the functor $H^0(\{U_i \to U \}, \cdot) : P \to Ab$, defined by:

This functor is additive and left exact, and we define Cech cohomology groups associated to the covering $\{U_i \to U \}$, with values in $F$, by

Sheaf cohomology

If $F$ is a sheaf, the sheaf condition implies that $H^0$ above is equal to $F(U)$. Hence any section functor $\Gamma_U: S\to Ab$ factors through $P$ via $H^0(\{U_i \to U \}, \cdot)$ and the natural inclusion functor $i$. Such a section functor is left exact, and its right derived functors $H^q(U, \cdot)$ are the cohomology groups with values in abelian sheaves.

A spectral sequence

Recall Grothendieck’s general spectral sequence for composition of left exact functors. When it exists (see Tamme p. 33 - for what topologies are these conditions satisfied?), it takes the following form in the current situation:

The Cech cohomology groups can be determined by means of Cech cochains, see Tamme p. 33.

Cech cohomology

We can define a category of coverings of an object $U$ by taking morphisms to be refinement maps (Tamme p. 37). We define Cech cohomology of $U$ with values in the abelian group $F$, by

The limit is taken over the category of coverings. As on might expect, we have the following theorem: The functor $\check{H}^0(U, \cdot): P \to Ab$ is left exact and additive. Its right derived functors are given by the Cech cohomology groups.

Note: $\Gamma_U = \check{H}^0(U, \cdot) \circ i$

---

Sheaf cohomology

The Leray spectral sequences

Let $f: T \to T'$ be a morphism of topologies. Then the right derived functors $R^q f^s: S' \to S$ exist.

Prop: For each abelian sheaf $F' \in S'$, there is an isomorphism:

In other words, $R^q f^s (F')$ is the sheaf associated to the presheaf $U \mapsto H^q(f(U), F')$ on $T$.

Since $f_s: S \to S'$ is not necessarily exact, we do not know that $f^s$ sends injectives to injectives. However, $f^s$ sends flabby sheaves to flabby sheaves.

Cor: Let $T'' {}_{\to}^{\ g} T {}_{\to}^{\ f} T'$ be morphisms of topologies. For each flabby $F'$ on $T'$, the sheaf $f^s F'$ on $T$ is a $g^s$-acyclic object. In other words, $R^q g^s (f^s F') = 0$ for $q >0$.

Thm: (Leray spectral sequence) Let $T'' {}_{\to}^{\ g} T {}_{\to}^{\ f} T'$ be morphisms of topologies. For all abelian sheaves $F'$ on $T'$, there is a spectral sequence:

which is functorial in $F'$.

Thm: (Special case of Leray) Let $f: T \to T'$ be a morphism of topologies, and let $U \in T$. For all abelian sheaves $F'$ on $T'$, there is a spectral sequence:

which is functorial in $F'$.

As a special case, we get the Hochschild-Serre spectral sequence in group cohomology. The relation between Tate cohomology and ordinary cohomology of a profinite group is also described by a Leray spectral sequence.

---

Sheaf cohomology

We begin by developing some theory of sheaves and presheaves. All sheaves are sheaves of abelian groups, unless otherwise specified. Let $T$ be a Grothendieck topology, and $P$, $S$ the categories of abelian presheaves and sheaves, respectively.

Presheaves

Consider two topologies $T$ and $T'$, and a functor $f: T \to T'$ on the underlying cats. Any abelian presheaf $F'$ on $T'$ defines a presheaf $F$ on $T$ by $F(U) = F' (f(U))$.

Thm: This construction defines a functor $f^p: P' \to P$ which is additive and exact, and commutes with inductive limits. It admits a left adjoint $f_p$, which is right exact, additive, and commutes with inductive limits. In case $f_p$ is exact, then $f^p$ maps injective objects to injective objects. (for proof, see Tamme p. 42)

Remark: Similar result for presheaves of sets.

The functor $f_p$ applied to a representable presheaf gives something which is representable by the obvious object.

Example: $Ab$ can be identified with the category of presheaves on the topology with underlying category $pt$ consisting of one object and one identity morphism. Let $U$ be an object in a topology $T$, and consider the functor $pt \to T$ which maps the unique object to $U$. We get what must be the constant presheaf functor from $Ab$ to $P$.

Sheaves

Thm: The inclusion functor $S \to P$ admits a left adjoint, the “sheafification functor”.

Proof: The proof uses the left exact functor $L: P \to P$, defined by $LF(U) = \check(H)^0(U, F)$. This functor sends presheaves to separated presheaves (meaning the first part of the sheaf condition sequence is injective) and separated presheaves to sheaves.

Thm: The category $S$ is an abelian category satisfying Ab5, and it has generators. The inclusion functor $i: S \to P$ is left exact and the sheafification functor is exact.

Proof: As for the Zariski topology, one shows that a presheaf kernel is in fact a sheaf, and that the sheafification of a presheaf cokernel is a cokernel in $S$. Direct sum is constructed by sheafifying the presheaf direct sum. Likewise, generators are obtained by sheafifying generators for $P$.

Cor: The category $S$ has enough injectives.

Cor: Let $I$ be a category. Every inductive limit (indexed by $I$) exists in $S$. It is equal to the sheafification of the presheaf $U \mapsto \lim_{\to} F_i(U)$. The functor $\lim_{\to}: Hom(I, S) \to S$ is right exact. It is exact if $I$ is pseudofiltered.

Cohomology

Any section functor $\Gamma_U: S \to Ab$ is left exact, being the composition of the left exact inlusion functor, and the exact section functor on $P$. Since $S$ has enough injectives, the right derived functors exist, and we define the q-th cohomology group of $U$ with values in $F$ by

Alternative notation includes the topology T among the arguments, or the final object of $T$, in case it exists. Question: Is cohomology functorial in $T$???

Spectral sequences for Cech cohomology

Recall the factorization $\Gamma_U = \check{H}^0(U, \cdot) \circ i$ of the section functor $S \to Ab$. We introduce the notation $\mathcal{H}^q( - ) : S \to P$ for $R^q i$.

Prop: For each abelian sheaf, we have a canonical isomorphism: $\mathcal{H}^q(F)(U) \cong H^q(U, F)$.

Prop: Recall the functor $L$ from the proof above. For each abelian sheaf $F$ we have $L \mathcal{H}^q(F) = 0$ for $q > 0$. More explicitly, we have $\check{H}^0(U, \mathcal{H}^q(F) ) = 0$.

Thm: Let $\{U_i \to U \}$ be a covering. For each $F \in S$, there is a spectral sequence,

functorial in $F$. The same statement holds if replace the initial term by $\check{H}^p(U, \mathcal{H}^q(F) )$.

Cor: If $F$ is a sheaf and $\{U_i \to U \}$ is a covering such that $H^q(U_{i_0} \times_U U_{i_1} \times_U \ldots \times_U U_{i_r}, F ) = 0$ for all $q>0$ and all finite products of $U_{i_k}$‘s, then the edge morphisms

are IMs for all $p$.

Cor: For all abelian sheaves $F$ the edge morphism $\check{H}^p(U, F) \to H(U, F)$ is bijective for $p = 0,1$ and injective for $p = 2$. This can be generalized (“shifted”).

Remark: The above spectral sequence yields as a special case the Hochschild-Serre spectral sequence for an open normal subgroup of a profinite group (Tamme p. 61).

Flabby sheaves

Def: An abelian sheaf is called flabby if $H^q( \{U_i \to U \}, F) = 0$ for all coverings and all $q > 0$.

Prop: If a direct sum $F \oplus G$ of abelian sheaves is flabby, then $F$ is also flabby. Injective sheaves are flabby. Let $0 \to F' \to F \to F'' \to 0$ be a short exact sequence in $S$. Then: If $F'$ is flabby, the sequence is also exact in $P$. If $F'$ and $F$ are flabby, then so is $F''$.

Cor: TFAE:

• $F$ is flabby
• For all $q > 0$, we have $\mathcal{H}^q(F) = 0$, and therefore $H^q(U, F) = 0$

In particular, flabby resolutions in $S$ can be used to compute $\mathcal{H}^q(-)$ and $H^q(U, -)$.

On a given topology $T$, all abelian sheaves are flabby iff the inclusion functor $i$ is exact.

Morphisms between topologies

Consider a morphism $f: T \to T'$ of topologies. We can define additive functors

and

(actually the sheafification $\#$ is not needed in the first definition.) These functors can also be defined for sheaves of sets.

Prop: The functor $f_s$ is left adjoint to $f^s$. Hence $f^s$ is left exact and $f_s$ is right exact, and $f_s$ commutes with inductive limits. If $f_s$ is exact, then $f^s$ maps injectives to injectives.

Example: A continuous map $\pi: X' \to X$ of topological spaces, induces a morphism of topologies $\pi^{-1}: T \to T'$. The functor $(\pi^{-1})^s$ is usually known as the direct image functor $\pi_*$, and the functor $(\pi^{-1})_s$ is the inverse image functor.

Thm: Let $T$ and $T'$ be topologies such that the underlying categories has final objects and finite fibre products. Let $f: T \to T'$ be a morphism of topologies, which respect final objects and finite fibre products. Then $f_s: S \to S'$ is exact.

Localization

Let $Z$ be an object in a topology $T$. Then we can form the topology $T/Z$ of objects over $Z$ in a natural way, and the natural functor $i: T/Z \to T$ is a morphism of topologies. Lemma: The functor $i^s$ is exact.

Cor: For all abelian sheaves $F$ on $T$, there are functorial isomorphisms:

The comparison lemma

Criterion for when a morphism of topologies induces an equivalence between their categories of abelian sheaves, and similar results, such as isomorphism results for the adjoint morphism $i^s i_s$, and exactness criterion for $i^s$. This yields isomorphism results on cohomology, for “pullback” and “pushforward”.

Noetherian topologies

Def: An object $U$ in a topology $T$ is called quasi-compact if each covering has a finite subcovering. If every object is quasi-compact, the topology is said to be noetherian.

For any topology $T$, we can definea new topology $T^f$ by allowing only finite coverings. If $T$ is noetherian, get isomorphic cohomology. Also other results on these finiteness issues. (Tamme p. 80)

Inductive limits of sheaves

In general, the canonical map $\lim_{\to} H^q (U, F_i) \to H^q(U, \lim_{\to} F_i)$ need not be an isomorphism. However, this is the case if $T$ is noetherian and the limit is over a pseudofiltered category. For example, if $T$ is noetherian, then cohomology commutes with direct sum of sheaves.

---

Sheaf cohomology

MathSciNet

Google Scholar

Google

arXiv: Experimental full text search

arXiv: Abstract search

---

Sheaf cohomology

AG (Algebraic geometry), CT (Category theory), AT (Algebraic topology)

---

Sheaf cohomology

Pure, Mixed

---

Sheaf cohomology

Sheaf cohomology can mean several different things. For sheaf cohomology in the sense of Hartshorne, see Zariski cohomology. Here we will describe the more general notion of sheaf cohomology with respect to any Grothendieck topology. As a special case of this, one can also talk about sheaf cohomology of topological spaces (Cartan and others).

In the book project, sheaf cohomology is for the time being included in the “Pure” chapter, because of its usefulness in constructing some Weil cohomology theories. See also the chapter on basic tools/techniques for cosntructing cohomology theories.

Examples: Etale cohomology, Crystalline cohomology, Zariski cohomology, Flat cohomology, l-adic cohomology, Nisnevich cohomology, cdh-cohomology. and cohomology with respect to any other Grothendieck topology, such as the h-topology or the qfh-topology.

See also Cech cohomology and Cohomology with compact supports

---

Sheaf cohomology

There is a result about finiteness of sheaf cohomology of general (quasi-∞)coherent analytic sheaves.

---

Sheaf cohomology

The main source for this page is Tamme: Introduction to Etale Cohomology.

Grothendieck’s Tohoku paper is really nice

A pre-Grothendieck standard reference on sheaf cohomology and spectral sequences is Godement 1958: Topologie Algebriques et Theorie des Faisceaux.

B. Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986.

M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematichen Wissenschaften 292, Springer-Verlag, 1990.

Various other things in the Homological algebra folder

---

Sheaf cohomology

See Stillman notes in Homol alg folder

---

Sheaf cohomology

One can express sheaf cohomology as Hom in the bounded derived category of abelian sheaves on $X$, from the constant sheaf Z to the sheaf $F[i]$ which is $F$ placed in degree $-i$ I think.

Is the sheaf cohomology of a topological space the same as the singular cohomology of the constant sheaf? I think yes.

Sheaves and cohomological functors on topological categories E E Skurikhin 1984 Russ. Math. Surv. 39 193-197

Thm (Tohoku page 170): Let $X$ be a Zariski space of (Krull) dimension $\leq n$. Then $H^i(X,F) = 0$ for every $i>n$ and every abelian sheaf $F$.

---

Sheaf cohomology

May on operads and sheaf cohomology

nLab, not much content as of March 2009. See also sheaf, and abelian sheaf cohom

---

Sheaf cohomology

People like Urs Schreiber at nLab seem to think that sheaf cohomology of all kinds is representable in some sense, probably something like $H^i(X, F) = \pi_i Hom (X, F)$ where identify X and F with their images in some suitable cat of simplical sheaves. It seems like there is something like this underlying the discussion of Voevodsky on pretheories in Homology of schemes II.

In some paper of Voevodsky (cd-structures, page 10), he states that for any site, we have $H^n(X, F) = Ext^n(Z(r(X)), F)$, where Z is the free abelian sheaf functor, and r probably is the sheaf associated with the representable presheaf.

---

Sheaf cohomology

http://mathoverflow.net/questions/29380/sheaf-cohomology-question

http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology

http://mathoverflow.net/questions/30609/idea-of-presheaf-cohomology-vs-sheaf-cohomology

http://mathoverflow.net/questions/55656/concrete-interpretations-of-higher-sheaf-cohomology-groups

http://mathoverflow.net/questions/11289/geometry-meaning-of-higher-cohomology-of-sheaves

http://mathoverflow.net/questions/13413/interpretation-of-elements-of-h1-in-sheaf-cohomology

http://mathoverflow.net/questions/38966/what-is-sheaf-cohomology-intuitively

http://mathoverflow.net/questions/1151/sheaf-cohomology-and-injective-resolutions

http://mathoverflow.net/questions/79693/why-does-a-group-action-on-a-scheme-induce-a-group-action-on-cohomology

http://mathoverflow.net/questions/28386/the-cohomology-of-a-product-of-sheaves-and-a-plea

nLab page on Sheaf cohomology