Sheaf cohomology

General Introduction
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, ladic cohomology, Nisnevich cohomology, cdhcohomology. and cohomology with respect to any other Grothendieck topology, such as the htopology or the qfhtopology.
See also Cech cohomology and Cohomology with compact supports
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Search results
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Online References
May on operads and sheaf cohomology
nLab, not much content as of March 2009. See also sheaf, and abelian sheaf cohomology
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Paper References
The main source for this page is Tamme: Introduction to Etale Cohomology.
Grothendieck's Tohoku paper is really nice
A preGrothendieck standard reference on sheaf cohomology and spectral sequences is Godement 1958: Topologie Algebriques et Theorie des Faisceaux.
B. Iversen, Cohomology of sheaves, Universitext, SpringerVerlag, Berlin, 1986.
M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematichen Wissenschaften 292, SpringerVerlag, 1990.
Various other things in the Homological algebra folder
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Definition
We consider a Grothendieck topology .
The category of presheaves of abelian groups on is abelian and has enough injectives. Any section functor is representable. For a functor , the inductive limit exists, and is computed "pointwise". The functor is additive and left exact. If is pseudofiltered, it is exact.
By standard results of homological algebra, th right derived functors exist for each left exact functor from into an abelian category. Any section functor on is exact, so the derived functors vanish.
Cech cohomology associated to a covering
Let be a covering. Consider the functor , defined by:
This functor is additive and left exact, and we define Cech cohomology groups associated to the covering , with values in , by
Sheaf cohomology
If is a sheaf, the sheaf condition implies that above is equal to . Hence any section functor factors through via and the natural inclusion functor . Such a section functor is left exact, and its right derived functors 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 by taking morphisms to be refinement maps (Tamme p. 37). We define Cech cohomology of with values in the abelian group , by
The limit is taken over the category of coverings. As on might expect, we have the following theorem: The functor is left exact and additive. Its right derived functors are given by the Cech cohomology groups.
Note:
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Properties
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Standard theorems
We begin by developing some theory of sheaves and presheaves. All sheaves are sheaves of abelian groups, unless otherwise specified. Let be a Grothendieck topology, and , the categories of abelian presheaves and sheaves, respectively.
Presheaves
Consider two topologies and , and a functor on the underlying cats. Any abelian presheaf on defines a presheaf on by .
Thm: This construction defines a functor which is additive and exact, and commutes with inductive limits. It admits a left adjoint , which is right exact, additive, and commutes with inductive limits. In case is exact, then maps injective objects to injective objects. (for proof, see Tamme p. 42)
Remark: Similar result for presheaves of sets.
The functor applied to a representable presheaf gives something which is representable by the obvious object.
Example: can be identified with the category of presheaves on the topology with underlying category consisting of one object and one identity morphism. Let be an object in a topology , and consider the functor which maps the unique object to . We get what must be the constant presheaf functor from to .
Sheaves
Thm: The inclusion functor admits a left adjoint, the "sheafification functor".
Proof: The proof uses the left exact functor , defined by . 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 is an abelian category satisfying Ab5, and it has generators. The inclusion functor 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 . Direct sum is constructed by sheafifying the presheaf direct sum. Likewise, generators are obtained by sheafifying generators for .
Cor: The category has enough injectives.
Cor: Let be a category. Every inductive limit (indexed by ) exists in . It is equal to the sheafification of the presheaf . The functor is right exact. It is exact if is pseudofiltered.
Cohomology
Any section functor is left exact, being the composition of the left exact inlusion functor, and the exact section functor on . Since has enough injectives, the right derived functors exist, and we define the qth cohomology group of with values in by
Alternative notation includes the topology T among the arguments, or the final object of , in case it exists. Question: Is cohomology functorial in ???
Spectral sequences for Cech cohomology
Recall the factorization of the section functor . We introduce the notation for .
Prop: For each abelian sheaf, we have a canonical isomorphism: .
Prop: Recall the functor from the proof above. For each abelian sheaf we have for . More explicitly, we have .
Thm: Let be a covering. For each , there is a spectral sequence,
functorial in . The same statement holds if replace the initial term by .
Cor: If is a sheaf and is a covering such that for all and all finite products of 's, then the edge morphisms
are IMs for all .
Cor: For all abelian sheaves the edge morphism is bijective for and injective for . This can be generalized ("shifted").
Remark: The above spectral sequence yields as a special case the HochschildSerre spectral sequence for an open normal subgroup of a profinite group (Tamme p. 61).
Flabby sheaves
Def: An abelian sheaf is called flabby if for all coverings and all .
Prop: If a direct sum of abelian sheaves is flabby, then is also flabby. Injective sheaves are flabby. Let be a short exact sequence in . Then: If is flabby, the sequence is also exact in . If and are flabby, then so is .
Cor: TFAE:
 is flabby
 For all , we have , and therefore
In particular, flabby resolutions in can be used to compute and .
On a given topology , all abelian sheaves are flabby iff the inclusion functor is exact.
Morphisms between topologies
Consider a morphism 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 is left adjoint to . Hence is left exact and is right exact, and commutes with inductive limits. If is exact, then maps injectives to injectives.
Example: A continuous map of topological spaces, induces a morphism of topologies . The functor is usually known as the direct image functor , and the functor is the inverse image functor.
Thm: Let and be topologies such that the underlying categories has final objects and finite fibre products. Let be a morphism of topologies, which respect final objects and finite fibre products. Then is exact.
Localization
Let be an object in a topology . Then we can form the topology of objects over in a natural way, and the natural functor is a morphism of topologies. Lemma: The functor is exact.
Cor: For all abelian sheaves on , 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 , and exactness criterion for . This yields isomorphism results on cohomology, for "pullback" and "pushforward".
Noetherian topologies
Def: An object in a topology is called quasicompact if each covering has a finite subcovering. If every object is quasicompact, the topology is said to be noetherian.
For any topology , we can definea new topology by allowing only finite coverings. If is noetherian, get isomorphic cohomology. Also other results on these finiteness issues. (Tamme p. 80)
Inductive limits of sheaves
In general, the canonical map need not be an isomorphism. However, this is the case if is noetherian and the limit is over a pseudofiltered category. For example, if is noetherian, then cohomology commutes with direct sum of sheaves.
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Open Problems
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Connections to Number Theory
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Computations and Examples
See Stillman notes in Homol alg folder
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History and Applications
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Some Research Articles
Here is an interesting paper: Topological representation of sheaf cohomology of sites, by Carsten Butz and Ieke Moerdijk
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Other Information
http://mathoverflow.net/questions/29380/sheafcohomologyquestion
http://mathoverflow.net/questions/32689/howshouldahomotopytheoristthinkaboutsheafcohomology
http://mathoverflow.net/questions/30609/ideaofpresheafcohomologyvssheafcohomology
http://mathoverflow.net/questions/55656/concreteinterpretationsofhighersheafcohomologygroups
http://mathoverflow.net/questions/11289/geometrymeaningofhighercohomologyofsheaves
http://mathoverflow.net/questions/13413/interpretationofelementsofh1insheafcohomology
http://mathoverflow.net/questions/38966/whatissheafcohomologyintuitively
http://mathoverflow.net/questions/1151/sheafcohomologyandinjectiveresolutions
http://mathoverflow.net/questions/28386/thecohomologyofaproductofsheavesandaplea
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