Godement resolution

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The Godement resolution of a sheaf is a construction in homological algebra which allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement.

Godement construction

Given a topological space X (more generally, a topos X with enough points), and a sheaf F on X, the Godement construction for F gives a sheaf Gode(F) constructed as follows. For each point ${\displaystyle x\in X}$, let ${\displaystyle F_{x}}$ denote the stalk of F at x. Given an open set ${\displaystyle U\subset X}$, define

${\displaystyle \operatorname {Gode} (F)(U):=\prod _{x\in U}F_{x}.}$

An open subset ${\displaystyle U\subset V}$ clearly induces a restriction map ${\displaystyle \operatorname {Gode} (F)(V)\rightarrow \operatorname {Gode} (F)(U)}$, so Gode(F) is a presheaf. One checks the sheaf axiom easily. One also proves easily that Gode(F) is flabby, meaning each restriction map is surjective. Gode can be turned into a functor because a map between two sheaves induces maps between their stalks. Finally, there is a canonical map of sheaves ${\displaystyle F\to \operatorname {Gode} (F)}$ which sends each section to the product of its germs. This canonical map is a natural transformation between the identity functor and Gode.

Another way to view Gode is as follows. Let ${\displaystyle \textstyle Y=\coprod _{x\in X}\{x\}}$ be the disjoint union of the points of X. There is a continuous map p : YX. This induces adjoint pushforward and pullback functors p* and p*. Gode is the unit of this adjunction, that is, it is p*p*.

Because Gode is the unit of an adjunction, there is an associated monad on the category of sheaves on X. Using this monad there is a way to turn a sheaf F into a coaugmented cosimplicial sheaf. This coaugmented cosimplicial sheaf is associated to an augmented cochain complex which is defined to be the Godement resolution of F.

In more down-to-earth terms, let ${\displaystyle G_{0}(F)=\operatorname {Gode} (F)}$, and let ${\displaystyle d_{0}\colon F\rightarrow G_{0}(F)}$ denote the canonical map. For each ${\displaystyle i>0}$, let ${\displaystyle G_{i}(F)}$ denote ${\displaystyle \operatorname {Gode} (\operatorname {coker} (d_{i-1}))}$, and let ${\displaystyle d_{i}\colon G_{i-1}\rightarrow G_{i}}$ denote the canonical map. The resulting resolution is a flabby resolution of F, and its cohomology is the sheaf cohomology of F.

References

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