Godement resolution

The Godement resolution of a sheaf is a construction in homological algebra that 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 ${\displaystyle \operatorname {Gode} (F)}$ 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 ${\displaystyle U\subseteq X}$, define

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

An open subset ${\displaystyle U\subseteq V}$ clearly induces a restriction map ${\displaystyle \operatorname {Gode} (F)(V)\rightarrow \operatorname {Gode} (F)(U)}$, so ${\displaystyle \operatorname {Gode} (F)}$ is a presheaf. One checks the sheaf axiom easily. One also proves easily that ${\displaystyle \operatorname {Gode} (F)}$ is flabby, meaning each restriction map is surjective. The map ${\displaystyle \operatorname {Gode} }$ 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 ${\displaystyle F\to \operatorname {Gode} (F)}$ that sends each section to the 'product' of its germs. This canonical map is a natural transformation between the identity functor and ${\displaystyle \operatorname {Gode} }$.

Another way to view ${\displaystyle \operatorname {Gode} }$ is as follows. Let ${\displaystyle X_{\text{disc}}}$ be the set X with the discrete topology. Let ${\displaystyle p\colon X_{\text{disc}}\to X}$ be the continuous map induced by the identity. It induces adjoint direct and inverse image functors ${\displaystyle p_{*}}$ and ${\displaystyle p^{-1}}$. Then ${\displaystyle \operatorname {Gode} =p_{*}\circ p^{-1}}$, and the unit of this adjunction is the natural transformation described above.

Because of this 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 gives rise to an augmented cochain complex that 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

• Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR 0345092
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge University Press, doi:10.1017/CBO9781139644136, ISBN 978-0-521-55987-4, MR 1269324