Godement resolution

In algebraic geometry, the Godement resolution, named after Roger Godement, of a sheaf allows one to view all of its local information globally. It is useful for computing sheaf cohomology.

Godement replacement

Given a topological space "X" (more generally a site X with enough points), and a sheaf "F" on X, the Godement resolution of "F" is the sheaf "Gode(F)" constructed as follows. For each point $xin\; X$, let $F\_x$ denote the stalk of "F" at "x". Given an open set $Usubset\; X$, define

: $operatorname\{Gode\}(F)(U):=prod\_\{xin\; U\}\; F\_x.$

An open subset $Usubset\; V$ clearly induces a restriction map $operatorname\{Gode\}(F)(V)\; ightarrow\; operatorname\{Gode\}(F)(U)$, so Gode("F") is a presheaf. One checks the sheaf axiom easily. One also proves easily that Gode("F") is flasque (i.e. each restriction map is surjective). Finally, one checks that Gode is a functor, and that there is a canonical map of sheaves $F\; o\; operatorname\{Gode\}(F)$, sending each section to its collection of germs.

Godement resolution

Now, given a sheaf "F", let $G\_0(F)\; =\; operatorname\{Gode\}(F)$, and let $d\_0colon\; F\; ightarrow\; G\_0(F)$ denote the canonical map. For $i>0$, let $G\_i(F)$ denote $operatorname\{Gode\}(operatorname\{coker\}(d\_\{i-1\}))$, and let $d\_icolon\; G\_\{i-1\}\; ightarrow\; G\_i$ denote the obvious map. The resulting resolution is a flasque resolution of "F", and its cohomology is the sheaf cohomology of "F".