Godement resolution

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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.

[edit] 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 $x\in X$ , let $F x$ denote the stalk of F at x. Given an open set $U\subset X$ , define

$\operatorname{Gode}(F)(U):=\prod_{x\in U} F_x.$

An open subset $U\subset V$ clearly induces a restriction map $\operatorname{Gode}(F)(V)\rightarrow \operatorname{Gode}(F)(U)$ , so Gode(F) is a presheaf. One checks the sheaf axiom easily. One also proves easily that Gode(F) is 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 $F\to \operatorname{Gode}(F)$ , sending each section to its collection of germs.

[edit] Godement resolution

Now, given a sheaf F, let $G_0(F) = \operatorname{Gode}(F)$ , and let $d_0\colon F\rightarrow G_0(F)$ denote the canonical map. For $i > 0$ , let $G i (F)$ denote $\operatorname{Gode}(\operatorname{coker}(d_{i-1}))$ , and let $d_i\colon G_{i-1}\rightarrow G_i$ denote the obvious map. The resulting resolution is a flasque resolution of F, and its cohomology is the sheaf cohomology of F.