- Godement resolution
In
algebraic geometry , the Godement resolution, named afterRoger Godement , of asheaf allows one to view all of its local information globally. It is useful for computingsheaf 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 thesheaf 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 thesheaf cohomology of "F".
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