Holmstrom Godement resolution

Let $F$ be a sheaf of abelian groups on a site $S$ with enough points. The Godement resolution of $F$ is defined as follows:

$I(F) = \prod_{x \in S} i_{x*} F_x$

$Gd^0(F) = I(F)$

$Gd^1(F) = I(coker ( F \to Gd^0(F) ) )$

$Gd^{n+1}(F) = I(coker ( Gd^{n-1}(F) \to Gd^n(F) ) )$

This resolution is functorial under morphisms of sites.

nLab page on Godement resolution