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.
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
denote the stalk of F at x.
Given an open set
, define An open subset
clearly induces a restriction map
One checks the sheaf axiom easily.
One also proves easily that
is flabby, meaning each restriction map is surjective.
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
that sends each section to the 'product' of its germs.
This canonical map is a natural transformation between the identity functor and
be the set X with the discrete topology.
be the continuous map induced by the identity.
It induces adjoint direct and inverse image functors
, 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
denote the canonical map.
denote the canonical map.
The resulting resolution is a flabby resolution of F, and its cohomology is the sheaf cohomology of F.