Godement resolution

Summary

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.

Godement construction edit

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   constructed as follows. For each point  , let   denote the stalk of F at x. Given an open set  , define

 

An open subset   clearly induces a restriction map  , so   is a presheaf. One checks the sheaf axiom easily. One also proves easily that   is flabby, meaning each restriction map is surjective. The map   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  .

Another way to view   is as follows. Let   be the set X with the discrete topology. Let   be the continuous map induced by the identity. It induces adjoint direct and inverse image functors   and  . Then  , 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  , and let   denote the canonical map. For each  , let   denote  , and let   denote the canonical map. The resulting resolution is a flabby resolution of F, and its cohomology is the sheaf cohomology of F.

References edit

  • Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, ISBN 9782705612528, MR 0345092
  • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge University Press, doi:10.1017/CBO9781139644136, ISBN 978-0-521-55987-4, MR 1269324

External links edit

  • The Stacks Project authors. "20.30 Godement resolution".
  • Goresky, Mark. "Introduction to Perverse Sheaves §.4.1. Godement resolution" (PDF). Institute for Advanced Study.