Let X be a category fibered in groupoids over a finite field
Fqβ, and let k be an algebraically closed field containing
Fqβ. Denote by Οkβ:XkββXkβ the
arithmetic Frobenius of Xkβ/k and suppose that M is a
stack over Fqβ (not necessarily in groupoids). Then there is a
natural functor Ξ±M,Xβ:M(X)βM(Dkβ(X)), where M(Dkβ(X)) is the category of Οkβ-invariant maps XkββM. A version of Drinfeld's lemma states that if X is a projective scheme and M is the stack of quasi-coherent
sheaves of finite presentation, then Ξ±M,Xβ is an
equivalence.
We extend this result in several directions. For proper algebraic stacks or
affine gerbes X, we prove Drinfeld's lemma and deduce that
Ξ±M,Xβ is an equivalence for very general
algebraic stacks M.
For arbitrary X, we show that Ξ±M,Xβ is an equivalence when M is the stack of immersions, the
stack of quasi-compact separated \'etale morphisms or any quasi-separated
Deligne-Mumford stack with separated diagonal