A recognition principle for the existence of descent data

Abstract

Suppose $$R\rightarrow S$$ is a faithfully flat ring map. Given an $$S$$ -module $$N$$ , does there exists some $$R$$ -module $$M$$ such that $$S\otimes _R M\cong N$$ ? In this paper we work out (as a special case of a more general question about extensions of comonads) a criterion for the existence of such an $$R$$ -module $$M$$ , under some reasonable hypotheses on the map $$R\rightarrow S$$ .