A recognition principle for the existence of descent data

Abstract

Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twisted forms lets one compute, given an $R$-module $M$, how many isomorphism classes of $R$-modules $M^{\prime}$ satisfy $S\otimes_R M\cong S\otimes_R M^{\prime}$. This is really a uniqueness problem. But this theory does not help one to solve the corresponding existence problem: given an $S$-module $N$, does there exists {\em some} $R$-module $M$ such that $S\otimes_R M\cong N$? In this paper we work out (as a special case of a general theorem about existence of coalgebra structures over a comonad) a criterion for the existence of such an $R$-module $M$, under some reasonable hypotheses on the map $R\rightarrow S$.