A recognition principle for the existence of descent data

TL;DR

This paper establishes a criterion for the existence of descent data for modules over a faithfully flat ring extension, addressing the problem of when an S-module can be descended to an R-module.

Contribution

It provides a new recognition principle for the existence of descent data, extending the theory to include existence conditions under certain hypotheses.

Findings

Provides a criterion for the existence of descent data.

Extends the theory of twisted forms to solve existence problems.

Applies a general theorem about coalgebra structures over a comonad.

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 $R\rightarrow S$.