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Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twistedforms lets one compute, given an $R$-module $M$, how many isomorphism classesof $R$-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 solvethe 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 wework out (as a special case of a general theorem about existence of coalgebrastructures over a comonad) a criterion for the existence of such an $R$-module$M$, under some reasonable hypotheses on the map $R\rightarrow S$.

A recognition principle for the existence of descent data