Finiteness of formal pushforwards

Abstract

Under mild hypotheses, given a scheme $U$ and an open subset $V$ whose complement has codimension at least two, the pushforward of a torsion-free coherent sheaf on $V$ is coherent on $U$, and in particular is finite. We prove an analog of this finiteness assertion in the context of formal schemes over a complete discrete valuation ring, but show that coherence does not always hold. We then relate this to the problem of gluing formal functions, where the patches do not cover the entire scheme.