Finiteness of formal pushforwards

TL;DR

This paper proves a finiteness result for pushforwards of torsion-free sheaves in formal schemes over a DVR and applies it to formal function gluing where patches are partial.

Contribution

It extends classical finiteness results to formal schemes over a DVR and introduces new gluing techniques for formal functions with incomplete coverage.

Findings

Pushforward of torsion-free sheaves is coherent under certain conditions.

New methods for gluing formal functions over partial covers.

Application of finiteness results to formal scheme contexts.

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 $U$. We prove an analog of this result in the context of formal schemes over a complete discrete valuation ring. We then apply this to obtain a result about gluing formal functions, where the patches do not cover the entire scheme.