Finiteness of formal pushforwards

TL;DR

This work extends a classical coherence-pushforward phenomenon to formal schemes over a complete discrete valuation ring, proving that the pushforward of a torsion-free coherent sheaf across a codimension-two open-embedding remains torsion-free and coherent on the formal completion. The authors develop a robust patching framework at the level of formal schemes, proving key intersection lemmas for torsion-free modules and establishing that formal pushforwards behave well under two affine patches. The main contributions include a general finiteness/ coherence result (genl coh ext), a flat-case strengthening, and a systematic treatment of patching problems yielding a unique maximal torsion-free (and in the flat case unique flat) solution. These results enable gluing local formal data into global finite modules without requiring cocycle conditions, with potential applications to patching in Galois theory and local-global principles on higher-dimensional bases.

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.

Theorems & Definitions (57)

• Theorem : see Theorem \ref{['genl coh ext']}
• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 4.1
• proof
• Remark 4.2