A freeness criterion without patching for modules over local rings

TL;DR

This paper establishes a new criterion for module freeness over local rings without using patching methods, focusing on modules with bounded flat dimension over certain complete intersection homomorphisms.

Contribution

It introduces a freeness criterion for modules over local rings that does not rely on patching, extending previous results to a broader class of modules and ring homomorphisms.

Findings

Modules with bounded flat dimension are free over certain complete intersection homomorphisms.

The criterion applies to nonzero finitely generated modules over local rings.

It generalizes previous results related to flat modules and patching methods.

Abstract

It is proved that if $\varphi\colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated $B$-module $N$ whose flat dimension over $A$ is at most $\mathrm{edim}\, A - \mathrm{edim}\, B$, is free over $B$, and $\varphi$ is a special type of complete intersection. This result is motivated by a "patching method" developed by Taylor and Wiles, and a conjecture of de Smit, proved by the first author, dealing with the special case when $N$ is flat over $A$.