Ascent of module structures, vanishing of Ext, and extended modules

TL;DR

This paper investigates conditions under which modules over certain local rings can be extended or lifted to larger rings, revealing new insights into module structures, Ext vanishing, and the behavior of extended modules in different ring completions.

Contribution

It establishes a criterion linking Ext vanishing to the existence of compatible module structures over flat local homomorphisms and explores properties of extended modules over Henselizations and completions.

Findings

Modules over Henselizations are summands of extended modules.

The extension property fails for $ $-adic completions.

Conditions are provided for when a module in a short exact sequence is extended.

Abstract

Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that $\m S = \n$ and the induced map on residue fields $R/\m \to S/\n$ is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if and only if $\Ext^i_R(S,M)=0$ for each $i\ge 1$. We say that an $S$-module $N$ is {\it extended} if there is a finitely generated $R$-module $M$ such that $N\cong S\otimes_RM$. Given a short exact sequence $0 \to N_1\to N \to N_2\to 0$ of finitely generated $S$-modules, with two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of $R$ is a direct summand of an extended module, but that the analogous result fails for the $\m$-adic completion.