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

Abstract

Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let $\varphi :R\to S$ 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 $N\cong S{\otimes }_{R}M$. 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.

16 pages, AMS-TeX; final version to appear in Michigan Math. J.; corrected proof of Main Theorem and made minor editorial changes; v3 has dedication to Mel Hochster

References in corpus (1)

• Finite Gorenstein representation type implies simple singularity

Cited by in corpus (1)

• Constructing Big Indecomposable modules