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

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.Comment: 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 Hochste