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

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

2007-07-27

arxiv.org/abs/0707.4197v3

Mathematics

Commutative Algebra

16 pages, AMS-TeX; final version to appear in Michigan Math. J.; corrected proof of Main Theorem and made minor editorial chan

Scientific paper

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.

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