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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2007
- DOI:
- 10.48550/arXiv.0707.4197
- arXiv:
- arXiv:0707.4197
- Bibcode:
- 2007arXiv0707.4197F
- Keywords:
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- Mathematics - Commutative Algebra
- E-Print:
- 16 pages, AMS-TeX