Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x_1,...,x_t$ be an $M$-regular sequence and $I = (x_1,...,x_t)$. Is it true that $$\mathrm{Tor}_1^R(R/I^n, M) = 0$$ for all $n \geq 1$?
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asked Jan 9, 2013 at 18:56
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1$\begingroup$ It's true for $n=1$ if the sequence is also $R$-regular, since then $\mathrm{Tor}_{1}^{R}(R/I,M) \simeq \mathrm{Tor}_{1}^{R/I}(R/I,M/I)$. (see Lemma 18.2.iii in Matsumura's CRT.) $\endgroup$– David HansenCommented Jan 9, 2013 at 19:11
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1$\begingroup$ Lemma 18.2 in Matsumura need $x$ is both $R$-regular and $M$-regular. $\endgroup$– Pham Hung QuyCommented Jan 10, 2013 at 2:41
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$\begingroup$ So Lemma 18.2 applies at least when $M$ has finite projective dimension. $\endgroup$– Mahdi Majidi-ZolbaninCommented Jan 10, 2013 at 4:22
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$\begingroup$ $I^{n-1}/I^n$ is a free $R/I$-module, so the statement follows from a simple induction. $\endgroup$– AngeloCommented Jul 25, 2013 at 5:41
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$\begingroup$ I wonder if this is true even in the $n=1$ case? I know it is true if $t=1$, but I wonder if it would be true when $n=1$ and $t>1$? $\endgroup$– user521337Commented Aug 12, 2021 at 9:13
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1 Answer
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I've posted a proof here for the special case when $M$ is cyclic. Furthermore, I've mentioned that the result holds for finitely generated modules when the sequence is $R$-regular and $M$-regular.
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$\begingroup$ Very nice proof. Thanks you very much YACP. The case $M$ is cyclic is an exercise in the book about tight closure of C. Huneke. Your proof even better than its solution (in my opinion). $\endgroup$ Commented Jan 12, 2013 at 15:43