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Let $A$ be a regular local ring, and let $M$ and $N$ be two finitely generated $A$-modules such that $M\otimes N$ is of finite length, and let $i$ be the largest integer such that $\operatorname{Tor}_i(M,N)\neq 0$. Can we prove that $\operatorname{Tor}_i(M,N)$ is of finite length?

This appears on page 111 of Local Algebra by Serre, line 8. Perhaps one does not need the ring to be regular to prove the above statement.

Any help would be appreciated. Thanks.

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1 Answer 1

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It is enough to show that the support of $\operatorname{Tor}_i(M,N)$ consists only of maximal ideals. Let $\mathfrak p$ be a prime ideal such that $\operatorname{Tor}_i(M,N)_{\mathfrak p}\neq 0$, equivalently $\operatorname{Tor}_i(M_{\mathfrak p},N_{\mathfrak p})\neq 0$. In particular we have $M_{\mathfrak p}\neq 0$ and $N_{\mathfrak p}\neq 0$. Since $A$ is local we get $M_{\mathfrak p}\otimes_{R_{\mathfrak p}}N_{\mathfrak p}\neq 0$, equivalently $(M\otimes N)_{\mathfrak p}\neq 0$. Since $M\otimes N$ has finite length it follows that $\mathfrak p$ is maximal.

Remark. As one can see there is no need to assume that $A$ is regular.

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  • $\begingroup$ This is a great answer, and infact very easy. I should have thought more, but since last 2 days i have been struggling to finish that theorem and i finally did, and the above observation was the only thing missing. Thank you very much. I dont know how to upvote this answer, or else i would immediately do so. $\endgroup$
    – messi
    Apr 6, 2013 at 9:47
  • $\begingroup$ By the way this is messi, you had replied to a few questions from me before, unfortunately i lost that id on here. $\endgroup$
    – messi
    Apr 6, 2013 at 9:49

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