Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$-module. I question is:
Does there exist $n_0$ such that for all $n \geq n_0$, the short exact sequence $$I^n/I^{n+1} \otimes_R M \cong I^nM/I^{n+1}M$$
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asked Apr 18, 2012 at 4:19
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2$\begingroup$ That's not a short exact sequence. $\endgroup$– Steven LandsburgCommented Apr 18, 2012 at 6:13
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1 Answer
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I suppose you intend to ask whether there is isomorphism as in the formula. The answer is negative, in general. For example, take $R = k[x]$, $I = (x)$, $M = R/I^2$.
[Edit] What you may have in mind is that the result holds when $M$ is flat over $R$.
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$\begingroup$ Thanks you! In fact, I remember some result about exactness, powers of an ideal, tensor of L. Avramov (or W.V. Vasconcelos), but I can not remember what is it. $\endgroup$ Commented Apr 18, 2012 at 6:56