MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $$I^n/I^{n+1} \otimes_R M \cong I^nM/I^{n+1}M$$

share|cite|improve this question
That's not a short exact sequence. – Steven Landsburg Apr 18 '12 at 6:13
up vote 3 down vote accepted

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 $R$.

share|cite|improve this answer
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. – Pham Hung Quy Apr 18 '12 at 6:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.