Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


Let $A$ be a ring. it is a well known fact that if $Ext^v_A(M,A)=0$ for every $v>\mu$ and every left module $M$ then $inj.dim(A)\leq \mu$ (seeing $A$ as a left $A$-module).

Does the weaker hypothesis $Ext^v_A(M,A)=0$ for every $v>\mu$ and every finitely generated left module $M$ imply the same result, at least when $A$ is Noetherian?

Thank you in advance.

share|cite|improve this question

1 Answer 1

up vote 2 down vote accepted

Yes. One can even restrict to $M=A/I$ for ideals $I$, and if $A$ is Noetherian it is enough to consider $M=A/p$ for prime ideals $p$. This is Lemma 18.1 in Matsumura's book "Commutative ring theory".

share|cite|improve this answer
Thank you. Matsumura's book always refers to commutative rings, but his proof holds even for the non-commutative ones. – Nick Mar 9 '12 at 23:33

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.