Question

Hi,

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.

Answer 1

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".