Modules with small support have big depth - reference wanted

Question

Hello, I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:

Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. Let $M$ be a f.g. $A$-module. Then $Ext^i (M,A)=0$ for $i < codim(supp(M))$ (codimension of support).

The basic local input should be: for local $A$ as above, $Ext^i(k,A) = 0$ for $i < dim(A)$. But to use it, I localize the above situation to a component of $M$, and get that appropriate $Ext$'s have support "strictly less" than that of $A$ (contained in that of $A$, and does not contain any component of that of $A$). But it does not imply that they are zero.

Also, I tried to do some induction, using "cuts" by local parameters. The problem is that if the support of $M$ is not regular, I can not cut exactly to get it.

Thanks, Sasha

Answer 1

For $A$ local, your statement follows from a result of Ischebeck. What you want holds even if $A$ is just Cohen-Macaulay. More generally $\mathrm{Ext}^i_A(M,N)=0$ for $i<\mathrm{depth}\ N-\dim M$. Put $N=A$ and assume $A$ is Cohen-Macaulay, then you get $\mathrm{Ext}^i_A(M,A)=0$ for $i<\dim A-\dim M$.

You can find a proof of Ischebeck Theorem in Matsumura's Commutative Ring Theorey on page 133.