# Depth of a module over local ring and vanishing of Ext functor

I'm studying depth of $A$-modules, where $A$ is a noetherian ring, in Matsumura's Commutative Algebra text and I'm experiencing some trouble understanding the proof of a basic result.

I think all of you do know the following fact:

$$\mathrm{depth}_I (M):= \mathrm{min} \{i\in \mathbb{N}\mid \mathrm{Ext}^i_A(A/I,M)\neq 0\}$$

and I assume that nobody will be surprised if, in the case $A$ is local, I'll write simply $\mathrm{depth}(M)$ (or also $\mathrm{codim}(M)$ following Buchsbaum).

Coming to my trouble, in Matsumura §15.D there is a Lemma called Ischebeck's Lemma that states the following:

Let be $A$ a noetherian local ring and let be $M,N\neq 0$ two finitely generated $A$-modules. Then $\mathrm{Ext}^i_A(N,M)=0$ for every $i<\mathrm{depth}(M)-\mathrm{dim}(N)$.

I can't understand the proof stated in Matsumura, in particular I don't understand why one can reduce the proof to the case $N=A/\mathfrak{p}$ for some $\mathfrak{p}\in \mathrm{Spec}(A)$. It's stated that the existence of a filtration $$0=N_0\subset N_1\subset \cdots \subset N_{l-1} \subset N_l =N$$ with $N_j/N_{j-1}\simeq A/\mathfrak{p}_j$ for some $\mathfrak {p}_j\in\mathrm{Spec}(A)$ could help, but I don't see how, unless you can replace $N$ with his associated graded $$\mathrm{gr}(N)=\bigoplus_{k=1}^{l} N_k/N_{k-1}$$ but this seems to me really unlikely.

Any suggestion?

Hint:
$N_1\simeq A/p_1$.
Apply $Ext(-,M)$ on $0\to N_1\to N_2\to A/p_2\to0$. Assume you have the result for $N_1\simeq A/p_1$ and $A/p_2$; so you have the result for $N_2$.
proceed this way.

• Only a question: how do you build that sequence? If I assume $N_2\simeq A/p_2$ I can't tell it's exact! Maybe I'm missing something, but the idea is great! – Caligula Jan 8 '15 at 13:33
• $N_2/ N_1 \simeq A/p_2$ – user 1 Jan 8 '15 at 13:34