Theorem 1

Let $A$ be a ring, $J$ an ideal of $A$, $x = (x_i)_{i \in I}$ a generating family of $J$, $M$ an $A$-module. The depth of $M$ relative to $J$ is the greatest lower bound (in $\mathbf{N} \cup \{+\infty\}$) of the integers $n$ such that $H^n(x, M) \neq 0$.

Put $p = \operatorname{prof}_A(J; M)$. Consider the complex $K^\bullet(x, M)$. Its homology is annihilated by $J$ (A, X, p. 148, Corollary 2), and the depth relative to $J$ of each of the modules $K^i(x, M)$ is equal to $p$ or to $+\infty$ (No. 1, Remark 4). It then follows from Corollary 1 of No. 2 that we have $H^i(x, M) = 0$ for $i < p$. It remains to prove that $H^p(x, M)$ is non-zero when $p < +\infty$.

The case $p = 0$ being evident, suppose $0 < p < +\infty$ and $H^p(x, M) = 0$. Let $L$ be a free resolution of the $A$-module $A/J$; let $C$ denote the complex $\operatorname{Homgr}_A(L, M)$. The $A$-module $H^i(C)$ is isomorphic to $\operatorname{Ext}_A^i(A/J, M)$ (A, X, p. 100, Theorem 1); it is therefore zero for $i < p$. We then have for $i < p$ canonical exact sequences

The $A$-module $C^i$ is a product of $A$-modules isomorphic to $M$; therefore $H^s(x, C^i) = 0$ for $s \leq p$. From the preceding exact sequences and from A, X, p. 150, we deduce that the connecting homomorphism $\partial^s : H^s(x, B^{i+1}(C)) \longrightarrow H^{s+1}(x, B^i(C))$ is injective for $s \leq p$ and $i < p$; since $B^0(C) = 0$, it follows that $H^{p-i}(x, B^{i+1}(C))$ is zero for $i < p$. In particular, we have $H^1(x, B^p(C)) = 0$, so that the exact sequence

yields a surjection $H^0(x, Z^p(C)) \longrightarrow H^0(x, H^p(C))$. Since $H^p(C)$ is isomorphic to $\operatorname{Ext}_A^p(A/J, M)$, which is nonzero and annihilated by J, one has $H^0(x, H^p(C)) \neq 0$, whence $H^0(x, Z^p(C)) \neq 0$ and consequently $H^0(x, C^p) \neq 0$. But this implies $H^0(x, M) \neq 0$, contrary to the hypothesis. Hence $H^p(x, M) \neq 0$, which completes the proof.