Remark 4

Let $A$ be a noetherian ring, $N$ a finitely generated $A$-module. The greatest lower bound in $N \cup \{+\infty\}$ of the set of integers $n$ such that $\mathrm{Ext}_A^n(N, A)$ is nonzero is sometimes called the grade of $N$, and is denoted by grade$(N)$. By Prop. 9, this is also the depth of $A$ along the support of $N$, or again (No. 4, Theorem 2) the least upper bound of the set of lengths of $A$-regular sequences of elements of the annihilator of $N$. Since for every prime ideal $p$ of $A$ the annihilator of $N_p$ is equal to $\mathrm{Ann}(N)_p$ (II, § 2, No. 4, formula (9)), we deduce from Prop. 5 of No. 3 the equality

where $\Omega$ denotes the set of maximal ideals of $A$.