Corollary 3

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module of finite projective dimension, $i$ an integer $\geq 0$, $N$ a finitely generated $A$-module, and $F$ the support of the $A$-module $\operatorname{Ext}_A^i(M, N)$ (resp. $\operatorname{Tor}_i^A(M, N)$). Then $\operatorname{prof}_F(A) \geq i$.

In fact, let $p \in F$. We have $\operatorname{Ext}_A^i(M_p, N_p) \neq 0$ (resp. $\operatorname{Tor}_i^A(M_p, N_p) \neq 0$) by prop. 2 of No. 2, hence $i \leq dp_{A_p}(M_p) \leq dp_A(M) < +\infty$ (No. 2, prop. 3). Theorem 1 implies $\operatorname{prof}(A_p) \geq i$. Consequently ($§ 1$, No. 5, prop. 8)

With the terminology of $§ 1$, No. 5, remark 4, the conclusion of cor. 3 means that the modules $\operatorname{Ext}_A^i(M, N)$ and $\operatorname{Tor}_i^A(M, N)$ are of grade $\geq i$. It implies that the codimension of their support in $\operatorname{Spec}(A)$ is $\geq i$ ($§ 1$, No. 7, prop. 12).