Remark 1

Let r be an integer. By Proposition 2 of No. 1 and II, § 4, No. 4, Corollary 2 of Proposition 17, the inequality $\mathrm{prof}_F(M) \geqslant r$ is equivalent to the following property: for every finitely generated A-module N whose support is contained in F, one has $\mathrm{Ext}_A^i(N,M) = 0$ for $i < r$.