Remark 3

If q is a point of $\mathrm{Supp}(M)$, we therefore have $\mathrm{prof}_A(q;M) = \inf_{p \supseteq q} \mathrm{prof}_{A_p}(M_p)$. In particular, one has the inequality $\mathrm{prof}_A(q;M) \leqslant \mathrm{prof}_{A_q}(M_q)$; there is equality when q is maximal. In the general case, one may have $\mathrm{prof}_A(q;M) < \mathrm{prof}_{A_q}(M_q)$; one may also have $\mathrm{prof}_A(q;M) < \inf \mathrm{prof}_{A_m}(M_m)$ where m runs through the set of maximal ideals of A containing q. Let p, for example, be a nonmaximal prime ideal of A, containing q and distinct from q; put $M = A/p$. One has $\mathrm{prof}_A(q;M) = 0$, $\mathrm{prof}_{A_q}(M_q) = +\infty$ and $\mathrm{prof}_{A_m}(M_m) > 0$ for every maximal ideal m of A.