Exercise 18, § 1

Let $A$ be a noetherian ring, $J$ an ideal of $A$, $M$ and $N$ finite type $A$-modules, with $\mathrm{prof}_A(J; N) \geqslant 2$.

a) Prove the inequality $\mathrm{prof}_A(J; \mathrm{Hom}_A(M, N)) \geqslant 2$ (consider an $N$-regular sequence $(x, y)$ in $J$).

b) Suppose that $\operatorname{prof}_A(J; \operatorname{Hom}_A(M, N)) \geqslant 3$. Prove that one has $\operatorname{prof}_A(J; \operatorname{Ext}_A^1(M, N)) \geqslant 1$ (consider an exact sequence of extension modules associated with an N-regular element of $J$).