Exercise 12, § 3

Let $A$ be a local noetherian ring, $M$ and $N$ nonzero $A$-modules of finite type. Suppose that the projective dimension of $M$ is finite. Let $q$ be the largest integer such that $\mathrm{Tor}_q^A(M, N) \neq 0$.

a) Suppose that $\mathrm{prof}_A(\mathrm{Tor}_q^A(M, N)) = 0$. Prove the equality $\mathrm{prof}_A(N) + q = \mathrm{dp}_A(M)$ (one may reason by induction on $\mathrm{prof}_A(N)$, considering an element $x$ of $m_A$ which is not a divisor of 0 in $N$).

b) Suppose in addition that the $A$-module $M \otimes_A N$ is of finite length. Deduce from a) that $\mathrm{Tor}_i^A(M, N)$ is zero for $i > \mathrm{prof}(A) - \mathrm{prof}(M) - \mathrm{prof}(N)$, and nonzero when there is equality. In particular one has $\mathrm{prof}(M) + \mathrm{prof}(N) \leq \mathrm{prof}(A)$.