Exercise 16, § 1

Let $A$ be a local noetherian ring. For every finitely generated $A$-module $M$, denote by $t_A(M)$, or simply $t(M)$, the dimension of the $\kappa_A$-vector space $\mathrm{Ext}_A^p(\kappa_A, M)$, with $p = \mathrm{prof}(M)$.

a) Let $(x_1, \ldots, x_r)$ be a sequence of elements of $m_A$ completely transversal for $M$, generating an ideal $J$. We have $t(M) = t(M/JM)$.

b) Let $\rho : A \to B$ be a local homomorphism of local noetherian rings; let $M$ be a finitely generated $A$-module, and let $N$ be a finitely generated $B$-module flat over $A$. Prove the formula $t_B(M \otimes_A N) = t_A(M) t_B(\kappa_A \otimes_A N)$ (reduce, with the help of a), to the case where $\mathrm{prof}_B(M \otimes_A N) = \mathrm{prof}_A(M) = \mathrm{prof}_B(\kappa_A \otimes_A N) = 0$.