Exercise 1, § 3

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, $N$ a finitely generated $B$-module. We suppose that the $A$-modules $B$ and $N$ are flat.

a) Prove the equality $dp_B(N) = dp_{\kappa_A \otimes_A B}(\kappa_A \otimes_A N)$ (consider a resolution of $N$ by finitely generated free $B$-modules).

b) Let $M$ be a finitely generated $A$-module. Prove the equality

(Let $K$ (resp. $L$) be a resolution of the $A$-module $M$ (resp. of the $B$-module $N$) by finitely generated free modules; prove that $K \otimes_A L$ is a resolution of $M \otimes_A N$ by free $B$-modules. If $dp_A(M) = m$ and $dp_B(N) = n$, calculate $H_{m+n}(\kappa_B \otimes_B (K \otimes_A L))$ in terms of $H_m(\kappa_A \otimes_A K)$ and $H_n(\kappa_B \otimes_B L)$.)