Exercise 3, § 10

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, such that $\rho(m_A)B$ is a definition ideal of $B$.

a) Construct for every $B$-module $N$ a natural $A$-linear isomorphism of $H_A(N)$ onto $H_B(N)$. (Using an injective resolution of $N$, now prove that it suffices to prove that one has $H_A^i(J) = 0$ for all $i > 0$ and all indecomposable injective $B$-modules $J$. Let $q$ be the unique element of $\operatorname{Ass}_B(J)$, and $p = \rho^{-1}(q)$. If $p \neq m_A$, observe that there exists $a \in m_A$ such that the homothety $a_J$ is bijective. If $p = m_A$, and if $I$ is a minimal injective resolution of the $A$-module $J$, observe that one has $\operatorname{Ass}_A(I^n) = \{m_A\}$ for all $n$.)

b) Suppose in addition that $B$ is a flat $A$-module. Construct for every $A$-module $M$ a natural $B$-linear isomorphism of $B \otimes_A H_A(M)$ onto $H_B(B \otimes_A M)$.