Exercise 8, § 4

Let $A$ be an integrally closed Noetherian domain and $M_1, M_2$ two finitely generated $A$-modules. (a) Show that the $A$-modules $\operatorname{Tor}_i^A(M_1, M_2)$, $\operatorname{Ext}_A^i(M_1, M_2)$ are pseudo-zero for $i > 2$ (reduce it to the case where $A$ is a principal ideal domain). (b) If $M_i$ is torsion-free, show that $\operatorname{Tor}_1^A(M_1, M_i)$ and $\operatorname{Ext}_A^1(M_1, M_2)$ are pseudo-zero (same method). (c) If $M_i$ is a torsion $A$-module, show that (in the notation of no. 5):

(d) Show that for every ordered pair of finitely generated A-modules $M_1, M_2$: