Proposition 6

*Let B be a ring and N an (A, B)-bimodule, which is a noetherian B-module (resp. of finite length).

a) Suppose A right noetherian and let M be a finitely generated right A-module. Then the B-modules (X, p. 81) $\mathrm{Tor}_n^A(M, N)$ are noetherian (resp. of finite length).

b) Suppose A left noetherian and let M be a finitely generated left A-module. Then the B-modules (X, p. 98) $\mathrm{Ext}_A^n(M, N)$ are noetherian (resp. of finite length).

Choose a free resolution $p : L \to M$ such that each of the A-modules $L_n$ is of finite type (X, p. 53, Prop. 6), and let C be the complex of B-modules $L \otimes_A N$ in case $a$, $\mathrm{Homgr}_A(L, N)$ in case $b$. Each of the B-modules $C_n$ is isomorphic to a product of a finite number of copies of N, hence is noetherian (resp. of finite length); the same is therefore true of the modules $\mathrm{H}_n(C)$. But, by X, p. 100, Th. 1, these are isomorphic to the $\mathrm{Tor}_n^A(M, N)$ in case $a$, to the $\mathrm{Ext}_A^{-n}(M, N)$ in case $b$.