Corollary

Let $A$ be a ring, $M$ and $N$ $A$-modules, $i$ an integer.

a) The support of $\mathrm{Tor}_i^A(M, N)$ is contained in $\mathrm{Supp}(M) \cap \mathrm{Supp}(N)$, and the same is true of the support of $\mathrm{Ext}_A^i(M, N)$ if $A$ is noetherian and $M$ finitely generated.

b) Suppose that $A$ is noetherian, and that the modules $M$ and $N$ are finitely generated; if the $A$-module $M \otimes_A N$ is of finite length, the same is true of $\mathrm{Tor}_i^A(M, N)$ and of $\mathrm{Ext}_A^i(M, N)$.

If $p$ is a prime ideal of $A$ not belonging to $\mathrm{Supp}(M) \cap \mathrm{Supp}(N)$, one of the modules $M_p$ or $N_p$ is zero, which implies a) in view of Prop. 2.

In order that a finitely generated module over a noetherian ring be of finite length, it is necessary and sufficient that its support consist of maximal ideals (IV, § 2, No. 5, Prop. 7). Under assumption b), the $A$-modules $\mathrm{Tor}_i^A(M, N)$ and $\mathrm{Ext}_A^i(M, N)$ are finitely generated (A, X, p. 108, Cor.) ; the assertion b) therefore follows from a).