Example 3

Suppose that $A$ is a local noetherian ring, let $m$ denote its maximal ideal, and put $\kappa = A/m$. Let $M$ be a finitely generated $A$-module and $P$ a minimal projective resolution of $M$ (X, p. 54). For every $n \geqslant 0$, the $\kappa$-vector spaces $\mathrm{Tor}_n^A(\kappa, M)$ and $\mathrm{Ext}_A^n(M, \kappa)$ are finite-dimensional, of dimension equal to the rank of the free $A$-module $P_n$; in fact, the complexes $\kappa \otimes_A P$ and $\mathrm{Homgr}_A(P, \kappa)$ have zero differential.