Proposition 10

Suppose that A is a left noetherian local ring, and let M be a finitely generated A-module. Then M possesses a minimal resolution $(P, p)$; for every $n \geqslant 0$, $P_n$ is a finitely generated free module.

In fact, in the construction carried out in No. 5 (p. 53), one may, by virtue of VIII, loc. cit., take for $(L_0, p)$ a projective cover of M, and for $L_{n+1}$ a projective cover of $\operatorname{Ker} d_n$. The resolution thus obtained is then minimal.

*Remarks. -- 1) Let m denote the maximal ideal of A and put $k = A/m$. Let P be a minimal projective resolution of M, and put $b_n = \dim_k (k \otimes_A P_n)$. Then $P_n$ is a free A-module of rank $b_n$. It follows from the corollary to Prop. 8 that for every other projective resolution $P'$ of M, one has $\dim_k (k \otimes_A P'_n) \geqslant b_n$, and that equality holds if and only if $P'$ is minimal.

2) By Prop. 9, $b_n$ is the dimension over $k$ of $H_n(k \otimes_A P)$, in other words of $\operatorname{Tor}_n^A(k, M)$. It is also the dimension over $k$ of $\operatorname{Ext}_A^n(M, k)$ (cf. X, p. 103, Remark 3).