Remark 5

Let A be a local ring and $\mathfrak{r}$ its radical. Let M be a finitely generated A-module, P a finitely generated projective A-module, and $u: P\rightarrow$ M a homomorphism. By Corollary 6 of VIII, p. 36, the A-module P is free. Choose a basis $(e_i)_{i\in I}$ of P. Set $x_i=u(e_i)$, and denote the canonical image of $x_i$ in $M/\mathfrak{r}M$ by $\overline{x}_i$. The following properties are equivalent:

(i) The pair $(P, u)$ is a projective cover of M.

(ii) The family $(x_i)_{i\in I}$ is a minimal generating family of the A-module M.

(iii) The family $(\overline{x}_i)_{i\in I}$ is a basis of the vector space $M/\mathfrak{r}M$ over the field $A/\mathfrak{r}$.

We have (i) $\Rightarrow$ (ii) by Remark 2 of VIII, p. 161 and (iii) $\Rightarrow$ (i) by Corollary 2. Furthermore, if the family $(x_i)$ is a minimal generating family of the A-module M, then the family $(\overline{x}_i)$ is a minimal generating family, that is, a basis, of the vector space $M/\mathfrak{r}M$ over $A/\mathfrak{r}$ (VIII, p. 158, Corollary 1).