Lemma 47.4.3. Let $(R, \mathfrak m, \kappa )$ be a local ring. Any finite $R$-module has a projective cover.

Proof. Let $M$ be a finite $R$-module. Let $r = \dim _\kappa (M/\mathfrak m M)$. Choose $x_1, \ldots , x_ r \in M$ mapping to a basis of $M/\mathfrak m M$. Consider the map $f : R^{\oplus r} \to M$. By Nakayama's lemma this is a surjection (Algebra, Lemma 10.19.1). If $N \subset R^{\oplus r}$ is a proper submodule, then $N/\mathfrak m N \to \kappa ^{\oplus r}$ is not surjective (by Nakayama's lemma again) hence $N/\mathfrak m N \to M/\mathfrak m M$ is not surjective. Thus $f$ is an essential surjection. $\square$

There are also:

• 2 comment(s) on Section 47.4: Projective covers

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).