The Stacks project

47.4 Projective covers

In this section we briefly discuss projective covers.

Definition 47.4.1. Let $R$ be a ring. A surjection $P \to M$ of $R$-modules is said to be a projective cover, or sometimes a projective envelope, if $P$ is a projective $R$-module and $P \to M$ is an essential surjection.

Projective covers do not always exist. For example, if $k$ is a field and $R = k[x]$ is the polynomial ring over $k$, then the module $M = R/(x)$ does not have a projective cover. Namely, for any surjection $f : P \to M$ with $P$ projective over $R$, the proper submodule $(x - 1)P$ surjects onto $M$. Hence $f$ is not essential.

Lemma 47.4.2. Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover of $M$ exists, then it is unique up to isomorphism.

Proof. Let $P \to M$ and $P' \to M$ be projective covers. Because $P$ is a projective $R$-module and $P' \to M$ is surjective, we can find an $R$-module map $\alpha : P \to P'$ compatible with the maps to $M$. Since $P' \to M$ is essential, we see that $\alpha $ is surjective. As $P'$ is a projective $R$-module we can choose a direct sum decomposition $P = \mathop{\mathrm{Ker}}(\alpha ) \oplus P'$. Since $P' \to M$ is surjective and since $P \to M$ is essential we conclude that $\mathop{\mathrm{Ker}}(\alpha )$ is zero as desired. $\square$

Here is an example where projective covers exist.

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. $\square$


Comments (2)

Comment #2607 by Dario WeiƟmann on

Typo in lemma 45.4.3: instead of .


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08XX. Beware of the difference between the letter 'O' and the digit '0'.