Lemma 10.109.7. Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module. Let $d \geq 0$. The equivalent conditions (1) – (4) of Lemma 10.109.4, condition (5) of Lemma 10.109.5, and condition (6) of Lemma 10.109.6 are also equivalent to

1. there exists a resolution $0 \to F_ d \to F_{d - 1} \to \ldots \to F_0 \to M \to 0$ with $F_ i$ finite free.

Proof. This follows from Lemmas 10.109.4, 10.109.5, and 10.109.6 and because a finite projective module over a local ring is finite free, see Lemma 10.78.2. $\square$

Comment #2529 by Oleksandr Kravets on

typo: $P_i$ should be $F_i$

There are also:

• 1 comment(s) on Section 10.109: Rings of finite global dimension

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.

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 0CXF. Beware of the difference between the letter 'O' and the digit '0'.