
## 10.110 Auslander-Buchsbaum

The following result can be found in .

Proposition 10.110.1. Let $R$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module which has finite projective dimension $\text{pd}_ R(M)$. Then we have

$\text{depth}(R) = \text{pd}_ R(M) + \text{depth}(M)$

Proof. We prove this by induction on $\text{depth}(M)$. The most interesting case is the case $\text{depth}(M) = 0$. In this case, let

$0 \to R^{n_ e} \to R^{n_{e-1}} \to \ldots \to R^{n_0} \to M \to 0$

be a minimal finite free resolution, so $e = \text{pd}_ R(M)$. By Lemma 10.101.2 we may assume all matrix coefficients of the maps in the complex are contained in the maximal ideal of $R$. Then on the one hand, by Proposition 10.101.9 we see that $\text{depth}(R) \geq e$. On the other hand, breaking the long exact sequence into short exact sequences

\begin{align*} 0 \to R^{n_ e} \to R^{n_{e - 1}} \to K_{e - 2} \to 0,\\ 0 \to K_{e - 2} \to R^{n_{e - 2}} \to K_{e - 3} \to 0,\\ \ldots ,\\ 0 \to K_0 \to R^{n_0} \to M \to 0 \end{align*}

we see, using Lemma 10.71.6, that

\begin{align*} \text{depth}(K_{e - 2}) \geq \text{depth}(R) - 1,\\ \text{depth}(K_{e - 3}) \geq \text{depth}(R) - 2,\\ \ldots ,\\ \text{depth}(K_0) \geq \text{depth}(R) - (e - 1),\\ \text{depth}(M) \geq \text{depth}(R) - e \end{align*}

and since $\text{depth}(M) = 0$ we conclude $\text{depth}(R) \leq e$. This finishes the proof of the case $\text{depth}(M) = 0$.

Induction step. If $\text{depth}(M) > 0$, then we pick $x \in \mathfrak m$ which is a nonzerodivisor on both $M$ and $R$. This is possible, because either $\text{pd}_ R(M) > 0$ and $\text{depth}(R) > 0$ by the aforementioned Proposition 10.101.9 or $\text{pd}_ R(M) = 0$ in which case $M$ is finite free hence also $\text{depth}(R) = \text{depth}(M) > 0$. Thus $\text{depth}(R \oplus M) > 0$ by Lemma 10.71.6 (for example) and we can find an $x \in \mathfrak m$ which is a nonzerodivisor on both $R$ and $M$. Let

$0 \to R^{n_ e} \to R^{n_{e-1}} \to \ldots \to R^{n_0} \to M \to 0$

be a minimal resolution as above. An application of the snake lemma shows that

$0 \to (R/xR)^{n_ e} \to (R/xR)^{n_{e-1}} \to \ldots \to (R/xR)^{n_0} \to M/xM \to 0$

is a minimal resolution too. Thus $\text{pd}_ R(M) = \text{pd}_{R/xR}(M/xM)$. By Lemma 10.71.7 we have $\text{depth}(R/xR) = \text{depth}(R) - 1$ and $\text{depth}(M/xM) = \text{depth}(M) - 1$. Till now depths have all been depths as $R$ modules, but we observe that $\text{depth}_ R(M/xM) = \text{depth}_{R/xR}(M/xM)$ and similarly for $R/xR$. By induction hypothesis we see that the Auslander-Buchsbaum formula holds for $M/xM$ over $R/xR$. Since the depths of both $R/xR$ and $M/xM$ have decreased by one and the projective dimension has not changed we conclude. $\square$

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 090U. Beware of the difference between the letter 'O' and the digit '0'.