Auslander–Buchsbaum formula

Remarks

One proof goes by induction on $\operatorname{depth} R$, using a minimal free resolution of $M$. When the depths are positive, one divides by an element of $\mathfrak{m}$ that is a non-zero-divisor on both $R$ and $M$. Example: for $R = K[\![x, y]\!]$, which has depth $2$, the residue field $K$ has projective dimension $2$ and depth $0$, and $R/(x)$ has projective dimension $1$ and depth $1$. The hypothesis $\operatorname{pd}_R M < \infty$ is needed: for $R = K[x]/(x^2)$ and $M = K$, the depths are $0$ and $0$, but $\operatorname{pd}_R K = \infty$. In particular, a finitely generated module of finite projective dimension has projective dimension at most $\operatorname{depth} R$.

Going deeper

For a student

The formula is the main link between resolutions and regular sequences. Over a regular local ring, every finitely generated module has finite projective dimension, so the formula always applies there. It shows that a module of finite projective dimension over a Cohen–Macaulay local ring is maximal Cohen–Macaulay exactly when it is free.