nLab Auslander-Buchsbaum formula

Let $R$ be a commutative Noetherian local ring. Then for any nonzero finitely generated $R$-module $M$ of finite projective dimension $pd_R(M)$ the following formula holds

$$pd_R(M) + depth(M) = depth(R)$$

As a corollary a commutative ring is regular iff it has a finite global dimension. This is also proved by Serre.

This in turn is the essence of the fact that for any regular scheme, hence in particular for any smooth scheme over a field, the bounded derived category of coherent sheaves coincides with the full triangulated subcategory of perfect complexes. In other words, the triangulated category of singularities of a smooth scheme is trivial.

One should be warned that another result on regular rings is usually known as Auslander-Buchsbaum theorem? (from a later article in 1959), namely that every regular local ring is a unique factorization domain.

Literature

• Maurice Auslander, David A. Buchsbaum, Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (2): 390–405 (1957) doi MR0086822
• stacks project 10.111 Auslander-Buchsbaum tag:090U
• wikipedia:Auslander-Buchsbaum_formula