nLab Auslander-Buchsbaum formula


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

pd R(M)+depth(M)=depth(R) 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.


Last revised on October 6, 2023 at 08:21:26. See the history of this page for a list of all contributions to it.