On Auslander-Buchsbaum - type formulas

Auslander - Buchsbaum

A vanishing theorem is proved for Ext groups over non-commutative graded algebras. Along the way, an “infinite” version is proved of the non-commutative Auslander-Buchsbaum theorem.

An Auslander-buchsbaum Identity for Semi- Dualizing Modules

The Tor Game

Let M and N be non-zero finitely generated modules over a local (Noetherian) ring (R,m). Assume that TorRi (M,N) = 0 for i >> 0, and let q = qR(M,N) be the largest integer i for which TorRi (M,N) 6= 0. For the case q = 0, Auslander [A, Theorem 1.2] found a useful formula relating the depths of M , N and R. In view of the Auslander-Buchsbaum formula [BH,Thm. 1.3.3] Auslander’s theorem goes as fo...

On a Depth Formula for Modules over Local Rings

We prove that for modules M and N over a local ring R, the depth formula: depthR M + depthR N − depthR = depthR Tor R s (M,N) − s, where s = sup{i | Tor i (M,N) 6= 0}, holds under certain conditions. This adds to the list cases where the depth formula, which extends the classical Auslander-Buchsbaum equality, is satisfied.

A Generalization of the Auslander-buchsbaum Formula

Let R be a Noetherian local ring and Ω an arbitrary R-module of finite depth and finite projective dimension. The flat dimension of Ω is at least depth(R)−depth(Ω) with equality in the following cases: (i) Ω is finitely generated over some Noetherian local R-algebra S; (ii) dim(R) = 1; (iii) dim(R) = 2 and Ω is separated; (iv) R is CohenMacaulay, dim(R) = 3 and Ω is complete.