ABSTRACT
Let M and N be non-zero finitely generated modules over a local (Noetherian) ring (R,m). Assume that https://www.w3.org/1998/Math/MathML"> Tor i R ( M , N ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1929.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for i ≫ 0, and let q = q R (M, N) be the largest integer i for which https://www.w3.org/1998/Math/MathML"> Tor i R ( M , N ) ≠ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1930.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . For the case q = 0, Auslander [2, Theorem 1.2] found a useful formula relating the depths of M, N and R. In view of the Auslander-Buchsbaum formula [4, Theorem 1.3.3] Auslander’s theorem goes as follows: () https://www.w3.org/1998/Math/MathML"> depth ( M ) + depth ( N ) − depth ( R ) = depth ( M ⊗ R N ) provided M has finite projective dimension and q ( M , N ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1931.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
