Complete intersection dimensions and Foxby classes

AbstractLet R be a local ring and M a finitely generated R-module. The complete intersection dimension of M–defined by Avramov, Gasharov and Peeva, and denoted CI-dimR(M)–is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities G-dimR(N)⩽CI-dimR(N)⩽pdR(N).Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ∘φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class AC(R) for each semidualizing R-complex C