Complete intersection dimensions and Foxby classes

Abstract

Let 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 - dim R ( 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 - dim R ( N ) ⩽ CI - dim R ( N ) ⩽ pd R ( 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 A C ( R ) for each semidualizing R -complex C .