Complete intersection dimensions and Foxby classes

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 ${\text{cidim}}_R(M)$–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 ${\text{gdim}}_R(N)\le {\text{cidim}}_R(N)\le {\text{pd}}_R(N)$. 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 $\varphi \text{\hspace{0.17em}⁣}:R\to S$ and $\psi \text{\hspace{0.17em}⁣}:S\to T$ such that $\varphi$ has finite Gorenstein dimension, if $\psi$ has finite complete intersection dimension, then the composition $\psi \circ \varphi$ 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 $\text{catac}(R)$ for each semidualizing $R$-complex $C$.