Complete intersection dimensions and Foxby classes

TL;DR

This paper explores the properties of complete intersection dimensions and Foxby classes, establishing new relationships between homological invariants of modules over local rings and their compositions.

Contribution

It generalizes a theorem of Avramov and Foxby by linking finite Gorenstein and complete intersection dimensions through compositions of local ring homomorphisms.

Findings

Finite complete intersection dimension implies $C$-reflexivity.

Composition of homomorphisms with finite Gorenstein and complete intersection dimensions has finite Gorenstein dimension.

Modules with finite complete intersection dimension belong to the Auslander class for all semidualizing complexes.

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 $\cidim_R(M)$--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 $\gdim_R(N)\leq\cidim_R(N)\leq\pd_R(N)$. 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 $\phi\colon R\to S$ and $\psi\colon S\to T$ such that $\phi$ has finite Gorenstein dimension, if $\psi$ has finite complete intersection dimension, then the composition $\psi\circ\phi$ 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 $\catac(R)$ for each semidualizing $R$-complex $C$.