Cartan-Eilenberg complexes and Auslander categories

TL;DR

This paper extends the Foxby equivalence to Cartan-Eilenberg complexes over commutative noetherian rings with semi-dualizing modules, introducing new categories and criteria for Gorenstein dimensions.

Contribution

It introduces Cartan-Eilenberg analogues of Auslander categories and Gorenstein complexes, expanding the theoretical framework of homological algebra.

Findings

Defined C-E Auslander categories and Gorenstein complexes.

Established criteria for finiteness of C-E Gorenstein dimensions.

Extended Foxby equivalence to Cartan-Eilenberg complexes.

Abstract

Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $\xymatrix@C=50pt{\mathcal {A}_C(R) \ar@<0.4ex>[r]^{C\otimes^{\mathbf{L}}_{R} -} & \mathcal {B}_C(R) \ar@<0.4ex>[l]^{\mathbf{R}\mathrm{Hom}_{R}(C, -)}}$. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $\mathcal{W}$ complexes and C-E $\mathcal{W}$-Gorenstein complexes are introduced, where $\mathcal{W}$ denotes a self-orthogonal class of $R$-modules. Moreover, criteria for finiteness of C-E Gorenstein dimensions of complexes in terms of resolution-free characterizations are considered.