Higher Auslander correspondence for dualizing $R$-varieties

TL;DR

This paper extends the higher Auslander correspondence from Artin R-algebras to dualizing R-varieties, characterizing d-abelian and d-Auslander dualizing R-varieties through equivalences with d-cluster-tilting subcategories.

Contribution

It generalizes the higher Auslander correspondence to dualizing R-varieties, providing new characterizations and equivalences in this broader context.

Findings

Dualizing R-varieties are d-abelian iff they are d-Auslander dualizing R-varieties.

Such varieties are equivalent to d-cluster-tilting subcategories of finitely presented modules.

The results unify and extend existing higher Auslander correspondence theories.

Abstract

Let $R$ be a commutative artinian ring. We extend higher Auslander correspondence from Artin $R$-algebras of finite representation type to dualizing $R$-varieties. More precisely, for a positive integer $d$, we show that a dualizing $R$-variety is $d$-abelian if and only if it is a $d$-Auslander dualizing $R$-variety if and only if it is equivalent to a $d$-cluster-tilting subcategory of the category of finitely presented modules over a dualizing $R$-variety.