Auslander's Formula for contravariantly finite subcategories

TL;DR

This paper extends Auslander's formula to contravariantly finite subcategories, providing dual results, examples, and applications, including Morita equivalence implications for artin algebras.

Contribution

It introduces a relative version of Auslander's formula for contravariantly finite subcategories and explores its dual, with applications to Morita equivalence of artin algebras.

Findings

Established a relative Auslander's formula for contravariantly finite subcategories.

Provided examples and applications demonstrating the theory.

Showed Morita equivalence of relative Auslander algebras implies Morita equivalence of original algebras.

Abstract

A relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Dual version will be treated. Several examples and applications will be provided. In particular, we show that under certain circumstances, if relative Auslander algebras of artin algebras $\Lambda$ and $\Lambda'$ are Morita equivalent, then $\Lambda$ and $\Lambda'$ are also Morita equivalent.