Homological theory of orthogonal modules

TL;DR

This paper explores the homological properties of orthogonal modules over self-injective algebras, establishing new categorical frameworks and characterizations related to Tachikawa's and Nakayama's conjectures.

Contribution

It introduces a systematic approach to orthogonal generators via stable categories, establishes recollements, and links Tachikawa's conjecture to Gorenstein categories, also defining Gorenstein-Morita algebras.

Findings

Recollement of M-relative stable categories established

Equivalent characterizations of Tachikawa's second conjecture provided

Nakayama conjecture verified for Gorenstein-Morita algebras

Abstract

Tachikawa's second conjecture predicts that a finitely generated, orthogonal module over a finite-dimensional self-injective algebra is projective. This conjecture is an important part of the Nakayama conjecture. Our principal motivation of this work is a systematic understanding of finitely generated, orthogonal generators over a self-injective Artin algebra from the view point of stable module categories. As a result, for an orthogonal generator M, we establish a recollement of the M-relative stable categories, describe compact objects of the right term of the recollement, and give equivalent characterizations of Tachikawa's second conjecture in terms of M-Gorenstein categories. Further, we introduce Gorenstein-Morita algebras and show that the Nakayama conjecture holds true for them.