# Homological theory of orthogonal modules

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. In the talk, we introduce a systematic study of finitely generated, orthogonal generators over a self-injective Artin algebra from the view point of stable module categories. 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. This is joint work with Changchang Xi.