Higher Morita-Tachikawa correspondence

Abstract

Important correspondences in representation theory can be regarded as restrictions of the Morita--Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo-symmetric algebras. Explicitly, the Morita--Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite-dimensional algebras are exactly the finite-dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi-generators and quasi-cogenerators which generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita--Tachikawa correspondence that takes into account relative dominant dimension with respect to a self-orthogonal module with arbitrary projective and injective dimension. These new versions also hold over Noetherian algebras which are finitely generated and projective over a commutative Noetherian ring.