A generalization of the Nakayama functor

TL;DR

This paper generalizes the Nakayama functor for finite-dimensional algebras, characterizes it via monads and comonads, and develops a Gorenstein homological algebra theory for this generalization.

Contribution

It introduces a broad generalization of the Nakayama functor and establishes a Gorenstein homological algebra framework for it, extending classical results.

Findings

Characterization of the Nakayama functor via ambidextrous adjunctions

Development of Gorenstein homological algebra for the generalized functor

Application to module categories over finitely generated projective algebras

Abstract

In this paper we introduce a generalization of the Nakayama functor for finite-dimensional algebras. This is obtained by abstracting its interaction with the forgetful functor to vector spaces. In particular, we characterize the Nakayama functor in terms of an ambidextrous adjunction of monads and comonads. In the second part we develop a theory of Gorenstein homological algebra for such Nakayama functor. We obtain analogues of several classical results for Iwanaga-Gorenstein algebras. One of our main examples is the module category $\Lambda\text{-}\operatorname{Mod}$ of a $k$-algebra $\Lambda$, where $k$ is a commutative ring and $\Lambda$ is finitely generated projective as a $k$-module.