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A Generalization of the Nakayama Functor

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 Λ-Mod of a k-algebra Λ, where k is a commutative ring and Λ is finitely generated projective as a k-module.

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Acknowledgments

This is part of the authors PhD thesis. The author thanks Gustavo Jasso, Julian Kü lshammer, Rosanna Laking, and Jan Schröer for helpful discussions and comments and on a previous version of this paper. He would also like to thank the referee for helpful comments and suggestions. The work was made possible by the funding provided by the Bonn International Graduate School in Mathematics

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Correspondence to Sondre Kvamme.

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Kvamme, S. A Generalization of the Nakayama Functor. Algebr Represent Theor 23, 1319–1353 (2020). https://doi.org/10.1007/s10468-019-09891-7

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Keywords

  • Abelian category
  • Gorenstein homological algebra
  • Gorenstein ring
  • Homological algebra
  • Monad

Mathematics Subject Classification (2010)

  • 18E10
  • 16E65
  • 16D90