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.

中文翻译：

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中山函子的推广

在本文中，我们介绍了中山函子对有限维代数的推广。这是通过将其与健忘函子的交互抽象为向量空间而获得的。尤其是，我们根据单子和共母的灵巧附加来表征中山函子。在第二部分中，我们为中山函子建立了Gorenstein同源代数的理论。我们获得了Iwanaga-Gorenstein代数的几个经典结果的类似物。我们的主要示例之一是k-代数Λ的模块类别Λ-Mod ，其中k是交换环，Λ是作为k模有限射影生成的。