Comonads and Gorenstein Homological Algebra

Abstract

In this dissertation we study Gorenstein homological algebra and a generalization of the Nakayama functor for finite-dimensional algebras. <br /> In the first part we show that the global Gorenstein projective dimensions of the categories of finitely presented right and left modules are the same over a preadditive category with weak kernels and weak cokernels.<br /> In the main part of the thesis we introduce and investigate comonads with Nakayama functor on abelian categories. We prove that the Nakayama functor associated to a comonad is unique up to isomorphism. We also show that several statements in Gorenstein homological algebra for a finite-dimensional algebra can be generalized to this setting. In particular, we generalize the notion of finite-dimensional Iwanaga-Gorenstein algebra to Gorenstein comonads, and we give generalizations of finite-dimensional Gorenstein projective and injective modules. We also prove a generalization of the statement that the left and right injective dimension of a finite-dimensional Iwanaga-Gorenstein algebra is the same, and the fact that it is equal to the global Gorenstein projective and injective dimension of the module category.<br /> In the final part of the thesis we investigate functor categories with codomain an abelian category and domain a small, Hom-finite, locally bounded category. We show that such a category is equipped with a comonad with Nakayama functor, and we use this to investigate the Gorenstein projective objects in the category. Under some mild assumptions we obtain a simpler description of these objects, generalizing previous work by several authors.