The Resolution Quiver And Singularity Category Of A Nakayama Algebra

This is a doctoral dissertation on the homological properties of the Nakayama al-gebras. We study the resolution quivers, singularity categories, Gorenstein homological properties, and Gorenstein projective modules of the Nakayama algebras.The thesis is organised as follows.In Chapter1, we briefly recall the historical origins, developments and main ob-jectives in triangulated categories, singularity categories, and Gorenstein homological algebras. We also introduce the background of the research and main results of the thesis.In Chapter2, we discuss some basic facts on triangulated categories, singularity categories, and Gorenstein projective modules. We also recall other necessary tools which will be used later.In Chapter3, we recall the definition and some basic properties of resolution quiv-ers. Then we prove that the cycles in the resolution quiver of a connected Nakayama algebra have the same size. We introduce the notion of weight for a cycle in resolution quivers. We also prove that the cycles in the resolution quiver of a connected Nakayama algebra have the same weight. As an application, we show that the number of isoclass-es of simple modules of infinite projective dimension equals the number of isoclasses of simple modules of infinite injective dimension for a connected Nakayama algebra of infinite global dimension. We also study the Gorenstein homological properties of Nakayama algebras. We give several necessary and sufficient conditions for a Nakaya-ma algebra to be Gorenstein.In Chapter4, we construct a Frobenius subcategory in the module category for a given Nakayama algebra. The construction uses the resolution quivers. We prove that the Frobenius subcategory is an abelian category and is equivalent to the category of the finitely generated modules over a self-injective Nakayama algebra. The difference and relationship between the Frobenius subcategory and the Gorenstein core are also discussed. We show that the stable category of this Frobenius subcategory is triangle equivalent to the singularity category of the given Nakayama algebra; this provides a new approach to describe the singularity categories of Nakayama algebras. Using this Frobenius subcategory, we also prove that the singularity category of a Nakayama algebra is triangle dual to the singularity category of its opposite algebra. Using this triangle duality, we prove that the resolution quivers of a Nakayama algebra and its opposite algebra have the same number of cycles, and all cycles have the same size as well as the same weight. We also construct two radical square algebras which are opposite to each other, and show that there is no triangle duality between their singularity categories.In Chapter5, we study the Gorenstein projective modules of Nakayama algebras over fields. We introduce the notion of perfect paths, and then we use perfect paths to describe the Gorenstein projective modules. We prove that there is a one-to-one correspondence between perfect paths and indecomposable non-projective Gorenstein projective modules in such algebras. We also reobtain C. M. Ringel’s description of Gorenstein projective modules for Nakayama algebras.