Some Research On Finite Group Action On Categories

This dissertation is devoted to the study of the number of isomorphic connected components of the quiver of the skew group algebra for a finite dimensional connected basic algebra and a finite cyclic group acting on it,and of the equivariant categories of the categories of coherent sheaves on weighted projective lines with respect to the group action of a finite subgroup of Picard group,and of the recursive relation of the Auslander-Reiten quivers of the categories of maximal Cohen-Macaulay modules over the local ring of single singularity.The dissertation includes four chapters.In the first chapter,we give a brief introduction of the background and make a systemic exposition of our main results.In the second chapter,we study the number of isomorphic connected components of the quiver of the skew group algebra for a finite dimensional connected basic algebra and a finite cyclic group acting on it.Firstly,we give the quiver of the smash product for a finite dimensional connected basic algebra which has a cyclic group grading,and introduce a method to compute the number of its isomorphic connected components.Then we construct a quiver such that the quiver of smash product for its path algebra "covers" the quiver of the skew group algebra for a finite dimensional connected basic algebra and a cyclic group acting on it,and show the quivers of this smash product and this skew group algebra has the same number of connected components.Hence we give the number of connected components of the quiver of the corresponding skew group algebra.In the third chapter,we investigate the equivariant category of the category of coherent sheaves on weighted projective line with respect to the group action of a finite subgroup of Picard group.Firstly,we show that the equivariant category of the category of coherent sheaves on a weighted projective curve with respect to the action of a finite subgroup of its automorphism group is equivalent to the category of coherent sheaves on some weighted projective curve.Secondly,we use the equivariant relation between the tube categories with respect to the action of a finite abelian group to give the weight type of the weighted projective curve corresponding to the equivariant category of the category of coherent sheaves on a weighted projective line with respect to the action of a finite subgroup of Picard group,and calculate the genus of the curve.Finally,we give the "trichotomy" result about the equivariant category of the category of coherent sheaves on a weighted projective line with respect to the action of a finite subgroup of Picard group.In the fourth chapter,we study the recursive relation of the Auslander-Reiten quivers of the categories of maximal Cohen-Macaulay modules over the local ring of single singularity.In this chapter,we establish the equivariant relations between the category of matrix factorizations and the category of maximal Cohen-Macaulay modules by the suitable actions of a cyclic group with order 2.Moreover we present an alternative proof of the structure theorem for the Auslander-Reiten quiver of the category of maximal Cohen-Macaulay modules over 2-dimensional single singularity,by using the method of completion and the Auslander-Reiten quiver of the category of vector bundles over weighted projective line which is domestic type.