Equivariantization Of Monomorphism Categories

Let k be an algebraically closed field,given an Artin k-algebra A,the inverse of Auslander Reiter transformation in the submodule category S(A)can also be computed within A-mod by τS-1=KerMepiτA-1.If AR-quiver of A-mod contains postprojective components,we show that the postjective components of S(A)can be calculated by fixed steps by an example.Given an acycle finite quiver Q,similar to the representation of the quiver Q over the field k,the monomorphism category of Q over A-mod,noted as Mon(Q,A-mod),is generalized from submodule category S(A).If there is a finite group G with order n acting on A,a new algebra AG can be constructed,which is called skew group algebra.Because the action of G on A can be induced to A-mod(resp.Mon(Q,A-mod))and is a strict action,we can obtain the equivariantization(A-mod)G(resp.Mon(Q,A-mod)G).Then we prove the category equivalence between Mon(Q,(A-mod)G)and Mon(Q,A-mod)G.This paper is divided into three chapters.In Chapter 1,we introduce the background and development of the related researches and framework of this paper.In Chapter 2,we recall some basic results about Auslander-Reiten theory of the submodule categories for Artin algebras.Then we show that τS-1 can alse be computed within A-mod and summarize the methods of calculating the AR-quiver of S(A)with postjective components by an example.In Chapter 3,we first recall the monomorphism category,generated from submodule category,then introduce the concept of equivariantization closely related to group action on category.Given a skew group algebra AG,we discuss that the action of group G on algebra A can be induced to a strict action on Mon(Q,A-mod),and obtain the equivariantization Mon(Q,A-mod)G.Our main theorem proves the equivalence between categories Mon(Q,(A-mod)G)and Mon(Q,A-mod)G.Combining main theorem with results in[1],we show that Mon(Q,A-mod)G(?)Mon(Q,AG-mod).Finally,we supplement a corollary when G is a solvable group and illustrate how to use Mon(Q,AG-mod)to calculate Mon(Q,A-mod)G by an example.