Invariant Properties Of Representations Under Weak Excellent Extensions

The chief objective of the representation theory of Artinian algebras is to.charac-terize an algebra in terms of the properties of its module category. For this purpose, the representation invariants play an important role in the representation theory. In this dissertation, we devote to studying several representation invariants under excel-lent extensions of rings. In particular, in this dissertation, we introduce the notion of weak excellent extensions of rings and study the invariant properties of the represen-tation type, the CM-finite type under excellent extensions (Chapter 2). We discuss the representation dimension of Artinian algebras under weak excellent extensions. We firstly prove that excellent extensions of commutative Artinian rings preserve the representation dimensions. Moreover, we study the invariant properties of the represen-tation dimension of an Artinian algebra under a cleft extension for a finite-dimensional semisimple cosemisimple Hopf algebra, which is an important kind of weak excellent extensions (Chapter 3). Finally, we discuss the tilting invariant properties of cleft extensions for finite-dimensional semisimple cosemisimple Hopf algebras (Chapter 4).This paper is divided into four chapters.In Chapter 1, we give the backgrounds and preliminaries.In Chapter 2, we introduce the notion of weak excellent extensions of rings as a generalization of the excellent extensions of rings, and we prove that Gorenstein projective dimension is invariant under a weak excellent extension. And we study the invariant properties of the representation type of finite-dimensional algebras, the CM-finite type of Artinian algebras under weak excellent extensions. The following are main results in this chapter.Theorem 0.0.1 Let S be a weak excellent extension of an Artinian algebra A.If A is of finite representation type (resp. CM-finite, CM-free), then so is S; furthermore, if S is an excellent extension of A, then the converse also holds true.Theorem 0.0.2 Let B be a weak excellent extension of a finite-dimensional k-algebra A. If A is of tame type (resp. wild type), then so is B; furthermore, if B is an excellent extension of A, then the converse also holds true.In Chapter 3, we study the representation dimensions of Artinian algebras under weak excellent extensions. We firstly prove that the representation dimension is invari-ant under an excellent extension of a commutative Artinian ring. Moreover, in case k an algebraically closed field, we study the invariant property of the representation di-mension under a cleft extension for a finite-dimensional semisimple cosemisimple Hopf k-algebra, which is a special kind of weak excellent extensions. The following are main results in this chapter.Theorem 0.0.3 Let R be a commutative Artinian ring and S an R-algebra. If S is an excellent extension of R, then the representation dimensions of R and S are identical.Theorem 0.0.4 Let H be a finite-dimensional semisimple cosemisimple Hopf k-algebra, and let A be a finite-dimensional twisted H-module algebra andσan invertible cocycle. Then rep. dim (A)=rep. dim (A#σH).Corollary 0.0.5 Let k be an algebraically closed field of characteristic p, and P be a normal Sylow p-subgroup of a finite group G, then rep.dim(kG)=rep.dim(kP).In Chapter 4, we study the tilting invariant properties of the crossed products. The following are main results in this chapter.Theorem 0.0.6 Let H be a finite-dimensional semisimple cosemisimple Hopf k-algebra, and let A be a finite-dimensional twisted H-module algebra andσan invertible cocycle. Then T∈mod A is a separating (resp. splitting) tilting A-module if and only if (A#σH)(?)A T is a separating (resp. splitting) tilting A#σH-module.