The Hochschild Cohomology And Derived Center Of An Algebra

In this dissertation, we mainly study the properties of characteristic morphisms from Hochschild cohomology algebras of finite dimensional hereditary algebras over an algebraically closed field to the graded centers of their bounded derived categories. It skeletally consists of two aspects:On one hand, for the one-point extension algebra B= ( OKAM) of an algebra A over a field K, we consider the relations between the O-th component ZO(Db(B)) of Z*(Db(B)) and the center Z(B) of B, where the A-K-bimodule M is finitely generated as a left A-module. Under the condition of ZO(Db(A-mod))≌ Z(A), we prove that ZO(Db(B-mod))≌ Z(B).On the other hand, we prove that the characteristic morphisms of finite dimen-sional hereditary algebras over an algebraically closed field are monomorphic. For a finite dimensional elementary hereditary algebra A over a field K, we consider whether there exists an M ∈ Ob(D(A-mod)) such that the image of each basis ele-ment (?) ∈ HH1(A) under the composition morphism of the characteristic morphism X*:HH*(A)â†' Z*(D(A-mod)) with the evaluation morphism evM*:Z*(D(A-mod)) â†' ExtA*(M,M) isn’t equal to zero in ExtA*(M, M). At first, for a representative derivation d(?)β of C, we construct a module M corresponding to Q such that evM1 o xl((?)) ≠0 by the method of quiver. Next, we prove that A’s characteristic morphism is monomorphic by introducing a parameter. Finally, for each finite dimensional hereditary algebra over an algebraically closed field, we prove that its characteristic morphism is monomorphic by the fact that it is Morita equivalent to its adjoint basic algebra and other properties.