Some Studies On Dominant Dimensions,Derived Double Centralizer Property And Derived Picard Group Actions

Let A be a finite dimensional algebra over a field k.In Chapter 2,we study dominant dimension from the point of view of the idempotent ideals.The canonical A-bimodule V:=HomA(DA,A)was studied in[19,36,20],where D=Homk(-,k).Under certain condition,we give a new understanding that DA(?)A V is isomorphic to an idempotent ideal tensor product itself as A-bimodules and show that the double dual functor is defined by DA(?)A V.We also give a characterization of dominant dimension for A in terms of vanishing of certain extension groups over A,which just uses A and DA as ingredients and generalizes the corresponding characterizations given in[19,20].Moreover,we show that A is a Motita algebra if and only if there is a natural isomorphism υυυ-1≌υ on A-mod where υ is the Nakayama functor.In representation theory,the double centralizer property is an important property for a module(bimodule).It plays a fundamental role in many theories.In Chapter 3,we extend this property to complexes in derived categories of algebras,under the name derived double centralizer property.Let A be a finite dimensional K-algebra with K a algebraically closed field.Characterizations for complexes of finitely generated Amodules with the derived double centralizer property and for tilting complexes in the bounded derived category with the endomorphism algebras hereditary are given.In particular,when A is a upper triangular matrix algebra,all complexes of A-Abimodules with this property are classified.Derived Picard group is a derived invariant for an algebra which is formed by all the isomorphism classes of invertible complexes of bimodules.In Chapter 4,we study the derived Picard group action on the Hochschild cohomology and compute the case for symmetric Nakayama algebras.