Unbounded Ladders Induced By Gorenstein Algebras And Objective Functors

The“gluing functors”have a venerable history in algebraic geometry.It was an observation of A.Grothendieck for a sheaf on theétale site.The axiom of recollements of triangulated categories was given by A.A.Beilinson,J.N.Bernstein and P.Deligne.Now it has become a powerful tool in representation theory and algebraic geometry.In order to get more properties for triangulated categories,A.A.Beilinson,V.A.Ginsburg and V.V.Schechtman introduced ladder.Ladder is a generalization of recollement.Recollement is precisely a ladder of height one,and ladder of height no less than two provides more information.A fundamental question is the existence of non-splitting ladders of given height.In particular,when does there exist an unbounded ladder?The first part of this thesis is the construction of unbounded ladders.They are unbounded ladders induced by the derived categories of Gorenstein triangular matrix algebras,and the singularity categories of Gorenstein algebras.More precisely:We get a criterion for a ladder.Using Brown representability theorem,which is very useful in triangulated category theory,we get a trichotomy of adjoint functors between compactly generated triangulated categories.Using this trichotomy we give an unbounded ladder induced by the derived categories of Gorenstein triangular matrix algebras.We give a criterion for a periodic ladder.We use Serre functors to extend a left recollement of triangulated categories to an unbounded ladder of period one.As an application,we get an unbounded ladder of period one given by the singularity categories of Gorenstein algebras.We study the spilitting recollement,and prove that the recollement of Calabi-Yau categories is always spilitting.On the other hand,a full?resp.faithful or dense?functor has a fundamental meaning in category theory.C.M.Ringel and P.Zhang introduced objective functors.It has been shown that objectivity is also a fundamental property of a functor.Inspired by the fact that in an adjoint pair between triangulated categories,one functor is triangulated if and only if so is the other one,we study the objectivity of triangulated functors in adjoint pairs in the second part of the thesis.We give some sufficient conditions of when a triangulated functor in an adjoint pair is objective.In particular,a triangulated functor is objective if the adjoint functor?no matter left adjoint or right adjoint?is full or dense.Using derived categories we give an example to show that the adjoint functor of an objective functor may not be objective.This example also shows that a dense functor may not be objective.In the last part of the thesis,using combinatorial methods and representational tools,we give Ringel-Schmidmeier-Simson equivalences between separated monomorphism categories and separated epimorphism categories when the quiver is of type A₄with nonlinear orientation explicitly.