Differential Graded Categories And The Standard Derived Equivalence Conjecture

This thesis for Ph.D degree is on dg categories and their applications.It mainly consists of the following two parts.1.In Chapter 2,for any additive category A and a full additive subcategory χ,we recall the notion of lower extension group provided an χ-resolution or χcoresolution exists.The study of this notion is motivated by a canonical triangle functor Φ:Kb(A)/Kb(χ)Kb(A/[χ])which appears in the construction of a realization functor given by a bounded t-structure on an algebraic triangulated category.We relate the lower extension groups with three different quotient categories:the additive quotient A/[χ],the Verdier quotient Kb(A)/Kb(χ)and the dg quotient A/χ.Based on these,the main result of Chapter 2 proves that the canonical functor Φ is a triangle equivalence provided certain lower extension groups vanish.We also obtain a criterion which asserts that the canonical functorΦ is faithful if and only if it is a triangle equivalence.2.In Chapter 3,we use the theory of dg categories to study Rickard’s standard derived equivalence conjecture:any derived equivalence between module categories over finite dimensional algebras is necessarily standard,i.e.it is isomorphic to the derived tensor functor given by a two-sided tilting complex.We prove the following theorem:if a finite dimensional algebra A is derived equivalent to a smooth projective scheme,then Rickard’s standard derived equivalence conjecture holds for A.During the proof of the above theorem,we obtain the following three results.(1)between the derived categories of two module categories,dg liftable functors coincide with standard functors;(2)any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a dg liftable derived equivalence;Combining(1),(2)and Orlov’s well-known theorem on triangle autoequivalences over smooth projective schemes,we obtain the first proof of the above theorem.(3)the derived category of coherent sheaves on a certain class of projective schemes is triangle-objective,that is,any triangle autoequivalence on it,which preserves the isomorphism classes of all objects,is necessarily isomorphic to the identity functor.Using(3)and Lunts-Orlov’s result on the existence of ample sequences,we loose the smoothness condition and obtain a partial generalization of the above theroem.As an application,we obtain a class of algebras of infinite global dimension over an algebraically closed field and Rickard’s standard derived equivalence conjecture holds for algebras in this class.