Koszul Duality For DG Algebras And Its Applications

In this thesis,we mainly discuss the Koszul duality for DG algebras and its appli-cations.We introduce the Koszul duality theory,which is developed by Keller for DG categories,in the language of DG algebras.We apply Koszul duality to the study of DG algebras and A? algebras,and obtain the following results.Given a non-positive DG algebra A that has finite dimensional cohomology group and an elementary simple-minded collection in Dfd(A),we give a construction of silting objects in per(A);Given a finite positively graded quiver Q,we prove that all minimalA? path algebra structures on Q are derived equivalent to each other.The thesis is organised as follows.In chapter 1,we introduce the background of the research,main results and the structure of the thesis.In chapter 2,we first recall the definition and some basic properties of DG algebras,DG coalgebras.Then we introduce the Bar construction and the Cobar construction.In chapter 3,we introduce the derived categories of DG algebras and Keller's Koszul duality theory.In chapter 4,we introduce the definition of A? algebras,the minimal model theo-rem and the properties of A? module and the derived category of A? algebras.In chapter 5,we first recall some basic facts of silting objects,non-positive DG algebras,simple-minded collection,Nakayama functor and Auslander-Reiten formula.Then we study the positive strictly unital minimal A? algebras.In the end we introduce main results of this thesis.