The Classification Of Derived-discrete Algebras Over R

In this thesis,we classify derived-discrete algebras over real numbers.Our ap-proach is based on the well-known classification on derived-discrete algebras over al-gebraically closed field.First,we prove that finite field extension is compatible with derived-discrete algebras,and in particular,with piecewise hereditary algebras of Dynkin types.Second,we construct the quiver presentation of the complexification of a real al-gebra given by a modulated quiver and consider the relation between these two quivers.Finally,making use of the above relation,we describe the real algebra whose com-plexification admits a quiver presentation which is gentle one-cycle and without clock condition.In chapter 3,let k be an infinite field and ?:A B a k-algebra extension.We prove that if ? is split,then B is k-derived-discrete implies that so is A;if ? is separable with the right A-module BA projective,then A is k-derived-discrete implies that so is B.We also have a similar statement for piecewise hereditary algebras.In the case of finite separable field extension K/k,we prove that A is piecewise hereditary of Dynkin type if and only if so is A(?)k K.In chapter 4,we consider the complexification of a real algebra with modulated quiver presentation T(Q,M)/I.We obtain the presentation(?,J)of its complexifica-tion.The quiver ? is constructed directly,while the admissible ideal J is not easy to describe.For ?,we study its combinatorial property related to Q.For J,we define the vertex-unique-modulated quiver,and describle J in this situation.We also give some symmetric properties of J by considering a special automorphism ? of ?.In particular,we prove that for a path p,p belongs to J if and only if so is ?(p).In chapter 5,we give a definition of an algebra which is gentle one-cycle and without clock condition as follow.Let T(Q,M)/I be its quiver presentation.First,it should have the same modulation on vertices,so we can view paths as elements in the algebra.Second,I can be generated by paths with length two.Finally,(Q,I)is gentle one-cycle and without clock condition.Using the conclusions in chapter 4,we prove that a real algebra is gentle one-cycle and without clock condition if and only if so are the connected components of its complexification.The difficult of the proof is that the ideal J is not unique.We solve it by considering all the possible forms of J.Combining with the conclusions in chapter 3,we give a classification of derived-discrete algebras over R.