EXCEPTIONAL SEQUENCES AND DERIVED CATEGORIES

Let k be an algebraically closed field of characteristic 0. We denote by C the abelian k-category which has enough projectives (or enough injectives), and by Db(C) the bounded derived category of C. A complex E • ∈ Db(C) is called exceptional if RHom(E•;E•) ∼ = k, and a sequence = ( · · ·;E•i;E•i+1; · · · ) of exceptional complexes is called an exceptional sequence if RHom(E•i;E•j) = 0 for all i> j. Let C be a category modA of finitely generated modules of a hereditary k-algebra A, or a category coh(X) of coherent sheaves of a weighted projective line X over k. In this case, for any exceptional sequence , the length of is at most the rank n of Grothendieck group of C. An exceptional sequence is called complete if the length of is equal to n. It was shown by W. Crawley-Boevey (in the case of C = modA) and by H. Meltzer (in the case of C = coh(X)) that the braid group Bn on n strings acts transitively on the set of complete exceptional sequences. It was also shown that the complete exceptional sequence generates Db(C) as a triangulated category. In this lecture, I will talk about the case of C = modR (where R is a 1-dimensional N-graded Gorenstein ring of finite Cohen-Macaulay represen-tation type)