Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Abstract

Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the derived tensor product with T is an equivalence. The complex T is called a tilting complex. When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. So we can compute DPic(A) in these cases. Assume A is noetherian. Dualizing complexes over A were defined by the author some years ago. These are complexes of bimodules which generalize the commutative definition of Grothendieck. We prove that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. Finally we consider finite k-algebras. For the algebra A of upper triangular 2 x 2 matrices over k, we prove that t^{3} = s, where t, s are the classes in DPic(A) of Hom_{k}(A, k) and A[1] respectively. In the Appendix by Elena Kreines this result is generalized to upper triangular n x n matrices, and it is shown that the relation t^{n + 1} = s^{n - 1} holds.