  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.
