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

Two rings A and B are said to be derived Morita equivalent if the derived categories D b (Mod A ) and D b (Mod B ) are equivalent. If A and B are derived Morita equivalent algebras over a field k , then there is a complex of bimodules T such that the functor T ⊗ L A −:D b (Mod A ) → D b (Mod B ) 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 ). This group acts naturally on the Grothendieck group K o ( A ). It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic( A ) in these cases. Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic( A ) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes. Finally finite k ‐algebras are considered. For the algebra A of upper triangular 2×2 matrices over k , it is proved that t 3 = s , where t , s ∈ DPic( A ) are the classes of A *:= Hom k ( A , k ) and A [1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n × n matrices, and it is shown that the relation t n +1 = s n −1 holds.