Title: Dualizing complexes, Morita equivalence and the derived Picard group of a ring
Authors: Amnon Yekutieli

Publication status: J. London Math. Soc. 60 (1999) 723-746

Abstract:

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. 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 functor T \otimes^{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_{0}(A).

We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables us to compute DPic(A) in these cases.

Assume A is noetherian. Dualizing complexes over A were defined in [Ye]. These are complexes of bimodules which generalize the commutative definition of [RD]. We prove that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. We use this classification to deduce properties of rigid dualizing complexes, as defined by Van den Bergh.

Finally we consider finite k-algebras. For the algebra A of upper triangular 2 \times 2 matrices over k, we prove that t^{3} = s, where t, s in DPic(A) are the classes of A^{*} := Hom}_{k}(A, k) and A[1] respectively. In the Appendix, by Elena Kreines, the algebra of n \times n matrices is considered.

Electronic version:

Errata.

1. In Theorem 2.6 one should replace "with A commutative" by "with A commutative, and such that Spec A has finitely many connected components (e.g. A is noetherian)".

2. The proof of Theorem 2.6 (its last paragraph) is correct only when the ring A is noetherian. The correct proof of the general case (after reducing to the case when Spec A is connected), is in Theorem 1.9 of my paper "Derived Equivalences Between Associative Deformations", Journal of Pure and Applied Algebra 214 (2010) 1469-1476.

3. In Prop 3.5, again one must assume that Spec A has finitely many connected components (e.g. A is noetherian)".

(updated 29 Jan 2015)