**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:**

**Errat****a.**

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)".

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.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)