TY - JOUR

T1 - Dualizing complexes, Morita equivalence and the derived Picard group of a ring

AU - Yekutieli, Amnon

N1 - Funding Information:
Received 7 August 1997; final revision 9 September 1998. 1991 Mathematics Subject Classification 16D90 (primary), 18E30, 18G15, 14F05 (secondary). This work was partially supported by the US–Israel Binational Science Foundation. J. London Math. Soc. (2) 60 (1999) 723–746
Funding Information:
Acknowledgements. Work on the paper started in conversations with K. Ajitabh, during a visit funded by a grant from the US–Israel Binational Science Foundation. I am grateful to M. Van den Bergh, V. Hinich, I. Reiten, J. Rickard and B. Keller for their helpful comments, to E. Kreines who contributed the appendix, and to the referee for finding a delicate mistake in the original formulation of Lemma 2.5.

PY - 1999/1/1

Y1 - 1999/1/1

N2 - Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(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 ⊗LA-:Db(Mod A)→.Db(ModB) 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 K0(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 t3 = s, where t,s ∈ DPic(A) are the classes of A* := Homk(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 tn+1 = sn-1 holds.

AB - Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(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 ⊗LA-:Db(Mod A)→.Db(ModB) 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 K0(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 t3 = s, where t,s ∈ DPic(A) are the classes of A* := Homk(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 tn+1 = sn-1 holds.

UR - http://www.scopus.com/inward/record.url?scp=0000392848&partnerID=8YFLogxK

U2 - 10.1112/S0024610799008108

DO - 10.1112/S0024610799008108

M3 - Article

AN - SCOPUS:0000392848

SN - 0024-6107

VL - 60

SP - 723

EP - 746

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -