DUALIZING COMPLEXES, MORITA EQUIVALENCE AND THE DERIVED PICARD GROUP OF A RING

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. 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[otimes ]^L_A−[ratio ]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 t³ = 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 tⁿ⁺¹ = sⁿ⁻¹ holds.