Picard groups of derived categories

Abstract

We investigate the group Pic( D M ) of isomorphism classes of invertible objects in the derived category of O -modules for a commutative unital ringed Grothendieck topos ( E, O) with enough points. When the ring O p has connected prime ideal spectrum for all points p of E we show that Pic( D M ) is naturally isomorphic to the Cartesian product of the Picard group of O -modules and the additive group of continuous functions from the space of isomorphism classes of points of E to the integers Z . Also, for a commutative unital ring R, the group Pic( D R) is isomorphic to the Cartesian product of Pic( R) and the additive group of continuous functions from spec R to the integers Z .