process

Duality for topological abelian group stacks and T -duality 247as follows.Consider an exact complexK W 0 ! A 1 ! K 1 ! K 0 ! A 0 ! 0such that K 1 is injective, and let K 2 C.S/ be the complex0 ! K 1 ! K 0 ! 0Then we define .ch.K// WD Y.K/.We must show that is well-defined. Indeed, if ch.K/ Š ch.L/, then (see 2.16)K Š L by an equivalence which induces the identity on the level of cohomology. Itfollows that Y.K/ D Y.L/.Since chW C 0 .S/ ! PIC [ .S/ is an equivalence, hence in particular surjectiveon the level of equivalence classes, the map is well-defined. Since every elementof Ext 2 Sh Ab S .A 1;A 0 / can be written as Y.K/ for a suitable complex K asabove we conclude that is surjective. If .ch.K// Š .ch.L//, then there existsM 2 C.S/ with given isomorphisms H i .M / Š A i together with quasi-isomorphismsK M ! L inducing the identity on cohomology. But this diagram induces anequivalence ch.K/ Š ch.M / Š ch.L/.2.5.11 We continue with a further description of BF for F 2 Sh Ab S, compare 1.2.8.Note that the sheaf of groups of automorphisms of BF is F , whereas the group ofobjects is trivial. Therefore we can represent BF as ch.F Œ1/, where F Œ1 is thecomplex 0 ! F ! 0 ! 0 concentrated in degree 1.2.5.12 Let P 2 PIC.S/ be a Picard stack and G H 1 .P /. Let us assume thatP D ch.K/ for a suitable complex K W 0 ! K 1 ! K 0 ! 0. We have a naturalinjection G,! ker.K 1 ! K 0 /. We consider the quotient xK 1 defined by the exactsequence 0 ! G ! K 1 ! xK 1 ! 0 and obtain a new Picard stack xP Š ch. xK/,where xK W 0 ! xK 1 ! K 0 ! 0. The following diagram represents a sequence ofmorphisms of Picard stacksBG ! P ! xP;0 G 00 K1 K 0 0 0 0 xK 1 K0 0.The Picard stack xP can be considered as a quotient of P by BG. We will employ thisconstruction in 6.4.5.