process

Duality for topological abelian group stacks and T -duality 339Using the construction 2.5.12 we can form a quotient D.P / which fits into the sequenceof maps of Picard stacksBT jB ! D.P / ! D.P /;where H 0 .D.P // Š Z jB and H 1 .D.P // Š Z n jB. The fibre productRD.P / ¹1º Z jBdefines a gerbe R ! B with band Z n jB .6.4.6 By Lemma 6.15 there exists a unique isomorphism class yE ! B of aT n -bundle whose Z n jB -gerbe of Rn -reductions R y EZ n is isomorphic to R. We fix suchan isomorphism and obtain a canonical map of stacks can (see (54)) fitting into thediagramyH op zyH D.P /yE can R y EZ nŠRD.P /(69) B ¹1º Z jB .The gerbes yH op ! yE and z yH ! R are defined such that the squares become twocartesian(we omit to write the two-isomorphisms). In this way the Picard stack P 2Q E defines the second pair . yE; yH/ of the triple ‰.P/ D ..E; H /; . yE; yH/;u/ whoseconstruction has to be completed by providing u.Lemma 6.18. We have the equality Oc.P/ DOc.‰.P// in H 2 .BI Z n jB /.Proof. By the definition in 6.1.12 we have Oc.‰.P// D c. yE/. Furthermore, by (56)Ewe have c. yE/ D d.R y / D d.R/. By Lemma 5.14 we have .D.P// D D..P //,Z n jBwhere is the characteristic class (46), andD W Q E Š Ext 2 Sh Ab S=B .E; T jB / ! Ext 2 Sh Ab S=B .D.T jB /; D.E//is as in Lemma 5.14. The map Oc W Q E! H 2 .BI Z/ by its definition fits into the