process

342 U. Bunke, T. Schick, M. Spitzweck, and A. Thomwhere x i WD p i .x/, y i WD q i .y/ for the projections onto the components p i W T n !T and q i W BZ n Š .BZ/ n ! BZ. Finally we use that can .y i / D x i , wherecanW T n ! BZ n is the canonical map (54) from a second copy of the torus to itsgerbe of R n -reductions (after identification of this gerbe with BZ n ).This finishes the construction of ‰ which started in 6.4.4.6.4.8 In [BRS, 2.11] we have seen that the group H 3 .BI Z/ acts on Triple.B/ preservingthe subsets Triple E .B/ Triple.B/ for every T n -bundle E ! B. We will recallthe description of the action in the proof of Lemma 6.20 below. By (65) it also acts onQ E .Lemma 6.20. The map ‰ is H 3 .BI Z/-equivariant.The proof requires some preparations.6.4.9 Note that we have a canonical isomorphism T Š D.Z/. In order to work withcanonical identifications we are going to use D.Z/ instead of T.The isomorphisms classes of gerbes with band D.Z/ over a space B 2 S areclassified byH 2 .BI D.Z/ jB / Š Ext 2 Sh Ab S=B .Z jB ;D.Z/ jB/: (71)The latter group also classifies Picard stacks P with fixed isomorphisms H 0 .P / Š Z jBand H 1 .P / Š D.Z jB /.Given P the gerbe G ! B can be reconstructed as a pull-backG P¹1º jBZ jB .If we want to stress the dependence of G on P we will write G.P /.Recall that an object in P.T/ as a stack over S consists of a map T ! B and anobject of P.T ! B/. In a similar manner we interpret morphisms.For example, the stack ¹1º jBis the space B.6.4.10 Let P 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB / be a Picard stack P with a fixed isomorphismsH 0 .P / Š Z jB and H 1 .P / Š D.Z jB /. Let W Ext PIC.S=B/ .Z jB ;D.Z/ jB / !Ext 2 Sh Ab S=B .Z jB ;D.Z/ jB/ be the characteristic class.Lemma 6.21. .D.P// Š .P/.Proof. First of all note that we have canonical isomorphismsH 0 .D.P // Š D.H 1 .P / Š D.D.Z jB // Š Z jBandH 1 .D.P // Š D.H 0 .P // Š D.Z jB / Š D.Z/ jB :