process

332 U. Bunke, T. Schick, M. Spitzweck, and A. ThomNote that we can apply this lemma in our main example where H D R n andG D T n . In this case the diagram (55) is the equalityc.E/ D d.R E Z n jB/ (56)in H 2 .BI Z n /.6.3 Pairs and group stacks6.3.1 Let E be a principal T n -bundle over B, or equivalently by (53), an extensionE 2 Sh Ab S=B0 ! T n jB ! E ! Z jB ! 0 (57)of sheaves of abelian groups. Let Qc.E/ 2 Ext 1 Sh Ab S=B .Z jB I T n jB/ be the class of thisextension. Under the isomorphismExt 1 Sh Ab S=B .Z jB I T n jB / Š H 1 .BI T n / Š H 2 .BI Z n /it corresponds to the Chern class c.E/ of the principal T n -bundle introduced in 6.9.6.3.2 We let Q E D Ext PIC.S/ .E; T jB / (see Lemma 2.20 for the notation) denote theset of equivalence classes of Picard stacks P 2 PIC.B/ with isomorphismsBy Lemma 2.20 we have a bijectionH 0 .P / Š ! E; H 1 .P / Š ! T B :Ext 2 Sh Ab S=B .E; T jB / Š Q E:This bijection induces a group structure on Q E which we will use in the discussion oflong exact sequences below. We will not need a description of this group structure interms of the Picard stacks themselves.We apply Ext Sh Ab S=B .:::;T jB / to the sequence (57) and get the following segmentof a long exact sequenceExt 1 Sh Ab S=B .T n jB ; T jB / ˛! Ext 2 Sh Ab S=B .Z jB ; T jB / ! Q E! Ext 2 Sh Ab S=B .T n jB ; T jB / ˇ! Ext 3 Sh Ab S=B .Z jB ; T jB /: (58)The maps ˛; ˇ are given by the left Yoneda product with the classQc.E/ 2 Ext 1 Sh Ab S=B .Z jB I T n jB /: