process

Duality for topological abelian group stacks and T -duality 311Proof. Let 2 Hom Ab .V; W / and v 2 V . Then for all m 2 Z mult we havem k .v/ D ‰ m ..v// D .‰ m .v// D .m l v/ D m l .v/;i.e. each w 2 im./ W.l/ satisfies .m k m l /w D 0 for all m 2 Z mult . This set ofequations implies that w D 0, since W is torsion-free.Lemma 4.74. If V;W 2 Sh Ab S with W torsion-free and k; l 2 Z, k 6D l, thenHom ShZmult-modS.V .k/; W.l// D 0:We leave the easy proof of this sheaf version of Lemma 4.73 to the interested reader.Note that a subquotient of a sheaf of Z mult -modules of weight k also has weight k.4.6.9 By S lc S we denote the sub-site of locally compact objects. The restriction tolocally compact spaces becomes necessary because of the use of the Künneth (or basechange) formula below.The first page of the spectral sequence .E r ;d r / introduced in 4.2.12 is given byE q;p1D Ext p Sh Ab S .Z.Gq /; Z/:This sheaf is the sheafification of the presheafS 3 A 7! H p .A G q ; Z/ 2 Ab:In order to calculate this cohomology we use the Künneth formula [Brd97, II.18.2] andthat H .G; Z/ is torsion-free (4.67). We get for A 2 S lc thatH .A G q ; Z/ Š H .A; Z/ ˝Z .ƒ yG/˝Zqfor locally compact A. The sheafification of S lc 3 A ! H i .A; Z/ vanishes for i 1,and gives Z for i D 0. Since sheafification commutes with the tensor product with afixed group we get.E q;1 / jS lcŠ .ƒ yG/˝Zq : (42)4.6.10 We now consider the tautological action of the multiplicative semigroup Z multon G of weight 1. As before we write G.1/ for the group G with this action. Observethat this action is continuous. Applying the duality functor we get an action of Z multon the dual group yG which is also of weight 1. ThereforebG.1/ D yG.1/:The calculation of the cohomology (41) of the topological space G with Z-coefficientsis functorial in G. We conclude that H .G.1/; Z/ Š ƒ Z . yG.1// is a decomposableZ mult -module. By 3.5.5 the groupis of weight p.H p .G q .1/; Z/ Š Œ.ƒ yG.1//˝Zq p