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Duality for topological abelian group stacks and T -duality 3051. Ext 1 Sh Ab S .G ˝Z G; Z/ is torsion.2. Hom ShAb S .. p Z.F .G//=K/ ] ; Z/ vanishes.Let us start with 1. Let Z ! I be an injective resolution. We studyWe haveH 1 Hom ShAb S .G ˝Z G;I /:Hom ShAb S .G ˝Z G;I / Š Hom ShAb S .G; Hom Sh Ab S .G;I //:Let K D Hom ShAb S /. Since .G;I Hom ShAb S .G; Z/ Š Hom top-Ab.G; Z/ Š 0 by thecompactness of G the map d 0 W K 0 ! K 1 is injective. Let d 1 W K 1 ! K 2 be thesecond differential. NowH 1 Hom ShAb S .G ˝Z G;I / Š Hom ShAb S .G; ker.d 1 //=im.d 0 /;where d 0 W Hom Sh Ab S .G;K0 / ! Hom ShAb S .G; ker.d 1 // is induced by d 0 . Since d 0is injective we haveim.d 0 / D Hom Sh Ab S .G; im.d 0 //:Applying Hom ShAb S .G;:::/to the exact sequence0 ! K 0 ! ker.d 1 / ! ker.d 1 /=im.d 0 / ! 0and using ker.d 1 /=im.d 0 / Š Ext 1 Sh Ab S .G; Z/ we getIn particular,0 ! Hom ShAb S .G;K0 / d 0! Hom ShAb S .G; ker.d 1 // !! Hom ShAb S .G; Ext1 Sh Ab S .G; Z// ! Ext1 Sh Ab S .G;K0 / ! :Ext 1 Sh Ab S .G ˝Z G; Z/ Š Hom ShAb S .G; ker.d 1 //=Hom ShAb S .G;K0 / Hom ShAb S .G; Ext1 Sh Ab S .G; Z//:By Lemma 4.49 we know that Ext 1 Sh Ab S .G; Z/ Š yG. Since G is compact and profinite,the group yG is discrete and torsion. It follows by Lemma 4.48 thatHom ShAb S .G; Ext1 Sh Ab S .G; Z//is a torsion sheaf. A subsheaf of a torsion sheaf is again a torsion sheaf. This finishesthe argument for the first fact.We now show the fact 2.Note thatHom ShAb S .. p Z.F .G//=K/ ] ; Z/ Š Hom ShAb S .Z.F .G//=K] ; Z/: (39)