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302 U. Bunke, T. Schick, M. Spitzweck, and A. ThomWe consider the exact sequence0 ! K ! G p ! G ! C ! 0:The groups K and C are groups of Z=pZ-modules and therefore profinite by 4.58.We claim that the sequence of sheaves0 ! K ! G p ! G ! C ! 0is exact. We first show the claim in the case that G is profinite. Since C is profinite,by Lemma 4.39 and Lemma 3.4 we know that0 ! G=K ! G ! C ! 0is exact. Furthermore G=K is profinite so that the projection G ! G=K has localsections. This implies that G=K Š G=K, and hence the claim.We now discuss the case of a connected locally topologically divisible G. In thiscase C D¹1º. Since by assumption p W G ! G has local sections we can again useLemma 3.4 in order to conclude.We decompose this sequence into two short exact sequences0 ! K ! G ! Q ! 0; 0 ! Q ! G ! C ! 0:Now we apply the functor Ext Sh Ab S .:::;Z/ to the first sequence and study the associatedlong exact sequenceExt i 11Sh Ab S .G; Z/ ! ExtiSh Ab S .K; Z/ ! Exti Sh Ab S .Q; Z/ ! Exti Sh Ab S .G; Z/ ! :Note that Ext 2 Sh Ab S .K; Z/ Š 0 by Lemma 4.61 if p is odd, and by Lemma 4.17 if p D 2(the two-three condition ensures that K in this case is an at most finite product of copiesof Z=2Z.). This implies thatis injective.Next we show thatis injective. For this it suffices to see thatExt 3 Sh Ab S .Q; Z/ ! Ext3 Sh Ab S .G; Z/Ext 2 Sh Ab S .Q; Z/ ! Ext2 Sh Ab S .G; Z/Ext 1 Sh Ab S .G; Z/ ! Ext1 Sh Ab S .K; Z/is surjective. But this follows from the diagramExt 1 Sh Ab S .G; Z/ Ext 1 Sh Ab S .K; Z/ ;yGŠsurjective yKŠ