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304 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1. Ext i Sh Ab S .G; Z/ is torsion-free for i D 2; 32. Ext i Sh Ab S .G; Z/ Š 0 for i D 2; 3.Proof. It is clear that 2. implies 1. Therefore let us show that Ext i Sh Ab S .G; Z/ Š 0 fori D 2; 3 under the assumption that we already know that it is torsion-free.We consider the double complex Hom ShAb S .U ;I / introduced in 4.5.7. By Lemma4.51 we know that the cohomology sheaves H i .Hom ShAb S .U ;I // of the associatedtotal complex are torsion sheaves.The spectral sequence .F r ;d r / considered in 4.5.14 calculates the associated gradedsheaves of a certain filtration of H i .Hom ShAb S .U ;I //. The left lower corner of itssecond page was already evaluated in Lemma 4.60.The term F 2;12Š Ext 2 Sh Ab S .G; Z/ survives to the limit of the spectral sequence. Onthe one hand by our assumption it is torsion-free. On the other hand by Lemma 4.51,case i D 2, it is a subsheaf of a torsion sheaf. It follows thatExt 2 Sh Ab S .G; Z/ Š 0:This settles the implication 1: ) 2: in case i D 2 of Lemma 4.64.We now claim that Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ is a torsion sheaf. Let us assume the claimand finish the proof of Lemma 4.64. Since F 3;12Š Ext 3 Sh Ab S .G; Z/ is torsion-free byassumption the differential d 2 must be trivial by Lemma 4.46. Since also F 0;33Š 0 thesheaf Ext 3 Sh Ab S .G; Z/ survives to the limit of the spectral sequence. By Lemma 4.51,case i D 3, it is a subsheaf of a torsion sheaf and therefore itself a torsion sheaf. Itfollows that Ext 3 Sh Ab S .G; Z/ is a torsion sheaf and a torsion-free sheaf at the same time,hence trivial. This is the assertion 1: ) 2: of Lemma 4.64 for i D 3.We now show the claim. We start with some general preparations. If F;H aretwo sheaves on some site, then one forms the presheaf S 3 A 7! .F ˝pZH /.A/ WDF .A/ ˝Z H.A/ 2 Ab. The sheaf F ˝Z H is by definition the sheafification ofF ˝pZ H . We can write the definition of the presheaf ƒ2 ZG in terms of the followingexact sequence of presheaves0 ! K ! p Z.F .G// ˛! G ˝pZ G ! ƒ2 Z G ! 0;where K is by definition the kernel. For W 2 S the map ˛W W p Z.G.W // ! G.W /˝ZG.W / is defined on generators by ˛W .x/ D x ˝ x, x 2 G.W /. Sheafification is anexact functor and thus gives0 ! . p Z.F .G//=K/ ] ! G ˝Z G ! .ƒ 2 Z G/] ! 0:We now apply the functor Hom ShAb S .:::;Z/ and consider the following segment of theassociated long exact sequence! Hom ShAb S .G ˝Z G; Z/ ! Hom ShAb S .. p Z.F .G//=K/ ] ; Z/! Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ ! Ext 1 Sh Ab S .G ˝Z G; Z/ !The following two facts imply the claim:(38)