process

314 U. Bunke, T. Schick, M. Spitzweck, and A. ThomSince yG is torsion-free and hence a flat Z-module we haveH q . yG / Š H q .K ˝ZyG/ Š H q .K / ˝ZyG:Therefore H q . yG / D 0 for q 2. This implies the assertion of the lemma in thecase 2.Lemma 4.77. Let F 0 F 1 F k 1 F k D F be a filtered sheaf of Z mult -modules such that Gr l .F / has weight l, and such that F 0 D F 1 D 0. IfV F is atorsion-free sheaf of weight 1, then V D 0.Proof. Assume that V 6D 0. We show by induction (downwards) that F l \ V 6D 0 forall l 1. The case l D 1 gives the contradiction. Assume that l>1. We consider theexact sequence0 ! F l 1 \ V ! F l \ V ! Gr l .F /:First of all, by induction assumption, the sheaf V \ F l is non-trivial, and a as asubsheaf of V it is torsion-free of weight 1. Since Gr l .F / has weight l 6D 1, the mapV \ F l ! Gr l .F / can not be injective. Otherwise its image would be a torsion-freesheaf of two different weights l and 1, and this is impossible by Lemma 4.74. HenceF l 1 \ V 6D 0.4.6.14 We now show that Lemma 4.64 extends to connected compact groups.Lemma 4.78. Let G be a compact connected abelian group. Then the followingassertions are equivalent.1. Ext i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3.2. Ext i Sh Ab S lc.G; Z/ Š 0 for i D 2; 3.Proof. The non-trivial direction is that 1. implies 2. Therefore let us assume thatExt i Sh Ab S lc.G; Z/ is torsion-free for i D 2; 3. We now look at the spectral sequence.F r ;d r /. The left lower corner of its second page was calculated in 4.60. We seethat F 1;22D Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ has weight 2, while Ext 3 Sh Ab S lc.G; Z/ has weight 1.Since Ext 3 Sh Ab S lc.G; Z/ is torsion-free by assumption, it follows from Lemma 4.74 thatthe differential d 1;22W Ext 1 Sh Ab S lc..ƒ 2 Z G/] ; Z/ ! Ext 3 Sh Ab S lc.G; Z/ is trivial.We conclude that Ext 2 Sh Ab S lc.G; Z/ and Ext 3 Sh Ab S lc.G; Z/ survive to the limit of thespectral sequence. We see that Ext i Sh Ab S lc.G; Z/ are torsion-free sub-sheaves of weight1 of H iC1 for i D 2; 3. Using the structure of H iC1 given in 4.75 in conjunction withLemmas 4.76 and 4.77 we get Ext i Sh Ab S lc.G; Z/ D 0 for i D 2; 3.4.6.15 The combination of Lemma 4.62 (resp. Lemma 4.62) and Lemma 4.78 givesthe following result.Theorem 4.79. 1. A compact connected abelian group G which satisfies the two-threecondition is admissible on the site S lc-acyc .2. If G is in addition locally topologically divisible, then it is admissible on S lc .