process

Duality for topological abelian group stacks and T -duality 303where the vertical maps are the isomorphisms given by Lemma 4.49, and surjectivityof yG ! yK follows from the fact that Pontrjagin duality preserves exact sequences.Next we apply Ext Sh Ab S .:::;Z/ to the sequenceand get the long exact sequence0 ! Q ! G ! C ! 0Ext i Sh Ab S .C ; Z/ ! Exti Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/ ! :Again we use that Ext i Sh Ab S .C ; Z/ Š 0 for i D 2; 3 by Lemma 4.61 for p>3, andby Lemma 4.17 if p 2¹2; 3º (in this case C is an at most finite product of copies ofZ=pZ by the two-three condition) in order to conclude thatExt i Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/is injective for i D 2; 3. Therefore the compositionExt i Sh Ab S .G; Z/ ! Exti Sh Ab S .Q; Z/ ! Exti Sh Ab S .G; Z/is injective for i D 2; 3. This is what we wanted to show.Lemma 4.63. Let G be a compact connected group which satisfies the two-three condition.Then the sheaves Ext i Sh Ab S lc-acyc.G; Z/ are torsion-free for i D 2; 3.Proof. The only point in the proof of Lemma 4.62 where we have used the conditionof local topological divisibility was the exactness of the sequence0 ! K ! G p ! G ! 0of sheaves on S. We show that this sequence is exact with this condition if we considerthe sheaves on S lc-acyc (by restriction). Let us start with the dual sequence0 ! yG p ! yG ! yK ! 0of discrete groups. It gives an exact sequence of sheaves0 ! yG p ! yG ! yK ! 0:We apply Hom ShAb S lc-acyc.:::;T/ and get, using Lemma 3.5, the long exact sequence0 ! K ! G p ! G ! Ext 1 Sh Ab S lc-acyc. yK; T/ ! :By Theorem 4.28 we have Ext 1 Sh Ab S lc-acyc. yK; T/ D 0. Now we can argue as in the proofof 4.62.Lemma 4.64. Let G be a profinite abelian group. Then the following assertions areequivalent.