process

286 U. Bunke, T. Schick, M. Spitzweck, and A. Thom(and a similar equation for H ). Since H ! G is injective, its Pontrjagin dual yG ! yHis surjective. Because of the classification of discrete finitely generated abelian groups,yG and yH both are homeomorphic to finite unions of finite dimensional tori. BecauseU is acyclic, every map U ! yH lifts to yG. Therefore yG.U / ! yH .U / is surjective.Corollary 4.27. We have R p lim Hom ShAb S lc-acyc.A; T/ Š 0 for all p 1.Theorem 4.28. Every discrete abelian group is admissible on S lc-acyc .Proof. This follows from (28) and Corollary 4.27.4.3.9 We now present an example which shows that not every discrete group D isadmissible on S or S lc , using Corollary 4.41 to be established later.Lemma 4.29. Let I be an infinite set, 1 6D n 2 N and D WD ˚I Z=nZ. Then D is notadmissible on S or S lc .Proof. We consider the sequence0 ! Z=nZ ! T n ! T ! 0 (31)which has no global section, and the product of I copies of it0 ! Y IZ=nZ ! Y IT ! Y IT ! 0: (32)This sequence of compact abelian groups does not have local sections. In fact, an opensubset of Q I T always contains a subset of the form U D Q I 0 T V , where I 0 I iscofinite and V Q I nI 0 T is open. A section s W U ! Q IT would consist of sectionsof the sequence (31) at the entries labeled with I 0 .By Lemma 3.4 the sequence of sheaves associated to (32) is not exact. In view ofCorollary 4.41 below, the group Q 4I Z=nZ Š˚I Z=nZ is not admissible on S or S lc .4.4 Admissibility of the groups R n and T nTheorem 4.30. The group T is admissible.Proof. Since T is compact,Assumption 2:of Lemma 4.9 follows from Proposition 4.16.It remains to show the first assumption of Lemma 4.9. As we will see this follows fromthe following result.Lemma 4.31. We have Ext i Sh Ab S .R; Z/ D 0 for i D 1; 2; 3.Let us assume this lemma for the moment. We apply Ext Sh Ab S .:::;Z/ to the exactsequence0 ! Z ! R ! T ! 0