process

Duality for topological abelian group stacks and T -duality 233For these sites the Ext-functor commutes with restriction (we verify this propertyin 3.4). Admissibility thus becomes a weaker condition on a smaller site. We will refineour notion of admissibility by saying that a group G is admissible on the site S lc (orsimilarly for S lc-acyc ), if the corresponding restrictions of the extension sheaves vanish,e.g.Ext 1 Sh Ab S .G; T/ jS lcŠ Ext 2 Sh Ab S .G; T/ jS lcŠ 0in the case S lc .1.4.4 Some locally compact abelian groups are admissible on the site S. This appliese.g. to finitely generated groups like Z; Z=nZ, but also to T n and R n .In the case of profinite groups G we need the technical assumption that it does nothave too much two-torsion and three-torsion.Definition 1.8 (4.6). We say that the topological abelian group G satisfies the two-threecondition, if1. it does not admit Q n2NZ=2Z as a sub-quotient,2. the multiplication by 3 on the component G 0 of the identity has finite cokernel.We can show that a profinite abelian group which satisfies the two-three conditionis also admissible. We conjecture that it is possible to remove the condition using othertechniques.A compact connected abelian group is divisible. Hence, if p 2 N is a prime, thenthe multiplication p W G ! G is set-theoretically surjective.Definition 1.9. We say that G is locally topologically p-divisible, if the map p has acontinuous local section. The group G is locally topologically divisible, if it is locallytopologically p-divisable for all primes p.For a general connected compact group which satisfies the two-three condition wecan only show that it is admissible on S lc-acyc . If it is locally topologically divisible,then it is admissible on the larger site S lc .We think that the two-tree condition and the restriction to a locally compact site isof technical nature. The condition of local compactness enters the proof since at oneplace we want to calculate the cohomology of the sheaf Z on the space A G using aKünneth formula. For this reason we want that A is locally compact.As our counterexample above shows, a general (infinitely generated) discrete groupis not admissible unless we restrict to the site S lc-acyc . We do not know if the restrictionto the site S lc-acyc is really necessary for a general compact connected groups.Using that the class of admissible groups is closed under finite products and extensionswe get the following general theorem.Theorem 1.10 (4.8). 1. If G is a locally compact abelian group which satisfies thetwo-three condition, then it is admissible over S lc-acyc .2. Assume that