process

Duality for topological abelian group stacks and T -duality 271Proof. By [HM98, Theorem 7.57(i)] the group G has a splitting G Š H R n for somen 2 N 0 , where H has a compact open subgroup U . The quotient G=.U R n / Š H=Uis therefore discrete. Using Lemma 4.5 conclude that G is admissible over S lc if R nand H are so.Admissibility of R n follows from Theorem 4.33 (which we prove later) in conjunctionwith 4.5 and 4.4.Since D WD H=U is discrete, the exact sequence0 ! U ! H ! H=U ! 0has local sections and therefore induces an exact sequence of associated sheaves0 ! U ! H ! D ! 0by Lemma 3.4. If D is finitely generated, then it is admissible by Theorem 4.18,and hence admissible on S lc by Lemma 4.4. Otherwise it is admissible on S lc-acyc byTheorem 4.28.Therefore H is admissible on S lc (or S lc-acyc , respectively) by Lemma 4.5 if U isadmissible over S lc (or S lc-acyc , respectively). The compact group U fits into an exactsequence0 ! U 0 ! U ! P ! 0where P is profinite and U 0 is closed and connected. By assumption U , U 0 and P satisfythe two-three condition. The connected compact group U 0 is admissible on S lc (or onS lc without the assumption that it is locally topologically divisible), by Theorem 4.79,and the profinite P is admissible by Theorem 4.66, and hence admissible on S lc byLemma 4.4.Note that0 ! U 0 ! U ! P ! 0is exact by Lemma 4.39. Now it follows from Lemma 4.5 that U is admissible on S lc(or S lc-acyc , respectively).4.1.3 We conjecture that the assumption that G satisfies the two-three condition is onlytechnical and forced by our technique to prove admissibility of compact groups.In the remainder of this section, we will mainly be concerned with the proof of thestatements used in the proof of Theorem 4.8 above.4.1.4 Our proofs of admissibility for a sheaf F 2 Sh Ab S will usually be based on thefollowing argument.Lemma 4.9. Assume that F 2 Sh Ab S satisfies1. Ext i Sh Ab S .F; Z/ Š 0 for i D 2; 3;2. Ext i Sh Ab S .F; R/ Š 0 for i D 1; 2.Then F is admissible.