process

Duality for topological abelian group stacks and T -duality 2894.4.6 Finally, since F 0;33Š 0, we see that d 2 W F 1;22! F 3;12must be an isomorphism,i.e.d 2 W Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/ ! Ext 3 Sh Ab S .R; Z/:We will finish the proof of Lemma 4.31 for i D 3 by showing that d 2 D 0.4.4.7 We consider the natural action of Z mult on R and hence on R. This turns R intoa sheaf of Z mult -modules of weight 1 (see 3.5.1). It follows that Hom ShAb S / is a .R;Icomplex of sheaves of Z mult -modules of weight 1. Finally we see that Ext i Sh Ab S .R; Z/are sheaves of Z mult -modules of weight 1 for i 0.Now observe (see 3.5.5) that ƒ 2 Z R is a presheaf of Z mult-modules of weight 2. Hence.ƒ 2 Z R/] and thus Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/ are sheaves of Z mult -modules of weight 2.Since R ! U .R/ DW U is a functor we get an action of Z mult on U and henceon the double complex Hom ShAb S .U ;I /. This implies that the differentials of theassociated spectral sequences commute with the Z mult -actions (see 4.6.11 for moredetails). This in particular applies to d 2 . The equality d 2 D 0 now directly followsfrom the following lemma.Lemma 4.32. Let V;W 2 Sh Ab S be sheaves of Z mult -modules of weights k 6D l.Assume that W has the structure of a sheaf of Q-vector spaces. If d 2 Hom ShAb S.V; W /is Z mult -equivariant, then d D 0.Proof. Let ˛ W Z mult ! End ShAb S.V / and ˇ W Z mult ! End ShAb S.W / denote the actions.Then we have d ı ˛.q/ ˇ.q/ ı d D 0 for all q 2 Z mult . We consider q WD 2.Then .2 k 2 l / ı d D 0. Since W is a sheaf of Q-vector spaces and .2 k 2 l / 6D 0 thisimplies that d D 0.Theorem 4.33. The group R is admissible.Proof. The outer terms of the exact sequence0 ! Z ! R ! T ! 0are admissible by Theorem 4.30 and Theorem 4.18. We now apply Lemma 4.5, stabilityof admissibility under extensions.4.5 Profinite groupsDefinition 4.34. A topological group G is called profinite if there exists a small leftfiltered8 poset 9 I and a system F 2 Ab I such that1. for all i 2 I the group F.i/is finite,2. for all i j the morphism F.i/ ! F.j/is a surjection,3. there exists an isomorphism G Š lim i2I F.i/as topological groups.8 i.e. for every pair i;k 2 I there exists j 2 I with j i and j k9 A poset is considered here as a category.