process

Duality for topological abelian group stacks and T -duality 293Proof. We write H D colim n2N . n H/, where n H WD ker.H nŠ! H/. By Lemma 3.3we have H D colim n2N n H . Now colim n2N n H is the sheafification of the presheafp colim n2N n H of torsion groups and therefore a torsion sheaf.In general H is not a presheaf of torsion groups. Consider e.g. H WD ˚n2N Z=nZ.Then the element id 2 H .H / is not torsion.Definition 4.45. A sheaf F 2 Sh Ab S is called torsion-free if the group F .A/ is torsionfreefor all A 2 S.A sheaf F is torsion-free if and only if for all n 2 N the map F ! n F is injective.It suffices to test this for all primes.Lemma 4.46. If F 2 Sh Ab S is a torsion sheaf and E 2 Sh Ab S is torsion-free, thenHom ShAb S .F; E/ Š 0.Proof. We haveHom ShAb S .F; E/ Š Hom Sh Ab S .colim n2N. n F/;E/Š lim n2N Hom ShAb S . nF;E/:On the one hand, via the first entry multiplication by nŠ induces on Hom ShAb S . nF;E/the trivial map. On the other hand it induces an injection since E is torsion-free.Therefore Hom ShAb S . nF;E/ Š 0 for all n 2 N. This implies the assertion.4.5.5 If F 2 ShS and C 2 S, then we can form the sheaf R C .F / 2 ShS by theprescription R C .F /.A/ WD F.A C/ for all A 2 S (see 3.3.18). We will show thatR C preserves torsion sheaves provided C is compact.Lemma 4.47. If C 2 S is compact and H 2 Sh Ab S is a torsion sheaf, then R C .H / isa torsion sheaf.Proof. Given s 2 R C .H /.A/ D H.A C/ and a 2 A we must find n 2 N and aneighbourhood U of a such that .ns/ jU C D 0. Since C is compact and A 2 S iscompactly generated by assumption on S, the compactly generated topology (this is thetopology we use here) on the product AC coincides with the product topology ([Ste67,Theorem 4.3]). Since H is torsion there exists an open covering .W i D A i C i / i2Iof A C and a family of non-zero integers .n i / i2I such that .n i s/ jWi D 0.The set of subsets of C¹C i j i 2 I; a 2 A i ºforms an open covering of C . Using the compactness of C we choose a finite seti 1 ;:::;i r 2 I with a 2 A ik such that ¹C ik j k D 1;:::rº is still an open coveringof C . Then we define the open neighbourhood U of a 2 A by U WD T rkD1 A i k. Setn WD Q rkD1 n i k. Then we have .ns/ U C D 0.Lemma 4.48. Let H be a discrete torsion group and G 2 S be a compactly generated 10group. Then the sheaf Hom ShAb S .G;H/ is a torsion sheaf.10 i.e. generated by a compact subset