process

Duality for topological abelian group stacks and T -duality 2693.5.6 It makes sense to speak of a sheaf or presheaf of Z mult -modules on the site S.We let Sh Zmult -modS and Pr Zmult -mod S denote the corresponding abelian categories ofsheaves and presheaves.Definition 3.43. Let V 2 Sh Zmult -modS (or V 2 Pr Zmult -mod S) and k 2 Z. We say thatV is of weight k if the map V mk m! V vanishes for all m 2 Zmult .We define the functors .k/W Sh Ab S ! Sh Zmult -modS, F W Sh Zmult -modS ! Sh Ab S,and W k W Sh Zmult -modS ! Sh Ab S (and their presheaf versions) object-wise. We alsohave a pair of adjoint functors.k/ W Sh Ab S , Sh Zmult -modS W W k(and the corresponding presheaf version). A sheaf V 2 Sh Zmult -modS of Z mult -moduleshas weight k 2 Z if V Š F .V /.k/ Š W k .V /.k/.4 Admissibility of sheaves represented by topological abeliangroups4.1 Admissible sheaves and groups4.1.1 The main topic of the present paper is a duality theory for abelian group stacks(Picard stacks, see 2.4) on the site S. A Picard stack P 2 PIC.S/ gives rise to thesheaf of objects H 0 .P / and the sheaf of automorphisms of the neutral object H 1 .P /.These are sheaves of abelian groups on S.We will define the notion of a dual Picard stack D.P / (see 5.4). With the intentionto generalize the Pontrjagin duality for locally compact abelian groups to group stackswe study the question under which conditions the natural map P ! D.D.P // is anisomorphism. In Theorem 5.11 we see that this is the case if the sheaves are dualizable(see 5.2) and admissible (see 4.1). Dualizability is a sheaf-theoretic generalization ofthe classical Pontrjagin duality and is satisfied e.g. for the sheaves G for locally compactgroups G 2 S (see 5.3). Admissibility is more exotic and will be defined below (4.1).One of the main results of the present paper asserts that the sheaves G are admissiblefor a large (but not exhaustive) class of locally compact groups G 2 S, and for an evenlarger class if S is replaced by S lc-acyc .The main tool in our computations is a certain double complex, which will beintroduced in Section 4.2. Similar techniques were used by Lawrence Breen in [Bre69],[Bre76], [Bre78].Definition 4.1. We call a sheaf of groups F admissible if Ext i Sh Ab S .F; T/ Š 0 fori D 1; 2. A topological abelian group G 2 S is called admissible if G is an admissiblesheaf.We will also consider the sub-sites S lc-acyc S lc S of locally compact spaces.