process

232 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.3.5 Given the structureBH 1 .A/ ! A ! H 0 .A/of A 2 PIC S, we ask for a similar description of the dual D.A/ 2 PIC S.We will see under the crucial conditionthat D.A/ fits intoExt 1 Sh Ab S .H 0 .A/; T/ Š Ext 2 Sh Ab S .H 1 .A/; T/ Š 0;BD.H 0 .A// ! D.A/ ! D.H 1 .A//:Without the condition the description of H 1 .D.A// is more complicated, and we referto (45) for more details.1.3.6 This discussion now leads to one of the main results of the present paper.Definition 1.5. We call the sheaf F 2 Sh Ab S admissible, iffExt 1 Sh Ab S .F; T/ Š Ext2 Sh Ab S .F; T/ Š 0:Theorem 1.6 (Pontrjagin duality for Picard stacks). If A 2 PIC S is a Picard stacksuch that H i .A/ and D.H i .A// are dualizable and admissible for i D 1; 0, then Ais dualizable.1.4 Admissible groups1.4.1 Pontrjagin duality for locally compact abelian groups implies that the sheaf Gassociated to a locally compact abelian group G is dualizable. In order to apply Theorem1.6 to Picard stacks A 2 PIC S whose sheaves H i .A/, i D 1; 0 are represented bylocally compact abelian groups we must know for a locally compact group G whetherthe sheaf G is admissible.Definition 1.7. We call a locally compact group G admissible, if the sheaf G is admissible.1.4.2 Admissibility of a locally compact group is a complicated property. Not everylocally compact group is admissible, e.g. a discrete group of the form ˚n2N Z=pZ forsome integer p is not admissible (see 4.29).1.4.3 Admissibility is a vanishing conditionExt 1 Sh Ab S .G; T/ Š Ext2 Sh Ab S .G; T/ Š 0:This condition depends on the site S. In the present paper we shall also consider the subsitesS lc-acyc S lc S of locally compact locally acyclic spaces and locally compactspaces.