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Duality for topological abelian group stacks and T -duality 3195.1.11 If G is a locally compact group which together with its Pontrjagin dual belongsto S, then by 5.3 the sheaf G is dualizable. By Theorem 4.8 we know a large class oflocally compact groups which are admissible on S or at least after restriction to S lc orS lc-acyc .We get the following result.Theorem 5.12. Let G 0 ;G 1 2 S be two locally compact abelian groups. We assumethat their Pontrjagin duals belong to S, and that G 0 and DG 1 are admissible on S. IfP 2 PIC.S/ has H i .P / Š G i for i D 1; 0, then P is dualizable.Let us specialize to the case which is important for the application to T -duality.Note that a group of the form T n R m F for a finitely generated abelian group Fis admissible by Theorem 4.8. This class of groups is closed under forming Pontrjaginduals.Corollary 5.13. If P 2 PIC.S/ has H i .P / Š T n i R m i F i for some finitelygenerated abelian groups F i for i D 1; 0, then P is dualizable.For more general groups one may have to restrict to the sub-site S lc or even toS lc-acyc .5.2 Duality and classification5.2.1 Let A; B 2 Sh Ab S. By Lemma 2.20 we know that the isomorphism classesExt PIC.S/ .A; B/ of Picard stacks with H 0 .P / Š B and H 1 .P / Š A are classified bya characteristic class W Ext PIC.S/ .B; A/ ! Ext 2 Sh Ab S .B; A/: (46)Lemma 5.14. If A is dualizable and D.A/; B are admissible, then there is a naturalisomorphismD W Ext 2 Sh Ab S .B; A/ ! Ext 2 Sh Ab S .D.A/; D.B//such that.D.P// D D..P // for all P 2 Ext PIC.S/ .A; B/: (47)Proof. In order to define D we use the identificationsExt 2 Sh Ab S .B; A/ Š Hom D C .Sh Ab S/.B; AŒ2/;Ext 2 Sh Ab S .D.A/; D.B// Š Hom D C .Sh Ab S/.D.A/; D.B/Œ2/:We choose an injective resolution T ! I. For a complex of sheaves F we defineRD.F / WD Hom ShAb S .F; I/. The map T ! I induces a map D.F / ! RD.F /.