process

Duality for topological abelian group stacks and T -duality 245Lemma 2.18. Let K; L 2 C.S/. Then we have an isomorphismfor i D 1; 0.H i .HOM PIC.S/ .ch.K/; ch.L/// Š R i Hom ShAb S .K; L/Proof. First observe that by the discussion above the left hand side, and by the definitionof R Hom the right side both only depend on the quasi-isomorphism type of the complexL of length 2. Without loss of generality we can therefore assume that L 1 is injective.We now choose an injective resolution I W 0 ! L 1 ! I 0 ! I 1 ::: of L startingwith the choice of an embedding L 0 ! I 0 . Then we have Hom.K; I / Š RHom.K; L/.We now observe that H i Hom.K; I / Š H i Hom.K; L/ for i D 0; 1. While the casei D 1 is obvious, for i D 0 observe that a 0-cycle in Hom.K; I / is a morphism ofcomplexes and necessarily factors over L ! I . We thus have for i 2¹ 1; 0ºR i Hom.K; L/ Š H i Hom.K; L/ Š H i .Hom.ch.K/; ch.L///:2.5.9 For A; B 2 Sh Ab S we have canonical isomorphismsExt 2 Sh Ab S .B; A/ Š R0 Hom ShAb S.B; AŒ2/ Š Hom D.ShAb S/.B; AŒ2/:In the following we recall two eventually equivalent ways how an exact complexrepresents an elementK W 0 ! A ! X ! Y ! B ! 0Y.K/ 2 Hom D.ShAb S/.B; AŒ2/ Š Ext 2 Sh Ab S .B; A/(the letter Y stands for Yoneda who investigated this construction first). Let K A be thecomplexK A W 0 ! X ! Y ! B ! 0;where B sits in degree 0. The obvious inclusion ˛ W AŒ2 ! K A induced by A ! Xis a quasi-isomorphism. Furthermore, we have a canonical map ˇ W B ! K A . Theelement Y.K/ 2 Hom D.ShAb S/.B; AŒ2/ is by definition the compositionWe can also consider the complex K B given byY.K/W B ˇ! K A˛ 1! AŒ2: (10)0 ! A ! X ! Y ! 0where A is in degree 2. The projection Y ! B induces a quasi-isomorphism W K B ! B. We furthermore have a canonical map ı W K B ! AŒ2. We consider thecomposition Y 0 .K/ 2 Hom D.ShAb S/.B; AŒ2/Y 0 .K/W B 1 ! K Bı! AŒ2: (11)