process

Duality for topological abelian group stacks and T -duality 321Let T ! I 0 ! I 1 ! ::: be the injective resolution I of T. We define J WDker.I 1 ! I 2 / and I WD I 0 . Then we have BT D ch.L/ with LW 0 ! I ! J ! 0with J in degree zero. Note that I is injective. Then by Lemma 2.17 we haveD.P / Š ch.H /; (48)where H WD 0 Hom ShAb S .K; L/. LetQ WD ker.Hom ShAb S .X; I / ˚ Hom Sh Ab S .Y; J / ! Hom Sh Ab S .X; J //:Then H is the complexH W 0 ! Hom ShAb S .Y; I / ! Q ! 0:There is a natural map D.B/ ! Hom ShAb S .Y; I / (induced by T ! I and Y ! B),and a projection Q ! D.A/ induced by A ! X and passage to cohomology. SinceB is admissible the complexH W 0 ! D.B/ ! Hom ShAb S .Y; I / d ! Q ! D.A/ ! 0is exact. Note that ker.d/ D H 1 Hom ShAb S .K; I / and coker.d/ D H 0 Hom ShAb S .K; I /.We get.D.P// .48)D .ch.H // (12) Lemma 2.19D Y.H/ D Y 0 .H/:Explicitly, in view of (11) the map Y 0 .H/ 2 Hom D C .Sh Ab S/.D.A/; D.B/Œ2/ is givenby the compositionY 0 .H/W D.A/ ! 1 ıH D.A/ ! D.B/Œ2;whereH D.A/ W 0 ! D.B/ ! Hom ShAb S .Y; I / ! Q ! 0with Q in degree 0, the map W H D.A/ ! D.A/ is the quasi-isomorphism inducedby the projection Q ! D.A/, and ı W H D.A/ ! D.B/Œ2 is the canonical projection.Since B is admissible we have R 1 Hom ShAb S .B; T/ Š 0 and hence a quasi-isomorphismH D.A/ ! H 0 fitting into the following larger diagram.D.A/H D.A/ W0 D.B/ Hom ShAb S .Y; I / Q 0H 0 D.A/ W0 Hom ShAb S .B; I / Hom Sh Ab S .B; J /˚Hom ShAb S .Y; I / Q0RD.K A /W 0 Hom ShAb S .B; I / Hom Sh Ab S .B; I 1 /˚ Hom2Sh Ab S .K A; I/Hom ShAb S .Y; I /:::