process

Duality for topological abelian group stacks and T -duality 281We claim that this R I A ˝ P Š L i2I.i/ ˝ U.i/. Since U.i/ is flat for all i 2 IAthis is a sum of exact sequences and hence exact. Since J is injective we conclude thatHom ShAb S. R I A ˝P;J/is exact. So Hom HomCat .I op ;Sh Ab S/.:::; I Hom ShAb S .P; J // preservesexactness, hence I Hom ShAb S .P; J / is an injective object in Hom Cat.I op ; Sh Ab S/.In order to finish the proof it remains to show the claim. We expand the push-outdiagram by inserting the structure of P .Li!j.j / ˝ LA l!i U.l/ aWDid˝.i!j/ Ll!j.j / ˝ U.l/AbWDA .i!j/˝idL l!i.i/ ˝ U.l/AdWDA .l!i/˝id L l2I.l/ ˝ U.l/.AIt suffices to show that the lower right corner has the universal property. The lowerhorizontal map has a split s W L l2I.l/ ˝ U.l/ ! L A l!i.i/ ˝ U.l/ given by theAinclusion of summands induced by l 7! l ! id l. Two maps l ; r from the lower leftand upper right corner to some sheaf V which satisfy l ı b D r ı a must inducea unique map W L l2I.l/ ˝ U.l/ ! V such that ı d D A l and ı c D r .We have now other choice than to define WD l ı s, and this map has the requiredproperties as can be checked by an easy diagram chaise.Lemma 4.20. Every sheaf F 2 Hom Cat .I; Sh Ab S/ has an I -free resolution.Proof. It suffices to show that there exists a surjection P ! F from an I -free sheaf.We start with the surjection Z.F / ! F and note that Z.F /.i/ is flat for all i 2 I . Thenwe define the I -free sheaf P by P.i/ WD ˚l!i Z.F /.l/, and the surjection P ! Z.F /by .l ! i/ W Z.F /.l/ ! Z.F /.i/ on the summand of P.i/with index .l ! i/.Lemma 4.21. We have a natural isomorphism in D C .Sh Ab S/RHom ShAb S .colim.F /; H / Š R limRI Hom ShAb S .F; H /:Proof. Let F 2 Hom Cat .I; Sh Ab S/. By Lemma 4.20 we can choose an I -free resolutionP of F . Let H 2 Sh Ab S and H ! I be an injective resolution. Then we haveRHom ShAb S .colim F;H/ Š Hom Sh Ab S .colim F;I /:Since the category Sh Ab S is a Grothendieck abelian category [Tam94, Theorem I.3.2.1]the functor colim is exact [Tam94, Theorem 0.3.2.1]. Therefore colimP .F / ! colimFis a quasi-isomorphism. It follows thatHom ShAb S .colim F;I / Š Hom ShAb S .colim P .F /; I /: