process

Duality for topological abelian group stacks and T -duality 263Let Œs 2 H i .I .AW//be represented by s 2 I i .AW/, and a 2 A. Then we mustfind a neighbourhood U A of a such that Œs jU W D 0, i.e. s jU W D dt for somet 2 I i 1 .U W/, where d W I i 1 ! I i is the boundary operator of the resolution.The ring structure of R induces on R n the structure of a sheaf of rings. In order todistinguish this sheaf of rings from the sheaf of groups R n we will use the notation C.Note that R n is in fact a sheaf of C-modules.The forgetful functor resW Sh C-mod S ! Sh Ab S fits into an adjoint pairind W Sh Ab S , Sh C-mod S W res;where ind is given by Sh Ab S 3 V ! V ˝Z C 2 Sh C-mod . Since C is a torsion-freesheaf it is flat. It follows that ind is exact and res preserves injectives.We can now choose an injective resolution R n ! J in Sh C-mod S and assume thatI D res.J /.Since the complex of sheaves I is exact we can find an open covering .V r / r2R ofA W such that s jVr D dt r for some t r 2 I i 1 .V r /. Since W is compact (locallycompact suffices), by [Ste67, Theorem 4.3]) the product topology on A W is thecompactly generated topology used for the product in S. Hence after refining thecovering .V r / we can assume that V r D A r W r for open subsets A r A andW r W for all r 2 R.We define R a WD ¹r 2 R j a 2 A r º. The family .W r / r2Ra is an open coveringof W . Since W is compact we can choose a finite set r 1 ;:::;r k 2 R a such thatW WD .W r1 ;:::;W rk / is still an open covering of W . The subset U WD T kj D1 A r jisan open neighbourhood of a 2 A.Since I i 1 is injective we can choose 5 extensions Qt r 2 I i 1 .A W/such that.Qt r / jVr D t r .We choose a partition of unity . 1 ;:::; v / subordinate to the finite covering W.We take advantage of the fact that I D res.J / which implies that we can multiplysections by continuous functions, and that d commutes with this multiplication. WedefinevXt WD k .Qt rk / jU W 2 I i 1 .U W/:kD1Note that k .s d Qt rk / jU W D 0. In fact we have k .s d Qt rk / j.U W/\Vrk D k .sdt rk / j.U W/\Vrk D 0. Furthermore, there is a neighbourhood Z of the complement of.U W/\ V rk in U W where k vanishes. Therefore the restrictions k .s d Qt rk /vanish on the open covering ¹Z; .U W/\ V rk º of U W , and this implies the5 Let U X be an open subset. Then we have an injection U ! X and hence an injection Z.U / !Z.X/. For an injective sheaf I we get a surjection Hom ShAb S.Z.X/; I / ! Hom ShAb S.Z.U /; I /. In othersymbols, I.X/ ! I.U/ is surjective.