process

Duality for topological abelian group stacks and T -duality 273Lemma 4.11. Every sheaf F 2 Sh ZŒH -mod S is a quotient of a flat sheaf.Proof. Indeed, the counits of the adjoint pairs .Z.:::/;F / and .ind; res/ induce asurjection ind.Z.F .res.F //// ! F . Explicitly it is given by the composition of thesum and action map (omitting to write some forgetful functors)ZŒH ˝Z ZŒF ! ZŒH ˝ F ! F:Since ZŒH is a sheaf of unital rings this action is surjective. Moreover, for A 2Sh ZŒH -mod S we haveA ˝ZŒH .ZŒH ˝Z ZŒF / Š A ˝Z ZŒF :Since ZŒF is a torsion-free sheaf of abelian groups the operation A ! A ˝ZŒH .ZŒH ˝Z ZŒF / preserves exact sequences in Sh ZŒH -mod S. Therefore ZŒH ˝Z ZŒF is a flat sheaf of ZŒH -modules.Lemma 4.12. The class of flat ZŒH -modules is coinv-acyclic.Proof. Let F be an exact lower bounded homological complex of flat ZŒH -modules.We choose a flat resolution P ! Z in Sh ZŒH -mod S which exists by Lemma 4.11 6 .Since F consists of flat modules the induced mapF ˝ZŒH P ! F ˝ZŒH Z D coinv.F /is a quasi-isomorphism. Since P consists of flat modules, tensoring by P commuteswith taking cohomology, so that we haveH .F ˝ZŒH P / Š H .H .F / ˝ZŒH P / Š 0:Therefore coinv.:::/ maps acyclic complexes of flat ZŒH -modules to acyclic complexesof Z-modules.Corollary 4.13. We can calculate L coinv.Z/ using a flat resolution.4.2.4 We will actually work with a very special flat resolution of Z. The bar constructionon the sheaf of groups H gives a sheaf H of simplicial sets with an action of H .We let C .H / WD C.Z.H // be the sheaf of homological chain complexes associatedto the sheaf of simplicial groups Z.H /. The H -action on H induces an H -action onC .H / and therefore the structure of a sheaf of ZŒH -modules. In order to understandthe structure of C .H / we first consider the presheaf version p C .H / WD C. p Z.H //.In fact, we can writeH i Š H H„Hƒ‚ …;i factorsas a sheaf of H -sets, and thereforep C i .H / Š p ZŒH ˝pZp Z.H„Hƒ‚ …/ (20)i factors6 Note that P is a homological complex, i.e. the differentials have degree 1.