process

Duality for topological abelian group stacks and T -duality 275Š Hom ShZŒH -mod S .C ; coind.res.I ///Š Hom ShAb S .coinv.C /; res.I //Š RHom ShAb S .Lcoinv.Z/; W /:4.2.6 In general, the coinvariants functor coinv.:::/D˝ZŒH Z can be written interms of the tensor product in the sense of presheaves composed with a sheafification.Furthermore, C .H / is the sheafification of p C .H /. Using the fact that the tensorproduct of presheaves commutes with sheafificationU i D coinv.C i /Š .C i .H / ˝pZŒH Z/] (Definition 4.14)D .. p C i .H // ] ˝p. p Z/ ] / ]. p ZŒH / ]Š . p C i .H / ˝pp ZŒH p Z/ ] (22)Š . p ZŒH ˝pZp Z.H„Hƒ‚ …/ ˝pp Z/ ] ZŒH (Equation (20))i factorsŠ . p Z.H„Hƒ‚ …// ]i factorsD . p D i / ]withIn particular, we havep D i WD p Z.H„Hƒ‚ …/: (23)i factorsU i D Z.H i /: (24)4.2.7 Let G be a group. Then we can form the standard reduced bar complex for thegroup homology with integer coefficientsG !Z.G n / ! Z.G n 1 / !!Z ! 0:The abelian group Z.G n / (sitting in degree n) is freely generated by the underlying setof G n , and we write the generators in the form Œg 1 j :::jg n . The differential is givenbynXd D . 1/ i d i W Z.G n / ! Z.G n 1 /;whereiD08ˆ< Œg 2 j :::jg n ; i D 0;d i Œg 1 j :::jg n WD Œg 1 j :::jg i g iC1 j :::jg n ; 1 i n 1;ˆ:Œg 1 j :::jg n 1 ; i D n:The cohomology of this complex is the group homology H .GI Z/.