process

Duality for topological abelian group stacks and T -duality 297Lemma 4.55. Every Z mult -module of weight 2 or of weight 3 is a weight 2-3-extension.The class of weight 2-3-extensions is closed under extensions.Proof. This is a simple diagram chase, using the fact that for a Z mult -module W ofweight k, ‰ v x D v k x for all k 2 Z mult and x 2 W .Lemma 4.56. Let n 2 N and V be a torsion-free Z-module. For i 2¹0; 2; 3º thecohomology H i ..Z=pZ/ n .1/I V/ is a Z=pZ-module whose weight is given by thefollowing table.Moreover, H 1 ..Z=pZ/ n I V/Š 0.i 0 2 3 4weight 0 1 2 2-3Proof. We first calculate H i ..Z=pZ/ n I Z/ using the Künneth formula and inductionby n 1. The start is Lemma 4.53. Let us assume the assertion for products with lessthan n factors. The cases i D 0; 1; 2 are straightforward. We further getH 3 ..Z=pZ/ n .1/I Z/ Š H 2 .Z=pZ.1/I Z/ Z H 2 ..Z=pZ/ n1 .1/I Z/:By Lemma 4.52 the Z -product of two modules of weight 1 is of weight 2. Similarly,we have an exact sequence0 !4MH j .Z=pZ n 1 .1/I Z/ ˝ H 4 j .Z=pZI Z/ ! H 4 ..Z=pZ/ n .1/I Z/j D0!3MH j .Z=pZ.1/I Z/ Z H 5 j ..Z=pZ/ n 1 .1/I Z/ ! 0:j D2By Lemma 4.55 and induction we conclude that H 4 ..Z=pZ/ n .1/I Z/ is a weight 2-3-extension.We can calculate the cohomology H i .GI V/of a group G in a trivial G-moduleby the standard complex C .GI V/ WD C.Map.G ;V//. If G is finite, then we haveC .GI V/Š C .GI Z/ ˝Z V .IfV is torsion-free, it is a flat Z-module and thereforeH i .C .GI Z/ ˝Z V/ Š H i .C .GI Z// ˝Z V . Applying this to G D .Z=pZ/ n weget the assertion from the special case Z D V .4.5.11 An abelian group G is a Z-module. Let p 2 Z be a prime. If pg D 0 for everyg 2 G, then we say that G is a Z=pZ-module.Definition 4.57. A sheaf F 2 Sh Ab S is a sheaf of Z=pZ-modules if F .A/ is a Z=pZmodulefor all A 2 S.