process

Duality for topological abelian group stacks and T -duality 267Corollary 3.37. 1. The restriction g W ShS ! ShS lc-acyc is exact.2. For F 2 Sh Ab S lc-acyc and G 2 Sh Ab S we haveandg RHom ShAb S .g F;G/ Š RHom ShAb S lc-acyc.F; g G/g Ext k Sh Ab S .g F;G/ Š Ext k Sh Ab S lc-acyc.F; g G/for all k 0.Corollary 3.38. Let G; H 2 S lc-acyc be locally acyclic abelian groups. We havefor all k 0.g Ext k Sh Ab S .G;H/ Š Extk Sh Ab S lc-acyc.G;H/Note that the product Q NT is compact but not locally acyclic.3.5 Z mult -modules3.5.1 We consider the multiplicative semigroup Z mult of non-zero integers. Everyabelian group G (written multiplicatively) has a tautological action of Z mult by homomorphismsgiven byZ mult G ! G; .n; g/ 7! g n :Definition 3.39. When we consider G with this action we write G.1/.The semigroup Z mult will therefore act on all objects naturally constructed from anabelian group G. We will in particular use the action of Z mult on the group-homologyand group-cohomology of G.If G is a topological group, then Z mult acts by continuous maps. It therefore alsoacts on the cohomology of G as a topological space. Also this action will play a rolelater.The remainder of the present subsection sets up some language related to Z mult -actions.Definition 3.40. A Z mult -module is an abelian group with an action of Z mult by homomorphisms.We will write this action as Z mult G 3 .m; g/ 7! ‰ m .g/ 2 G. Thus in the caseof G.1/ we have ‰ m .g/ D g m .3.5.2 We let Z mult -mod denote the category of Z mult -modules. An equivalent descriptionof this category is as the category of modules under the commutative semigroupring ZŒZ mult . The category Z mult -mod is an abelian tensor category.We have an exact inclusion of categoriesAb ,! .Z mult -mod/;G 7! G.1/