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268 U. Bunke, T. Schick, M. Spitzweck, and A. ThomByF W .Z mult -mod/ ! Abwe denote the forgetful functor.Definition 3.41. Let G be an abelian group. For k 2 Z we let G.k/ 2 Z mult -mod denotethe Z mult -module given by the action Z mult G 3 .p; g/ 7! ‰ p .g/ WD g pk 2 G.Observe that for abelian groups V;W we have a natural isomorphism3.5.3 Let V be a Z mult -module.V.k/˝Z W.l/ Š .V ˝Z W /.k C l/: (19)Definition 3.42. We say that V has weight k if there exists an isomorphism V ŠF .V /.k/ of Z mult -modules.If V has weight k, then every sub-quotient of V has weight k. Note that aZ mult -module can have many weights. We have e.g. isomorphisms of Z mult -modules.Z=2Z/.1/ Š .Z=2Z/.k/ for all k 6D 0.3.5.4 Let V 2 Z mult -mod. We say that v 2 V has weight k if it generates a submoduleZ V of weight k. Fork 2 N we let W k W .Z mult -mod/ ! Ab be the functorwhich associates to V 2 Z mult -mod its subgroup of vectors of weight k. Then we havean adjoint pair of functors.k/ W Ab , .Z mult -mod/ W W k :Observe that the functor W k is not exact. Consider for example a prime number p 2 Nand the sequence0 ! Z.1/ ! p Z.1/ ! .Z=pZ/.p/ ! 0:The projection map is indeed Z mult -equivariant since m p m mod p for all m 2 Z.Then0 Š W p .Z.1// ! W p ..Z=pZ/.p// Š Z=pZis not surjective.3.5.5 Let V 2 Ab and V .1/ 2 Z mult -mod. Then we can form the graded tensor algebraT Z .V .1// D Z ˚ V .1/ ˚ V .1/ ˝Z V .1/ ˚ :We see that T k .V .1// has weight k. The elements x ˝ x 2 V .1/ ˝ V .1/ generate ahomogeneous ideal I . Hence the graded algebra ƒ Z .V .1// WD T Z .V .1//=I has theproperty that ƒ k Z.V .1// has weight k.