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296 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.5.8 For abelian groups V;W let V Z W WD Tor Z 1 .V; W / denote the Tor-product. If Vand W in addition are Z mult -modules then V W is a Z mult -module by the functorialityof the Tor-product.Lemma 4.52. If V;W are Z mult -modules of weight k; l, then V ˝Z W and V Z Ware of weight k C l.Proof. The assertion for the tensor product follows from (19). Let P ! V 0 andQ ! W 0 be projective resolutions of the underlying Z-modules V 0 ;W 0 of V;W .Then P .k/ ! V and Q .l/ ! W are Z mult -equivariant resolutions of V and W .Wehave by (19)V Z W Š H 1 .P .k/ ˝Z Q .l// Š H 1 .P ˝Z Q /.k C l/:4.5.9 The tautological action of Z mult on an abelian group G (we write G.1/ forthis Z mult -module) (see 3.5.1) induces an action of Z mult on the bar complex Z.G / bydiagonal action on the generators, and therefore on the group cohomology H .G.1/I Z/of G.In the following lemma we calculate the cohomology of the group Z=pZ.1/ as aZ mult -module.Lemma 4.53. Let p 2 N be a prime. Then we have8ˆ< Z.0/; i D 0;H i .Z=pZ.1/I Z/ Š 0; i odd,ˆ:Z=pZ.k/; i D 2k 2:Proof. The group cohomology of Z=pZ can be identified as a ring with the ring Z ˚cZ=pZŒc, where c has degree 2. Furthermore, for every group there is a canonicalmap yG ! H 2 .GI Z/ which in the case G Š Z=pZ happens to be an isomorphism.This implies that H 2 .Z=pZ.1/I Z/ has weight 1 as a Z mult -module. Since the cupproduct in group cohomology is natural it is Z mult -equivariant. Therefore, the powerc k generates a module of weight k. Hence H 2k .Z=pZ.1/I Z/ has weight k.4.5.10 For a Z mult -module V (with Z mult -action ‰) let P v k be the operator x 7! ‰v xv k x. Note that this is a commuting family of operators. We let M23 v be the monoidgenerated by P2 v;Pv 3 .Definition 4.54. A Z mult -module V is called a weight 2-3-extension if for all x 2 Vand all v 2 Z mult there is P v 2 M23 v such thatP v x D 0: (37)A sheaf of Z mult -modules is called a weight 2-3-extension, if every section locallysatisfies Equation (37) (with P v depending on the section and the neighborhood).In the tables below we will mark weight 2-3-extensions with attribute weight 2-3.