process

298 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.5.12 Below we consider profinite abelian groups which are also Z=pZ modules.The following lemma describes their structure.Lemma 4.58. Let p be a prime number. If G is compact and a Z=pZ-module, thenthere exists a set S such that G Š Q SZ=pZ. In particular, G is profinite.Proof. The dual group yG of the compact group G is discrete and also a Z=pZ-module,hence an F p -vector space. Let S yG be an F p -basis. Then we can write yG Š˚SZ=pZ. Pontrjagin duality interchanges sums and products. We getG Š yG Š 3˚SZ=pZ Š Y S1Z=pZ Š Y SZ=pZ:4.5.13 Assume that G is a compact group and a Z=pZ-module. Note that the constructionG ! U WD U .G/ is functorial in G (see 4.6.11 for more details). Thetautological action of Z mult on G induces an action of Z mult on U . We can improveLemma 4.51 as follows.Lemma 4.59. Let Z ! I be an injective resolution. For i 2¹2; 3; 4º the sheavesH i Hom ShAb S .U ;I /are sheaves of Z=pZ-modules and Z mult -modules whose weights are given by thefollowing table.i 0 2 3 4Moreover, H 1 Hom ShAb S .U ;I / Š 0.weight 0 1 2 2-3Proof. We argue with the spectral sequence as in the proof of 4.51. Since E p;q1D 0for q 1, it suffices to show that the sheaves E i;02are sheaves of Z=pZ-modulesand Z mult -modules of weight k, where k corresponds to i as in the table (and with theappropriate modification for i D 4).We have for A 2 SE i;01 .A/ Š Z.A Gi / Š Hom S .G i ; Map.A; Z// D Ccont i .GI Map.A; Z//;where for a topological G-module V the complex C cont .GI V/denotes the continuousgroup cohomology complex. We now consider the presheaf zX i defined byA 7! zX i .A/ WD Hcont i .G; Map.A; Z//:Then by definition E i;02WD X i WD . zX i / ] is the sheafification of zX i .The action of Z mult on G induces an action q 7! Œq on the presheaf zX i , whichdescends to an action of Z mult on the associated sheaf X i .We fix i 2¹0; 1; 2; 3º and let k be the associated weight as in the table. For i 2we must show that for each section s 2 Hcont i .GI Map.A; Z// and a 2 A there exists a