process

306 U. Bunke, T. Schick, M. Spitzweck, and A. ThomConsider A 2 S and 2 Hom ShAb S .Z.F .G//=K] ; Z/.A/. We must show that eacha 2 A has a neighbourhood U A such that jU D 0. Pre-composing with theprojection Z.F .G// ! Z.F .G//=K ] we get an elementN 2 Hom ShAb S .Z.F .G//; Z/.A/ Š Hom Lemma 3.9Sh S .F .G/; Z/.A/ Š Z.A G/:We are going to show that N D 0 after restriction to a suitable neighbourhood ofa 2 A using the fact that it annihilates K ] . Let us start again on the left-hand side of(39). We haveHom ShAb S .. p Z.F .G//=K/ ] ; Z/.A/ Š Hom PrAb S .. p Z.F .G//=K/; Z/.A/Š Hom PrAb S=A. p Z.F .G jA //=K jA ; Z jA /:For .A G ! A/ 2 S=A the morphism gives rise to a group homomorphismy W Z.F .G.A G///=K.A G/ ! Z.A G/:The symbol F .G.A G// denotes the underlying set of the group G.A G/. Wehave.pr G W A G ! G/ 2 G.A G/ and by the explicit description of the element N givenafter the proof of Lemma 3.9 we see N D y .¹Œpr G º/, where Œpr G 2 Z.F .G.AG///denotes the generator corresponding to pr G 2 G.A G/, and ¹::: º indicates that wetake the class modulo K.A G/.The homomorphism y 2 Hom Ab .ZF .G.A G//=K.A G/;Z.A G// is representedby a homomorphismQ W ZF .G.A G// ! Z.A G/;and we have N D Q.Œpr G /. By definition of K we have the exact sequence0 ! K.AG/ ! Z.F .G.AG/// ! G.AG/˝ZG.AG/ ! ƒ 2 ZG.AG/ ! 0:For x 2 G.A G/ let Œx 2 Z.F .G.A G/// denote the corresponding generator.Since .nx/ ˝ .nx/ D n 2 .x ˝ x/ in G.A G/ ˝Z G.A G/ we have n 2 Œx Œnx 2K.A G/ for n 2 Z. It follows that Q must satisfy the relation Q.n 2 Œx/ D Q.Œnx/for all n 2 Z and x 2 G.A G/. Let us now apply this reasoning to x WD Œpr G .Forevery n 2 Z we have a map A G id AnDW n! A G, and by naturality the map Qrespects this action. Moreover, the element Œnx 2 Z.F .G.A G/// is obtained fromŒx via this action, sinceA Gn A Gcommutes. It follows thatGpr Gn Gpr Gn 2 N D Q.n 2 Œpr G / D Q.Œnpr G / D Q.ZF . xG. n //.Œpr G // D Z. n /. N/ D n N;