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GHHXH,SpacempmrhomPiBsubseteqUperiodSpaceGinB,SpaceGinBperiodSpace(tG)p,Space(tG)pperiodSpaceExt(tGcomma

GcongExt(G, X)congExt(tG, X),SpacetGcongbigoplusPinPtHom((tG)p, Q/Z)Space(tG)pcongtprodirinfty((C(pi))mPi)period

B(rho)=B1oplusldotsoplusBioplusldotsSpacesemicolonSpaceBi=(C(pi))(mPi),SpaceGcongI|rhoZ(p)mrhooplusprodipin=1infty((C(pi))mPi)comma

Xotimes(Q/Zrangle)congprodpinPHom((tG)rho, xotimes(Q/Z))inftyHom((tG)p, C(pinfty))congZ(p)mpoplusII1((C(pi)mPi)i=

UinGammaIPZ(p)/Xcong(prodpinPZ(p)/bigopluspinPZ(p))/(x/bigopluspinPZ(p))

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