process

272 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. We apply the functor Ext Sh Ab S .F;:::/to the sequence0 ! Z ! R ! T ! 0and get the following segments of the long exact sequence!Ext i Sh Ab S .F; R/ ! Exti Sh Ab S .F; T/ ! ExtiC1Sh Ab S.F; Z/ ! :We see that the assumptions on F imply that Ext i Sh Ab S .F; T/ Š 0 for i D 1; 2.4.1.5 A space W 2 S is called profinite if it can be written as the limit of an inversesystem of finite spaces. Lemma 3.29 has as a special case the following theorem.Theorem 4.10. If W 2 S is profinite, then Z.W / is admissible.4.2 A double complex4.2.1 Let H 2 Sh Ab S be a sheaf of groups with underlying sheaf of sets F .H / 2ShS. Applying the linearization functor Z (see 3.2.6) we get again a sheaf of groupsZ.F .H // 2 Sh Ab S. The group structure of H induces on Z.F .H // a ring structure.We denote this sheaf of rings by ZŒH . It is the sheafification of the presheaf p ZŒH which associates to A 2 S the integral group ring p ZŒH .A/ of the group H.A/.We consider the category Sh ZŒH -mod S of sheaves of ZŒH -modules. The trivialaction of the sheaf of groups H on the sheaf Z induces the structure of a sheaf ofZŒH -modules on Z. With the notation introduced below we could (but refrain fromdoing this) write this sheaf of ZŒH -modules as coind.Z/.4.2.2 The forgetful functor resW Sh ZŒH -mod S ! Sh Ab S fits into an adjoint pair offunctorsind W Sh Ab S , Sh ZŒH -mod S W res:Explicitly, the functor ind is given byind.V / WD ZŒH ˝Z V:Since ZŒH is a torsion-free sheaf and therefore a sheaf of flat Z-modules the functorind is exact. Consequently the functor res preserves injectives.4.2.3 We let coinv W Sh ZŒH -mod S ! Sh Ab S denote the coinvariants functor given bySh ZŒH -mod S 3 V 7! coinv.V / WD V ˝ZŒH Z 2 Sh Ab S:This functor fits into the adjoint paircoinv W Sh ZŒH -mod S , Sh Ab S W coindwith the coinduction functor which maps W 2 Sh Ab S to the sheaf of ZŒH -modulesinduced by the trivial action of H in W , formally this can be written ascoind.W / WD Hom ShAb S .Z;W/: