process

274 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThe cohomology of the complex p C .H / is given by´ pZ;H i . p i D 0;C .H // Š0; i 1;where p Z 2 Pr Ab S denotes the constant presheaf with value Z. Since sheafification i ]is an exact functor and by definition C .H / D i ]p C .H / we get´H i Z; i D 0;.C .H // Š0; i 1:Furthermore,C i .H / Š ZŒH ˝Z Z.H„Hƒ‚ …/ D ind.Z.H„Hƒ‚ …// (21)i factorsi factorsshows that C i .H / is flat. Let us write C WD C .H / D ind.D / withD i WD Z.H„Hƒ‚ …/:i factorsDefinition 4.14. For a sheaf H 2 Sh Ab S we define the complex U coinv.C /.WD U .H / WDIt follows from the construction that U depends functorially on H . In particular,for ˛ W H ! H 0 we have a map of complexes U .˛/W U .H / ! U .H 0 /.4.2.5 The main tool in our proofs of admissibility of a sheaf F is the study of thesheavesR Hom ShZŒH -mod S .Z; coind.W //for W D Z; R. Let us write this in a more complicated way using the special flatresolution C .H / ! Z constructed in 4.2.4. We choose an injective resolutioncoind.W / ! I in Sh ZŒH -mod S. Using that res ı coind D id, and that res.I / isinjective in Sh Ab S,weget 7RHom ShZŒH -mod S .Z; coind.W // Š Hom Sh ZŒH -mod S / .Z;IŠ Hom ShZŒH -mod S .C .H /; I /Š Hom ShZŒH -mod S /; I /.ind.DŠ Hom ShAb S ; res.I //.DD Hom ShAb S ; res.coind.res.I ////.DD Hom ShZŒH -mod S /; coind.res.I ///.ind.D7 Note that D is considered in this calculation as a sequence of sheaves, not as a complex. The differentialsin the intermediate steps involving D are still induced via the isomorphisms from the differentials of C and I .