process

308 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.6.3 Fix a discrete group H . Let p H be the constant presheaf with values H . Notethat then the sheafification p H ] is isomorphic to H as defined in 3.2.5. MoreoverLC .XI p H / Š LC .X; H /: (40)In general Čech cohomology differs from sheaf cohomology. The relation betweenthese two is given by the Čech cohomology spectral sequence .E r ;d r / (see [Tam94,3.4.4]) converging to H .XI F/. Let H .F / WD R i.F/ denote the right derivedfunctors of the inclusion i W Sh Ab S ! Pr Ab S. Then the second page of the spectralsequence is given byE p;q2Š HLp .XI H q .F //:By [Tam94, 3.4.7] the edge homomorphismL H p .XI F/! H p .XI F/is an isomorphism for p D 0; 1 and injective for p D 2.4.6.4 We now observe that Čech cohomology transforms strict inverse limits of compactspaces into colimits of cohomology groups.Lemma 4.68. Let H be a discrete abelian group. If .X i / i2I is an inverse system ofcompact spaces in S such that X D lim i2I X i , thenHLp .XI H / Š colim i2I HLp .X i I H /:Proof. We first show that the system of open coverings of X contains a cofinal systemof coverings which are pulled back from the quotients p i W X ! X i . Let U D .U r / r2Rbe a covering by open subsets. For each r there exists a family I r I and subsetsU r;i X i , i 2 I r such that U r D[ i2Ir pi 1 .U r;i /. The set of open subsets ¹pi1 .U r;i / jr 2 R; i 2 I r º is an open covering of X. Since X is compact we can choosea finite subcovering V WD ¹U r1 ;i 1;:::U rk ;i kº which can naturally be viewed as arefinement of U. Since I is left filtered we can choose j 2 I such that j < i dfor d D 1;:::;k. For j i let p ji W X j ! X i be the structure map of the systemwhich are all surjective by the strictness assumption on the inverse system. ThereforeV 0 WD ¹pj;i 1d.U rd ;i d/ j d D 1;:::;kº is an open covering of X j , and V D pj 1 .V 0 /.Now we observe that LC .p V 0 I p H / Š LC .V 0 I p H /. ThereforeLC .XI H / Š LC .XI p H / (by Equation (40))Š colim ULC .UI p H /D colim i2I colim coverings U of XiLC .UI p H /Š colim i2ILC .X i I p H /Š colim i2ILC .X i I H / (by Equation (40))since one can interchange colimits. Since filtered colimits are exact and thereforecommute with taking cohomology this implies the lemma.