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Duality for topological abelian group stacks and T -duality 3094.6.5 Next we show that Čech cohomology has a Künneth formula.Lemma 4.69. Let H be a discrete ring of finite cohomological dimension. Assume thatX; Y are compact. Then there exists a Künneth spectral sequence with second termE 2 p;q WDMiCj Dqwhich converges to L H pCq .X Y I H /.Tor H p . L H i .XI H /; L H j .Y I H //Proof. Since X is compact the topology on X Y is the product topology [Ste67,Theorem 4.3]. Using again the compactness of X and Y we can find a cofinal systemof coverings of X Y of the form p U \ q V for coverings U of X and V of Y ,where p W X Y ! X and q W X Y ! Y denote the projections, and the intersectionof coverings is the covering by the collection of all cross intersections. We haveLC .p U \ q VI p H / Š LC .p UI p H / ˝HLC .q VI p H /:Since the tensor product over H commutes with colimits we getand hence by (40)LC .X Y I p H / Š LC .XI p H / ˝HLC .Y I p H /;LC .X Y I H / Š LC .XI H / ˝HLC .Y I H /:The Künneth spectral sequence is the spectral sequence associated to this doublecomplex of flat (as colimits of free) H -modules.4.6.6 We now recall that sheaf cohomology also transforms strict inverse limits ofspaces into colimits, and that it has a Künneth spectral sequence. The following is aspecialization of [Brd97, II.14.6] to compact spaces.Lemma 4.70. Let .X i / i2I be an inverse system of compact spaces and X D lim i2I X i ,and H let be a discrete abelian group. ThenH .XI H / Š colim i2I H .X i I H /:For simplicity we formulate the Künneth formula for the sheaf Z only. The followingis a specialization of [Brd97, II.18.2] to compact spaces and the sheaf Z.Lemma 4.71. For compact spaces X; Y we have a Künneth spectral sequence withsecond termEp;q 2 DMTorp Z .H i .XI Z/; H j .Y I Z//iCj Dqwhich converges to H pCq .X Y I Z/.