The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some twisted DeRham complex.
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2$\begingroup$ Have you heard about crystalline cohomology? (see en.wikipedia.org/wiki/Crystalline_cohomology) It is not really a resolution of the local system $k$, though... $\endgroup$– Leo AlonsoCommented Mar 31, 2011 at 8:27
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2$\begingroup$ It fails even in char.0, because the Poincare lemma works only formally locally and the de Rham complex is not exact in positive degrees. On the other hand, the cohomology of any constant sheaf vanishes on an irreducible topological space, so anyway taking resolutions of $k$ would not lead no anything of interest. $\endgroup$– Piotr AchingerCommented Mar 31, 2011 at 13:05
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$\begingroup$ in char o the deRham is a resolution of C , which implies $H(X,C) \cong H_{Dr}(X)$ $\endgroup$– chemaidaCommented Mar 31, 2011 at 15:16
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1$\begingroup$ chemaida, to be clear, the Poincar\'e lemma holds for holomorphic or $C^\infty$ forms, but it can fail for the algebraic de Rham complex even in characteristic $0$ (which is what Piotr was referring to). Nevertheless the hypercohomology of the algebraic de Rham complex does give the correct answer, although the reasons are more subtle. $\endgroup$– Donu ArapuraCommented Mar 31, 2011 at 16:00
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$\begingroup$ Ok..thank you for your responses..I would be interested in the subtle reason that Donu alludes to or even a reference $\endgroup$– chemaidaCommented Apr 1, 2011 at 7:25
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