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.

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.
–
Piotr AchingerMar 31 '11 at 13:05

in char o the deRham is a resolution of C , which implies $H(X,C) \cong H_{Dr}(X)$
–
chemaidaMar 31 '11 at 15:16

1

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.
–
Donu ArapuraMar 31 '11 at 16:00

Ok..thank you for your responses..I would be interested in the subtle reason that Donu alludes to or even a reference
–
chemaidaApr 1 '11 at 7:25

holomorphicor $C^\infty$ 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. – Donu Arapura Mar 31 '11 at 16:00