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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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