Overconvergent De Rham-witt Cohomology
نویسندگان
چکیده
The goal of this work is to construct, for a smooth variety X over a perfect field k of finite characteristic, an overconvergent de Rham-Witt complex W ΩX/k as a suitable subcomplex of the de RhamWitt complex of Deligne-Illusie. This complex, which is functorial in X, is a complex of étale sheaves and a differential graded algebra over the ring W (OX) of overconvergent Witt-vectors. If X is affine one proves that there is a canonical isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent de Rham-Witt cohomology. In general, we compare this cohomology with the rigid cohomology of X, on which we find a natural lattice, i.e. an integral structure. Résumé. Le but de ce travail est de construire, pour X une variété lisse sur un corps parfait k de charactéristique finie, un complexe de de Rham-Witt surconvergent W ΩX/k comme un sous-complexe convenable du complexe de de Rham-Witt de Deligne-Illusie. Ce complexe qui est fonctoriel en X est un complexe des faisceaux étales et une algèbre différentielle graduée sur l’anneau W (OX) des vecteurs de Witt surconvergents. Lorsque X est affine, on démontre qu’il existe un isomorphisme canonique entre la cohomologie de Monsky-Washnitzer et la cohomologie (rationelle) de de Rham-Witt surconvergente. En général, on compare cette cohomologie avec la cohomologie rigide de X sur laquelle on trouve un réseau naturel, c’est à dire une structure intégrale.
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