Overconvergent de Rham-Witt cohomology

Christopher Davis; Andreas Langer; Thomas Zink

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 2, page 197-262
  • ISSN: 0012-9593

Abstract

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The goal of this work is to construct, for a smooth variety X over a perfect field k of finite characteristic p > 0 , an overconvergent de Rham-Witt complex W † Ω X / k as a suitable subcomplex of the de Rham-Witt 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 † ( 𝒪 X ) of overconvergent Witt-vectors. If X is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent de Rham-Witt cohomology. Finally we define for a quasiprojective X an isomorphism between the rational overconvergent de Rham-Witt cohomology and the rigid cohomology.

How to cite

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Davis, Christopher, Langer, Andreas, and Zink, Thomas. "Overconvergent de Rham-Witt cohomology." Annales scientifiques de l'École Normale Supérieure 44.2 (2011): 197-262. <http://eudml.org/doc/272170>.

@article{Davis2011,
abstract = {The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p &gt; 0$, an overconvergent de Rham-Witt complex $W^\{\dag \}\Omega _\{X/k\}$ as a suitable subcomplex of the de Rham-Witt 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 $W^\{\dag \}(\mathcal \{O\}_X)$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent de Rham-Witt cohomology. Finally we define for a quasiprojective $X$ an isomorphism between the rational overconvergent de Rham-Witt cohomology and the rigid cohomology.},
author = {Davis, Christopher, Langer, Andreas, Zink, Thomas},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {rigid cohomology; de Rham-Witt complex},
language = {eng},
number = {2},
pages = {197-262},
publisher = {Société mathématique de France},
title = {Overconvergent de Rham-Witt cohomology},
url = {http://eudml.org/doc/272170},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Davis, Christopher
AU - Langer, Andreas
AU - Zink, Thomas
TI - Overconvergent de Rham-Witt cohomology
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 2
SP - 197
EP - 262
AB - The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p &gt; 0$, an overconvergent de Rham-Witt complex $W^{\dag }\Omega _{X/k}$ as a suitable subcomplex of the de Rham-Witt 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 $W^{\dag }(\mathcal {O}_X)$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent de Rham-Witt cohomology. Finally we define for a quasiprojective $X$ an isomorphism between the rational overconvergent de Rham-Witt cohomology and the rigid cohomology.
LA - eng
KW - rigid cohomology; de Rham-Witt complex
UR - http://eudml.org/doc/272170
ER -

References

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  9. [9] K. S. Kedlaya, More étale covers of affine spaces in positive characteristic, J. Algebraic Geom.14 (2005), 187–192. Zbl1065.14020MR2092132
  10. [10] A. Langer & T. Zink, De Rham-Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu3 (2004), 231–314. Zbl1100.14506MR2055710
  11. [11] A. Langer & T. Zink, Gauss-Manin connection via Witt-differentials, Nagoya Math. J.179 (2005), 1–16. Zbl1101.14020MR2164399
  12. [12] S. Lubkin, Generalization of p -adic cohomology: bounded Witt vectors. A canonical lifting of a variety in characteristic p ≠ 0 back to characteristic zero, Compositio Math. 34 (1977), 225–277. Zbl0368.14009MR453745
  13. [13] D. Meredith, Weak formal schemes, Nagoya Math. J.45 (1972), 1–38. Zbl0207.51502MR330167
  14. [14] P. Monsky & G. Washnitzer, Formal cohomology. I, Ann. of Math. 88 (1968), 181–217. Zbl0162.52504MR248141

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