A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 3, page 441-458
  • ISSN: 0037-9484

Abstract

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If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ∞ ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ∞ ) on an 𝒪 X -module ℰ is equivalent to giving an infinite sequence of 𝒪 X -modules descending ℰ via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X → S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that p belongs to an ideal of definition. In [12], dos Santos used this result to lift 𝒟 X ( ∞ ) -modules from characteristic p to characteristic 0 with control of the differential Galois group.

How to cite

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Berthelot, Pierre. "A note on Frobenius divided modules in mixed characteristics." Bulletin de la Société Mathématique de France 140.3 (2012): 441-458. <http://eudml.org/doc/272597>.

@article{Berthelot2012,
abstract = {If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\mathcal \{D\}_\{X\}^\{(\infty )\}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $\mathcal \{D\}_\{X\}^\{(\infty )\}$ on an $\{\mathcal \{O\}\}_X$-module $\{\mathcal \{E\}\}$ is equivalent to giving an infinite sequence of $\{\mathcal \{O\}\}_X$-modules descending $\{\mathcal \{E\}\}$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \rightarrow S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an ideal of definition. In [12], dos Santos used this result to lift $\mathcal \{D\}_\{X\}^\{(\infty )\}$-modules from characteristic $p$ to characteristic $0$ with control of the differential Galois group.},
author = {Berthelot, Pierre},
journal = {Bulletin de la Société Mathématique de France},
keywords = {$D$-modules; Frobenius morphism; descent theory; deformation theory},
language = {eng},
number = {3},
pages = {441-458},
publisher = {Société mathématique de France},
title = {A note on Frobenius divided modules in mixed characteristics},
url = {http://eudml.org/doc/272597},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Berthelot, Pierre
TI - A note on Frobenius divided modules in mixed characteristics
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 3
SP - 441
EP - 458
AB - If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\mathcal {D}_{X}^{(\infty )}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $\mathcal {D}_{X}^{(\infty )}$ on an ${\mathcal {O}}_X$-module ${\mathcal {E}}$ is equivalent to giving an infinite sequence of ${\mathcal {O}}_X$-modules descending ${\mathcal {E}}$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \rightarrow S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an ideal of definition. In [12], dos Santos used this result to lift $\mathcal {D}_{X}^{(\infty )}$-modules from characteristic $p$ to characteristic $0$ with control of the differential Galois group.
LA - eng
KW - $D$-modules; Frobenius morphism; descent theory; deformation theory
UR - http://eudml.org/doc/272597
ER -

References

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  2. [2] —, « 𝒟 -modules arithmétiques. II. Descente par Frobenius », Mém. Soc. Math. Fr. (N.S.) 81 (2000). Zbl0948.14017
  3. [3] —, « Introduction à la théorie arithmétique des 𝒟 -modules », Astérisque279 (2002), p. 1–80. Zbl1098.14010
  4. [4] P. Berthelot & A. Ogus – Notes on crystalline cohomology, Princeton Univ. Press, 1978. Zbl0383.14010MR491705
  5. [5] D. Gieseker – « Flat vector bundles and the fundamental group in non-zero characteristics », Ann. Scuola Norm. Sup. Pisa Cl. Sci.2 (1975), p. 1–31. Zbl0322.14009MR382271
  6. [6] A. Grothendieck – « Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III », Publ. Math. I.H.É.S. 28 (1966). Zbl0144.19904
  7. [7] —, « Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV », Publ. Math. I.H.É.S. 32 (1967). Zbl0135.39701
  8. [8] —, « Crystals and the de Rham cohomology of schemes », in Dix Exposés sur la Cohomologie des Schémas, North-Holland, 1968, p. 306–358. Zbl0215.37102MR269663
  9. [9] L. Illusie & M. Raynaud – « Les suites spectrales associées au complexe de de Rham-Witt », Publ. Math. I.H.É.S. 57 (1983), p. 73–212. Zbl0538.14012MR699058
  10. [10] N. M. Katz – « Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin », Publ. Math. I.H.É.S. 39 (1970), p. 175–232. Zbl0221.14007MR291177
  11. [11] B. H. Matzat – « Integral p -adic differential modules », in Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, 2006, p. 263–292. Zbl1158.13009MR2316354
  12. [12] J. P. dos Santos – « Lifting 𝒟 -modules from positive to zero characteristic », preprint. Zbl1233.13009

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