The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

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

top
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

top

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

top
  1. [1] P. Berthelot – « 𝒟 -modules arithmétiques. I. Opérateurs différentiels de niveau fini », Ann. Sci. École Norm. Sup.29 (1996), p. 185–272. Zbl0886.14004MR1373933
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.