Lifting D -modules from positive to zero characteristic

João Pedro P. dos Santos

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 2, page 193-242
  • ISSN: 0037-9484

Abstract

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We study liftings or deformations of D -modules ( D is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic D -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given D -module in positive characteristic. At the end we compare the problems of deforming a D -module with the problem of deforming a representation of a naturally associated group scheme.

How to cite

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dos Santos, João Pedro P.. "Lifting $D$-modules from positive to zero characteristic." Bulletin de la Société Mathématique de France 139.2 (2011): 193-242. <http://eudml.org/doc/272332>.

@article{dosSantos2011,
abstract = {We study liftings or deformations of $D$-modules ($D$ is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic $D$-modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given $D$-module in positive characteristic. At the end we compare the problems of deforming a $D$-module with the problem of deforming a representation of a naturally associated group scheme.},
author = {dos Santos, João Pedro P.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {$D$-modules; differential Galois theory; group schemes in mixed characteristic; monoidal categories; deformation theory},
language = {eng},
number = {2},
pages = {193-242},
publisher = {Société mathématique de France},
title = {Lifting $D$-modules from positive to zero characteristic},
url = {http://eudml.org/doc/272332},
volume = {139},
year = {2011},
}

TY - JOUR
AU - dos Santos, João Pedro P.
TI - Lifting $D$-modules from positive to zero characteristic
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 2
SP - 193
EP - 242
AB - We study liftings or deformations of $D$-modules ($D$ is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic $D$-modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given $D$-module in positive characteristic. At the end we compare the problems of deforming a $D$-module with the problem of deforming a representation of a naturally associated group scheme.
LA - eng
KW - $D$-modules; differential Galois theory; group schemes in mixed characteristic; monoidal categories; deformation theory
UR - http://eudml.org/doc/272332
ER -

References

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  1. [1] S. Anantharaman – « Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1 », Mémoires de la SMF33 (1973), p. 5–79. Zbl0286.14001
  2. [2] M. Artin, A. Grothendieck & J.-L. Verdier – « Théorie des topos et cohomologie étale des schémas », Lecture Notes in Math. 269, 270, 305 (1971). 
  3. [3] P. Berthelot – « 𝒟 -modules arithmétiques. I. Opérateurs différentiels de niveau fini », Ann. Sci. École Norm. Sup.29 (1996), p. 185–272. Zbl0886.14004MR1373933
  4. [4] —, « 𝒟 -modules arithmétiques. II. Descente par Frobenius », Mém. Soc. Math. Fr. (N.S.) 81 (2000). Zbl0948.14017
  5. [5] —, « Introduction à la théorie arithmétique des 𝒟 -modules », Astérisque279 (2002), p. 1–80. Zbl1098.14010
  6. [6] —, « A note on Frobenius divided modules in mixed characteristics », preprint arXiv:1003.2571. Zbl1277.14016
  7. [7] P. Berthelot & A. Ogus – Notes on crystalline cohomology, Princeton Univ. Press, 1978. Zbl0383.14010MR491705
  8. [8] P. Deligne – « Le groupe fondamental de la droite projective moins trois points », in Galois groups over 𝐐 (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, 1989, p. 79–297. Zbl0742.14022MR1012168
  9. [9] —, « Catégories tannakiennes », in The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser, 1990, p. 111–195. 
  10. [10] M. Demazure & A. Grothendieck (éds.) – Schémas en groupes. I: Propriétés générales des schémas en groupes. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Math., vol. 151, Springer, 1970. Zbl0207.51401
  11. [11] M. Demazure – « Schémas en groupes réductifs », Bull. Soc. Math. France93 (1965), p. 369–413. Zbl0163.27402MR197467
  12. [12] M. Demazure & P. Gabriel – Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, 1970. Zbl0203.23401MR302656
  13. [13] 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
  14. [14] J. Giraud – Cohomologie non abélienne, Grundl. math. Wissensch., vol. 179, Springer, 1971. Zbl0226.14011MR344253
  15. [15] W. M. Goldman & J. J. Millson – « The deformation theory of representations of fundamental groups of compact Kähler manifolds », Publ. Math. I.H.É.S. 67 (1988), p. 43–96. Zbl0678.53059MR972343
  16. [16] A. Grothendieck (éd.) – Revêtements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math., vol. 224, Springer, 1971. Zbl0234.14002MR354651
  17. [17] —, « Technique de descente et théorèmes d’éxistence en géométrie algébrique », Séminaire Bourbaki: t. 12, n. 190, 1959/60; t. 12, n. 195, 1959/60; t. 13, n. 212, 1960/61; t. 13, n. 221, 1960/61; t. 14, n. 232, 1961/62; t. 14, n. 236, 1961/62. 
  18. [18] A. Grothendieck & J. Dieudonné – « Éléments de géométrie algébrique », Publ. Math. IHÉS 8, 11 (1961); 17 (1963); 20 (1964); 24 (1965); 28 (1966); 32 (1967). Zbl0203.23301
  19. [19] L. Illusie – « Grothendieck’s existence theorem in formal geometry », in Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., 2005, p. 179–233. Zbl1085.14001MR2223409
  20. [20] J. C. Jantzen – Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press Inc., 1987. Zbl0654.20039MR899071
  21. [21] 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
  22. [22] S. Mac Lane – « Categorical algebra », Bull. Amer. Math. Soc.71 (1965), p. 40–106. Zbl0161.01601MR171826
  23. [23] —, Categories for the working mathematician, second éd., Graduate Texts in Math., vol. 5, Springer, 1998. Zbl0906.18001MR1712872
  24. [24] M. Manetti – « Deformation theory via differential graded Lie algebras », in Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), Scuola Norm. Sup., 1999, p. 21–48. MR1754793
  25. [25] H. Matsumura – Commutative ring theory, second éd., Cambridge Studies in Advanced Math., vol. 8, Cambridge Univ. Press, 1989. Zbl0666.13002MR1011461
  26. [26] 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
  27. [27] B. H. Matzat & M. van der Put – « Iterative differential equations and the Abhyankar conjecture », J. reine angew. Math. 557 (2003), p. 1–52. Zbl1040.12010MR1978401
  28. [28] B. Mazur – « Deforming Galois representations », in Galois groups over 𝐐 (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16, Springer, 1989, p. 385–437. Zbl0714.11076MR1012172
  29. [29] J. S. Milne – Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton Univ. Press, 1980. Zbl0433.14012MR559531
  30. [30] D. Mumford – Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. Zbl0223.14022MR282985
  31. [31] M. V. Nori – « On the representations of the fundamental group », Compositio Math.33 (1976), p. 29–41. Zbl0337.14016MR417179
  32. [32] J. P. Pridham – « Deformation theory and the fundamental group », Thèse, University of Cambridge, 2004. 
  33. [33] —, « Deformations via simplicial deformation complexes », preprint arXiv:math/0311168. 
  34. [34] M. van der Put & M. F. Singer – Galois theory of linear differential equations, Grund. Math. Wiss., vol. 328, Springer, 2003. Zbl1036.12008MR1960772
  35. [35] N. Saavedra Rivano – Catégories tannakiennes, Lecture Notes in Math., vol. 265, Springer, 1972. Zbl0241.14008MR338002
  36. [36] J. P. P. dos Santos – « Fundamental group schemes for stratified sheaves », J. Algebra 317 (2007), p. 691–713. Zbl1130.14032MR2362937
  37. [37] —, « The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach », J. reine angew. Math. 637 (2009), p. 63–98. Zbl1242.12005MR2599082
  38. [38] M. Schlessinger – « Functors of Artin rings », Trans. Amer. Math. Soc.130 (1968), p. 208–222. Zbl0167.49503MR217093
  39. [39] C. A. Weibel – An introduction to homological algebra, Cambridge Studies in Advanced Math., vol. 38, Cambridge Univ. Press, 1994. Zbl0797.18001MR1269324
  40. [40] O. Zariski – « Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields », in Collected papers of O. Zariski, vol. II, MIT Press, 1973. Zbl0045.24001

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