Comparison theorems between algebraic and analytic De Rham cohomology (with emphasis on the -adic case)
Yves André[1]
- [1] Institut de Mathématiques 175 rue du Chevaleret F-75013 Paris
Journal de Théorie des Nombres de Bordeaux (2004)
- Volume: 16, Issue: 2, page 335-355
- ISSN: 1246-7405
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topAndré, Yves. "Comparison theorems between algebraic and analytic De Rham cohomology (with emphasis on the $p$-adic case)." Journal de Théorie des Nombres de Bordeaux 16.2 (2004): 335-355. <http://eudml.org/doc/249275>.
@article{André2004,
abstract = {We present a panorama of comparison theorems between algebraic and analytic De Rham cohomology with algebraic connections as coefficients. These theorems have played an important role in the development of $\mathcal\{D\}$-module theory, in particular in the study of their ramification properties (irregularity...). In part I, we concentrate on the case of regular coefficients and sketch the new proof of these theorems given by F. Baldassarri and the author, which is of elementary nature and unifies the complex and $p$-adic theories. In the $p$-adic case, however, the comparison theorem was expected to extend to irregular coefficients, and this has recently been proved in [AB]. The proof of this extension follows the same pattern as in the regular case, but involves in addition a detailed study of irregularity in several variables. In part II, we give an overview of this proof which can serve as a guide to the book [AB].added on proofs: a second (revised) edition of [AB] is in preparation.},
affiliation = {Institut de Mathématiques 175 rue du Chevaleret F-75013 Paris},
author = {André, Yves},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {335-355},
publisher = {Université Bordeaux 1},
title = {Comparison theorems between algebraic and analytic De Rham cohomology (with emphasis on the $p$-adic case)},
url = {http://eudml.org/doc/249275},
volume = {16},
year = {2004},
}
TY - JOUR
AU - André, Yves
TI - Comparison theorems between algebraic and analytic De Rham cohomology (with emphasis on the $p$-adic case)
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 2
SP - 335
EP - 355
AB - We present a panorama of comparison theorems between algebraic and analytic De Rham cohomology with algebraic connections as coefficients. These theorems have played an important role in the development of $\mathcal{D}$-module theory, in particular in the study of their ramification properties (irregularity...). In part I, we concentrate on the case of regular coefficients and sketch the new proof of these theorems given by F. Baldassarri and the author, which is of elementary nature and unifies the complex and $p$-adic theories. In the $p$-adic case, however, the comparison theorem was expected to extend to irregular coefficients, and this has recently been proved in [AB]. The proof of this extension follows the same pattern as in the regular case, but involves in addition a detailed study of irregularity in several variables. In part II, we give an overview of this proof which can serve as a guide to the book [AB].added on proofs: a second (revised) edition of [AB] is in preparation.
LA - eng
UR - http://eudml.org/doc/249275
ER -
References
top- Y. André, F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties. Progress in Mathematics 189, Birkäuser (2001). Zbl0995.14003MR1807281
- M. Artin, Comparaison avec la cohomologie classique. Cas d’un schéma lisse. In: M. Artin, A. Grothendieck, J.L. Verdier, Théorie des Topos et Cohomologie Étale des Schémas. Tome 3. Lecture Notes in Math. 305, Springer-Verlag (1973). Zbl0267.14008MR354654
- F. Baldassarri, Differential modules and singular points of -adic differential equations. Advances in Math. 44 (1982), 155–179. Zbl0493.12030MR658539
- F. Baldassarri, Comparaison entre la cohomologie algébrique et la cohomologie -adique rigide à coefficients dans un module différentiel I. Invent. Math. 87 (1987), 83–99 . Zbl0586.14009MR862713
- F. Baldassarri, Comparaison entre la cohomologie algébrique et la cohomologie -adique rigide à coefficients dans un module différentiel II. Math. Ann. 280 (1988), 417–439. Zbl0651.14012MR936321
- J. Bernstein, Lectures on -modules. Mimeographed notes.
- A. Borel, Algebraic -modules. Perspectives in Mathematics, Vol.2, Academic Press (1987). Zbl0642.32001MR882000
- B. Chiarellotto, Sur le théorème de comparaison entre cohomologies de De Rham algébrique et -adique rigide. Ann. Inst. Fourier 38 (1988), 1–15. Zbl0644.14006MR978239
- D. Clark, A note on the -adic convergence of solutions of linear differential equations. Proc. Amer. Math. Soc. 17 (1966), 262–269. Zbl0147.31101MR186895
- P. Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163, Springer-Verlag (1970). Zbl0244.14004MR417174
- A. Dimca, F. Maaref, C. Sabbah, M. Saito, Dwork cohomology and algebraic -modules. Math. Ann. 318 (2000), 107–125. Zbl0985.14007MR1785578
- H. Grauert, R. Remmert, Komplexe Räume. Math. Ann. 136 (1958), 245–318. Zbl0087.29003MR103285
- A. Grothendieck, On the De Rham cohomology of algebraic varieties. Publications Mathématiques IHES 29 (1966), 93–103. Zbl0145.17602MR199194
- N. Katz, Nilpotent connections and the monodromy theorem. Applications of a result of Turrittin. Publ. Math. IHES 39 (1970), 175–232. Zbl0221.14007MR291177
- N. Katz, The regularity theorem in algebraic geometry. Actes du Congrès Intern. Math. 1970, T.1, 437–443. Zbl0235.14006MR472822
- N. Katz, A simple algorithm for cyclic vectors. Amer. J. of Math. 109 (1987), 65–70. Zbl0621.13003MR878198
- N. Katz, T. Oda, On the differentiation of De Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8, 2 (1968), 199–213. Zbl0165.54802MR237510
- R. Kiehl, Die De Rham Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper. Publ. Math. IHES 33 (1967), 5–20. Zbl0159.22404MR229644
- Y. Laurent, Z. Mebkhout, Pentes algébriques et pentes analytiques d’un -module. Ann. Scient. Ecole Normale Sup. (4) 32 (1999), 39–69. Zbl0944.14007MR1670595
- Yu. Manin, Moduli fuchsiani. Annali Scuola Normale Sup. Pisa III 19 (1965), 113–126. Zbl0166.04301MR180581
- Z. Mebkhout, Le théorème de positivité de l’irrégularité pour les -modules. In Grothendieck Festschrift vol. III, 83–132 , Progress in Mathematics, Birkhäuser (1990). Zbl0731.14007MR1106912
- Z. Mebkhout, Le théorème de comparaison entre cohomologies de De Rham d’une variété algébrique complexe et le théorème d’existence de Riemann. Publ. Math. IHES 69 (1989), 47–89. Zbl0709.14015MR1019961
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