Canonical integral structures on the de Rham cohomology of curves

Bryden Cais[1]

  • [1] McGill University Department of Mathematics 805 Sherbrooke Street West Montréal, QC. H3A 2K6 (Canada)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 6, page 2255-2300
  • ISSN: 0373-0956

Abstract

top
For a smooth and proper curve X K over the fraction field K of a discrete valuation ring R , we explain (under very mild hypotheses) how to equip the de Rham cohomology H dR 1 ( X K / K ) with a canonical integral structure: i.e., an R -lattice which is functorial in finite (generically étale) K -morphisms of X K and which is preserved by the cup-product auto-duality on H dR 1 ( X K / K ) . Our construction of this lattice uses a certain class of normal proper models of X K and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper R -model of X K and that the index for this inclusion of lattices is a numerical invariant of X K (we call it the de Rham conductor). Using work of Bloch and of Liu-Saito, we prove that the de Rham conductor of X K is bounded above by the Artin conductor, and bounded below by the efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of X K is affected by finite extension of scalars.

How to cite

top

Cais, Bryden. "Canonical integral structures on the de Rham cohomology of curves." Annales de l’institut Fourier 59.6 (2009): 2255-2300. <http://eudml.org/doc/10454>.

@article{Cais2009,
abstract = {For a smooth and proper curve $X_K$ over the fraction field $K$ of a discrete valuation ring $R$, we explain (under very mild hypotheses) how to equip the de Rham cohomology $H^1_\{\mathrm\{dR\}\}(X_K/K)$ with a canonical integral structure: i.e., an $R$-lattice which is functorial in finite (generically étale) $K$-morphisms of $X_K$ and which is preserved by the cup-product auto-duality on $H^1_\{\mathrm\{dR\}\}(X_K/K)$. Our construction of this lattice uses a certain class of normal proper models of $X_K$ and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper $R$-model of $X_K$ and that the index for this inclusion of lattices is a numerical invariant of $X_K$ (we call it the de Rham conductor). Using work of Bloch and of Liu-Saito, we prove that the de Rham conductor of $X_K$ is bounded above by the Artin conductor, and bounded below by the efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of $X_K$ is affected by finite extension of scalars.},
affiliation = {McGill University Department of Mathematics 805 Sherbrooke Street West Montréal, QC. H3A 2K6 (Canada)},
author = {Cais, Bryden},
journal = {Annales de l’institut Fourier},
keywords = {de Rham cohomology; $p$-adic local Langlands; curve; rational singularities; arithmetic surface; Grothendieck duality; Artin conductor; efficient conductor; simultaneous resolution of singularities; -adic local Langlands; arithmetic surfaces},
language = {eng},
number = {6},
pages = {2255-2300},
publisher = {Association des Annales de l’institut Fourier},
title = {Canonical integral structures on the de Rham cohomology of curves},
url = {http://eudml.org/doc/10454},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Cais, Bryden
TI - Canonical integral structures on the de Rham cohomology of curves
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2255
EP - 2300
AB - For a smooth and proper curve $X_K$ over the fraction field $K$ of a discrete valuation ring $R$, we explain (under very mild hypotheses) how to equip the de Rham cohomology $H^1_{\mathrm{dR}}(X_K/K)$ with a canonical integral structure: i.e., an $R$-lattice which is functorial in finite (generically étale) $K$-morphisms of $X_K$ and which is preserved by the cup-product auto-duality on $H^1_{\mathrm{dR}}(X_K/K)$. Our construction of this lattice uses a certain class of normal proper models of $X_K$ and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper $R$-model of $X_K$ and that the index for this inclusion of lattices is a numerical invariant of $X_K$ (we call it the de Rham conductor). Using work of Bloch and of Liu-Saito, we prove that the de Rham conductor of $X_K$ is bounded above by the Artin conductor, and bounded below by the efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of $X_K$ is affected by finite extension of scalars.
LA - eng
KW - de Rham cohomology; $p$-adic local Langlands; curve; rational singularities; arithmetic surface; Grothendieck duality; Artin conductor; efficient conductor; simultaneous resolution of singularities; -adic local Langlands; arithmetic surfaces
UR - http://eudml.org/doc/10454
ER -

References

top
  1. Shreeram Abhyankar, Simultaneous resolution for algebraic surfaces, Amer. J. Math. 78 (1956), 761-790 Zbl0073.37902MR82722
  2. Shreeram Abhyankar, Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) (1965), 111-152, Harper & Row, New York Zbl0147.20503MR200272
  3. Michael Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira) (1969), 21-71, Univ. Tokyo Press, Tokyo Zbl0205.50402MR260746
  4. Michael Artin, Lipman’s proof of resolution of singularities for surfaces, Arithmetic geometry (Storrs, Conn., 1984) (1986), 267-287, CornellGaryG., New York Zbl0602.14011MR861980
  5. Spencer Bloch, de Rham cohomology and conductors of curves, Duke Math. J. 54 (1987), 295-308 Zbl0632.14018MR899399
  6. Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, Non-Archimedean analysis, 261 (1984), Springer-Verlag, Berlin Zbl0539.14017MR746961
  7. Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud, Néron models, 21 (1990), Springer-Verlag, Berlin Zbl0705.14001MR1045822
  8. Bryden Cais, Canonical extensions of Néron models of Jacobians, (2008) Zbl1193.14058
  9. Brian Conrad, Grothendieck duality and base change, 1750 (2000), Springer-Verlag, Berlin Zbl0992.14001MR1804902
  10. Brian Conrad, Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu 6 (2007), 209-278 Zbl1140.14018MR2311664
  11. Brian Conrad, Bas Edixhoven, William Stein, J 1 ( p ) has connected fibers, Doc. Math. 8 (2003), 331-408 (electronic) Zbl1101.14311MR2029169
  12. Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), 5-57 Zbl0219.14007MR498551
  13. Pierre Deligne, Luc Illusie, Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math. 89 (1987), 247-270 Zbl0632.14017MR894379
  14. Pierre Deligne, David Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), 75-109 Zbl0181.48803MR262240
  15. Pierre Deligne, Michael Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (1973), 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin Zbl0281.14010MR337993
  16. Renée Elkik, Rationalité des singularités canoniques, Invent. Math. 64 (1981), 1-6 Zbl0498.14002MR621766
  17. Matthew Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (2006), 1-84 Zbl1090.22008MR2207783
  18. Alexander Grothendieck, Jean Dieudonné, Éléments de géométrie algébrique, (1960–7), Inst. Hautes Études Sci. Publ. Math. Zbl0203.23301
  19. Robin Hartshorne, Residues and duality, (1966), Springer-Verlag, Berlin MR222093
  20. Stephen Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380-405 Zbl0194.22101MR230724
  21. Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969), 195-279 Zbl0181.48903MR276239
  22. Joseph Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), 151-207 Zbl0349.14004MR491722
  23. Qing Liu, Conducteur et discriminant minimal de courbes de genre 2 , Compositio Math. 94 (1994), 51-79 Zbl0837.14023MR1302311
  24. Qing Liu, Algebraic geometry and arithmetic curves, 6 (2002), Oxford University Press, Oxford Zbl0996.14005MR1917232
  25. Qing Liu, Stable reduction of finite covers of curves, Compos. Math. 142 (2006), 101-118 Zbl1108.14020MR2196764
  26. Qing Liu, Dino Lorenzini, Models of curves and finite covers, Compositio Math. 118 (1999), 61-102 Zbl0962.14020MR1705977
  27. Qing Liu, Takeshi Saito, Inequality for conductor and differentials of a curve over a local field, J. Algebraic Geom. 9 (2000), 409-424 Zbl0992.14008MR1752009
  28. Hideyuki Matsumura, Commutative ring theory, 8 (1989), Cambridge University Press, Cambridge Zbl0666.13002MR1011461
  29. Barry Mazur, William Messing, Universal extensions and one dimensional crystalline cohomology, (1974), Springer-Verlag, Berlin Zbl0301.14016MR374150
  30. Barry Mazur, Ken Ribet, Two-dimensional representations in the arithmetic of modular curves, Astérisque (1991), 6, 215-255 (1992) Zbl0780.14015MR1141460
  31. Michel Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. (1970), 27-76 Zbl0207.51602MR282993
  32. Michel Raynaud, Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1-89 Zbl0227.14010MR308104
  33. Peter Schneider, Jeremy Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359-380 Zbl1006.46053MR1900706

NotesEmbed ?

top

You must be logged in to post comments.