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

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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

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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 -

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