## 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_{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.

Original language | English (US) |
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Pages (from-to) | 2255-2300 |

Number of pages | 46 |

Journal | Annales de l'Institut Fourier |

Volume | 59 |

Issue number | 6 |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

## Keywords

- Arithmetic surface
- Artin conductor
- Curve
- De Rham cohomology
- Efficient conductor
- Grothendieck duality
- P-adic local Langlands
- Rational singularities
- Simultaneous resolution of singularities

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology