Canonical integral structures on the de rham cohomology of curves

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

For a smooth and proper curve XK over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology HdR1(XK/K) with a canonical integral structure: i.e., an R-lattice which is functorial in finite (generically étale) K-morphisms of XK and which is preserved by the cup-product auto-duality on HdR1(XK/K). Our construction of this lattice uses a certain class of normal proper models of XK 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 XK and that the index for this inclusion of lattices is a numerical invariant of XK (we call it the de Rham conductor). Using work of Bloch and of Liu-Saito, we prove that the de Rham conductor of XK 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 XK is affected by finite extension of scalars.

Original languageEnglish (US)
Pages (from-to)2255-2300
Number of pages46
JournalAnnales de l'Institut Fourier
Volume59
Issue number6
DOIs
StatePublished - 2009
Externally publishedYes

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

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