Canonical integral structures on the de Rham cohomology of curves

TL;DR

This paper constructs a canonical integral structure on the de Rham cohomology of curves over discretely valued fields, relating it to models, duality, and conductors, and analyzing its behavior under scalar extensions.

Contribution

It introduces a new canonical integral lattice in de Rham cohomology of curves, compatible with morphisms and duality, and relates it to existing invariants like conductors.

Findings

The lattice contains the de Rham complex lattice of a regular model.

The index of the lattice is a numerical invariant called the de Rham conductor.

The de Rham conductor is bounded by the Artin and Efficient conductors.

Abstract

For a smooth and proper curve X 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_{dR}(X/K) with a canonical integral structure: i.e. an R-lattice which is functorial in finite (generically etale) K-morphisms of X and which is preserved by the cup-product auto-duality on H^1_{dR}(X/K). Our construction of this lattice uses a certain class of normal proper models of X 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 and that the index for this inclusion of lattices is a numerical invariant of X (we call it the de Rham conductor). Using work of Bloch and Liu-Saito, we prove that the de Rham conductor of X 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 is affected by finite extension of scalars.