Canonical Integral Structures on the de Rham Cohomology of Curves

Bryden Cais

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 H1dR(XK/K) with a canonical integral structure: i.e. an R-lattice which is functorial in finite (generically etale) K-morphisms of XK and which is preserved by the cup-product auto-duality on H1dR(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 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.

This paper strengthens and extends the work of Chapter 2 of my PhD thesis.