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Bryden Cais
Canonical integral structures on the de Rham cohomology of curves
(Structures entières canoniques sur la cohomologie de de Rham d’un courbe)
Annales de l'institut Fourier, 59 no. 6 (2009), p. 2255-2300, doi: 10.5802/aif.2490
Article: subscription required (your ip address: 54.82.186.169) | Reviews MR 2640920 | Zbl pre05673896
Class. Math.: 14F40, 11G20, 14F30, 14G20, 14H25
Keywords: de Rham cohomology, $p$-adic local Langlands, curve, rational singularities, arithmetic surface, Grothendieck duality, Artin conductor, efficient conductor, simultaneous resolution of singularities

Résumé - 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.

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