Canonical integral structures on the de Rham cohomology of curves

[Structures entières canoniques sur la cohomologie de de Rham d’un courbe]

Cais, Bryden

Annales de l'Institut Fourier, Tome 59 (2009), p. 2255-2300 / Harvested from Numdam

Résumé

Soit $R$ un anneau de valuation discrète de corps de fractions $K$ et soit $X_K$ une courbe propre et lisse sur $K$. Nous montrons qu’on peut munir (sous certaines hypothèses faibles) la cohomologie de de Rham de $X_K$ sur $K$ d’une structure entière canonique : c’est-à-dire, d’un sous-$R$-réseau qui est fonctoriel pour les morphismes finis (et génériquement étales) de courbes sur $K$, et qui est son propre dual par rapport au cup-produit sur $H^1_{\mathrm{dR}}(X_K/K)$. Notre construction de ce réseau utilise une classe de $R$-modèles normaux et propres de $X_K$ et les faisceaux dualisants relatifs. Nous montrons que notre réseau contient le réseau fourni par le complexe de de Rham (tronqué) d’un $R$-modèle propre et régulier de $X_K$. L’indice pour cette inclusion est un invariant numérique de $X_K$, qu’on appelle le conducteur de de Rham. Partant d’un travail de Bloch et de Liu-Saito, nous prouvons que le conducteur de de Rham est majoré par le conducteur d’Artin, et minoré par le conducteur efficace. Nous étudions ensuite comment la position de notre réseau canonique varie sous les extensions finies de scalaires.

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.

Détail

Publié le : 2009-01-01

MR 2640920 | Zbl pre05673896

DOI : https://doi.org/10.5802/aif.2490
Classification:  14F40,  11G20,  14F30,  14G20,  14H25
Mots clés: cohomologie de de Rham, le programme de Langlands $p$-adique, courbe, singularités rationnelle, surface arithmétique, conducteur d’Artin, conducteur efficace, résolution simultanée des singularités

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