Canonical integral structures on the de Rham cohomology of curves

Cais, Bryden

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2255-2300.

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 $H_{\mathrm{dR}}^1(X_K/K)$. 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.

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 $H_{\mathrm{dR}}^1(X_K/K)$ 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 $H_{\mathrm{dR}}^1(X_K/K)$. 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.

[1] Abhyankar, Shreeram Simultaneous resolution for algebraic surfaces, Amer. J. Math., Volume 78 (1956), pp. 761-790 | DOI | MR | Zbl

[2] Abhyankar, Shreeram Resolution of singularities of arithmetical surfaces, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 111-152 | MR | Zbl

[3] Artin, Michael Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21-71 | MR | Zbl

[4] Artin, Michael; Cornell, Gary; Silverman, Joseph H. Lipman’s proof of resolution of singularities for surfaces, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 267-287 (Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30–August 10, 1984) | MR | Zbl

[5] Bloch, Spencer de Rham cohomology and conductors of curves, Duke Math. J., Volume 54 (1987) no. 2, pp. 295-308 | DOI | MR | Zbl

[6] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261, Springer-Verlag, Berlin, 1984 (A systematic approach to rigid analytic geometry) | MR | Zbl

[7] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21, Springer-Verlag, Berlin, 1990 | MR | Zbl

[8] Cais, Bryden Canonical extensions of Néron models of Jacobians (2008) (Submitted)

[9] Conrad, Brian Grothendieck duality and base change, Lecture Notes in Mathematics, 1750, Springer-Verlag, Berlin, 2000 | MR | Zbl

[10] Conrad, Brian Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 209-278 | DOI | MR | Zbl

[11] Conrad, Brian; Edixhoven, Bas; Stein, William $J_1\left(p\right)$ has connected fibers, Doc. Math., Volume 8 (2003), p. 331-408 (electronic) | EuDML | MR | Zbl

[12] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 | DOI | EuDML | Numdam | MR | Zbl

[13] Deligne, Pierre; Illusie, Luc Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math., Volume 89 (1987) no. 2, pp. 247-270 | DOI | EuDML | MR | Zbl

[14] Deligne, Pierre; Mumford, David The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 75-109 | DOI | EuDML | Numdam | MR | Zbl

[15] Deligne, Pierre; Rapoport, Michael Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, p. 143-316. Lecture Notes in Math., Vol. 349 | MR | Zbl

[16] Elkik, Renée Rationalité des singularités canoniques, Invent. Math., Volume 64 (1981) no. 1, pp. 1-6 | DOI | EuDML | MR | Zbl

[17] Emerton, Matthew On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math., Volume 164 (2006) no. 1, pp. 1-84 | DOI | MR | Zbl

[18] Grothendieck, Alexander; Dieudonné, Jean Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math., 1960–7 no. 4,8,11,17,20,24,28,37

[19] Hartshorne, Robin Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966 | EuDML | MR

[20] Lichtenbaum, Stephen Curves over discrete valuation rings, Amer. J. Math., Volume 90 (1968), pp. 380-405 | DOI | MR | Zbl

[21] Lipman, Joseph Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. (1969) no. 36, pp. 195-279 | DOI | EuDML | Numdam | MR | Zbl

[22] Lipman, Joseph Desingularization of two-dimensional schemes, Ann. Math. (2), Volume 107 (1978) no. 1, pp. 151-207 | DOI | MR | Zbl

[23] Liu, Qing Conducteur et discriminant minimal de courbes de genre $2$, Compositio Math., Volume 94 (1994) no. 1, pp. 51-79 | EuDML | Numdam | MR | Zbl

[24] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, Oxford, 2002 (Translated from the French by Reinie Erné, Oxford Science Publications) | MR | Zbl

[25] Liu, Qing Stable reduction of finite covers of curves, Compos. Math., Volume 142 (2006) no. 1, pp. 101-118 | DOI | MR | Zbl

[26] Liu, Qing; Lorenzini, Dino Models of curves and finite covers, Compositio Math., Volume 118 (1999) no. 1, pp. 61-102 | DOI | MR | Zbl

[27] Liu, Qing; Saito, Takeshi Inequality for conductor and differentials of a curve over a local field, J. Algebraic Geom., Volume 9 (2000) no. 3, pp. 409-424 | MR | Zbl

[28] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989 (Translated from the Japanese by M. Reid) | MR | Zbl

[29] Mazur, Barry; Messing, William Universal extensions and one dimensional crystalline cohomology, Springer-Verlag, Berlin, 1974 (Lecture Notes in Mathematics, Vol. 370) | MR | Zbl

[30] Mazur, Barry; Ribet, Ken Two-dimensional representations in the arithmetic of modular curves, Astérisque (1991) no. 196-197, p. 6, 215-255 (1992) Courbes modulaires et courbes de Shimura (Orsay, 1987/1988) | MR | Zbl

[31] Raynaud, Michel Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. (1970) no. 38, pp. 27-76 | DOI | EuDML | Numdam | MR | Zbl

[32] Raynaud, Michel; Gruson, Laurent Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | EuDML | MR | Zbl

[33] Schneider, Peter; Teitelbaum, Jeremy Banach space representations and Iwasawa theory, Israel J. Math., Volume 127 (2002), pp. 359-380 | DOI | MR | Zbl