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CONSTRUCTIBILITÉ ET MODÉRATION UNIFORMES EN COHOMOLOGIE ÉTALE Fabrice Orgogozo Version CONSTRUCTIBILITÉ ET MODÉRATION UNIFORMES EN COHOMOLOGIE ÉTALE Fabrice Orgogozo‡ version : 69b422d 2017-03-30 18:28:46 +0200 Résumé. Soient 푆 un schéma nœthérien et 푓 ∶ 푋 → 푆 un morphisme propre. D’après SGA 4 XIV, pour tout faisceau constructible ℱ de ℤ/푛ℤ-modules sur 푋, 푖 les faisceaux de ℤ/푛ℤ-modules 횁 푓⋆ℱ , obtenus par image directe (pour la to- pologie étale), sont également constructibles : il existe une stratification 픖 de 푆 telle que ces faisceaux soient localement constants constructibles sur les strates. À la suite de travaux de N. Katz et G. Laumon, ou L. Illusie, dans le cas particu- lier où 푆 est génériquement de caractéristique nulle ou bien les faisceaux ℱ sont constants (de torsion inversible sur 푆), on étudie ici la dépendance de 픖 en ℱ . On montre qu’une condition naturelle de constructibilité et modération « uniforme » satisfaite par les faisceaux constants, introduite par O. Gabber, est stable par les 푖 foncteurs 횁 푓⋆. Si 푓 n’est pas supposé propre, ce résultat subsiste sous réserve de modération à l’infini, relativement à 푆. On démontre aussi l’existence de bornes uniformes sur les nombres de Betti, qui s’appliquent notamment pour les fibres 푖 ℓ des faisceaux 횁 푓⋆픽ℓ, où parcourt les nombres premiers inversibles sur 푆. Abstract. Let 푆 be a Noetherian scheme and 푓 ∶ 푋 → 푆 a proper morphism. By SGA 4 XIV, for any constructible sheaf ℱ of ℤ/푛ℤ-modules on 푋, the sheaves 푖 of ℤ/푛ℤ-modules 횁 푓⋆ℱ obtained by direct image (for the étale topology) are themselves constructible, that is, there is a stratification 픖 of 푆 on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which 푆 is generically of characteristic zero or the sheaves ℱ are constant (with invertible torsion on 푆), here we study the dependency of 픖 on ℱ . We show that a natural “uniform” tameness and constructibility condition satisfied by constant sheaves, which was 푖 introduced by O. Gabber, is stable under the functors 횁 푓⋆. If 푓 is not proper, this result still holds assuming tameness at infinity, relative to 푆. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the 푖 ℓ sheaves 횁 푓⋆픽ℓ, where ranges through all prime numbers invertible on 푆. Table des matières Introduction 2 1. Définitions, sorites et préliminaires topologiques 4 2. Lemmes d’Abhyamkar et courbes nodales 14 3. Image directe par un morphisme propre 22 4. Image directe par une immersion ouverte 33 5. Six opérations 40 6. Applications 43 ‡CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France [email protected] 2 CONSTRUCTIBILITÉ ET MODÉRATION UNIFORMES EN COHOMOLOGIE ÉTALE Introduction 0.1. Soient 푆 un schéma et 푓 ∶ 푋 → 푆 un morphisme propre de présentation finie. D’après un théorème de finitude classique, dû à Michael Artin et Alexander Grothendieck ([SGA 4 XIV, 1.1]), pour tout faisceau constructible ℱ de ℤ/푛ℤ- 푖 modules sur 푋, les faisceaux de ℤ/푛ℤ-modules 횁 푓⋆ℱ , obtenus par image directe, sont également constructibles : (lorsque 푆 est cohérent, par exemple nœthérien) il 푖 existe une stratification 픖 de 푆 telle que les faisceaux 횁 푓⋆ℱ soient localement constants et de présentation finie sur les strates. (Ils sont également nuls pour 푖 ≫ 0 indépendamment de ℱ .) À 푓 fixé, il est naturel de s’interroger sur la dépendance de 픖 en ℱ , notam- ment sous la forme suivante, qui n’est qu’un cas particulier des questions que l’on souhaite aborder ici : Existe-t-il une stratification 픖 qui convienne pour tous les fais- 푖 ceaux 횁 푓⋆ℤ/푛ℤ, où 푛 parcourt les entiers inversibles sur 푆 ? (Notons que si l’entier 푛 parcourt les puissances d’un même nombre premier ℓ, comme cela est naturel du point de vue de la cohomologie ℓ-adique, la réponse à la question précédente est trivialement « oui ».) 0.2. Si 푆 est de type fini sur le corps ℂ des nombres complexes, l’existence d’une telle stratification est bien connue en vertu du résultat analogue pour le morphisme d’espaces topologiques 푋(ℂ) → 푆(ℂ) — cf. par exemple [Borel 1984, V.10.16] ou [Beilinson, Bernstein et Deligne 1982, p. 153, premier paragraphe] — et du théo- rème de comparaison entre cohomologie de Betti et cohomologie étale de M. Artin et A. Grothendieck ([SGA 4 XVI, 4.1]). 0.3. Bien entendu, tant pour les applications que pour les démonstrations, il est utile de ne pas se restreindre à des faisceaux constants mais de considérer des fais- ceaux — et même des complexes — constructibles le long d’une stratification 픛 de 푋 (au sens évident rappelé en 1.1.3). Cependant, en cohomologie étale, des phéno- mènes de ramification sauvage font que l’on ne peut s’attendre à ce que, donnés un morphisme propre 푓 ∶ 푋 → 푆 et 픛 une stratification de 푋, il existe une stratification 픖 de 푆 telle que les images directes supérieures par 푓 de faisceaux sur 푋 constructibles le long de 픛 soient constructibles le long de 픖. En effet, si par exemple ℓ est un nombre premier fixé, 푘 un corps algébriquement clos de ca- × ractéristique 푝 > 0 divisant ℓ − 1 et 휓 ∶ ℤ/푝ℤ → 픽ℓ un caractère non-trivial, les 1 1 ℓ-faisceaux d’Artin-Schreier sur sont lisses pour 픽 ℒ휆 ≔ ℒ휓 ((푥 − 휆)푦) 퐀푥/픽푝 × 퐀푦/픽푝 chaque mais les faisceaux image directe exceptionnelle 2 pr ne sont 휆 ∈ 푘 횁 1! ℒ휆 pas constructibles le long d’une même stratification de 1 , indépendante de : le 퐀푥/픽푝 휆 faisceau 2 pr est non nul et à support dans . (Voir [Laumon 1987, 1.2.2.2] 횁 1! ℒ휆 {휆} pour l’analogue ℓ-adique et [Orgogozo 2003, p. 139—140] pour une discussion du même phénomène dans une autre perspective.) La flèche 2 pr 2 pr 횁 1! ℒ휆 → 횁 1⋆ ℒ휆 d’oubli des supports étant un isomorphisme ([Laumon 1987, 1.3.1.1]), on voit éga- lement que l’on ne dispose pas d’un analogue uniforme « naïf » du théorème [Th. finitude, 1.9 (ii)] de commutation au changement de base générique : en tout point CONSTRUCTIBILITÉ ET MODÉRATION UNIFORMES EN COHOMOLOGIE ÉTALE 3 au voisinage duquel 2 pr commute au changement de base, sa fibre est né- 횁 1⋆ ℒ휆 cessairement nulle. Cependant, comme l’a démontré Luc Illusie — motivé par une question de Jean-Pierre Serre —, la situation est aussi bonne que possible si l’on considère des coefficients constants (voir [Illusie 2010]). La formule de Grothendieck-Ogg-Šafarevič ([SGA 5 X, 7.2]) montre également que l’on ne peut obtenir de bornes uniformes sur les nombres de Betti, comme celles rappelées au paragraphe suivant, sans hypothèse de modération. 0.4. Dans [Katz et Laumon 1985, §3] (voir aussi par exemple [Fouvry et Katz 2001, §2] et [Katz 1990, §13]) sont établis des résultats intermédiaires entre l’énon- cé topologique rappelé ci-dessus et des énoncés, à démontrer, tenant compte de la ramification sauvage. Nicholas Katz et Gérard Laumon établissent notamment le résultat suivant, sous l’hypothèse technique supplémentaire que 푆 est de type fini sur un schéma nœthérien régulier de dimension inférieure ou égale à 1 ([Katz et Laumon 1985, 3.1.2]) : il existe un entier 푁 ⩾ 1 et une stratification 픖 de 푆 tels que 푏 pour chaque nombre premier ℓ, chaque 풦 ∈ Ob 햣푐(푋[1/푁ℓ], ℚℓ) constructible le 푏 long de 픛[1/푁ℓ], le complexe 횁푓⋆풦 ∈ Ob 햣푐(푆[1/푁ℓ], ℚℓ) est constructible le long de 픖[1/푁ℓ]. (« Constructible le long de … » : voir 1.1.3.) De plus, les auteurs majorent uniformément la taille des images directes : il existe une constante 퐶 telle que pour chaque 푠 ∈ 푆[1/푁ℓ] on ait l’inégalité ‖횁푓!풦‖(푠) ⩽ 퐶 ⋅ sup ‖풦‖(푥), 푥∈푓 −1(푠) où dim 푖 pour un choix arbitraire de point géométrique ‖풦‖(푥) ≔ ∑푖 ℚℓ 홷 (풦)푥 푥 au-dessus de 푥 ∈ 푋. Ce résultat s’appuie sur la résolution des singularités d’Hironaka et n’est inté- ressant que si la caractéristique générique de 푆 est nulle. Cette hypothèse permet également, grâce au lemme d’Abhyaṁkar relatif (cf. [ibid., §3.4.2] et [SGA 1 XIII, 5.5]), d’ignorer la ramification sauvage. 0.5. L’objectif de ce texte est de s’affranchir de ces restrictions : les « fibrations élémentaires » ([SGA 4 XI, §3]) utilisées dans [Katz et Laumon 1985] sont rem- placées par des courbes nodales, dont l’ubiquité a été démontrée par Aise Johann de Jong ([de Jong 1996 ; de Jong 1997]). Armé de cet outil et familier des tech- niques de dévissages en cohomologie étale, la principale difficulté est de trouver un énoncé de stabilité de familles « (uniformément) constructibles et modérées ». On le doit à Ofer Gabber, dont la contribution à cet article ne s’arrête pas là. Un point clef est que « uniformément » doit être interprété localement pour la topolo- gie propre — engendrée par la prétopologie dont les morphismes couvrants sont les morphismes propres et surjectifs — (ou la topologie ℎ), idée que l’on trouve déjà dans une lettre de Richard Pink à N. Katz ([Pink 1995]), pourtant antérieure aux travaux de A. J. de Jong. (L’insuffisance d’une localisation pour la topologie finie est expliquée en 3.2.) Les méthodes employées ici font que la plupart des résultats de cet article sont de démonstration plus simple dans le cas particuliers des schémas de type fini sur un anneau de Dedekind mais nous les énonçons pour des schémas quasi-excellents, 4 CONSTRUCTIBILITÉ ET MODÉRATION UNIFORMES EN COHOMOLOGIE ÉTALE quitte à en donner la démonstration en deux temps.
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