Algebra & Number Theory

Calculabilité de la cohomologie étale modulo $\ell$

David Madore and Fabrice Orgogozo

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Abstract

Soient X un schéma algébrique sur un corps algébriquement clos et un nombre premier inversible sur X. D’après le théorème 1.1 de (SGA 41 2, Th. finitude), les groupes de cohomologie étale Hi(X, ) sont de dimension finie. Utilisant une variante -adique des bons voisinages d’Artin et des résultats élémentaires sur la cohomologie des pro- groupes, on exprime la cohomologie de X comme colimite bien contrôlée de celle de topos construits sur des BG, où les G sont des -groupes finis calculables. On en déduit que les nombres de Betti modulo  de X sont algorithmiquement calculables (au sens de Church–Turing). La première partie du texte est consacrée à la démonstration de ce fait et de quelques compléments naturels. Elle s’appuie sur les outils de la seconde partie, dédiée à la géométrie algébrique effective.

Let X be an algebraic scheme over an algebraically closed field and a prime number invertible on X. According to Theorem 1.1 of (SGA 41 2, Th. finitude), the étale cohomology groups Hi(X, ) are finite-dimensional. Using an -adic variant of Artin’s good neighborhoods and elementary results on the cohomology of pro- groups, we express the cohomology of X as a well controlled colimit of that of toposes constructed on BG where the G are computable finite -groups. From this, we deduce that the Betti numbers modulo  of X are algorithmically computable (in the sense of Church and Turing). The proof of this fact, along with certain related results, occupies the first part of this paper. This relies on the tools collected in the second part, which deals with computational algebraic geometry.

Article information

Source
Algebra Number Theory, Volume 9, Number 7 (2015), 1647-1739.

Dates
Received: 16 October 2014
Revised: 16 April 2015
Accepted: 28 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842393

Digital Object Identifier
doi:10.2140/ant.2015.9.1647

Mathematical Reviews number (MathSciNet)
MR3404650

Zentralblatt MATH identifier
1327.14095

Subjects
Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 03D99: None of the above, but in this section 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50] 12Y05: Computational aspects of field theory and polynomials 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 14A20: Generalizations (algebraic spaces, stacks) 14F35: Homotopy theory; fundamental groups [See also 14H30] 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 20E18: Limits, profinite groups 55P20: Eilenberg-Mac Lane spaces 55T05: General

Keywords
cohomologie étale cohomologie galoisienne descente cohomologique suite spectrale schéma simplicial groupe profini espace d'Eilenberg–MacLane voisinage d'Artin champ algébrique gerbe géométrie algébrique effective calculabilité étale cohomology Galois cohomology cohomological descent spectral sequence simplicial scheme profinite group Eilenberg–MacLane space Artin's neighborhood stack effective algebraic geometry computability

Citation

Madore, David; Orgogozo, Fabrice. Calculabilité de la cohomologie étale modulo $\ell$. Algebra Number Theory 9 (2015), no. 7, 1647--1739. doi:10.2140/ant.2015.9.1647. https://projecteuclid.org/euclid.ant/1510842393


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