Calculabilité de la cohomologie étale modulo $\ell$
| Autor: | David Madore, Fabrice Orgogozo |
|---|---|
| Přispěvatelé: | Mathématiques discrètes, Codage et Cryptographie (MC2), Laboratoire Traitement et Communication de l'Information (LTCI), Institut Mines-Télécom [Paris] (IMT)-Télécom Paris-Institut Mines-Télécom [Paris] (IMT)-Télécom Paris, Département Informatique et Réseaux (INFRES), Télécom ParisTech, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | francouzština |
| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
profinite group Étale cohomology 010103 numerical & computational mathematics effective algebraic geometry 12Y05 01 natural sciences 18G30 champ algébrique Mathematics - Algebraic Geometry spectral sequence 12G05 Mathematics 14F20 55T05 simplicial scheme Eilenberg–MacLane space 16. Peace & justice cohomological descent 03D99 étale cohomology Spectral sequence schéma simplicial espace d'Eilenberg–MacLane voisinage d'Artin Stack (mathematics) 20E18 Galois cohomology 14F35 descente cohomologique Gerbe Artin's neighborhood suite spectrale groupe profini calculabilité 0101 mathematics 13P10 cohomologie étale géométrie algébrique effective computability Algebra and Number Theory Profinite group 010102 general mathematics gerbe stack 14A20 cohomologie galoisienne 55P20 [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] |
| Zdroj: | Algebra & Number Theory Algebra & Number Theory, Mathematical Sciences Publishers 2015, 9 (7), pp.1647-1739. ⟨10.2140/ant.2015.9.1647⟩ Algebra & Number Theory, Mathematical Sciences Publishers 2015, 9 (7), pp.1647-1739 Algebra Number Theory 9, no. 7 (2015), 1647-1739 |
| ISSN: | 1937-0652 |
| DOI: | 10.2140/ant.2015.9.1647⟩ |
| Popis: | Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$ a prime number invertible on $X$. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the \'etale cohomology groups $\mathrm{H}^i(X,\mathbb{Z}/\ell\mathbb{Z})$ are finite-dimensional. Using an $\ell$-adic variant of M. Artin's good neighborhoods and elementary results on the cohomology of pro-$\ell$ 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 $\ell$-groups. From this, we deduce that the Betti numbers modulo $\ell$ of $X$ are algorithmically computable (in the sense of Church-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. Finally, in the third part, we present a "universal" formalism for computation on the elements of a field. Comment: In French. v2 has been considerably reworked and expanded. v3 incorporates slight corrections and simplifications and a few additions (notably: computability of the morphism from hyper\v{c}ech cohomology, graded algebra structure, and a worked out example); submitted for publication |
| Databáze: | OpenAIRE |
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