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Le problème de la schématisation de Grothendieck revisité

Bertrand Toën

Abstract

The objective of this work is to reconsider the schematization problem of [6], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type. We deduce from this several results on the behaviour of the schematization functor, which we propose as a solution to the schematization problem.

Le problème de la schématisation de Grothendieck revisité

Abstract

The objective of this work is to reconsider the schematization problem of [6], with a particular focus on the global case over Z. For this, we prove the conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of the schematization of a simply connected homotopy type. We deduce from this several results on the behaviour of the schematization functor, which we propose as a solution to the schematization problem.

Paper Structure

This paper contains 7 sections, 4 theorems, 86 equations.

Table of Contents

  1. Rappels sur les champs affines
  2. Le groupe additif de Hilbert $\mathbb{H}$
  3. L'affinisation de $K(\mathbb{Z},n)$
  4. Faisceaux d'homotopie de l'affinisation
  5. Propriétés d'injectivité du foncteur d'affinisation
  6. Le théorème d'Eilenberg-Moore
  7. Le topos $\mathrm{fpqc}$

Key Result

Proposition 1.1

Soit $X$ un espace topologique et $F$ un champ affine muni d'un morphisme $u : X \to F(\mathbb{Z})$$($ou de manière équivalente du champ constant $X$ vers $F)$. Alors les deux conditions suivantes sont équivalentes.

Theorems & Definitions (22)

  • Proposition 1.1
  • proof
  • Proposition 2.2
  • proof
  • proof
  • Proposition 3.1
  • proof
  • proof
  • proof
  • proof
  • ...and 12 more