Le probl\`eme de la sch\'ematisation de Grothendieck revisit\'e

TL;DR

This paper revisits Grothendieck's schematization problem over Z, proving a key conjecture to understand the homotopy groups of schematizations and proposing a new approach to the problem.

Contribution

It proves a conjecture relating to the homotopy groups of schematizations over Z, advancing the understanding of the schematization functor.

Findings

Proves the conjecture [Conj. 2.3.6] for homotopy groups.

Provides results on the behavior of the schematization functor.

Proposes a solution to the schematization problem.

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.