Un crit\`ere d'extension d'un foncteur d\'efini sur les sch\'emas lisses

TL;DR

This paper establishes a criterion for extending functors from smooth schemes to all separated schemes of finite type over a field of characteristic zero, using Hironaka's desingularization theorem.

Contribution

It introduces an extension criterion for functors based on their behavior on blow-ups, applicable to motives, Hodge-De Rham complexes, and rational homotopy theory.

Findings

Functor extension criterion proven using desingularization.

Applicable to motives, Hodge-De Rham complexes, and rational homotopy.

Enhances understanding of functor behavior on singular schemes.

Abstract

Let $k$ be a field of characteristic zero. By using Hironaka's desingularisation theorem, we prove an extension criterion for a functor defined on nonsingular k-schemes and taking values on a category of complexes. Roughly speaking, the criterion shows that if such a functor satisfies the standard exact sequence of a blowing-up, then the functor can be extended to all separated k-schemes of finite type. The result is applied to the Grothendieck's theory of motives, to the Hodge-De Rham filtered complex of an analytic space, and to the rational homotopy of k-schemes in algebraic De Rham theory.