Unstable motivic homotopy categories in Nisnevich and cdh-topologies

TL;DR

This paper demonstrates that, assuming resolution of singularities, the unstable motivic homotopy category of all schemes with the cdh-topology is nearly equivalent to that of smooth schemes with the Nisnevich topology, bridging different scheme categories.

Contribution

It proves an equivalence between motivic homotopy categories of all schemes and smooth schemes under certain conditions, extending the applicability of motivic homotopy theory.

Findings

The cdh-topology satisfies conditions for Brown-Gersten approach.

Unstable motivic homotopy categories are nearly equivalent under resolution of singularities.

Standard cd-topologies meet the necessary conditions for homotopy theory methods.

Abstract

The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category built out of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is very desirable to be able to work with all schemes instead of the smooth ones. In this paper we prove that, under the resolution of singularities assumption, the unstable motivic homotopy category of all schemes over a field with respect to the cdh-topology is almost equivalent to the unstable motivic homotopy category of smooth schemes over the same field with respect to the Nisnevich topology. In order to do it we show that the standard cd-topologies on the category of Noetherian schemes, including the cdh-topology, satisfy certain conditions which allows one to use the generalized version of the Brown-Gersten approach to the homotopy theory of simplicial sheaves.