\'Etale motives of geometric origin

TL;DR

This paper characterizes étale motives of geometric origin over certain schemes using a categorical constructibility property, confirming their geometric nature and establishing key descent and excision properties.

Contribution

It provides a categorical characterization of étale motives of geometric origin, answering a longstanding question about their geometric origin and demonstrating their continuity, h-descent, and Milnor excision.

Findings

Étale motives of geometric origin are characterized by a categorical constructibility property.

They satisfy continuity, h-descent, and Milnor excision properties.

All constructible étale motives of geometric origin come from geometry.

Abstract

Over qcqs finite-dimensional schemes, we prove that \'etale motives of geometric origin can be characterised by a constructibility property which is purely categorical, giving a full answer to the question "Do all constructible \'etale motives come from geometry?" which dates back to Cisinski and D\'eglise's work. We also show that they afford the continuity property and satisfy h-descent and Milnor excision.