Geometric symmetric powers in the motivic homotopy category

TL;DR

This paper explores the extension and comparison of geometric symmetric powers within the motivic homotopy category, providing insights into their properties in both unstable and stable contexts.

Contribution

It introduces a method to extend symmetric powers to motivic spaces and compares different symmetric power constructions in motivic homotopy theory.

Findings

Geometric symmetric powers can be extended via left Kan extensions.

Comparison of geometric and other symmetric powers in motivic homotopy categories.

Insights into properties of symmetric powers over fields.

Abstract

Symmetric powers of quasi-projective schemes can be extended, in terms of left Kan extensions, to geometric symmetric powers of motivic spaces. In this paper, we study geometric symmetric powers and compare with various symmetric powers in the unstable and stable motivic homotopy category of schemes over a field.