Isotropic motivic fundamental groups

TL;DR

This paper develops a framework for isotropic motivic fundamental groups using motivic t-structures and Tannakian categories, providing computations and linking motives to affine derived group schemes.

Contribution

It introduces a novel approach to defining and computing isotropic motivic fundamental groups via motivic t-structures and derived algebraic geometry.

Findings

Defined isotropic motivic fundamental groups in the motivic setting

Computed these groups for punctured projective line and split tori

Established an equivalence between isotropic Tate motives and representations of affine derived group schemes

Abstract

The main goal of this paper is to study relative versions of the category of modules over the isotropic motivic Brown-Peterson spectrum, with a particular emphasis on their cellular subcategories. Using techniques developed by Levine, we equip these categories with motivic $t$-structures, whose hearts are Tannakian categories over ${\mathbb F}_2$. This allows to define isotropic motivic fundamental groups, and to interpret relative isotropic Tate motives in the heart as their representations. Moreover, we compute these groups in the cases of the punctured projective line and split tori. Finally, we also apply Spitzweck's derived approach to establish an identification between relative isotropic Tate motives and representations of certain affine derived group schemes, whose 0-truncations coincide with the aforementioned isotropic motivic fundamental groups.