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Isotropic motivic fundamental groups

Fabio Tanania

TL;DR

The paper develops a theory of isotropic motivic categories at the prime 2, constructing motivic t-structures on cellular isotropic MBP-modules and identifying their hearts as neutral Tannakian categories. It defines isotropic motivic fundamental groups as Tannaka groups extending $\mathbb{G}_m$ by pro-unipotent radicals, and computes them for $\mathbb{P}^1\setminus S$ and split tori, expressing the associated motivic categories as graded representations. A key bridge is built between isotropic Milnor K-theory and isotropic MBP-cohomology, including a Koszulity conjecture that governs when derived hearts recover module categories. The paper also uses Spitzweck’s derived framework to realize isotropic Tate motives as representations of affine derived group schemes, with a 0-truncation recovering the classical isotropic fundamental groups and connecting to the Koszulity question. Overall, it provides concrete computations, a derived perspective, and a clear link between motivic, Milnor K-theoretic, and Tannakian data in the isotropic setting.

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.

Isotropic motivic fundamental groups

TL;DR

The paper develops a theory of isotropic motivic categories at the prime 2, constructing motivic t-structures on cellular isotropic MBP-modules and identifying their hearts as neutral Tannakian categories. It defines isotropic motivic fundamental groups as Tannaka groups extending by pro-unipotent radicals, and computes them for and split tori, expressing the associated motivic categories as graded representations. A key bridge is built between isotropic Milnor K-theory and isotropic MBP-cohomology, including a Koszulity conjecture that governs when derived hearts recover module categories. The paper also uses Spitzweck’s derived framework to realize isotropic Tate motives as representations of affine derived group schemes, with a 0-truncation recovering the classical isotropic fundamental groups and connecting to the Koszulity question. Overall, it provides concrete computations, a derived perspective, and a clear link between motivic, Milnor K-theoretic, and Tannakian data in the isotropic setting.

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 -structures, whose hearts are Tannakian categories over . 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.

Paper Structure

This paper contains 9 sections, 30 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Levine's categories of Tate type
  4. Relative isotropic motivic categories
  5. Isotropic Milnor K-theory
  6. Isotropic Tate motives over finite extensions
  7. Isotropic motivic fundamental groups
  8. The isotropic motivic fundamental groups of $\mathop{\mathrm{{\mathbb P}^1}}\nolimits \setminus S$ and $\mathop{\mathrm{\mathbb G}_m^{\times n}}\nolimits$
  9. Derived isotropic motivic fundamental groups

Key Result

Theorem 1.1

For every smooth isotropic variety $X$ over $k$, there is a non-degenerate $t$-structure on $\mathop{\mathrm{\mathrm{MBP}}}\nolimits^{iso}_X-\mathop{\mathrm{\mathrm{Mod}}}\nolimits_{cell}^{\omega}$ whose heart $\mathop{\mathrm{\mathrm{MBP}}}\nolimits^{iso}_X-\mathop{\mathrm{\mathrm{Mod}}}\nolimits_{

Theorems & Definitions (90)

  • Theorem 1.1: Theorem \ref{['main']}
  • Conjecture 1.2: Conjecture \ref{['kos']}
  • Theorem 1.3: Theorems \ref{['pones']} and \ref{['uisogmn']}
  • Theorem 1.4: Theorem \ref{['der']}
  • Definition 3.1: 3, Definition 1.1
  • Remark 3.2
  • Theorem 3.3: 3, Lemma 1.2 and Definition 1.3
  • Definition 3.4
  • Theorem 3.5: 3, Theorem 1.4
  • Remark 3.6
  • ...and 80 more