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K-theory and matrix transfers

Grigory Garkusha

TL;DR

The paper develops matrix transfers as a noncommutative analogue of Voevodsky framed transfers to model bivariant algebraic K-theory. It constructs symmetric and big symmetric matrix motives via spectral categories, and proves a network of equivalences linking matrix transfers to Waldhausen K-theory and to classical K-motives, providing a local, non-$\mathbb{A}^1$-localized framework for motivic K-theory. A central outcome is that symmetric matrix motives of varieties are stably equivalent to $K$-motives, thereby recovering Quillen’s $K$-theory through motivic homotopy techniques and allowing a clean passage to $KGL$-modules after inverting the exponential characteristic. The framework yields a robust toolkit for working with $K$-theory motivically, including a cancellation theorem and a concrete model for $\mathsf{kgl}$-modules, situating bivariant K-theory firmly inside enriched motivic homotopy theory. Overall, the work unifies transfers, matrix motives, and motivic K-theory into a coherent spectral-motive formalism with strong functorial and monoidal properties.

Abstract

We introduce and study matrix transfers to achieve elementary models for bivariant $K$-theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties introduced in this paper. Symmetric matrix motives recover $K$-motives and fit in a closed symmetric monoidal triangulated category of symmetric matrix motives constructed in this paper by using methods of enriched motivic homotopy theory.

K-theory and matrix transfers

TL;DR

The paper develops matrix transfers as a noncommutative analogue of Voevodsky framed transfers to model bivariant algebraic K-theory. It constructs symmetric and big symmetric matrix motives via spectral categories, and proves a network of equivalences linking matrix transfers to Waldhausen K-theory and to classical K-motives, providing a local, non--localized framework for motivic K-theory. A central outcome is that symmetric matrix motives of varieties are stably equivalent to -motives, thereby recovering Quillen’s -theory through motivic homotopy techniques and allowing a clean passage to -modules after inverting the exponential characteristic. The framework yields a robust toolkit for working with -theory motivically, including a cancellation theorem and a concrete model for -modules, situating bivariant K-theory firmly inside enriched motivic homotopy theory. Overall, the work unifies transfers, matrix motives, and motivic K-theory into a coherent spectral-motive formalism with strong functorial and monoidal properties.

Abstract

We introduce and study matrix transfers to achieve elementary models for bivariant -theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties introduced in this paper. Symmetric matrix motives recover -motives and fit in a closed symmetric monoidal triangulated category of symmetric matrix motives constructed in this paper by using methods of enriched motivic homotopy theory.

Paper Structure

This paper contains 9 sections, 33 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The category of matrix transfers $\mathop{\mathrm{\mathsf{Mtr}}}\nolimits_*(\mathop{\mathrm{Alg}}\nolimits^u_k)$
  3. The spectral category $\mathop{\mathrm{\mathcal{M}\mathsf{tr}}}\nolimits_*(k)$
  4. The spectral category $\mathbb K$
  5. Additivity Theorem
  6. Comparison Theorem
  7. Symmetric matrix motives
  8. The triangulated category of symmetric matrix motives
  9. The triangulated category of big symmetric matrix motives

Key Result

Lemma 2.5

For any $A\in\mathop{\mathrm{Alg}}\nolimits^u_k$ the functor $\mathop{\mathrm{\mathsf{Mtr}}}\nolimits(A,-)$ is additive. Moreover, $\mathop{\mathrm{\mathsf{Mtr}}}\nolimits(A,B)=\mathop{\mathrm{Hom}}\nolimits_{{\mathop{\mathrm{Alg}}\nolimits_{k}}}(A,M_\infty (B))$, where $B\in\mathop{\mathrm{Alg}}\no

Theorems & Definitions (65)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Theorem 2.8
  • ...and 55 more