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Trace methods for stable categories I: The linear approximation of algebraic K-theory

Yonatan Harpaz, Thomas Nikolaus, Victor Saunier

Abstract

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods.

Trace methods for stable categories I: The linear approximation of algebraic K-theory

Abstract

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods.

Paper Structure

This paper contains 21 sections, 58 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Acknowledgements
  4. Laced categories
  5. Bimodules on a stable category
  6. Laced categories as the tangent bundle of CatEx
  7. The symmetric monoidal structure on CatLace
  8. Tensors and cotensors by unstable laced categories
  9. The K-theory of laced categories
  10. Laced semi-orthogonal decompositions
  11. The universal property of laced K-theory
  12. Trace-like functors and topological Hochschild homology
  13. Topological Hochschlid homology and the cyclic bar construction
  14. Trace-like functors
  15. The universal property of THH
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a stable category, then the functor exhibits $\mathrm{BiMod}(\mathcal{C})$ as the stabilization of $\mathrm{Cat}^{\mathrm{ex}}_{/\mathcal{C}}$. In particular, it induces an equivalence under which the functor $\mathrm{Lace}(\mathcal{C},-)$ identifies with $\Omega^{\infty}_{/\mathcal{C}}$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7: Stable K-theory is THH
  • Example 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 109 more