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Purity in chromatically localized algebraic $K$-theory

Markus Land, Akhil Mathew, Lennart Meier, Georg Tamme

Abstract

We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\geq 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \dots \oplus T(n)$-localization of $R$. This complements a classical result of Waldhausen in rational $K$-theory. Combining our result with work of Clausen--Mathew--Naumann--Noel, one finds that $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $\mathrm{TC}(τ_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$.

Purity in chromatically localized algebraic $K$-theory

Abstract

We prove a purity property in telescopically localized algebraic -theory of ring spectra: For , the -localization of only depends on the -localization of . This complements a classical result of Waldhausen in rational -theory. Combining our result with work of Clausen--Mathew--Naumann--Noel, one finds that in fact only depends on the -localization of , again for . As consequences, we deduce several vanishing results for telescopically localized -theory, as well as an equivalence between and after -localization for .

Paper Structure

This paper contains 15 sections, 46 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries from chromatic homotopy theory
  4. Some localization functors
  5. Localizing invariants and K-theory
  6. Proof of the Purity Theorem
  7. The highly connective case
  8. A truncating property
  9. The general case
  10. Consequences and examples
  11. Direct consequences
  12. Examples of vanishing results
  13. Examples of purity
  14. Consequences for K(1)-local K-theory
  15. Consequences for $T(n)$-local $K$-theory for $n\geq 2$

Key Result

Theorem 1

Let $A$ be a ring spectrum.

Theorems & Definitions (98)

  • Theorem 1
  • Theorem 1.1: CMNN2
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 88 more