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Algebraic $K$-theory of the two-periodic first Morava $K$-theory

Haldun Özgür Bayındır

Abstract

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of $T(2)_*\text{K}(ku)$ for $p>3$. Through this, we also produce a new algebraic $K$-theory computation; namely we obtain $T(2)_*\text{K}(ku/p)$, where $ku/p$ is the $2$-periodic Morava $K$-theory spectrum of height $1$.

Algebraic $K$-theory of the two-periodic first Morava $K$-theory

Abstract

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of for . Through this, we also produce a new algebraic -theory computation; namely we obtain , where is the -periodic Morava -theory spectrum of height .
Paper Structure (16 sections, 28 theorems, 116 equations)

This paper contains 16 sections, 28 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Graded ring spectra
  4. Complex $K$-theory spectrum as a $p-1$-graded $E_\infty$-ring
  5. Complex $K$-theory spectrum
  6. Adjoining roots to ring spectra
  7. Complex $K$-theory spectrum via root adjunction
  8. Two periodic Morava K-theory as a $p-1$-graded $ku_p$-algebra
  9. Graded THH, $\mathop{\mathrm{\textup{TC}^-}}\nolimits$ and TC
  10. Weight zero splitting of THH for root adjunctions.
  11. Logarithmic THH of the complex $K$-theory spectrum
  12. Topological cyclic homology of complex $K$-theory
  13. Higher Bott element
  14. Topological cyclic homology
  15. Algebraic $K$-theory of the $2$-periodic Morava $K$-theory
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $p>3$ be a prime. There is an isomorphism of $\mathbb{F}_p[b]$-algebras: where $\lvert b \rvert = 2p+2$.

Theorems & Definitions (65)

  • Theorem 1.1: ausoni2010kthryofcomplexkthry, Theorem \ref{['theo restated Ausoni algebraic k theory of ku']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Theorem \ref{['theo restated kth of ku mod p']}
  • Proposition 4.3
  • proof
  • Theorem 4.5
  • ...and 55 more