Algebraic K-theory of real topological K-theory

TL;DR

This work determines the $A(1)$-homotopy of $\mathrm{TC}(\mathrm{ko})$ at the prime $2$ by applying the prismatic and syntomic filtrations for $\mathbb{E}_\infty$-rings, extending the Bhatt–Morrow–Scholze framework via Hahn–Raksit–Wilson. It provides a precise associated graded description $\mathrm{gr}_{\mathrm{mot}}^* A(1)_* \mathrm{TC}(\mathrm{ko})$ as a free $\mathbb{F}_2[v_2^4]$-module of rank $52$ and derives the $A(1)_*$-structure on $\mathrm{TC}(\mathrm{ko})$, $\mathrm{K}(\mathrm{ko})$, and $\mathrm{K}(\mathrm{ko})^{\wedge}_2$ from detailed motivic/décent analyses. The paper also computes weight-2 prismatic cohomology and syntomic cohomology with $A(1)$-coefficients, tracks the differentials in the motivic spectral sequence, and uses cyclotomic trace and descent results to establish the height-$2$ telescope conjecture for $\mathrm{K}(\mathrm{ko})$, $\mathrm{K}(\mathrm{ko})^{\wedge}_2$, and $\mathrm{TC}(\mathrm{ko})$. Together these results quantify the chromatic redshift phenomena in this real-topological K-theory context and provide explicit algebraic models for $A(1)_*\mathrm{TC}(\mathrm{ko})$ and related $K$-theory groups. These findings have implications for understanding the chromatic and motivic structure of ring spectra at height $2$ in the $2$-local setting.

Abstract

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.

Figures (13)

• Figure 1.1: $\mathbb{F}_2[v_2]$-basis for the syntomic cohomology modulo $(2, \eta, v_1)$ of $\mathrm{ko}$, with lines of slope $-1$, $1$ and $1/3$ indicating multiplication by $\partial$, $\eta$ and $\nu$, respectively
• Figure 2.1: $\eta$-Bockstein $E_{1}\Longrightarrow \overline{V}(1)_{*}\mathop{\mathrm{gr}}\nolimits_{\mathrm{mot}}^{*}\mathop{\mathrm{THH}}\nolimits(\mathrm{ko})$
• Figure 4.1: Adams $E_2$-terms for $A(1)[00]$, $A(1)[10]$, $A(1)[01]$ and $A(1)[11]$ (from top to bottom)
• Figure 4.2: $v_2$-Bockstein $E_\infty \Longrightarrow \mathop{\mathrm{Ext}}\nolimits_{\mathrm{BP}_*\mathrm{BP}}(\mathrm{BP}_*, \mathrm{BP}_*/I_2)$
• Figure 4.3: Adams $E_2$-term for $C2$

Theorems & Definitions (129)

• Theorem A: Theorem \ref{['thm:A1htpyTCko']}
• Theorem B: Theorem \ref{['thm: syntomic A(1)']}
• Theorem C: Theorems \ref{['thm:k2-complete']} and \ref{['thm:k']}
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6