[Paper Review] A homological approach to chromatic complexity of algebraic K-theory

Gabriel Angelini‐Knoll, J. D. Quigley|arXiv (Cornell University)|Aug 24, 2019

Homotopy and Cohomology in Algebraic Topology39 references4 citations

TL;DR

This paper establishes that topological periodic cyclic homology (TP) of the Thom spectra $y(n)$ exhibits chromatic complexity $n+1$, while relative algebraic K-theory, topological cyclic homology (TC), and topological negative cyclic homology (TC⁻) at least preserve chromatic complexity. Using homological spectral sequences and the Greenlees filtration, the authors prove that $K(m)_*(TP(y(n))) \cong 0$ for $1 \leq m \leq n$, providing strong evidence for a chromatic red-shift in TP, supporting a variant of the Ausoni–Rognes red-shift conjecture at all chromatic heights.

ABSTRACT

The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood.

Motivation & Objective

• To investigate whether topological periodic cyclic homology (TP) shifts chromatic height for $E_1$ ring spectra, particularly for the family $y(n)$.
• To extend the chromatic red-shift conjecture beyond algebraic K-theory and topological cyclic homology to include TP and TC⁻.
• To analyze the vanishing of Morava K-theory in TP, TC⁻, TC, and relative K-theory of $y(n)$ using spectral sequences and filtration techniques.
• To provide evidence for a generalized red-shift phenomenon in TP at all chromatic heights, using $y(n)$ as a test case.

Proposed method

• Utilizes the Greenlees filtration of $TP(y(n))$ to construct a spectral sequence for computing its homotopy groups.
• Applies the homological Tate and homotopy fixed point spectral sequences to analyze the $E_1$-page and convergence of the filtration.
• Employs the $K(m)$-based Bökstedt spectral sequence to compute $K(m)_*THH(y(n))$ and deduce vanishing results.
• Uses the fiber sequence $\Sigma THH(y(n))_{h\mathbb{T}} \to TC^-(y(n)) \to TP(y(n))$ to relate TC⁻ and TP via homotopy fixed points.
• Applies the Dundas-Goodwillie-McCarthy theorem to relate $TC(y(n), H\mathbb{F}_2)$ to $K(y(n), H\mathbb{F}_2)$, enabling K-theory computations.
• Establishes bounds on comodule primitive degrees in $H_*(TP(y(n))[i])$ using coaction structure and filtration arguments.

Experimental results

Research questions

• RQ1 Does topological periodic cyclic homology (TP) of $y(n)$ exhibit chromatic complexity $n+1$?
• RQ2 Does TP shift chromatic height at all chromatic levels, supporting a red-shift conjecture?
• RQ3 Do relative algebraic K-theory, TC⁻, and TC preserve chromatic complexity for $y(n)$?
• RQ4 Can the vanishing of $K(m)_*$ for $1 \leq m \leq n$ in $TP(y(n))$ be established via spectral sequences and filtration?
• RQ5 Is there a uniform bound on the degree of comodule primitives in $H_*(TP(y(n))[i])$?

Key findings

• For $1 \leq m \leq n$, $K(m)_*(TP(y(n))) \cong 0$, proving that TP increases chromatic complexity from $n$ to $n+1$.
• The degree of comodule primitives in $H_*(TP(y(n))[i])$ is uniformly bounded by $2^{n+1}$, independent of $i$.
• For $1 \leq m \leq n$, $K(m)_*(TC^-(y(n), H\mathbb{F}_2)) \cong 0$, showing TC⁻ preserves chromatic complexity.
• For $1 \leq \ell \leq n-1$, $K(\ell)_*(TC^-(y(n))) \cong 0$, confirming vanishing in TC⁻ at lower chromatic levels.
• Relative algebraic K-theory $K(y(n), H\mathbb{F}_2)$ satisfies $K(m)_*(K(y(n), H\mathbb{F}_2)) \cong 0$ for $0 \leq m \leq n-1$, preserving chromatic complexity.
• The fiber sequence $TC(y(n), H\mathbb{F}_2) \to TC^-(y(n), H\mathbb{F}_2) \to TP(y(n), H\mathbb{F}_2)$ and the Dundas-Goodwillie-McCarthy theorem imply $TC(y(n), H\mathbb{F}_2) \simeq K(y(n), H\mathbb{F}_2)$, with vanishing $K(m)$-homology for $0 \leq m \leq n-1$.