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Algebraic K-theory of elliptic cohomology

Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver, Eva Höning, John Rognes

TL;DR

The paper computes the mod$(p,v_1,v_2)$-localized homotopy $V(2)_*TC(BP\langle2\rangle)$ for primes $p\ge7$, showing it is a finitely generated free $P(v_3)$-module on $12p+4$ generators in degrees from $-1$ to $2p^3+2p^2+2p-3$, explicitly described via generators $\partial,\lambda_1,\lambda_2,\lambda_3$ and the families $\Xi_{i,d}$ with $0<d<p$. This yields a corresponding description of $V(2)_*K(BP\langle2\rangle_p)$ and demonstrates chromatic redshift from $v_2$-periodicity to $v_3$-periodicity in a quantitative, computable setting, consistent with the Nikolaus–Scholze reformulation of TC and with the recent motivic filtration perspectives. The authors develop and employ $E_2$ and $E_3$ power operations, detailed filtrations, and $C_p$, $C_{p^n}$-Tate spectral sequences, together with the $\mathbb{T}$-Tate and $\mathbb{T}$-fixed point frameworks, to achieve complete control of the $V(2)$-homotopy groups and to transfer information to algebraic K-theory via the cyclotomic trace. These results support a descent picture for the motivic-like cohomology governing TC and K-theory of elliptic cohomology, and illuminate the structure of $TC(BP\langle2\rangle)$ in high chromatic layers. The methodology provides templates for future higher-n calculations and reinforces the redshift paradigm in a precise, computable manner.

Abstract

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.

Algebraic K-theory of elliptic cohomology

TL;DR

The paper computes the mod-localized homotopy for primes , showing it is a finitely generated free -module on generators in degrees from to , explicitly described via generators and the families with . This yields a corresponding description of and demonstrates chromatic redshift from -periodicity to -periodicity in a quantitative, computable setting, consistent with the Nikolaus–Scholze reformulation of TC and with the recent motivic filtration perspectives. The authors develop and employ and power operations, detailed filtrations, and , -Tate spectral sequences, together with the -Tate and -fixed point frameworks, to achieve complete control of the -homotopy groups and to transfer information to algebraic K-theory via the cyclotomic trace. These results support a descent picture for the motivic-like cohomology governing TC and K-theory of elliptic cohomology, and illuminate the structure of in high chromatic layers. The methodology provides templates for future higher-n calculations and reinforces the redshift paradigm in a precise, computable manner.

Abstract

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.
Paper Structure (12 sections, 52 theorems, 307 equations, 1 figure)

This paper contains 12 sections, 52 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Smith--Toda and truncated Brown--Peterson spectra
  3. Topological Hochschild homology
  4. Cyclotomic nomenclature
  5. Homotopy power operations
  6. Some $V(0)$- and $V(1)$-homotopy classes
  7. Approximate homotopy fixed points
  8. The $C_p$-Tate spectral sequence
  9. The $C_{p^2}$-Tate spectral sequence
  10. The $C_{p^n}$-Tate spectral sequences
  11. The $\mathbb{T}$-Tate spectral sequence
  12. Topological cyclic homology and algebraic $K$-theory

Key Result

Theorem 1.1

Let $p\ge7$. There is a preferred isomorphism of $P(v_3) \otimes E(\lambda_1, \lambda_2, \lambda_3)$-modules. This is a finitely generated and free $P(v_3)$-module on $12p+4$ explicit generators in degrees $-1 \le * \le 2p^3+2p^2+2p-3$.

Figures (1)

  • Figure 7.1: $E^3 \Longrightarrow V(0)_* F(S^{2p^2+1}, THH(BP))^{\mathbb{T}}$ in vertical degrees $* < 4p^2+2p-5$, with all $d^{2p}$-differentials (red) and selected $d^{4p-2}$-differentials (blue).

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Proposition 3.1: MS93*Rem. 4.3, AR05*Thm. 5.12
  • ...and 114 more