Algebraic K-theory of elliptic cohomology

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

doi:10.2140/gt.2025.29.619

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Algebraic $K\mskip-2.5mu$-theory of elliptic cohomology

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

Geometry & Topology 29 (2025) 619–686

DOI: 10.2140/gt.2025.29.619

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 $p \geq 7$ , and show that it is a finitely generated and free $𝔽_{p} [v_{3}]$ -module on $12 p + 4$ generators in explicit degrees within the range $- 1 \leq * \leq 2 p^{3} + 2 p^{2} + 2 p - 3$ . 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.

Keywords

algebraic $K$-theory, cyclotomic trace map, topological cyclic homology, elliptic cohomology, truncated Brown–Peterson spectrum, Smith–Toda complex, $v_n$-periodicity, chromatic redshift, power operation, homotopy fixed point, Tate construction

Mathematical Subject Classification

Primary: 19D50, 19D55, 55P43, 55Q51

Secondary: 55N20, 55N34, 55N91, 55Q10, 55T25

References

Publication

Received: 22 December 2022

Revised: 24 October 2023

Accepted: 25 November 2023

Published: 12 March 2025

Proposed: Mark Behrens

Seconded: Stefan Schwede, Jesper Grodal

Authors

Gabriel Angelini-Knoll
Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539 Villetaneuse France

Christian Ausoni
Université Sorbonne Paris Nord, LAGA, CNRS, UMR 7539 Villetaneuse France

Dominic Leon Culver
Max Planck Institute for Mathematics Bonn Germany

Eva Höning
Department of Mathematics Radboud University Nijmegen The Netherlands

John Rognes
Department of Mathematics University of Oslo Oslo Norway

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 2 (2025)