On higher real $K$-theories and finite spectra

TL;DR

This paper explores higher chromatic analogues of real K-theory spectra, their properties, and implications for algebraic K-theory and finite spectra, revealing new constraints on Moore spectra valuations.

Contribution

It introduces and studies the spectra eo_h as fp spectra of type h, and connects these to algebraic K-theory and Moore spectra valuations.

Findings

eo_h spectra are fp spectra of type h

Partial answer to Levy's question on algebraic K-theory of finite spectra

Valuation constraints on Moore spectra products

Abstract

We study higher chromatic height analogues $eo_h$ of the connective real $K$-theory spectrum $ko$. We show that $eo_h$ is an fp spectrum of type $h$ in the sense of Mahowald--Rezk. We use these to study an Euler characteristic for fp spectra introduced by Ishan Levy, and give a partial answer to a question of Levy regarding the algebraic $K$-theory of the category of finite type $h$ spectra. As a corollary, we prove that if the generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ exists, then the $2$-adic valuation of $\prod i_j$ must exceed that of the height $h$.