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Remarks on the motivic sphere without $\mathbb A^1$-invariance

Marc Hoyois

Abstract

We generalize several basic facts about the motivic sphere spectrum in $\mathbb A^1$-homotopy theory to the category $\mathrm{MS}$ of non-$\mathbb A^1$-invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace $\mathbf 1_\mathbb Q^+$ of the rational motivic sphere is the rational motivic cohomology spectrum $\mathrm H\mathbb Q$, which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of $\mathrm H\mathbb Q$-modules in $\mathrm{MS}$: a rational motivic spectrum is an $\mathrm H\mathbb Q$-module iff it is orientable, iff the involution $\langle -1\rangle$ is the identity, iff the Hopf map $η$ is zero, iff it satisfies étale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over $\mathbb Z[ζ_n]$ for any $n\geq 3$.

Remarks on the motivic sphere without $\mathbb A^1$-invariance

Abstract

We generalize several basic facts about the motivic sphere spectrum in -homotopy theory to the category of non--invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace of the rational motivic sphere is the rational motivic cohomology spectrum , which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of -modules in : a rational motivic spectrum is an -module iff it is orientable, iff the involution is the identity, iff the Hopf map is zero, iff it satisfies étale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over for any .

Paper Structure

This paper contains 7 sections, 26 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The multiplicative structure on the Tate circle
  3. The Hopf map
  4. The Steinberg relation
  5. The Grothendieck–Witt relations
  6. The positive eigenspace of the rational motivic sphere
  7. Beilinson motives

Key Result

Theorem 1.1

The following relations hold in the graded ring $\pi_0\mathrm{Map}(\mathbf 1,(\Sigma^{-1}\mathbb P^1)^{\otimes *})$:

Theorems & Definitions (57)

  • Theorem 1.1: The Milnor–Witt K-theory relations, Theorems \ref{['thm:easy-MW']} and \ref{['thm:steinberg']}
  • Theorem 1.2: The Grothendieck–Witt relations, Corollary \ref{['cor:GW']}
  • Theorem 1.3: Motivic Serre finiteness, Corollary \ref{['cor:Q+']}
  • Theorem 1.4: Characterizations of Beilinson motives, Theorem \ref{['thm:beilinson']}
  • Remark 2.1: Equivariant $\mathbb P$-homotopies
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4: The Hopf relations
  • ...and 47 more