Motivic spectra and universality of $K$-theory

TL;DR

This paper develops a broad theory of motivic spectra without assuming $ ext{A}^1$-homotopy invariance and demonstrates that $K$-theory of schemes is a universal Zariski sheaf of spectra with specific properties.

Contribution

It introduces a general framework for motivic spectra and establishes the universality of $K$-theory as a Zariski sheaf with additional structure.

Findings

$K$-theory is a universal Zariski sheaf of spectra

$K$-theory admits an action of the Picard stack

$K$-theory satisfies the projective bundle formula

Abstract

We develop a theory of motivic spectra in a broad generality; in particular $\mathbb{A}^1$-homotopy invariance is not assumed. As an application, we prove that $K$-theory of schemes is a universal Zariski sheaf of spectra which is equipped with an action of the Picard stack and satisfies projective bundle formula.