Motivic spectra and universality of $K$-theory

TL;DR

This work develops a general theory of motivic spectra by inverting ${\mathbb P}^1$ in the Zariski topos and applies it to algebraic $K$-theory. It shows that non-connective $K$-theory is the universal oriented Zariski sheaf of spectra with Picard stack action satisfying the projective bundle formula, and it extends this universality to Selmer $K$-theory with étale descent. The construction hinges on a robust framework of motivic spectra, fundamental stability, and an orientation calculus together with Chern classes and formal group laws, culminating in a precise description of $K$-theory as a universal object among pbf-local motivic theories. The results provide a structural lens for motivic operations, periodicity, and filtrations on $K$-theory, with potential implications for motivic homotopy theory and arithmetic geometry.

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.

Theorems & Definitions (174)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Proposition 1.2.2
• proof
• Remark 1.2.4
• Lemma 1.2.5
• proof
• Remark 1.3.2