The Morel-Voevodsky localization theorem in spectral algebraic geometry

TL;DR

This paper extends the Morel-Voevodsky localization theorem to spectral algebraic spaces, leading to a derived nilpotent invariance result that simplifies the homotopy groups of certain ring spectra.

Contribution

It introduces a spectral algebraic geometry analogue of the localization theorem and establishes a new invariance property for connective commutative ring spectra.

Findings

Proved the localization theorem in spectral algebraic geometry.

Derived a nilpotent invariance result for A^1-homotopy.

Showed higher homotopy groups vanish under certain conditions.

Abstract

We prove an analogue of the Morel-Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a "derived nilpotent invariance" result which, informally speaking, says that A^1-homotopy invariance kills all higher homotopy groups of a connective commutative ring spectrum.