Motivic stable homotopy groups

Abstract

Motivic homotopy theory was developed by Morel and Voevodsky in the 1990s. The original motivation for the theory was to import homotopical techniques into algebraic geometry. This chapter introduces the motivic Adams spectral sequence, which is one of the key tools for computing stable motivic homotopy groups. The precise relationship is that the motivic stable homotopy groups are the global sections of the motivic stable homotopy sheaves. For cellular motivic spectra, the motivic stable homotopy groups do detect equivalences, and the most commonly studied motivic spectra are typically cellular. So a thorough understanding of motivic stable homotopy groups over arbitrary fields leads back to complete information about the sheaves as well. The chapter considers motivic stable homotopy groups over larger classes of fields. Naturally, specific information is harder to obtain when the base field is allowed to vary widely.