Thick ideals in equivariant and motivic stable homotopy categories

TL;DR

This paper explores the structure of thick ideals in the stable motivic homotopy category over various fields, establishing connections with classical and equivariant homotopy theories and analyzing motivic Morava K-theories.

Contribution

It identifies thick ideals in motivic homotopy categories via comparison functors and proves a motivic analogue of the Bousfield class decomposition for Morava K-theories.

Findings

Thick ideals in SH(k) relate to classical and equivariant theories.

Decomposition of Bousfield classes for motivic Morava K-theories AK(n).

AK(n)-acyclicity implies AK(n-1)-acyclicity for p>2.

Abstract

We study thick ideals in the stable motivic homotopy category SH(k) and in its subcategories of compact and of finite cellular objects. If k is a subfield of the complex or even the real numbers, then using comparison functors we find thick ideals corresponding to thick ideals in classical or Z/2-equivariant stable homotopy theory, respectively. We also study motivic Morava K-theories AK(n), for which we prove the motivic analogue of the decomposition of the Bousfield class of E(n) into Bousfield classes of K(i)'s over the complex numbers if p>2. In that case we also prove that AK(n)-acyclicity implies AK(n-1)-acyclicity.