Relations between slices and quotients of the algebraic cobordism spectrum

TL;DR

This paper investigates the relationship between slices and quotients of the algebraic cobordism spectrum, proving a relative version of Voevodsky's conjecture under certain conditions and discussing implications for K-theory and rational slices.

Contribution

It establishes a relative statement about the slices of the algebraic cobordism spectrum and connects it to a conjecture by Voevodsky, extending understanding of algebraic cobordism.

Findings

Proves a relative statement about slices of MGL spectrum.

Shows the map to a quotient of MGL corresponds to the zero-slice.

Outlines implications for K-theory and rational slices.

Abstract

We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.