On the Orientability of the Slice Filtration

TL;DR

This paper proves that the layers of the slice filtration in motivic stable homotopy are strict modules over algebraic cobordism, and shows the zero slice is an oriented ring spectrum with an additive formal group law, confirming Voevodsky's conjecture.

Contribution

It demonstrates the orientability of slices in the motivic stable homotopy category and establishes their structure as motives with transfers over certain schemes.

Findings

Slices are strict modules over algebraic cobordism spectrum.

Zero slice of any commutative ring spectrum is oriented with additive formal group law.

With rational coefficients, slices are motives with transfers.

Abstract

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\stablehomotopy$ are strict modules over Voevodsky's algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\stablehomotopy$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we get that with rational coefficients the slices are in fact motives in the sense of Cisinski-D{\'e}glise \cite{mixedmotives}, and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky \cite[conjecture 11]{MR1977582}.