On the Orientability of the Slice Filtration

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{é}glise \cite{mixedmotives}, and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky \cite[conjecture 11]{MR1977582}.

9 pages, Following a remark of Denis-Charles Cisinski the condition on the base scheme being geometrically unibranch has been removed

References in corpus (2)

• Triangulated categories of mixed motives
• On the zero slice of the sphere spectrum