Slices and Transfers

TL;DR

This paper investigates the slice filtration for S^1-spectra over a field, revealing that higher slices relate to effective motives, while the 0th slice generally does not admit transfers, raising new questions.

Contribution

It demonstrates that higher slices in the filtration are associated with effective motives, and shows the 0th slice typically lacks transfer structures, advancing understanding of slice filtrations.

Findings

Higher slices are Eilenberg-MacLane spectra of effective motives.

The 0th slice generally does not admit transfers.

Raises questions about properties of the slice filtration.

Abstract

We study the slice filtration for S^1-spectra over a field k, and raise a number of questions regardings its properties. We show that the slices, except for the 0th slice, admit a further filtration whose layers are in a natural way the Eilenberg-Maclane spectra associated to a homotopy invariant complex of Nisnevic sheaves with transfer, i.e., an effective motive. We give an example to show that in general the 0th slice does not admit transfers.