Multiplicative Properties of the Slice Filtration

TL;DR

This paper demonstrates that the slice filtration in motivic homotopy theory is compatible with monoidal structures, leading to ring and module structures on slices and implications for spectral sequences.

Contribution

It establishes the compatibility of the slice filtration with the symmetric monoidal structure in motivic spectra, showing slices form ring and module spectra and affecting spectral sequence pairings.

Findings

Zero slice of the sphere spectrum is a ring spectrum.

Slices of spectra are modules over the zero slice.

Compatibility leads to pairings in the motivic Atiyah-Hirzebruch spectral sequence.

Abstract

We show that the slice filtration introduced by Voevodsky is compatible in a suitable sense with the symmetric monoidal structure in the category of motivic symmetric T-spectra constructed by Jardine. It follows from this compatibility that the zero slice of the sphere spectrum s_{0}(1) is a ring spectrum and that for every symmetric T-spectrum X, and every integer n; the n-slice s_{n}(X) is a module over s_{0}(1). In particular, if the base scheme is a field of characteristic zero, we have that all the slices s_{n}(X) are big motives in the sense of Voevodsky. We also get as a corollary that the smash product of symmetric T-spectra induces pairings in the motivic Atiyah-Hirzebruch spectral sequence.