Multiplicative Properties of the Slice Filtration

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.