# Multiplicative Properties of the Slice Filtration

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*Pablo Pelaez*

A publication of the Société Mathématique de France

Let \(S\) be a Noetherian separated scheme of finite Krull dimension,
and \(\mathcal {SH}(S)\) be the motivic stable homotopy category of
Morel-Voevodsky. In order to get a motivic analogue of the Postnikov tower,
Voevodsky (MR 1977582) constructs the slice filtration by filtering
\(\mathcal {SH}(S)\) with respect to the smash powers of the
multiplicative group \(\mathbb G_{m}\).

The author shows that the slice
filtration is compatible with the smash product in Jardine's category
\(\mathrm {Spt}_{T}^{\Sigma }\mathcal {M}_{\ast}\) of motivic symmetric
\(T\)-spectra (MR 1787949) and describes several interesting
consequences that follow from this compatibility. Among the
consequences that follow from this compatibility is that over
a perfect field all the slices \(s_{q}\) are in a canonical way modules
in \(\mathrm {Spt}_{T}^{\Sigma }\mathcal {M}_{\ast }\) over the motivic
Eilenberg-MacLane spectrum \(H\mathbb Z\), and if the field has
characteristic zero it follows that the slices \(s_{q}\) are big motives
in the sense of Voevodsky. This relies on the work of Levine (MR 2365658),
Röndigs-Østvær (MR 2435654), and Voevodsky (MR 2101286).
It also follows that the smash product in \(\mathrm {Spt}_{T}^{\Sigma
}\mathcal {M}_{\ast }\) induces pairings in the motivic Atiyah-Hirzebruch
spectral sequence.

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#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.