**Astérisque**

Volume: 335;
2011;
291 pp;
Softcover

MSC: Primary 14; 18;
**Print ISBN: 978-2-85629-305-8
Product Code: AST/335**

List Price: $90.00

Individual Member Price: $72.00

# Multiplicative Properties of the Slice Filtration

Share this page
*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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Table of Contents

# Table of Contents

## Multiplicative Properties of the Slice Filtration

#### Readership

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