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2018 A motivic Grothendieck–Teichmüller group
Ismaël Soudères
Algebr. Geom. Topol. 18(2): 635-685 (2018). DOI: 10.2140/agt.2018.18.635


We prove the Beilinson–Soulé vanishing conjecture for motives attached to the moduli spaces 0 , n of curves of genus 0 with n marked points. As part of the proof, we also show that these motives are mixed Tate. As a consequence of Levine’s work, we thus obtain a well-defined category of mixed Tate motives over the moduli space of curves 0 , n . We furthermore show that the morphisms between the moduli spaces 0 , n obtained by forgetting marked points and by embedding boundary components induce functors between the associated categories of mixed Tate motives. Finally, we explain how tangential base points fit into these functorialities.

The categories we construct are Tannakian, and therefore have attached Tannakian fundamental groups, connected by morphisms induced by those between the categories. This system of groups and morphisms leads to the definition of a motivic Grothendieck–Teichmüller group.

The proofs of the above results rely on the geometry of the tower of the moduli spaces 0 , n . This allows us to treat the general case of motives over Spec ( ) with coefficients in , working in Spitzweck’s category of motives. From there, passing to coefficients, we deal with the classical Tannakian formalism and explain how working over Spec ( ) yields a more concrete description of the Tannakian groups.


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Ismaël Soudères. "A motivic Grothendieck–Teichmüller group." Algebr. Geom. Topol. 18 (2) 635 - 685, 2018.


Received: 13 April 2015; Revised: 31 August 2017; Accepted: 30 October 2017; Published: 2018
First available in Project Euclid: 22 March 2018

zbMATH: 06859600
MathSciNet: MR3773734
Digital Object Identifier: 10.2140/agt.2018.18.635

Primary: 14F42, 14J10, 19E15
Secondary: 14F05

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.18 • No. 2 • 2018
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