Schur Functors and Motives

Abstract

In this article we study the class of Schur-finite motives, that is, motives which are annihilated by a Schur functor. We compare this notion to a similar one due to Kimura. In particular, we show that the motive of any curve is Kimura-finite. This last result has also been obtained by V. Guletskĭi. We conclude with an example by O’Sullivan of a non Kimura-finite motive which is Schur-finite. If λ is a partition of n, the Schur functor Sλ sends a motive X to the direct summand of X⊗n determined by λ. We say that X is Schur-finite if it is annihilated by some Schur functor. This definition is due to Deligne ([Del02]) who related Schur-finiteness to super-Tannankian categories. In this paper we study Schur-finite objects in several categories, including motives. Kimura and O’Sullivan have independently defined a stronger notion which we will call Kimura-finiteness. Kimura showed in [Kim] that if a motive M is Kimura-finite, then any endomorphism of M is either nilpotent or detected by cohomology. Kimura-finiteness has been examined further by Guletskĭi and Pedrini (see [GP02] and [GP03]) and by Andre and Kahn (see [AK02]). In the first part of this paper we define Schur-finiteness and study its properties in the setting of a Q-linear tensor category. In particular, we investigate its behavior with respect to tensor functors and triangles in the derived category of an abelian category with tensor. In the second part we apply this formalism to the category of classical motives and to Voevodsky’s category DM Nis (k,Q). We conclude by showing that the motives of all curves are Kimura-finite. This last result has also been obtained by Guletskĭi in [Gulb]. Using a result by O’Sullivan, we produce a motive which is Schur-finite but not Kimura-finite. The author would like to thank Chuck Weibel for everything he did. Much credit is due to Pierre Deligne, who introduced the definition. We are also grateful to Claudio Pedrini, Luca Barbieri Viale, and Bruno Kahn for precious conversations and their comments. The author would like also to thank the Instituto de Matematicas of UNAM in Morelia, Mexico for hospitality while this manuscript was prepared. ∗Partially supported by INdAM “Borse Di Studio Per L’Estero”