Schur functors and motives

In [Kim], Kimura introduced the notion of a “finite dimensional” motive (which we will refer to as “Kimura-finite” motive) and he conjectured that all Q-linear motives modulo rational equivalence are Kimura-finite. The same notion was introduced independently in a different context by O’Sullivan. Kimura-finiteness has been the subject of several articles recently ([GP02], [GP], [AK02]). In [GP], Guletskiı̆ and Pedrini proved that if X is a smooth projective surface with pg = 0, then the motive of X is Kimura-finite if and only if Bloch’s conjecture holds for X , i.e., the kernel of the Albanese map vanishes. In this dissertation we introduce the notion of Schur-finite motives, that is, motives which are annihilated by a Schur functor. We study its relation to Kimura-finiteness and in particular we show that this new notion is more flexible than Kimura’s. Moreover, we show that the motive of any curve is Kimura-finite. In the first chapter we first introduce some basic notions coming from representation theory, such as Schur functors and Tannakian categories. Then, we recall the constructions of the categories of classical motives, and also of Voevodsky’s triangulated category DM. In the second chapter, we define the central notion of Schur-finiteness. We study its basic properties and its relations with Kimura-finiteness in the most general setting. We then proceed to study more particular examples, i.e., how Schur-finiteness behaves with respect to short exact sequences in abelian categories and triangles in derived categories. The third and last chapter analyzes the class of Schur-finite objects in the categories of classical motives and in the category DM. We prove that Schur-finiteness has the two out of three property for triangles in DM, and this allows us to prove that the motive of every curve is Kimura-finite. (This last result has also been obtained by Guletskiı̆.) We close with an example due to O’Sullivan of a Schur-finite motive which is not Kimura-finite. I would like to thank Chuck Weibel for reasons too numerous to list. I am deeply indebted to both Luca Barbieri Viale and Filippo De Mari for being my mentors during the early stages of my career. I am grateful to Wolmer Vasconcelos, Siddhartha Sahi and Claudio Pedrini for their valuable comments and for serving as members of the defense committee. I would also like to thank Friedrich Knop and Bruno Kahn for the exchanges of ideas. On a more personal note, I thank Marco, Jooyoun, Manuela, Tony, Andrea, Daniela, Davide and Alina for helping me survive grad school. This dissertation is dedicated to my parents.