On the Multiplicities of a Motive

The aim of this article, in the vein of [1], is to study abstractly the properties of categories of pure motives and to make clear(er) which of them are formal and which are of a more arithmetic-geometric nature. We work with a rigid tensor category A such that K = End(1) is a field of characteristic 0. Our main tool, which was the motivation for this work, is the multiplicities of an object M ∈ A when A is semi-simple: they are a collection of central scalars which relates the categorical trace with the ring-theoretic trace (Proposition 1.2). It turns out that the proprety for these multiplicities to be integers or, better, to be so after extending scalars from K to its algebraic closure, is very well-behaved and is satisfied in many important cases. Namely: • The full subcategory Aint of A formed by such objects is thick, tensor, rigid, contains the finite-dimensional objects in the sense of Kimura-O’Sullivan and is preserved under tensor functors to another semi-simple rigid category (Corollary 2.5). • Aint = A if A is of “homological origin” (Theorem 4.5). The category of pure motives over a field modulo numerical equivalence is semi-simple thanks to Jannsen’s theorem [4], and of homological origin.