  Let k be a finite base field. In this note, making use of topological
periodic cyclic homology and of the theory of noncommutative motives, we prove
that the numerical Grothendieck group of every smooth proper dg k-linear
category is a finitely generated free abelian group. Along the way, we prove
moreover that the category of noncommutative numerical motives over k is
abelian semi-simple, as conjectured by Kontsevich. Furthermore, we show that
the zeta functions of endomorphisms of noncommutative Chow motives are rational
and satisfy a functional equation.
