  Already in the 1960s Grothendieck understood that one could obtain an almost
entirely satisfactory theory of motives over a finite field when one assumes
the full Tate conjecture. In this note we prove a similar result for motivic
complexes. In particular Beilinson's Q-algebra of "correspondences at the
generic point" is then defined for all connected varieties. We compute this for
all smooth projective varieties (hence also for varieties birational to such a
variety).
