  Over a Noetherian, local ring R of prime characteristic p, the Frobenius
functor F induces a diagonalizable map on certain quotients of rational
Grothendieck groups. This leads to an explicit formula for the Dutta
multiplicity, and it is shown that a weaker version of Serre's vanishing
conjecture holds if only chi(F(X)) = p^{dim R}chi(X) for all bounded complexes
X of finitely generated, projective modules with finite length homology.
