Diagonalizing the Frobenius

Over a Noetherian, local ring R of prime characteristic p, the Frobenius functor FR 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 Serreʹs vanishing conjecture holds whenever χ(FR(X))=pdimRχ(X) for all bounded complexes X of finitely generated, projective modules with finite length homology.