Diagonalizing the Frobenius

TL;DR

This paper explores the diagonalization of the Frobenius functor on Grothendieck groups over Noetherian local rings of prime characteristic, deriving formulas for Dutta multiplicity and relating to Serre's vanishing conjecture.

Contribution

It provides an explicit formula for Dutta multiplicity and links Frobenius diagonalization to a weaker form of Serre's vanishing conjecture.

Findings

Frobenius induces a diagonalizable map on certain Grothendieck group quotients

Derived an explicit formula for Dutta multiplicity

Established a partial result towards Serre's vanishing conjecture

Abstract

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.