On Deligne's functorial Riemann-Roch theorem in positive characteristic

TL;DR

This paper provides a simplified, functorial proof of Deligne's Riemann-Roch theorem in positive characteristic, leveraging ideas from Pink and Rössler's proof of the Adams-Riemann-Roch theorem, and offers partial compatibility with Mumford's isomorphism.

Contribution

It introduces a new, more direct proof of the functorial Deligne-Riemann-Roch theorem in positive characteristic, based on alternative methods from Pink and Rössler.

Findings

Proof is valid for any positive characteristic.

Method simplifies the classical proof approach.

Partial compatibility with Mumford's isomorphism.

Abstract

In this note, we give a proof for a variant of the functorial Deligne-Riemann-Roch theorem in positive characteristic based on ideas appearing in Pink and R\"ossler's proof of the Adams-Riemann-Roch theorem in positive characteristic (see \cite{Pi}). The method of their proof appearing in \cite{Pi}, which is valid for any positive characteristic and which is completely different from the classical proof, will allow us to prove the functorial Deligne-Riemann-Roch theorem in a much easier and more direct way. Our proof is also partially compatible with Mumford's isomorphism.