The Deligne pairing and a functorial Riemann Roch theorem in positive characteristic

TL;DR

This paper proves a functorial Riemann-Roch theorem in positive characteristic for smooth, projective morphisms of any relative dimension, generalizing previous results and connecting to Deligne pairings and Adams Riemann-Roch.

Contribution

It extends the functorial Riemann-Roch theorem to all relative dimensions in positive characteristic, building on Deligne pairings and Adams Riemann-Roch.

Findings

Established a functorial Riemann-Roch theorem in positive characteristic for arbitrary relative dimension.

Derived an analogue of the Knudsen-Mumford extension from the main theorem.

Connected the result to extended Deligne pairings and Adams Riemann-Roch in positive characteristic.

Abstract

In this paper, we prove the functorial Riemann-Roch theorem in positive characteristic for a smooth and projective morphism with any relative dimension. In the case of relative dimension $1$, we have given an analogue with Deligne's functorial Riemann Roch theorem in previous author's paper. For any relative dimension, our result can deduce an analogue to the Knudsen-Mumford extension. The present result is a generalization, which mainly originated from the extended Deligne pairing by S. Zhang and the Adams Riemann Roch theorem in positive characteristic by R. Pink and D. R\"{o}ssler.