Universal additive Chern classes and a GRR-type theorem

TL;DR

This paper introduces a universal functor for Chern classes with additive first Chern class, linking it to the Grothendieck ring via a GRR-type theorem, advancing algebraic geometry's understanding of characteristic classes.

Contribution

It constructs a universal functor for Chern classes with additive first Chern class, providing a foundational framework in algebraic geometry.

Findings

Defines a functor from schemes to graded rings

Establishes a GRR-type theorem relating the functor to the Grothendieck ring

Provides a universal property for Chern classes with additive first Chern class

Abstract

We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor associates to $X$ is related to the associated graded ring of the $\gamma$-filtration on the Grothendieck ring of finite rank locally free sheaves on $X$ via a Grothendieck-Riemann-Roch type theorem.