Norm functors and effective zero cycles

TL;DR

This paper compares two definitions of effective zero cycles, showing their equivalence in characteristic zero and introducing a new norm functor approach in general settings.

Contribution

It establishes the equivalence of trace-based and norm-based definitions in characteristic zero and introduces a novel norm functor framework for zero cycles.

Findings

Definitions agree in characteristic zero

Norm functor extends to categories of line bundles

Provides a new approach to families of zero cycles

Abstract

We compare two known definitions for a relative family of effective zero cycles, based on traces and norms of functions, respectively. In characteristic zero we show that both definitions agree. In the general setting, we show that the norm map on functions can be expanded to a norm functor between certain categories of line bundles, therefore giving a third approach to families of zero cycles.