Two Formulas for the BR Multiplicity

TL;DR

This paper introduces two formulas for the Buchsbaum--Rim multiplicity, extending classical multiplicity formulas through intersection theory and projection techniques, with implications for algebraic geometry and module theory.

Contribution

It presents a projection formula and an expansion formula for the Buchsbaum--Rim multiplicity, generalizing classical multiplicity results and simplifying proofs using intersection theory.

Findings

Projection formula expresses relative multiplicity via algebra extensions.

Expansion formula generalizes additivity of multiplicity.

Proofs rely on intersection numbers and Chern class projections.

Abstract

We prove a projection formula, expressing a relative Buchsbaum--Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula.