Intersection Multiplicity of Serre in the Unramified Case

TL;DR

This paper advances understanding of Serre's Intersection Multiplicity Conjecture by establishing a lower bound for intersection multiplicity in unramified regular local rings, paralleling the equicharacteristic case, with geometric insights.

Contribution

It proves that in unramified regular local rings, the intersection multiplicity is bounded below by the product of Hilbert-Samuel multiplicities, extending known results to the unramified case.

Findings

Lower bound for intersection multiplicity in unramified regular local rings.

Geometric interpretation via blowup of Spec(A) at the closed point.

Parallel to the equicharacteristic case results.

Abstract

We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is an unramified regular local ring, then just as in the equicharacteristic case, the intersection multiplicity of two complimentary-dimensional modules is bounded below by the product of their Hilbert-Samuel multiplicities. We also explain, in terms of the blowup of Spec(A) at the closed point, the geometric significance of achieving this lower bound.