Chris Skalit : Intersection Multiplicity of Serre in the Unramified Case

Posted by Kevin Tucker , part of the Algebraic Geometry Seminar.

At

Oct. 15, 2014, 3 p.m.

In

SEO 636

Abstract

Suppose that $(A, \mathfrak{m})$ is a regular local ring whose $\mathfrak{m}$-adic completion is a power series over a discrete valuation ring. For properly-meeting, closed subschemes of complimentary dimension $Y, Z \subseteq \mathrm{Spec} (A)$, we show that the Serre intersection multiplicity $\chi^A(\mathcal{O}_Y,\mathcal{O}_Z) := \sum_{i=0}^{\infty}{(-1)^i \ell(\mathrm{Tor}_i^A(\mathcal{O}_Y,\mathcal{O}_Z))}$ is bounded below by the product of the Hilbert-Samuel multiplicities of $Y$ and $Z$. We also investigate the geometric significance of achieving this lower bound by examining the proper transforms of $Y$ and $Z$ under the blowup of $\mathrm{Spec} (A)$.