normal irreducible varieties are nonsingular in codimension 1

Theorem 1.

Let $X$ be a normal irreducible variety. The singular set $S\mathrm{\subset }X$ has codimension 2 or more.

Proof.

Assume not. We may assume $X$ is affine, since codimension is local. Now let $𝔲$ be the ideal of functions vanishing on $S$. This is an ideal of height 1, so the local ring of $Y$, ${𝒪}_S=A{(X)}_{𝔲}$, where $A(X)$ is the affine ring of $X$, is a 1-dimensional local ring, and integrally closed, since $X$ is normal. Any integrally closed 1-dimensional local domain is a DVR, and thus regular. But $S$ is the singular set, so its local ring is not regular, a contradiction. ∎

Title	normal irreducible varieties are nonsingular in codimension 1
Canonical name	NormalIrreducibleVarietiesAreNonsingularInCodimension1
Date of creation	2013-03-22 13:20:20
Last modified on	2013-03-22 13:20:20
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	7
Author	archibal (4430)
Entry type	Theorem
Classification	msc 14A99