normal irreducible varieties are nonsingular in codimension 1

Theorem 1.

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

Proof.

Assume not. We may assume $X$ is affine, since codimension is local. Now let $\\mathfrak{u}$ be the ideal of functions vanishing on $S$ . This is an ideal of height 1, so the local ring of $Y$ , $\\mathcal{O}_{S}=A(X)_{\\mathfrak{u}}$ , 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 aDVR, and thus regular. But $S$ is the singular set, so its local ring is not regular, a contradiction.∎

单词	NormalIrreducibleVarietiesAreNonsingularInCodimension1
释义	normal irreducible varieties are nonsingular in codimension 1 Theorem 1. Let $X$ be a normal irreducible variety. The singular set $S\\subset X$ has codimension 2 or more. Proof. Assume not. We may assume $X$ is affine, since codimension is local. Now let $\\mathfrak{u}$ be the ideal of functions vanishing on $S$ . This is an ideal of height 1, so the local ring of $Y$ , $\\mathcal{O}_{S}=A(X)_{\\mathfrak{u}}$ , 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 aDVR, and thus regular. But $S$ is the singular set, so its local ring is not regular, a contradiction.∎
随便看	RestrictedDirectProduct restricted direct product RestrictedDirectProductOfAlgebraicSystems restricted direct product of algebraic systems RestrictedDivisorsOfTheFirst1000PositiveIntegers restricted divisors of the first 1000 positive integers RestrictedHomomorphism restricted homomorphism RestrictedLieAlgebra restricted Lie algebra RestrictionOfAContinuousMappingIsContinuous restriction of a continuous mapping is continuous RestrictionOfAFunction restriction of a function RestrictionRepresentation restriction representation Resultant resultant ResultantalternativeTreatment resultant (alternative treatment) ResultOnQuadraticResidues result on quadratic residues Retract retract Reversal