# 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 $\U0001d532$ be the ideal of functions vanishing on $S$. This is an ideal of height 1, so the local ring^{} of $Y$, ${\mathcal{O}}_{S}=A{(X)}_{\U0001d532}$, 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 |