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Discrete valuation rings are regular local

PDF p. 15: Codimension of the singular locus

Statement

A discrete valuation ring is a regular local ring.

Here a discrete valuation ring is a nonfield noetherian local domain of dimension one with principal maximal ideal. Regular local means that the dimension of $m / m^{2}$ over the residue field equals the Krull dimension.

Facts & Assumptions

Given: A discrete valuation ring $R$ with maximal ideal $m$ .

Goal: Prove that $R$ is regular local.

[F1]

A discrete valuation ring is a one-dimensional noetherian local domain whose maximal ideal is principal (Equivalent characterizations of a DVR).

[F2]

A noetherian local ring is regular exactly when its embedding dimension equals its Krull dimension (embedding dimension and regular local ring).

Proof

1.1F1givenalgebra

By [F1], $m = (π)$ for some nonzero $π$ , and $\dim R = 1$ . The residue class of $π$ spans $m / m^{2}$ . It is nonzero: if $π = a π^{2}$ , cancellation in the domain gives $1 = a π$ , contrary to $π \in m$ .

2.1F2step 1.1∎

Therefore the embedding dimension is one, equal to $\dim R$ . The criterion [F2] shows that $R$ is regular local.

Depends on

Used by · 1

Smooth density and normal codimension over a perfect field

Dependency tree · 3 levels

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