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- Tags: - #AG/basics - Refs: - #todo/add-references - Links: - smooth scheme - closed point - Cohen-Macaulay - Noetherian scheme
regular scheme
A scheme \(X\) is regular iff every local ring \({\mathcal{O}}_{X, x}\) is a regular ring.
Note that it is sufficient to check this at closed points \(x\in{\left\lvert {X} \right\rvert}\) by Serre's criterion.
Used in the definition of smoothness.
Idea: Dimension 1 regular Noetherian local rings are like germs of smooth curves.
Examples
smooth implies regular but not conversely: let \(k\) be an imperfect field of positive odd prime characteristic \(p\) and let \(a\in k \setminus k^p\) be an element which is not a \(p\)th power. Then the curve $X = V(y^2=x^p-a) \subseteq {\mathbf{A}}^2_{/ {k}} $ is a Dedekind scheme but its base change \(X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } } = V(y^2=z^p)\) where \(z\coloneqq x-a^{1\over p}\) is not regular at the origin.
regular in codimension one
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A scheme \(X\) is regular in codimension 1 iff every local ring \(R\) of dimension 1 is regular (so \(\dim_k {\mathfrak{m}}_R/{\mathfrak{m}}_R^2 = 1\) for the unique \({\mathfrak{m}}_R \in \operatorname{mSpec}R\)).
- Often translates to smooth in codimension 1.
- Technical condition needed to make sense of Weil divisors.