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regular ring
Idea: regular local rings correspond to smooth points on schemes. In general, regular rings \(\subseteq\) Gorenstein rings \(\subseteq\) Cohen-Macaulay rings.
Regular local rings
Let \(R\) be a local Noetherian ring and let \(n\) be the minimal number of generators of its maximal ideal \({\mathfrak{m}}_R\). By Krull’s intersection theorem, \begin{align*}n\geq \operatorname{krulldim}(R),\end{align*} so define \(R\) to be regular iff this is an equality. Note that a minimal set of generators is a regular system.
Equivalently, \(R\) is regular iff its residue fields \(\kappa(R) \coloneqq R/{\mathfrak{m}}\) satisfy \begin{align*}\dim_{\kappa(R)} {\mathfrak{m}}/{\mathfrak{m}}^2 = \operatorname{krulldim}(R).\end{align*}
Equivalently, \(R\) is regular iff \(R\) has finite global dimension.
Regular rings
An arbitrary commutative ring \(R\) is regular iff the prime localizations \(R \left[ { \scriptstyle { { ({{\mathfrak{p}}}^c) }^{-1}} } \right]\) are regular local rings (as above) for every \({\mathfrak{p}}\in \operatorname{Spec}R\).
Examples
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Regular rings:
- Fields are regular local.
- Formal power series rings \(k{\left[\left[ t_1, t_2,\cdots, t_m \right]\right] }\) in finitely many indeterminates over a field are regular local.
- Every DVR is regular local.
- The p-adic integers \({ {\mathbf{Z}}_{\widehat{p}} }\) are DVRs and thus regular local.
- Formal power series rings \(A{\left[\left[ x_0, \cdots, x_n \right]\right] }\) over a regular local ring are again regular local.
- Any localization of a regular ring.
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Non-regular rings:
- \(k[x]/\left\langle{x^2}\right\rangle\) #todo/exericse
Results
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See Serre’s criterion for regularity.
- Localizations and completions of regular local rings are regular.
- Consequence for schemes: a point $x\in X\in{\mathsf{Sch}}_{/ {k}} $ is a smooth point iff the stalk \({\mathcal{O}}_{X, x}\) is a regular local ring.