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    Regular local rings, regular rings, and DVRs

    D-DEFREGLR

    (regular local ring)

    A Noetherian local ring \((A,\mm)\) is a regular local ring if \[ \dim A = \dim_{A/\mm} \mm/\mm^2 . \] A Noetherian ring \(A\) is a regular ring if \(A_\mfp\) is a regular local ring for every prime \(\mfp\).

    (valuation)

    Let \(K\) be a field and \(G\) a totally ordered abelian group. A valuation of \(K\) with values in \(G\) is a group homomorphism \(v \colon K^\times \to G\) such that \(v(x+y) \geq \min(v(x), v(y))\) for all \(x, y \in K^\times\) with \(x + y \neq 0\). Its valuation ring is \(R = \ts{x \in K^\times \st v(x) \geq 0} \cup \ts{0}\). A domain \(R\) is a valuation ring if it is the valuation ring of some valuation of \(\operatorname{Frac} R\), and the valuation is discrete if \(G = \ZZ\). [@Har10a, §I.6]

    By Krull’s height theorem, \(\dim A \leq \dim_{A/\mm}\mm/\mm^2\) for every Noetherian local ring, and \(\dim_{A/\mm}\mm/\mm^2\) is the minimal number of generators of \(\mm\). If \(A\) is regular, then \(\mm\) is generated by a regular sequence, and \(A\) is Cohen–Macaulay.

    A regular local ring is a domain, is a UFD by the Auslander–Buchsbaum theorem, and is therefore integrally closed. By Serre’s theorem, a Noetherian local ring is regular if and only if its residue field has finite projective dimension, if and only if it has finite global dimension; consequently every localization of a regular local ring at a prime is regular.

    For a Noetherian local domain of dimension one, the following are equivalent: it is regular; it is a DVR; it is integrally closed; \(\mm\) is principal. Hence on a Noetherian integral scheme regular in codimension one, the local ring at the generic point of each prime divisor \(Y\) is a DVR, and its valuation \(v_Y\) gives the coefficients of the principal Weil divisor \(\operatorname{div}(f)\) (D-5PQ5W).

    When is a Noetherian local ring regular?
    Characterise discrete valuation rings among Noetherian local rings.
    Is a regular local ring a domain? Is it integrally closed?
    Show that $R$ is a discrete valuation ring if and only if $R$ is a local PID that is not a field, and define a uniformizer.
    Show that the ring of germs at $0$ of holomorphic functions on a disc is a discrete valuation ring with uniformizer $z$.
    For the local ring $R$ of a curve at a point, over $k = \bar k$, identify the closed and generic points of $\Spec R$ and their residue fields, and decide which maps between $\Spec R$ and $\Spec \operatorname{Frac} R$ are morphisms of schemes.

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    • D-SCHCMGORCohen--Macaulay, Gorenstein and local complete intersection schemesuses

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