# Regular local ring

A **regular local ring** is a commutative Noetherian local ring \((R, \mathfrak m)\) of [Krull dimension](https://www.edgepedia.org/krull-dimension) \(d\) whose maximal ideal \(\mathfrak m\) can be generated by \(d\) elements; such a generating set is called a regular system of parameters<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>. Equivalently, the minimal number of generators of \(\mathfrak m\), which by Nakayama's lemma equals \(\dim_\kappa(\mathfrak m/\mathfrak m^2)\) for the residue field \(\kappa\), equals the dimension of the ring<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>. Regular local rings are the algebraic local models of smooth points on algebraic varieties, and they carry unusually strong homological properties<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \((R, \mathfrak m)\) Noetherian local of dimension \(d\) with \(\mathfrak m = (x_1, \ldots, x_d)\); the \(x_i\) form a regular system of parameters<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> |
| Fundamental inequality | \(\dim(R) \le \dim_\kappa(\mathfrak m/\mathfrak m^2)\) always holds; regularity is exactly equality<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> |
| Associated graded ring | \(\bigoplus_n \mathfrak m^n/\mathfrak m^{n+1} \cong \kappa[X_1, \ldots, X_d]\)<sup>[3](https://stacks.math.columbia.edu/tag/00NN)</sup> |
| Ring-theoretic strength | A regular local ring is a domain, its parameter generators form a regular sequence, and it is Cohen-Macaulay<sup>[3](https://stacks.math.columbia.edu/tag/00NN)</sup> |
| Homological criterion | \(R\) is regular if and only if it has finite global dimension; then \(\mathrm{gl\,dim}(R) = \dim(R)\)<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup><sup> • </sup><sup>[4](https://stacks.math.columbia.edu/tag/065U)</sup> |
| Factorization | Every regular local ring is a unique factorization domain (Auslander and Buchsbaum, 1959)<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup> |
| Positive characteristic test | \(R\) of characteristic \(p\) is regular if and only if the Frobenius map \(R \to R\) is flat (Kunz, 1969)<sup>[5](https://raw.githubusercontent.com/kschwede/FrobeniusSingularitiesBook/main/FrobeniusSingularitiesBook.pdf)</sup> |

## Definition and first properties

For any Noetherian local ring \((R, \mathfrak m, \kappa)\), the maximal ideal cannot be generated by fewer than \(\dim(R)\) elements, giving the fundamental inequality

\[ \dim(R) \le \dim_\kappa(\mathfrak m/\mathfrak m^2). \]

The rings where equality holds are the regular local rings<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>. By Krull's principal ideal theorem the maximal ideal of a \(d\)-dimensional local ring cannot be generated by fewer than \(d\) elements, so regular local rings have a minimal number of generators for their maximal ideal<sup>[6](https://planetmath.org/regularlocalring)</sup>.

Several structural facts follow directly. The [associated graded ring](https://www.edgepedia.org/associated-graded-ring) \(\bigoplus_n \mathfrak m^n/\mathfrak m^{n+1}\) of a \(d\)-dimensional regular local ring is isomorphic to the polynomial ring \(\kappa[X_1, \ldots, X_d]\)<sup>[3](https://stacks.math.columbia.edu/tag/00NN)</sup>. Every regular local ring is a domain<sup>[3](https://stacks.math.columbia.edu/tag/00NN)</sup>, and a minimal set of generators \(x_1, \ldots, x_d\) of \(\mathfrak m\) is a regular sequence, so \(R\) is Cohen-Macaulay and each quotient \(R/(x_1, \ldots, x_c)\) is again a regular local ring of dimension \(d - c\)<sup>[3](https://stacks.math.columbia.edu/tag/00NN)</sup>. Localizations behave well: if \(R\) is a regular local ring, then \(R_\mathfrak p\) is a regular local ring for every prime \(\mathfrak p\)<sup>[4](https://stacks.math.columbia.edu/tag/065U)</sup>.

## The homological characterization

In 1956, [Maurice Auslander](https://www.edgepedia.org/maurice-auslander) and [David Buchsbaum](https://www.edgepedia.org/david-buchsbaum) and, independently, [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre) used homological methods to characterize regular local rings as exactly those Noetherian local rings of finite global dimension, where the global dimension is the supremum of projective dimensions of modules<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>. Serre proved that \(\mathrm{Tor}^R_n(\kappa, \kappa) \ne 0\) when \(\dim_\kappa(\mathfrak m/\mathfrak m^2) = n\), so \(\mathrm{gl\,dim}(R) \ge n\), while the Auslander-Buchsbaum theorem bounds finite projective dimensions by the depth of the ring; if all projective dimensions are finite, the two bounds force equality:

\[ \mathrm{gl\,dim}(R) = \dim_\kappa(\mathfrak m/\mathfrak m^2) = \dim(R). \]

In particular, finite global dimension implies regularity<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>. The Stacks project formulation records the equivalent chain: the residue field \(\kappa\) has finite projective dimension as an \(R\)-module if and only if \(R\) has finite global dimension if and only if \(R\) is regular<sup>[4](https://stacks.math.columbia.edu/tag/065U)</sup>.

The **Auslander-Buchsbaum formula** states that for a finitely generated module \(E\) of finite projective dimension over a local ring \(A\),

\[ \mathrm{pd}(E) + \mathrm{depth}(E) = \mathrm{depth}(A). \]

Over a regular local ring of dimension \(n\), where \(\mathrm{depth}(A) = \dim(A)\) and every finitely generated module has finite projective dimension, this becomes \(\mathrm{pd}(E) = n - \mathrm{depth}(E)\)<sup>[7](https://www.math.stonybrook.edu/~roblaz/LSGAV.Prelim.Draft.pdf)</sup>.

Two years after the homological characterization, Auslander and Buchsbaum proved in 1959 that every regular local ring is a unique factorization domain<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup><sup> • </sup><sup>[8](https://stacks.math.columbia.edu/tag/0FJH)</sup>.

## Geometric meaning

Regular local rings became important in algebraic geometry because they are the local coordinate rings of smooth algebraic varieties: a point \(P\) on a variety is smooth precisely when the local ring \(\mathcal O_{X,P}\) is regular<sup>[2](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup><sup> • </sup><sup>[9](https://calwoo.github.io/notes/concepts/algebraic-geometry/shafarevich-ch2-local-properties/index.html)</sup>. At a smooth point the associated graded of the local ring is a polynomial ring in the local parameters, the algebraic counterpart of coordinates near a point<sup>[9](https://calwoo.github.io/notes/concepts/algebraic-geometry/shafarevich-ch2-local-properties/index.html)</sup>.

Regularity can be checked by derivatives. The **Jacobian criterion** says that over a perfect field \(K\), for an integral domain \(A\) of dimension \(d\) presented as a quotient of a polynomial ring, the local rings are regular exactly where the Jacobian matrix of the defining equations has the expected rank<sup>[10](https://www.math.mcgill.ca/darmon/courses/18-19/algebra2/kunz.pdf)</sup>. Over a field of characteristic zero, a Noetherian local \(k\)-algebra \(A\) is regular if and only if \(k \to A\) is formally smooth in the \(\mathfrak m\)-adic topology; in characteristic \(p\) the corresponding statement characterizes geometric regularity<sup>[11](https://stacks.math.columbia.edu/tag/07EK)</sup>.

## Completion and the Cohen structure theorem

Completion does not disturb regularity: a local ring is regular if and only if its completion is regular, because completing changes neither the Krull dimension nor the embedding dimension \(\dim_\kappa(\mathfrak m/\mathfrak m^2)\), and it leaves the associated graded ring unchanged<sup>[12](https://people.math.sc.edu/kustin/teaching/746/HochsterNotesOnCohenStructureTheorem.pdf)</sup>.

This reduction makes the classification of complete regular local rings decisive. In the equicharacteristic case, a complete regular local ring containing a field is simply a formal power series ring in finitely many variables over a field<sup>[12](https://people.math.sc.edu/kustin/teaching/746/HochsterNotesOnCohenStructureTheorem.pdf)</sup>. In mixed characteristic, where the ring has characteristic zero but residue field characteristic \(p > 0\), the complete regular ring \(R\) of dimension \(d\) with coefficient ring \(V\) is either a formal power series ring \(V[[x_1, \ldots, x_{d-1}]]\), or it has the form \(T/(p - f)\), where \(T = V[[x_1, \ldots, x_d]]\) and \(f \in \mathfrak m_T^2\)<sup>[12](https://people.math.sc.edu/kustin/teaching/746/HochsterNotesOnCohenStructureTheorem.pdf)</sup>. Here \(V\) is a complete discrete valuation ring whose maximal ideal is generated by \(p\), and \(V/pV \to R/\mathfrak m\) is an isomorphism on residue fields<sup>[12](https://people.math.sc.edu/kustin/teaching/746/HochsterNotesOnCohenStructureTheorem.pdf)</sup>.

The general backdrop is [Irving Kaplansky](https://www.edgepedia.org/irving-kaplansky)-era student I. S. Cohen's 1946 structure theorem: every complete local ring is a homomorphic image of a complete regular local ring of power-series type, with coefficients from a field or a valuation ring<sup>[13](https://doi.org/10.1090/s0002-9947-1946-0016094-3)</sup>. Since a formal power series ring over a complete DVR is itself regular, every complete local ring is a homomorphic image of a regular local ring<sup>[14](https://dlk53.github.io/teaching/Math%20831%20Spring%202026/cohen2.html)</sup>.

## Positive characteristic and the Frobenius

In characteristic \(p\), regularity admits a test with no counterpart in characteristic zero. Ernst Kunz proved in 1969 that a [Noetherian ring](https://www.edgepedia.org/noetherian-ring) \(R\) of prime characteristic \(p\) is regular if and only if the Frobenius endomorphism \(F \colon R \to R\), \(x \mapsto x^p\), makes \(F_*R\) a flat \(R\)-module<sup>[5](https://raw.githubusercontent.com/kschwede/FrobeniusSingularitiesBook/main/FrobeniusSingularitiesBook.pdf)</sup>. For \(F\)-finite schemes this says \(X\) is regular if and only if the coherent sheaf \(F_*\mathcal O_X\) is locally free<sup>[5](https://raw.githubusercontent.com/kschwede/FrobeniusSingularitiesBook/main/FrobeniusSingularitiesBook.pdf)</sup>. Kunz's result was the first indication that the Frobenius map detects singularities, and it launched the study of Frobenius techniques in commutative algebra<sup>[15](https://arxiv.org/abs/math/0301208)</sup>.

In a modern mixed-characteristic counterpart of this criterion, Bhargav Bhatt, Srikanth B. Iyengar, and Linquan Ma showed that a Noetherian ring that is complete p-adically is regular if and only if there is a faithfully flat map from it to a perfectoid ring, and they also proved a variant in which flatness is weakened to almost flatness<sup>[16](https://arxiv.org/abs/1803.03229)</sup>.

## Further consequences

Because localizations of a regular local ring are regular, a Noetherian ring all of whose local rings at primes are regular is called a regular ring; a regular ring of finite dimension \(n\) has finite global dimension \(n\), though a regular ring of infinite dimension need not have finite global dimension<sup>[4](https://stacks.math.columbia.edu/tag/065U)</sup>. Regularity also underwrites results elsewhere in mathematics: the Gersten conjecture for [Milnor K-theory](https://www.edgepedia.org/milnor-k-theory) holds for regular schemes over a field, and the Grothendieck-Serre conjecture on principal \(G\)-bundles is formulated over regular local rings<sup>[17](https://link.springer.com/article/10.1007/s00029-024-00963-1)</sup><sup> • </sup><sup>[18](https://arxiv.org/abs/1406.0247)</sup>.

## References

1. [Section 10.60 (00KD): Dimension, The Stacks Project](https://stacks.math.columbia.edu/tag/00KD)
2. [Regular Sequence, an overview, ScienceDirect Topics (C. A. Weibel, History of Homological Algebra)](https://www.sciencedirect.com/topics/mathematics/regular-sequence)
3. [Section 10.106 (00NN): Regular local rings, The Stacks Project](https://stacks.math.columbia.edu/tag/00NN)
4. [Section 10.110 (065U): Regular rings and global dimension, The Stacks Project](https://stacks.math.columbia.edu/tag/065U)
5. [Frobenius and Regularity (Schwede, FrobeniusSingularitiesBook)](https://raw.githubusercontent.com/kschwede/FrobeniusSingularitiesBook/main/FrobeniusSingularitiesBook.pdf)
6. [regular local ring, PlanetMath](https://planetmath.org/regularlocalring)
7. [LSGAV Prelim Draft, R. Lazarsfeld, Stony Brook](https://www.math.stonybrook.edu/~roblaz/LSGAV.Prelim.Draft.pdf)
8. [Section 15.123 (0FJH): A regular local ring is a UFD, The Stacks Project](https://stacks.math.columbia.edu/tag/0FJH)
9. [Local Properties of Varieties (Shafarevich Ch. 2 notes), C. Woo](https://calwoo.github.io/notes/concepts/algebraic-geometry/shafarevich-ch2-local-properties/index.html)
10. [§1. Regular points of varieties. Regular local rings (Kunz, Introduction to Commutative Algebra and Algebraic Geometry)](https://www.math.mcgill.ca/darmon/courses/18-19/algebra2/kunz.pdf)
11. [Section 15.41 (07EK): Geometric regularity and formal smoothness, The Stacks Project](https://stacks.math.columbia.edu/tag/07EK)
12. [The structure theory of complete local rings, M. Hochster course notes, University of South Carolina](https://people.math.sc.edu/kustin/teaching/746/HochsterNotesOnCohenStructureTheorem.pdf)
13. [I. S. Cohen (1946), On the structure and ideal theory of complete local rings, Transactions of the AMS](https://doi.org/10.1090/s0002-9947-1946-0016094-3)
14. [A Guide to Cohen's Structure Theorem for Complete Local Rings](https://dlk53.github.io/teaching/Math%20831%20Spring%202026/cohen2.html)
15. [C. Miller (2003), The Frobenius endomorphism and homological dimensions, arXiv](https://arxiv.org/abs/math/0301208)
16. [B. Bhatt, S. B. Iyengar, L. Ma (2018), Regular rings and perfect(oid) algebras, arXiv](https://arxiv.org/abs/1803.03229)
17. [Zero-cycles in families of rationally connected varieties, Selecta Mathematica (2024)](https://link.springer.com/article/10.1007/s00029-024-00963-1)
18. [Proof of Grothendieck–Serre conjecture on principal G-bundles over regular local rings containing a finite field, arXiv](https://arxiv.org/abs/1406.0247)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Regular rings and homological properties*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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