Difference between revisions of "Regular ring (in commutative algebra)"
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| − | + | A [[Noetherian ring|Noetherian ring]] $ A $ | |
| + | whose localizations (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]) $ A _ {\mathfrak p } $ | ||
| + | are all regular (here $ \mathfrak p $ | ||
| + | is a prime ideal in $ A $). | ||
| + | A local Noetherian ring $ A $( | ||
| + | cf. [[Local ring|Local ring]]) with maximal ideal $ \mathfrak m $ | ||
| + | is called regular if $ \mathfrak m $ | ||
| + | is generated by $ n $ | ||
| + | elements, where $ n = \mathop{\rm dim} A $, | ||
| + | that is, if the tangent space $ \mathfrak m / \mathfrak m ^ {2} $( | ||
| + | as a vector space over the field of residues) has dimension equal to $ \mathop{\rm dim} A $. | ||
| + | This is equivalent to the absence of singularities in the [[Scheme|scheme]] $ \mathop{\rm Spec} A $. | ||
| + | A regular local ring $ A $ | ||
| + | is always integral and normal, and also factorial (cf. [[Factorial ring|Factorial ring]]; the Auslander–Buchsbaum theorem), and its depth is equal to $ \mathop{\rm dim} A $( | ||
| + | cf. [[Depth of a module|Depth of a module]]). The associated graded ring | ||
| − | Fields and Dedekind rings are regular rings. If | + | $$ |
| + | G _ {\mathfrak m } ( A) = \ | ||
| + | \oplus _ {i \geq 0 } \mathfrak m ^ {i} / \mathfrak m ^ {i+} 1 | ||
| + | $$ | ||
| + | |||
| + | is isomorphic to the polynomial ring $ k [ X _ {1} \dots X _ {n} ] $. | ||
| + | A local Noetherian ring $ A $ | ||
| + | is regular if and only if its completion $ \widehat{A} $ | ||
| + | is regular; in general, if $ A \subset B $ | ||
| + | is a flat extension of local rings and $ B $ | ||
| + | is regular, then $ A $ | ||
| + | is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form $ R [ [ X _ {1} \dots X _ {n} ] ] $, | ||
| + | where $ R $ | ||
| + | is a field or a discrete valuation ring. Any module of finite type over a regular local ring has a finite free resolution (see [[Hilbert syzygy theorem]]); the converse also holds (see [[#References|[2]]]). | ||
| + | |||
| + | Fields and Dedekind rings are regular rings. If $ A $ | ||
| + | is regular, then the ring of polynomials $ A [ X _ {1} \dots X _ {n} ] $ | ||
| + | and the ring of formal power series $ A [ [ X _ {1} \dots X _ {n} ] ] $ | ||
| + | over $ A $ | ||
| + | are also regular. If $ a \in A $ | ||
| + | is a non-invertible element of a local regular ring, then $ A / aA $ | ||
| + | is regular if and only if $ a \notin m ^ {2} $. | ||
====References==== | ====References==== | ||
Latest revision as of 08:10, 6 June 2020
A Noetherian ring $ A $
whose localizations (cf. Localization in a commutative algebra) $ A _ {\mathfrak p } $
are all regular (here $ \mathfrak p $
is a prime ideal in $ A $).
A local Noetherian ring $ A $(
cf. Local ring) with maximal ideal $ \mathfrak m $
is called regular if $ \mathfrak m $
is generated by $ n $
elements, where $ n = \mathop{\rm dim} A $,
that is, if the tangent space $ \mathfrak m / \mathfrak m ^ {2} $(
as a vector space over the field of residues) has dimension equal to $ \mathop{\rm dim} A $.
This is equivalent to the absence of singularities in the scheme $ \mathop{\rm Spec} A $.
A regular local ring $ A $
is always integral and normal, and also factorial (cf. Factorial ring; the Auslander–Buchsbaum theorem), and its depth is equal to $ \mathop{\rm dim} A $(
cf. Depth of a module). The associated graded ring
$$ G _ {\mathfrak m } ( A) = \ \oplus _ {i \geq 0 } \mathfrak m ^ {i} / \mathfrak m ^ {i+} 1 $$
is isomorphic to the polynomial ring $ k [ X _ {1} \dots X _ {n} ] $. A local Noetherian ring $ A $ is regular if and only if its completion $ \widehat{A} $ is regular; in general, if $ A \subset B $ is a flat extension of local rings and $ B $ is regular, then $ A $ is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form $ R [ [ X _ {1} \dots X _ {n} ] ] $, where $ R $ is a field or a discrete valuation ring. Any module of finite type over a regular local ring has a finite free resolution (see Hilbert syzygy theorem); the converse also holds (see [2]).
Fields and Dedekind rings are regular rings. If $ A $ is regular, then the ring of polynomials $ A [ X _ {1} \dots X _ {n} ] $ and the ring of formal power series $ A [ [ X _ {1} \dots X _ {n} ] ] $ over $ A $ are also regular. If $ a \in A $ is a non-invertible element of a local regular ring, then $ A / aA $ is regular if and only if $ a \notin m ^ {2} $.
References
| [1] | O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001 |
| [2] | J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603 |
| [3] | A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique. I. Le langage des schémas" Publ. Math. IHES , 4 (1964) MR0173675 Zbl 0118.36206 |
How to Cite This Entry:
Regular ring (in commutative algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_ring_(in_commutative_algebra)&oldid=48484
Regular ring (in commutative algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_ring_(in_commutative_algebra)&oldid=48484
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article