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A [[Noetherian ring|Noetherian ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808201.png" /> whose localizations (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808202.png" /> are all regular (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808203.png" /> is a prime ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808204.png" />). A local Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808205.png" /> (cf. [[Local ring|Local ring]]) with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808206.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808207.png" /> is generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808208.png" /> elements, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r0808209.png" />, that is, if the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082010.png" /> (as a vector space over the field of residues) has dimension equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082011.png" />. This is equivalent to the absence of singularities in the [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082012.png" />. A regular local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082013.png" /> is always integral and normal, and also factorial (cf. [[Factorial ring|Factorial ring]]; the Auslander–Buchsbaum theorem), and its depth is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082014.png" /> (cf. [[Depth of a module|Depth of a module]]). The associated graded ring
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is isomorphic to the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082016.png" />. A local Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082017.png" /> is regular if and only if its completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082018.png" /> is regular; in general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082019.png" /> is a flat extension of local rings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082020.png" /> is regular, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082021.png" /> is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082023.png" /> 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 theorem|Hilbert theorem]] on syzygies); the converse also holds (see [[#References|[2]]]).
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A [[Noetherian ring|Noetherian ring]]  $  A $
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whose localizations (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]])  $  A _ {\mathfrak p }  $
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are all regular (here  $  \mathfrak p $
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is a prime ideal in  $  A $).  
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A local Noetherian ring $  A $(
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cf. [[Local ring|Local ring]]) with maximal ideal  $  \mathfrak m $
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is called regular if $  \mathfrak m $
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is generated by  $  n $
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elements, where  $  n = \mathop{\rm dim}  A $,
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that is, if the tangent space  $  \mathfrak m / \mathfrak m  ^ {2} $(
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as a vector space over the field of residues) has dimension equal to  $  \mathop{\rm dim}  A $.  
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This is equivalent to the absence of singularities in the [[Scheme|scheme]]  $  \mathop{\rm Spec}  A $.  
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A regular local ring $  A $
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082024.png" /> is regular, then the ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082025.png" /> and the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082026.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082027.png" /> are also regular. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082028.png" /> is a non-invertible element of a local regular ring, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082029.png" /> is regular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082030.png" />.
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$$
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G _ {\mathfrak m }  ( A)  = \
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\oplus _ {i \geq  0 } \mathfrak m  ^ {i} / \mathfrak m  ^ {i+} 1
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$$
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is isomorphic to the polynomial ring  $  k [ X _ {1} \dots X _ {n} ] $.
 +
A local Noetherian ring  $  A $
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is regular if and only if its completion  $  \widehat{A}  $
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is regular; in general, if  $  A \subset  B $
 +
is a flat extension of local rings and  $  B $
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is regular, then  $  A $
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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} ] $
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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique. I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1964) {{MR|0173675}} {{ZBL|0118.36206}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique. I. Le langage des schémas" ''Publ. Math. IHES'' , '''4''' (1964) {{MR|0173675}} {{ZBL|0118.36206}} </TD></TR>
 +
</table>

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=24549
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article