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Difference between revisions of "Regular ring (in commutative algebra)"

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082015.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080820/r08082015.png" /></td> </tr></table>
  
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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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 syzygy theorem]]); the converse also holds (see [[#References|[2]]]).
  
 
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" />.
 
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" />.
  
 
====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>
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<table>
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<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>
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<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>
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<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>
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</table>

Revision as of 16:33, 20 August 2016

A Noetherian ring whose localizations (cf. Localization in a commutative algebra) are all regular (here is a prime ideal in ). A local Noetherian ring (cf. Local ring) with maximal ideal is called regular if is generated by elements, where , that is, if the tangent space (as a vector space over the field of residues) has dimension equal to . This is equivalent to the absence of singularities in the scheme . A regular local ring is always integral and normal, and also factorial (cf. Factorial ring; the Auslander–Buchsbaum theorem), and its depth is equal to (cf. Depth of a module). The associated graded ring

is isomorphic to the polynomial ring . A local Noetherian ring is regular if and only if its completion is regular; in general, if is a flat extension of local rings and is regular, then is also regular. For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form , where 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 is regular, then the ring of polynomials and the ring of formal power series over are also regular. If is a non-invertible element of a local regular ring, then is regular if and only if .

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=39058
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article