View source for Regular ring (in commutative algebra)

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

<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]]]).

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====
<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>