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 | + | 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> | + | <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> |
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 |
Regular ring (in commutative algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_ring_(in_commutative_algebra)&oldid=24549