Regular ring (in commutative algebra)
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
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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 theorem on syzygies); 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 |
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