Regular ring (in commutative algebra)

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