# Excellent ring

A commutative Noetherian ring satisfying the three axioms stated below. It is known that a geometric ring possesses several qualitative properties not inherent in arbitrary Noetherian rings. The concept of an excellent ring makes it possible to take the most important properties of geometric rings axiomatically into account.

Axioms of an excellent ring .

A1. The ring is a universal chain ring. (A ring is said to be a chain ring if for any two prime ideals of it the lengths of any two saturated chains of prime ideals are the same. A ring is said to be a universal chain ring if any polynomial ring is a chain ring.)

A2. The formal fibres of are geometrically regular, i.e. for any prime ideal and any homomorphism from into a field , the ring is regular. Here is the completion of the local ring .

A3. For any integral finite -algebra there is a non-zero element such that the ring of fractions, , is regular.

Excellent rings possess the following properties:

1) For an excellent ring , the set of regular (normal) points of the scheme is open.

2) If an excellent local ring is reduced (normal or equi-dimensional), then so is the completion .

3) The integral closure of an excellent ring in a finite extension of the field of fractions of is a finite -algebra.

4) If a ring is excellent, then any -algebra of finite type is also an excellent ring.

Two important examples of excellent rings are the complete local rings (or analytic rings) and the Dedekind rings with field of fractions of characteristic zero. Therefore, the class of excellent rings is sufficiently large and contains, in particular, all algebras of finite type over a field or over the ring of integers.

The excellency of a ring is closely connected with the possibility of resolution of singularities of the scheme (cf. [1] and [2]).

#### References

[1] | A. Grothendieck, J. Dieudonne, "Eléments de géométrie algébrique" Publ. Math. IHES , 2 (1965) MR0199181 Zbl 0135.39701 |

[2] | H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero. I" Ann. of Math. , 79 : 1 (1964) pp. 109–203 MR0199184 Zbl 0122.38603 |

#### Comments

A chain ring is also called a catenarian ring. A sequence of prime ideals is saturated if there is no prime ideal and integer such that , with both inclusions strict. An excellent ring is a universal Japanese ring. An integer domain is if for every finite extension of its field fractions the integral closure of in is a finite -module. A ring is universal Japanese if it is Noetherian and if is for every prime ideal of . Other terminology for universal Japanese: Nagata ring, pseudo-geometric ring. Cf. also Geometric ring.

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Excellent ring.

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