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 $ A $.

A1. The ring $ A $ is a universal chain ring. (A ring $ A $ is said to be a chain ring if for any two prime ideals $ \mathfrak p \neq \mathfrak p ^ \prime $ of it the lengths of any two saturated chains $ \mathfrak p = \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \mathfrak p _ {n} = \mathfrak p ^ \prime $ of prime ideals are the same. A ring $ A $ is said to be a universal chain ring if any polynomial ring $ A [ T _ {1} \dots T _ {k} ] $ is a chain ring.)

A2. The formal fibres of $ A $ are geometrically regular, i.e. for any prime ideal $ \mathfrak p \subset A $ and any homomorphism from $ A $ into a field $ K $, the ring $ \widehat{A} _ {\mathfrak p } \otimes _ {A} K $ is regular. Here $ \widehat{A} _ {\mathfrak p } $ is the completion of the local ring $ A _ {\mathfrak p } $.

A3. For any integral finite $ A $- algebra $ B $ there is a non-zero element $ b \in B $ such that the ring of fractions, $ B [ b ^ {-} 1 ] $, is regular.

Excellent rings possess the following properties:

1) For an excellent ring $ A $, the set of regular (normal) points of the scheme $ \mathop{\rm Spec} A $ is open.

2) If an excellent local ring $ A $ is reduced (normal or equi-dimensional), then so is the completion $ \widehat{A} $.

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

4) If a ring $ A $ is excellent, then any $ A $- 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 $ \mathbf Z $ of integers.

The excellency of a ring $ A $ is closely connected with the possibility of resolution of singularities of the scheme $ \mathop{\rm Spec} A $( 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 $ \mathfrak p _ {0} \subset \dots \subset \mathfrak p _ {n} $ is saturated if there is no prime ideal $ \mathfrak q $ and integer $ i \in \{ 0 \dots n - 1 \} $ such that $ \mathfrak p _ {i} \subset \mathfrak q \subset \mathfrak p _ {i+} 1 $, with both inclusions strict. An excellent ring is a universal Japanese ring. An integer domain $ A $ is $ N _ {2} $ if for every finite extension $ L / K $ of its field fractions $ K $ the integral closure $ A _ {L} $ of $ A $ in $ L $ is a finite $ A $- module. A ring $ B $ is universal Japanese if it is Noetherian and if $ B / \mathfrak p $ is $ N _ {2} $ for every prime ideal $ \mathfrak p $ of $ B $. Other terminology for universal Japanese: Nagata ring, pseudo-geometric ring. Cf. also Geometric ring.