Examples
Most naturally occurring commutative rings in number theory or algebraic geometry are excellent. In particular:
- All complete Noetherian local rings, and in particular all fields, are excellent.
- All Dedekind domains of characteristic 0 are excellent. In particular the ring Z of integers is excellent. Dedekind domains over fields of characteristic greater than 0 need not be excellent.
- The rings of convergent power series in a finite number of variables over R or C are excellent.
- Any localization of an excellent ring is excellent.
- Any finitely generated algebra over an excellent ring is excellent.
Here is an example of a regular local ring A of dimension 1 and characteristic p>0 which is not excellent. If k is any field of characteristic p with = ∞ and R=k] and A is the subring of power series Σaixi such that is finite then the formal fibers of A are not all geometrically regular so A is not excellent. Here kp denotes the image of k under the Frobenius morphism a→ap.
Any quasi-excellent ring is a Nagata ring.
Read more about this topic: Excellent Ring
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