Excellent ring

In mathematics, in the fields of commutative algebra and algebraic geometry, an excellent ring is a Noetherian commutative ring with many of the good properties of complete local rings. This class of rings was defined by Alexander Grothendieck (1965).

Most Noetherian rings that occur in algebraic geometry or number theory are excellent, and excellence of a ring is closely related to resolution of singularities of the associated scheme (Hironaka (1964)).

Definitions

• A ring R containing a field k is called geometrically regular over k if for any finite extension K of k the ring R⊗_kK is regular.
• A homomorphism of rings from R to S is called regular if it is flat and for every p∈Spec(R) the fiber S⊗_Rk(p) is geometrically regular over the residue field k(p) of p.
• A ring R is called a G-ring (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any p∈Spec(R), the map from the local ring R_p to its completion is regular in the sense above.
• A ring R is called quasi-excellent if it is a G-ring and for every finitely generated R-algebra S, the singular points of Spec(S) form a closed subset.
• A ring is called excellent if it is quasi-excellent and universally catenary.

In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.

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 [k:k^p] = ∞ and R=k[[x]] and A is the subring of power series Σa_ixⁱ such that [k^p(a₀,a₁,...):k^p ] is finite then the formal fibers of A are not all geometrically regular so A is not excellent. Here k^p denotes the image of k under the Frobenius morphism a→a^p.

Any quasi-excellent ring is a Nagata ring.

Resolution of singularities

Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivation for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964) proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring R then the ring R is quasi-excellent.

References

• V.I. Danilov (2001), "Excellent ring", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=e/e036760 
• A. Grothendieck, J. Dieudonne, Eléments de géométrie algébrique IV Publ. Math. IHES 24 (1965), section 7
• Hironaka, Heisuke Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 1964 205-326.
• H. Matsumura, Commutative algebra ISBN 0-8053-7026-9, chapter 13.