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 Rp 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.
Read more about this topic: Excellent Ring
Famous quotes containing the word definitions:
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)