Recall that a ring homomorphism A->B is geometrically regular if for all primes p of A, the fiber of B over p is geometrically regular over k(p). A Grothendieck ring (or, G-ring) is one for which A\_p->A\_p\* is regular for all primes p. These are the maps from the local rings of A to their completions.


If A is an order in a number field, is A a G-ring? Equivalently (in this special case), is A excellent?


I've heard it said that 'all' rings that appear in algebraic geometry are excellent. Since an order A corresponds to a singular curve, I guess I expect A to be excellent as well.



Are non-maximal orders in number fields Grothendieck rings?
Yes: if R is excellent, so is any finite type R-algebra (apply this to Z and A).


