G-ring

• A (Noetherian) ring R containing a field k is called geometrically regular over k if for any finite extension K of k the ring R ⊗k K is a regular ring. • A homomorphism of rings from R to S is called regular if it is flat and for every p ∈ Spec(R) the fiber S ⊗R k(p) is geometrically regular over the residue field k(p) of p. (see also Popescu's theorem.) • A ring is called a local G-ring if it is a Noetherian local ring and the map to its completion (with respect to its maximal ideal) is regular. • A ring is called a G-ring if it is Noetherian and all its localizations at prime ideals are local G-rings. (It is enough to check this just for the maximal ideals, so in particular local G-rings are G-rings.) ==Examples==

• Every field is a G-ring. • Every complete Noetherian local ring is a G-ring. • Every ring of convergent power series in a finite number of variables over R or C is a G-ring. • Every Dedekind domain in characteristic 0, and in particular the ring of integers, is a G-ring, but in positive characteristic there are Dedekind domains (and even discrete valuation rings) that are not G-rings. • Every localization of a G-ring is a G-ring. • Every finitely generated algebra over a G-ring is a G-ring. This is a theorem due to Grothendieck. Here is an example of a discrete valuation ring A of characteristic p>0 which is not a G-ring. If k is any field of characteristic p with [k : kp] = ∞ and R = kx and A is the subring of power series Σaixi such that [kp(a0,a1,...) : kp] is finite then the formal fiber of A over the generic point is not geometrically regular so A is not a G-ring. Here kp denotes the image of k under the Frobenius morphism a→ap. ==References==