G-ring


In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular. Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck.
A ring that is a both G-ring and a J-2 ring is called a quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring.

Definitions

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 = ∞ and R=kx and A is the subring of power series Σaixi such that 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.