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 both a 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
- A 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 the fiber S ⊗R k is geometrically regular over the residue field k of p.
- A ring is called a local G-ring if it is a Noetherian local ring and the map to its completion is regular.
- A ring is called a G-ring if it is Noetherian and all its localizations at prime ideals are local 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 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.