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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 (defined below). 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== *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.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「G-ring」の詳細全文を読む スポンサード リンク
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