|
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but the positive characteristic case is (as of 2013) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent. ==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 regular. *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''. *A ring ''R'' is called a G-ring (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any ''p''∈Spec(''R''), the map from the local ring ''R''''p'' to its completion is regular in the sense above. *A ring is called a J-2 ring if for every finitely generated ''R''-algebra ''S'', the singular points of Spec(''S'') form a closed subset. *A ring ''R'' is called quasi-excellent if it is a ''G''-ring and a J-2 ring. *A ring is called excellent if it is quasi-excellent and universally catenary. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings. *A scheme is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property. 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「excellent ring」の詳細全文を読む スポンサード リンク
|