# Geometrically regular ring

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In [algebraic geometry](/source/algebraic_geometry), a '''geometrically regular ring''' is a [Noetherian ring](/source/Noetherian_ring) over a [field](/source/field_(algebra)) that remains a [regular ring](/source/regular_ring) after any [finite extension](/source/finite_extension) of the base field. Geometrically regular [schemes](/source/scheme_(mathematics)) are defined in a similar way. In older terminology, points with regular [local ring](/source/local_ring)s were called '''simple points''', and points with geometrically regular local rings were called '''absolutely simple points'''. Over fields that are of characteristic 0, or algebraically closed, or more generally [perfect](/source/perfect_field), geometrically regular rings are the same as regular rings. Geometric regularity originated when [Claude Chevalley](/source/Claude_Chevalley) and [André Weil](/source/Andr%C3%A9_Weil) pointed out to  {{harvs|txt|last=Zariski|first=Oscar|authorlink=Oscar Zariski|year=1947}} that, over non-perfect fields, the [Jacobian criterion](/source/Jacobian_criterion) for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.

A Noetherian local ring containing a field ''k'' is geometrically regular over ''k'' if and only if it is [formally smooth](/source/formally_smooth) over ''k''.

==Examples==

{{harvtxt|Zariski|1947}} gave the following two examples of local rings that are regular but not geometrically regular. 

#Suppose that ''k'' is a field of characteristic ''p''&nbsp;>&nbsp;0 and ''a'' is an element of ''k'' that is not a ''p''th power. Then every point of the curve ''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>''p''</sup>&nbsp;=&nbsp;''a'' is regular. However over the field ''k''[''a''<sup>1/''p''</sup>], every point of the curve is singular. So the points of this curve are regular but not geometrically regular.
#In the previous example, the equation defining the curve becomes reducible over a finite extension of the base field. This is not the real cause of the phenomenon: Chevalley pointed out to Zariski that the curve ''x''<sup>''p''</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;''a'' (with the notation of the previous example) is absolutely irreducible but still has a point that is regular but not geometrically regular.

== See also ==
*[Regular scheme](/source/Regular_scheme)

==References==

*{{EGA|book=IV-2}}
*{{citation|mr=0021694|last=Zariski|first= Oscar|authorlink=Oscar Zariski|title=The concept of a simple point of an abstract algebraic variety. 
|journal=[Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society) |volume=62|year=1947|issue=1 |pages= 1–52|jstor=1990628|doi=10.1090/s0002-9947-1947-0021694-1|doi-access=free}}

Category:Commutative algebra
Category:Algebraic geometry

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