algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a geometrically regular ring is a

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

that remains a

regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...

after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, points with regular

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...

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, geometrically regular rings are the same as regular rings. Geometric regularity originated when

Claude Chevalley Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foundin ...

and

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

pointed out to that, over non-perfect fields, the

Jacobian criterion In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant * Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler ...

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 In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) mean ...

over ''k''.

Examples

gave the following two examples of local rings that are regular but not geometrically regular. #Suppose that ''k'' is a field of characteristic ''p'' > 0 and ''a'' is an element of ''k'' that is not a ''p''th power. Then every point of the curve ''x''^''p'' + ''y''^''p'' = ''a'' is regular. However over the field ''k'' 'a''^1/''p'' 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''^''p'' + ''y''² = ''a'' (with the notation of the previous example) is absolutely irreducible but still has a point that is regular but not geometrically regular.

References

* *{{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 The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...

, volume=62, year=1947, pages= 1–52, jstor=1990628, doi=10.1090/s0002-9947-1947-0021694-1, doi-access=free Commutative algebra Algebraic geometry

Examples

See also

References