algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, 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-Noethe ...

over a field 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 any Noetherian local ring with unique maxi ...

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 mathematics, more specifically in 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 algebraic varieties or manifolds, or of ...

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 found ...

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 one of the most influential mathematicians of the twentieth century. His influence is du ...

pointed out to that, over non-perfect fields, the 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 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 pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

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

Examples

See also

References