In
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 regular scheme is a locally Noetherian
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
whose
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 are
regular everywhere. Every
smooth scheme In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smoo ...
is regular, and every regular scheme of finite type over a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k'' is ...
is smooth.
[.]
For an example of a regular scheme that is not smooth, see
Geometrically regular ring#Examples.
See also
*
Étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy th ...
*
Dimension of an algebraic variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutati ...
*Glossary of scheme theory
*Smooth completion
References
Algebraic geometry
Scheme theory
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