Smooth scheme

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A generalization of the concept of a non-singular algebraic variety. A scheme of (locally) finite type over a field is called a smooth scheme (over ) if the scheme obtained from by replacing the field of constants with its algebraic closure is a regular scheme, i.e. if all its local rings are regular. For a perfect field the concepts of a smooth scheme over and a regular scheme over are identical. In particular, a smooth scheme of finite type over an algebraically closed field is a non-singular algebraic variety. In the case of the field of complex numbers a non-singular algebraic variety has the structure of a complex analytic manifold.

A scheme is smooth if and only if it can be covered by smooth neighbourhoods. A point of a scheme is called a simple point of the scheme if in a certain neighbourhood of it is smooth; otherwise the point is called a singular point. A connected smooth scheme is irreducible. A product of smooth schemes is itself a smooth scheme. In general, if is a smooth scheme over and is a smooth morphism, then is a smooth scheme over .

An affine space and a projective space are smooth schemes over ; any algebraic group (i.e. a reduced algebraic group scheme) over a perfect field is a smooth scheme. A reduced scheme over an algebraically closed field is smooth in an everywhere-dense open set.

If a scheme is defined by the equations

in an affine space , then a point is simple if and only if the rank of the Jacobi matrix is equal to , where is the dimension of at (Jacobi's criterion). In a more general case, a closed subscheme of a smooth scheme defined by a sheaf of ideals is smooth in a neighbourhood of a point if and only if there exists a system of generators of the ideal in the ring for which form part of a basis of a free -module of the differential sheaf .


[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian)
[2] A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967)
[3] O. Zariski, "The concept of a simple point of an abstract algebraic variety" Trans. Amer. Math. Soc. , 62 (1947) pp. 1–52



[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2
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Smooth scheme. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article