Smooth scheme

A generalization of the concept of a non-singular algebraic variety. A scheme $X$ of (locally) finite type over a field $k$ is called a smooth scheme (over $k$) if the scheme obtained from $X$ by replacing the field of constants $k$ with its algebraic closure $\overline{k}$ is a regular scheme, i.e. if all its local rings are regular. For a perfect field $k$ the concepts of a smooth scheme over $k$ and a regular scheme over $k$ 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 $X$ is called a simple point of the scheme if in a certain neighbourhood of it $X$ 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 $Y$ is a smooth scheme over $k$ and $f: \ X \rightarrow Y$ is a smooth morphism, then $X$ is a smooth scheme over $k$.

An affine space $A _{k} ^{n}$ and a projective space $\mathbf P _{k} ^{n}$ are smooth schemes over $k$; 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 $X$ is defined by the equations $$F _{i} (X _{1} \dots X _{m} ) = 0, i = 1 \dots n,$$ in an affine space $A _{k} ^{m}$, then a point $x \in X$ is simple if and only if the rank of the Jacobi matrix $\| {\partial F _{i} / \partial X _ j} (x) \|$ is equal to $m - d$, where $d$ is the dimension of $X$ at $x$( Jacobi's criterion). In a more general case, a closed subscheme $X$ of a smooth scheme $Y$ defined by a sheaf of ideals $I$ is smooth in a neighbourhood of a point $x$ if and only if there exists a system of generators $g _{1} \dots g _{n}$ of the ideal $I _{x}$ in the ring ${\mathcal O} _{X,x}$ for which $dg _{1} \dots dg _{n}$ form part of a basis of a free $O _{X,x}$- module of the differential sheaf $\Omega _{X/k,x}$.

How to Cite This Entry:
Smooth scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smooth_scheme&oldid=44319
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