User:Maximilian Janisch/latexlist/Algebraic Groups/Smooth scheme

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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 $k$ 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 $A _ { k } ^ { \prime \prime }$ and a projective space $P _ { k } ^ { \prime }$ 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

\begin{equation} F _ { i } ( X _ { 1 } , \ldots , X _ { m } ) = 0 , \quad i = 1 , \ldots , n \end{equation}

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 $a$ is the dimension of $x$ at $\pi$ (Jacobi's criterion). In a more general case, a closed subscheme $x$ of a smooth scheme $Y$ defined by a sheaf of ideals $1$ is smooth in a neighbourhood of a point $\pi$ if and only if there exists a system of generators $g 1 , \ldots , g _ { x }$ of the ideal $I _ { X }$ in the ring $O X , X$ for which $d g _ { 1 } , \ldots , d g _ { r }$ form part of a basis of a free $O x , x$-module of the differential sheaf $\Omega X / k , x$.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967) MR0238860 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901
[3]	O. Zariski, "The concept of a simple point of an abstract algebraic variety" Trans. Amer. Math. Soc. , 62 (1947) pp. 1–52 MR0021694 Zbl 0031.26101

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001