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 $\Omega _{X/k,x}$.

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