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