In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.
First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space A<sup>n</sup> over k for some natural number n. Then X is the closed subscheme defined by some equations g<sub>1</sub> = 0, ..., g<sub>r</sub> = 0, where each g<sub>i</sub> is in the polynomial ring k[x<sub>1</sub>,..., x<sub>n</sub>]. The affine scheme X is smooth of dimension m over k if X has dimension at least m in a neighborhood of each point, and the matrix of derivatives (âÂÂg<sub>i</sub>/âÂÂx<sub>j</sub>) has rank at least nâÂÂm everywhere on X. (It follows that X has dimension equal to m in a neighborhood of each point.) Smoothness is independent of the choice of immersion of X into affine space.
The condition on the matrix of derivatives is understood to mean that the closed subset of X where all (nâÂÂm) à(n â m) minors of the matrix of derivatives are zero is the empty set. Equivalently, the ideal in the polynomial ring generated by all g<sub>i</sub> and all those minors is the whole polynomial ring.
In geometric terms, the matrix of derivatives (âÂÂg<sub>i</sub>/âÂÂx<sub>j</sub>) at a point p in X gives a linear map F<sup>n</sup> â F<sup>r</sup>, where F is the residue field of p. The kernel of this map is called the Zariski tangent space of X at p. Smoothness of X means that the dimension of the Zariski tangent space is equal to the dimension of X near each point; at a singular point, the Zariski tangent space would be bigger.
More generally, a scheme X over a field k is smooth over k if each point of X has an open neighborhood which is a smooth affine scheme of some dimension over k. In particular, a smooth scheme over k is locally of finite type.
There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme X is smooth over a field k if and only if the morphism X â Spec k is smooth.
A smooth scheme over a field is regular and hence normal. In particular, a smooth scheme over a field is reduced.
Define a variety over a field k to be an integral separated scheme of finite type over k. Then any smooth separated scheme of finite type over k is a finite disjoint union of smooth varieties over k.
For a smooth variety X over the complex numbers, the space X(C) of complex points of X is a complex manifold, using the classical (Euclidean) topology. Likewise, for a smooth variety X over the real numbers, the space X(R) of real points is a real manifold, possibly empty.
For any scheme X that is locally of finite type over a field k, there is a coherent sheaf é<sup>1</sup> of differentials on X. The scheme X is smooth over k if and only if é<sup>1</sup> is a vector bundle of rank equal to the dimension of X near each point. In that case, é<sup>1</sup> is called the cotangent bundle of X. The tangent bundle of a smooth scheme over k can be defined as the dual bundle, TX = (é<sup>1</sup>)<sup>*</sup>.
Smoothness is a geometric property, meaning that for any field extension E of k, a scheme X is smooth over k if and only if the scheme X<sub>E</sub> := X ÃÂ<sub>Spec k</sub> Spec E is smooth over E. For a perfect field k, a scheme X is smooth over k if and only if X is locally of finite type over k and X is regular.
A scheme X is said to be generically smooth of dimension n over k if X contains an open dense subset that is smooth of dimension n over k. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth.