Smooth scheme

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 An over k for some natural number n. Then X is the closed subscheme defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. 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 (∂gi/∂xj) has rank at least n−m 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 (n−m) × (n − m) minors of the matrix of derivatives are zero is the empty set. Equivalently, the ideal in the polynomial ring generated by all gi and all those minors is the whole polynomial ring. In geometric terms, the matrix of derivatives (∂gi/∂xj) at a point p in X gives a linear map Fn → Fr, 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.