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, ..., g_r = 0, where each g_i is in the polynomial ring k[x₁,..., x_n]. 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 (∂g_i/∂x_j) 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 embedding 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 g_i and all those minors is the whole polynomial ring.

In geometric terms, the matrix of derivatives (∂g_i/∂x_j) at a point p in X gives a linear map Fⁿ → F^r, 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.

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