tangent space

At a point of a smooth curve you can draw a tangent line, at a point of a smooth surface a tangent plane. The tangent space is the best linear approximation to a variety at a point — the flat space that hugs it most closely there. Algebraic geometry must build this without calculus, using only the ring structure, so that it works over any field.

The algebraic definition uses the local ring O_p at the point, with maximal ideal m. The cotangent space is the k-vector space m/m^2, and the Zariski tangent space is its dual, the space of k-linear maps m/m^2 to k, where k is the residue field. Equivalently a tangent vector is a derivation, or a k-point of the doubled-point scheme Spec k[e]/(e^2) mapping to the point. For a hypersurface f = 0 it is cut out by the linear equation given by the vanishing of the gradient's differential, the Jacobian at the point.

The dimension of the tangent space always satisfies dim T_p X greater than or equal to dim_p X, with equality exactly at smooth (regular) points; a strict inequality signals a singularity such as a node or cusp. This makes the tangent space a clean local test for smoothness and explains why the local ring is called regular precisely when the minimal number of generators of m equals the Krull dimension.