Smooth point of a variety
= Smooth point of a variety
Over a <perfect field>, a point of a <variety> is smooth when its <local ring> is a <regular local ring>. For an <irreducible variety> of dimension $n$ at a <closed point>, this is equivalent to its <Zariski tangent space> having dimension $n$. The <Jacobian criterion> expresses the equality as a matrix-rank condition.