Codimension of an algebraic subvariety (${\mathrm{codim}}_{X}Y=\dim X-\dim Y$)

For an irreducible closed subvariety $Y$ of an irreducible variety $X$, its codimension is $\dim X-\dim Y$. In an affine chart it also equals the height of the corresponding prime ideal, by the dimension formula for finite-type domains over a field.