Codimension of an irreducible closed subvariety

For a nonempty irreducible closed subvariety $Z$ of an irreducible classical variety $X$, define ${\mathrm{codim}}_{X}Z=\dim X-\dim Z$. These are finite integers. In a reducible ambient variety a difference of global dimensions must not be substituted for the height of a local prime; the containing component matters.

Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.