Dimension of an algebraic set ($\dim X$)

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

The Krull dimension of ${\mathcal{O}}_{X,x}$ at a point $x$. For a closed point of an irreducible affine variety it equals $\dim X$, by the closed-point dimension lemma for affine domains. At a generic point of a codimension-one subvariety the local dimension is one.

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.