Dimension of a topological space by irreducible chains ($\dim X$)

This is the supremum of lengths of strict chains of nonempty irreducible closed subsets. Closedness is relative to the given space. This is the relevant dimension for locally closed subsets of algebraic varieties, rather than a general Hausdorff-space notion of dimension.

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.

An algebraic variety whose irreducible components all have the same dimension of an algebraic set. The components of a generic fibre in the fiber dimension theorem have this property even when that fibre is reducible.

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.