I'm reading the book Algebraic Geometry, an Introduction (Daniel Perrin) and it introduces a notion of dimension of a topological space as the maximal length of ascending chains of irreducible closed subsets.
It looks like the same notion of topological dimension is introduced in Hartshorne and called the combinatorial dimension (see this question).
Using this definition on an algebraic variety equipped with the Zarizki topology, the dimension can be defined algebraically (e.g. on an affine variety it corresponds to the Krull dimension of the coordinate ring).
My question is: how does the combinatorial dimension relate to the Lebesgue covering dimension, which is what is usually used on a generic topological space? Also, how do we see in the case $K = \mathbb{C}$ and when the variety $X$ is non-singular, that it corresponds to the dimension of $X$ as a manifold?