We shall be working in affine $n$-space $\mathbb{A}^n$
Fix an algebraic set, say, $V$
Now, the dimension of $V$ is defined as the supremum of all distinct, irreducible, Zariski-closed subspaces of $V$. (Technically it’s the $\sup-1$)
Now, suppose we have a chain with maximal element $V$, but with length strictly lesser than the $\sup$. Is is always possible to extend this to a chain having length equal to the supremum?
Basic example to illustrate the premise: Consider the unit sphere, and the following chain (a point on the sphere):
$V(x, y-1,z)\subset V(x^2+y^2+z^2-1)$
It is possible to extend it like so (Insert the unit circle contained in the sphere which contains the point):
$V(x, y-1,z)\subset V(x^2+y^2-1, z) \subset V(x^2+y^2+z^2-1)$
As far as what ‘progress’ I have made, I have made the observation that in our fixed algebraic set, it is enough to consider an irreducible component of maximum degree, which contains the immediately preceding subvariety in the chain because of the irreducible restriction. So the question reduces to answering the question for varieties.