Equivalence of dimension notions for affine domain

Statement
Let $$A$$ be an affine domain over a field $$k$$, i.e. a finitely generated algebra over $$k$$, that also happens to be an integral domain. Then, the following are equal:


 * The Krull dimension of $$A$$
 * The Krull dimension of the localization of $$A$$ at any maximal ideal (which is the same as that obtained using the Hilbert-Samuel polynomial)
 * The transcendence degree of the field of fractions of $$A$$, over $$k$$

Facts used

 * Proving that the Krull dimension of the polynomial ring in $$n$$ variables, is equal to exactly $$n$$
 * Noether normalization theorem
 * Going up theorem
 * Going down for integral extensions of normal domains