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 equivalent:

• 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 $n$
• Going up theorem
• Going down theorem

Proof

Fill this in later