Going-up and going-down theorems: motivation I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book.
I'm wondering if anyone could give me some basic facts/examples to help me understand why these two theorems are so important.
Neither of these theorems are actually used in Atiyah & Macdonald, and yet I don't find the statements to be so obviously remarkable that they should deserve a special name.

1) Suppose $A\subset B$ is an integral ring extension.
 Then the affine scheme $\text{Spec} (B)$ has the same Krull dimension as $\text{Spec} (A)$.
This results from the going-up theorem 5.11, the lying-over theorem 5.10 and the incomparability theorem 5.9  
2) Does this situation arise in practice? You bet!
For example Noether's normalization theorem states that any finitely generated algebra $B$ over a field $k$ (which is not assumed algebraically closed) is an integral extension of a purely transcendental ring in $n$ indeterminates $A=k[t_1,\cdots,t_n]$, and is  thus a finitely generated module over $A$: cf. Remark page 60.
Geometrically this means that any affine variety $X$ of dimension $n$ over $k$ admits of a finite ramified covering $X\to\mathbb A^n_k$ onto affine $n$-space . What a wonderful  result!
