Going-up and going-down theorems: motivation

Question

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.

Answer 1

Algebraic geometry makes many facts like this more compelling. For example, the going-up property for a ring map $R\to S$ is equivalent to $\operatorname{Spec} S \to \operatorname{Spec} R$ being a closed map. Also, if $R\to S$ has finite presentation and the going-down property, then $\operatorname{Spec} S \to \operatorname{Spec} R$ is open.

So going-up is important in the study of proper morphisms (which are the algebraic geometry version of compact maps of topological spaces). It is used in the proof that finite morphisms of schemes are proper.

Answer 2

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!