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Integral extensions lift finite prime chains from the base

Statement

Assume the Axiom of Choice.

Let $f : A \to B$ be an integral ring map, let $p_{0} \subseteq p_{1} \subseteq \dots \subseteq p_{n}$ be a finite chain of prime ideals in $A$ , and let $q_{0}$ be a prime ideal of $B$ with $f^{- 1} (q_{0}) = p_{0}$ . Then there exist prime ideals $q_{0} \subseteq q_{1} \subseteq \dots \subseteq q_{n}$ of $B$ such that $f^{- 1} (q_{i}) = p_{i}$ for every $i$ .

Facts & Assumptions

Given: An integral ring map $f : A \to B$ , a finite prime chain $p_{0} \subseteq \dots \subseteq p_{n}$ in $A$ , and a prime $q_{0}$ of $B$ over $p_{0}$ .

[L1]

Assuming the Axiom of Choice, the going-up theorem lifts one prime extension step at a time (Going up for integral ring maps).

Proof

technique · induction on the chain length

1.1L1basegiven

For $n = 0$ , the given prime $q_{0}$ already lifts the chain.

1.2ihgiven

Fix $n \geq 0$ and assume the statement for chains of length $n$ . Let $p_{0} \subseteq \dots \subseteq p_{n} \subseteq p_{n + 1}$ be a chain of length $n + 1$ . By the induction hypothesis, there are primes $q_{0} \subseteq \dots \subseteq q_{n}$ over $p_{0}, \dots, p_{n}$ .

2.1L1step 1.2

Apply [L1] to the inclusion $p_{n} \subseteq p_{n + 1}$ and the prime $q_{n}$ . This yields a prime $q_{n + 1} \supseteq q_{n}$ with contraction $p_{n + 1}$ .

3.1step 1.1step 1.2step 2.1discharge-induction∎

Step 1.1 is the base case, and steps 1.2 and 2.1 provide the induction step. Therefore every finite prime chain in $A$ lifts to one in $B$ once the first prime upstairs is fixed.

Depends on

Going up for integral ring maps

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, Corollary 7.7 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Theorem (14.3)(4) (standard reference, not scraped)