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The parameter power-series map is injective by dimension

Statement

Assume the Axiom of Dependent Choice.

Let $(A, m)$ be a complete equicharacteristic Noetherian local domain of dimension $d$ , let $k \subseteq A$ be a coefficient field, and let $x_{1}, \dots, x_{d}$ be a system of parameters. Then the continuous map $ϕ : k ⟦ X_{1}, \dots, X_{d} ⟧ \to A, X_{i} \mapsto x_{i},$ is injective.

Facts & Assumptions

Given: A complete equicharacteristic Noetherian local domain $(A, m)$ of dimension $d$ , a coefficient field $k$ , a system of parameters $x_{1}, \dots, x_{d}$ , and the Axiom of Dependent Choice.

[L1]

The map $ϕ$ makes $A$ finite over its image (Parameters make a complete local domain finite over the image of a power-series map).

[L2]

A system of parameters records that $\dim A = d$ (Systems of parameters and parameter ideals, Local dimension is the minimal number of generators of an ideal with maximal radical).

[L3]

The formal power-series ring $k ⟦ X_{1}, \dots, X_{d} ⟧$ is a Noetherian local domain of dimension $d$ (Stacks Project, Section 10.160, Remark 10.160.9).

[L4]

A strict chain of primes contracts to a strict chain along an integral injection (Strict prime chains contract strictly under integral extensions).

Proof

technique · a nonzero kernel would force the source quotient to have dimension less than the target

1.1

Let $B = k ⟦ X_{1}, \dots, X_{d} ⟧$ and suppose $P : = \ker (ϕ) \neq 0$ . Since $A$ is a domain, $P$ is prime. By [L1], $A$ is finite over the image $B / P$ , hence integral over $B / P$ .

L1givenassume-contraalgebra

2.1

By [L3], $B$ is a domain of dimension $d$ . Every strict chain of primes in $B / P$ lifts to a strict chain of primes of $B$ containing $P$ ; adjoining $P$ at the bottom if necessary, write it as $P = p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{ℓ} .$ This chain can be preceded by the strict inclusion $(0) ⊊ P$ . Hence $ℓ + 1 \leq d$ , and therefore $\dim (B / P) \leq d - 1$ .

L3step 1.1algebra

3.1

By [L4], every strict chain of primes in $A$ contracts to a strict chain in $B / P$ , so $\dim A \leq \dim (B / P)$ . Combining this with step 2.1 gives $d = \dim A \leq d - 1$ , contradicting [L2].

L2L4step 1.1step 2.1discharge-contradiction

4.1

Therefore $\ker (ϕ) = 0$ , so $ϕ$ is injective.

step 3.1∎

Depends on

Used by

A complete local domain is finite over a regular power-series ring Corollary

Dependency tree · two levels

21 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

Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)
The Stacks Project, Section 10.160: The Cohen structure theorem (standard reference, not scraped)