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CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passverified 2026-09-24 (gpt-6-sol)
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Quotienting by a first parameter lowers local dimension by one

Statement

Assume the Axiom of Choice.

Let (R,m) be a Noetherian local ring of positive dimension, and let (x1,…,xd) be a system of parameters. Then dim⁡(R/(x1))=d−1.

Facts & Assumptions

Given: The Axiom of Choice, a Noetherian local ring (R,m) of positive dimension and a system of parameters (x1,…,xd).

[L1]

By definition, x1 is the first member of a system of parameters (Systems of parameters and parameter ideals).

[L2]

Under the Axiom of Choice, a first parameter lowers the local dimension by exactly one (A first parameter lowers local dimension by exactly one).

Proof

technique · direct
1.1L1L2given

The tuple (x1,…,xd) satisfies the hypothesis of [L2] by [L1].

2.1L2step 1.1∎

Therefore dim⁡(R/(x1))=d−1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources