LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Henselian factor lifting descends to quotients

Statement

Let $(A, I)$ be a Henselian pair and let $J \subseteq A$ be an ideal. Then the quotient pair $(A / J, (I + J) / J)$ has the coprime monic factor-lifting property.

Facts & Assumptions

Given: A Henselian pair $(A, I)$ and an ideal $J \subseteq A$ .

[L1]

A Henselian pair uniquely lifts coprime monic factorizations modulo its defining ideal (Henselian pairs and Henselian local rings).

Proof

technique · use the integral quotient map and then unpack the definition

1.1

The quotient map $A \to A / J$ is integral because every element of $A / J$ satisfies a monic linear equation over the image of $A$ . The integral-base-change lemma for Henselian pairs in Stacks tag 09XD therefore applies to the Henselian pair $(A, I)$ and shows that the quotient pair $(A / J, (I + J) / J)$ is again Henselian.

givenalgebra

2.1

By [L1], every Henselian pair has the coprime monic factor-lifting property. Applying that definition to the pair from step 1.1 gives the claimed lifting property for $(A / J, (I + J) / J)$ .

L1step 1.1

3.1

Hence Henselian factor lifting descends to quotients.

step 2.1∎

Depends on

Henselian pairs and Henselian local rings

Used by

Quotients of Henselian local rings are Henselian Corollary

Dependency tree · two levels

2 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

The Stacks Project, Section 15.11: Henselian pairs (standard reference, not scraped)