LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A flat local map splits regular sequences into base and fibre parts

Statement

Let $(R, m, k) \to (S, n, ℓ)$ be a flat local homomorphism of Noetherian local rings. If $x_{1}, \dots, x_{r}$ is an $R$ -regular sequence and ${\overset{ˉ}{y}}_{1}, \dots, {\overset{ˉ}{y}}_{s}$ is regular on the closed fibre $S / m S$ , then arbitrary lifts $y_{j} \in n$ make $x_{1}, \dots, x_{r}, y_{1}, \dots, y_{s}$ an $S$ -regular sequence.

Facts & Assumptions

Given: a flat local map is faithfully flat.

Proof

technique · direct

1.1

Flat base change preserves the injectivity of multiplication by each $x_{i}$ on the successive source quotients, and faithful flatness preserves their nonzero terminal quotient. Thus the $x_{i}$ form an $S$ -regular sequence.

given

2.1

After quotienting by the $x_{i}$ , the induced map remains flat and has the same closed fibre. The local flatness criterion applied successively to the lifts $y_{j}$ promotes injectivity on the fibre quotients to injectivity on the corresponding $S$ -quotients. Nakayama preserves the nonzero terminal fibre quotient. Hence the concatenated sequence is regular.

step 1.1algebra∎

Depends on

Used by

Depth is additive for a flat local homomorphism Corollary

Dependency tree · two levels

9 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)