LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Flatness over a local filtered colimit appears at a Noetherian stage

Statement

Assume the Axiom of Choice. Let $(R_{i} \to S_{i}, M_{i})$ be a directed system of Noetherian local ring maps and finite $S_{i}$ -modules. For $i \leq j$ , suppose $S_{i} \otimes_{R_{i}} R_{j} \to S_{j}$ is a localization and $M_{i} \otimes_{S_{i}} S_{j} \to M_{j}$ is an isomorphism. Write $R = \underset{\to}{l i m} R_{i}$ , $S = \underset{\to}{l i m} S_{i}$ , and $M = \underset{\to}{l i m} M_{i}$ . If $M$ is flat over $R$ , then $M_{j}$ is flat over $R_{j}$ at some stage $j$ . In particular this applies to the localized approximation system of Finite presentation data descend to Noetherian algebra and module stages.

Facts & Assumptions

Given: The directed local system and the flat colimit module.

[F1]

The kernel of $J \otimes_{R} M \to M$ is ${Tor}_{1}^{R} (R / J, M)$ , and flatness makes this kernel zero (The long exact Tor sequence in the right-module variable, Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests).

[F2]

Tensor products and finite presentations commute with filtered colimits. In the localized finite-presentation system the transition maps have exactly the localization and module base-change forms in the Statement (Tensoring is right exact, Finite presentation data descend to Noetherian algebra and module stages).

[F3]

The Tor-killing local criterion applies to a square of Noetherian local maps whose target transition is a localization of the scalar extension. A flat closed quotient and a zero first-Tor map make the later finite target module flat (A killed first Tor obstruction yields flatness after local Noetherian base change).

Proof

technique · kill finitely many generators of the first Tor obstruction at a finite stage, then use the local flatness criterion

1.1F1

Fix a stage $i$ and write $m_{i}$ for the maximal ideal of $R_{i}$ . Since $R_{i}$ is Noetherian, $m_{i}$ is finite. The $S_{i}$ -module $m_{i} \otimes_{R_{i}} M_{i}$ is finite, as it is a quotient of a finite direct sum of copies of $M_{i}$ . Its submodule $T_{i} = \ker (m_{i} \otimes_{R_{i}} M_{i} \to M_{i}) = {Tor}_{1}^{R_{i}} (R_{i} / m_{i}, M_{i})$ is finite because $S_{i}$ is Noetherian. Choose finitely many $S_{i}$ -generators $ξ_{1}, \dots, ξ_{r}$ of $T_{i}$ .

2.1F1F2step 1.1

Consider, for $j \geq i$ , the ideal $J_{j} = m_{i} R_{j}$ . Its colimit is the ideal $J = m_{i} R \subseteq R$ . A fixed finite generating set for $m_{i}$ presents each $J_{j}$ as a quotient of a finite free $R_{j}$ -module. Every finite relation among those generators in $R$ already vanishes at a later stage; hence the natural map ${\underset{\to}{l i m}}_{j \geq i} (J_{j} \otimes_{R_{j}} M_{j}) ⟶ J \otimes_{R} M$ is an isomorphism. Equivalently, this follows by filtered-colimit exactness applied to the finite generating sequence and then by right exactness of tensor. Since $M$ is $R$ -flat, [F1] makes $J \otimes_{R} M \to M$ injective. Every $ξ_{a}$ has zero product in $M_{i}$ and thus has zero image in $J \otimes_{R} M$ .

3.1F1F2step 1.1step 2.1

By the colimit description in step 2.1, each $ξ_{a}$ maps to zero in $J_{j} \otimes_{R_{j}} M_{j}$ at some stage $j \geq i$ . Directedness and finiteness of the chosen generators give one common stage $j$ where all vanish. Since the natural map $T_{i} ⟶ {Tor}_{1}^{R_{j}} (R_{j} / J_{j}, M_{j}) = \ker (J_{j} \otimes_{R_{j}} M_{j} \to M_{j})$ is $S_{i}$ -linear, it is zero on all of $T_{i}$ . This is the precise Tor-killing hypothesis for [F3]; merely observing vanishing in the colimit would not suffice without this finite-stage argument.

4.1F3step 3.1

The quotient $M_{i} / m_{i} M_{i}$ is a vector space over the field $R_{i} / m_{i}$ , hence flat. Apply [F3] to the square $R_{i} \to S_{i}$ , $R_{j} \to S_{j}$ , the proper ideal $I = m_{i}$ , and $M_{i}$ . Its target-transition hypothesis is part of the Statement, its closed-quotient hypothesis is the vector-space flatness, and its Tor hypothesis is step 3.1. Therefore $M_{j}$ is flat over $R_{j}$ . This proves the eventual flatness assertion.

5.1

The final assertion follows by applying the argument to the localized system supplied by [F2]. No assertion is made that every stage becomes flat; the proof constructs one later flat stage. The Axiom of Choice covers the finite generator selections and the published Tor boundary. [F2, step 4.1] $□$

Depends on

Used by

Dependency tree · two levels

34 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, Algebra, Lemma 10.128.3 (tag 00R6), eventual flatness (standard reference, not scraped)
The Stacks Project, Algebra, Lemma 10.127.13 (tag 00QX), local approximation (standard reference, not scraped)