ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-01

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A finite product of principal localizations covering the spectrum is faithfully flat

Example

Assume the Axiom of Choice for the faithfully-flat characterization used below.

Let $f_{1}, \dots, f_{n} \in R$ generate the unit ideal. Then the product map

R ⟶ \prod_{i = 1}^{n} R_{f_{i}}

is faithfully flat.

Facts & Assumptions

Given: The Axiom of Choice, a commutative ring $R$ , and elements $f_{1}, \dots, f_{n} \in R$ with $(f_{1}, \dots, f_{n}) = R$ .

[L1]

Each localization $R_{f_{i}}$ is flat over $R$ (Every localization is flat, and localizing a flat module preserves flatness).

[L2]

A flat ring map is faithfully flat exactly when proper ideals remain proper (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

Verification

technique · direct

1.1

Each factor $R_{f_{i}}$ is flat by [L1], so the product ring $\prod_{i} R_{f_{i}}$ is flat over $R$ because finite direct products are finite direct sums as modules.

L1givenalgebra

1.2

Let $I ⊊ R$ be proper. If $I \prod_{i} R_{f_{i}} = \prod_{i} R_{f_{i}}$ , then for every $i$ some power of $f_{i}$ lies in $I$ , because $1 \in I R_{f_{i}}$ implies $f_{i}^{m_{i}} \in I$ for some $m_{i}$ . Since the $f_{i}$ generate the unit ideal, so do the powers $f_{i}^{m_{i}}$ , forcing $1 \in I$ , contradiction. Thus the extended ideal is proper.

L2algebra

2.1

By [L2], the product map is faithfully flat.

L2∎

Depends on

Used by

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Dependency tree · two levels

7 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

J. S. Milne, A Primer of Commutative Algebra, Proposition 11.22 (standard reference, not scraped)