CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Under the stated choice boundary, free modules are projective and hence flat

Statement

Let $R$ be a commutative ring and let $F$ be a free $R$ -module with basis indexed by $X$ .

Assuming the Axiom of Choice, $F$ is projective and therefore flat.
If $X$ is finite, only finite choice is needed for projectivity; if $X = \emptyset$ , no choice is needed.
Regardless of choice, $F$ is flat, because tensoring with $F$ is a direct sum of copies of the identity tensor functor.

Facts & Assumptions

Given: A commutative ring $R$ and a free $R$ -module $F$ with basis indexed by $X$ .

[L1]

Under AC every free module is projective; a finite basis requires only finite choice, and an empty basis requires none (Free modules are projective, with the exact choice boundary).

[L2]

Every projective module over a commutative ring is flat without choice (Every projective module over a commutative ring is flat).

[L3]

Tensor products commute with arbitrary direct sums in either variable; in particular $A \otimes_{R} ⨁_{x \in X} R ≅ ⨁_{x \in X} (A \otimes_{R} R)$ (Tensor products commute with arbitrary direct sums).

[L4]

The regular module is a tensor unit (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

Proof

technique · direct

1.1

Under AC, [L1] makes $F$ projective and [L2] then makes it flat. The refined finite and empty-basis choice bounds are exactly those stated in [L1].

givenL1L2

1.2

Independently of AC, write $F = ⨁_{x \in X} R$ . By [L3] and [L4], tensoring an exact sequence with $F$ gives the direct sum, over $X$ , of the original exact sequence; kernels and images are computed coordinatewise, so the result remains exact. Thus $F$ is flat without any choice principle.

L3L4algebra

2.1

Step 1.1 establishes the projective route with its precise choice boundary, while step 1.2 establishes flatness unconditionally; the two routes are logically distinct.

step 1.1step 1.2∎

Depends on

Used by

A short exact sequence with flat quotient remains short exact after tensoring Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

C. Dennis, Week 4 on tensor products and flatness (standard reference, not scraped)
W. Li, Commutative Algebra, Lectures 9-10 (standard reference, not scraped)