PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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.

Flatness is transitive under a flat change of rings

Statement

Let $R \to S$ be a flat homomorphism of commutative rings. If $N$ is a flat $S$ -module, then $N$ , restricted to an $R$ -module, is flat over $R$ . Consequently a composite of flat ring homomorphisms is flat.

The same assertions hold with "faithfully flat" throughout.

Facts & Assumptions

Given: A flat ring map $R \to S$ and a flat $S$ -module $N$ ; for the faithful assertion, assume both are faithfully flat.

[L1]

A ring map is flat or faithfully flat exactly when its target has that property as a module over its source (Flat and faithfully flat modules and ring homomorphisms).

[L2]

For every right $R$ -module $X$ , change of rings gives $X \otimes_{R} N ≅ (X \otimes_{R} S) \otimes_{S} N$ (Change of rings: $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ ).

[L3]

Restriction of scalars leaves the underlying groups and maps unchanged (Restriction of scalars and extension of scalars $S \otimes_{R} M$ along a ring homomorphism $R \to S$ ).

Proof

technique · direct

1.1givenL1

Let $A \to B \to C$ be an exact sequence of $R$ -modules. Flatness of $S$ over $R$ makes $A \otimes_{R} S \to B \otimes_{R} S \to C \otimes_{R} S$ exact as a sequence of $S$ -modules.

2.1givenstep 1.1L1

Flatness of $N$ over $S$ preserves the exactness of step 1.1 after tensoring over $S$ .

3.1step 2.1L2L3

By [L2], the sequence in step 2.1 is naturally isomorphic to $A \otimes_{R} N \to B \otimes_{R} N \to C \otimes_{R} N$ , so the restricted $R$ -module $N$ is flat.

3.2step 1.1step 2.1L1L2

If both functors are faithful on exactness, the implications in steps 1.1 and 2.1 may be read backwards as well; [L2] then shows that tensoring with $N$ over $R$ reflects exactness, so the restricted module is faithfully flat.

4.1step 3.1step 3.2L1∎

Taking $N$ to be the target ring of a second flat, respectively faithfully flat, ring map and using [L1] proves the corresponding composition statement.

Depends on

Used by

Dependency tree · two levels

10 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

Stacks Project, Lemma 10.39.4 (standard reference, not scraped)