ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 differential graded algebra as its regular left module

Example

Let $A$ be any cohomological DGA over a commutative ring $R$ . Let $M = A$ as a graded $R$ -module, let $A$ act on $M$ by left multiplication, and let $d_{M} = d_{A}$ .

Facts & Assumptions

Given: A cohomological DGA $A$ over $R$ with its multiplication and differential. Verify that its regular left module is a differential graded module.

[F1]

Cohomological differential graded algebra: The multiplication on $A$ is $R$ -bilinear, associative and unital, has degree zero, and has central $R$ -action.

[F2]

Cohomological differential graded algebra: For homogeneous $a \in A^{p}$ and $b \in A^{q}$ , the differential satisfies $d^{p + q} (a b) = d^{p} (a) b + (- 1)^{p} a d^{q} (b)$ .

[F3]

Cohomological left and right differential graded modules: A cohomological left differential graded $A$ -module is a graded unital left $A$ -module whose action has degree zero, together with a degree-one square-zero differential satisfying $d_{M} (a m) = d_{A} (a) m + (- 1)^{p} a d_{M} (m)$ for homogeneous $a \in A^{p}$ .

[F4]

Cohomological differential graded algebra: The differential consists of $R$ -linear maps $d^{n} : A^{n} \to A^{n + 1}$ with $d^{n + 1} d^{n} = 0$ .

Verification

1.1F1F3givenalgebra

Associativity and the unit of $A$ make left multiplication a unital left $A$ -action on $M = A$ , and degree preservation gives $A^{p} M^{q} \subseteq M^{p + q}$ ; zero acts by zero and the unit acts as the identity.

1.2F2F3F4given

The differential $d_{M} = d_{A}$ has degree $+ 1$ and squares to zero by [F4]. For homogeneous $a, m \in A$ , the required identity $d_{M} (a m) = d_{A} (a) m + (- 1)^{∣ a ∣} a d_{M} (m)$ is the DGA Leibniz identity [F2].

2.1F3step 1.1step 1.2∎

The action and differential therefore satisfy every condition in [F3], so $A$ is a left DG module over itself.

Depends on

Used by

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