LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The triple Massey expression is a cocycle

Statement

For every triple Massey datum in Triple Massey data in a cohomological differential graded algebra, the expression $ω = (- 1)^{p + 1} a v + u c$ is closed. Thus it determines a class in $H^{p + q + r - 1} (A)$ .

Facts & Assumptions

Given: A cohomological differential graded algebra $A$ and homogeneous closed $a, b, c$ of degrees $p, q, r$ , with $d u = a b$ and $d v = b c$ .

[F1]

Triple Massey data in a cohomological differential graded algebra: the expression is $ω = (- 1)^{p + 1} a v + u c$ , with $∣ u ∣ = p + q - 1$ and $∣ v ∣ = q + r - 1$ .

[F2]

Cohomological differential graded algebra: for homogeneous $x, y$ , $d (x y) = d (x) y + (- 1)^{∣ x ∣} x d (y)$ , and multiplication is associative.

Proof

1.1givenF1F2algebra

By [F2], closedness of $a, c$ and the defining equations in [F1], $d ω = (- 1)^{p + 1} (- 1)^{p} a d v + (d u) c = - a (b c) + (a b) c .$

2.1F2step 1.1∎

Associativity gives $a (b c) = (a b) c$ , so the right side of step 1.1 is zero. Therefore $ω$ is a cycle of degree $p + q + r - 1$ and has a class in the displayed cohomology group.

Depends on

Used by

Triple Massey product with a fixed cohomological sign convention Definition

Dependency tree · two levels

4 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

Bernhard Keller, Introduction to A-infinity algebras and modules, §3.3 (standard reference, not scraped)