DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Triple Massey data in a cohomological differential graded algebra

Definition

Let $(A, d, μ)$ be a cohomological differential graded algebra over a commutative ring, with the degree-one differential and signed Leibniz rule of Cohomological differential graded algebra. For $j \in Z$ , define

$H^{j} (A) = \ker (d : A^{j} \to A^{j + 1}) / im (d : A^{j - 1} \to A^{j}) .$

A triple Massey datum consists of homogeneous closed elements $a \in A^{p}$ , $b \in A^{q}$ , and $c \in A^{r}$ such that $a b$ and $b c$ are boundaries, together with elements

$u \in A^{p + q - 1}, v \in A^{q + r - 1}, d u = a b, d v = b c .$

The associated triple Massey expression is

$ω (a, b, c; u, v) = (- 1)^{p + 1} a v + u c \in A^{p + q + r - 1} .$

This item defines the data and expression only. The next lemma proves that the expression is closed, so its cohomology class is defined.

Depends on

Cohomological differential graded algebra

Used by

Dependency tree · two levels

3 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)
Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §9.2.5 (standard reference, not scraped)