TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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The wedge product is associative and graded commutative

Statement

For alternating covectors $α \in {Alt}^{k} (V)$ , $β \in {Alt}^{ℓ} (V)$ , and $γ \in {Alt}^{m} (V)$ ,

(α \land β) \land γ = α \land (β \land γ)

and

α \land β = (- 1)^{k ℓ} β \land α .

Facts & Assumptions

Given: Alternating covectors $α, β, γ$ of degrees $k, ℓ, m$ .

[F1]

The wedge product is the normalized alternation of the tensor product, equivalently the signed shuffle sum (The wedge product of alternating covectors).

[L1]

The wedge product is alternating and bilinear (The wedge product is alternating and bilinear).

Proof

technique · direct

1.1

Using [F1] twice and bilinearity from [L1], both $(α \land β) \land γ$ and $α \land (β \land γ)$ are the full alternation of the multilinear tensor $α \otimes β \otimes γ$ with the same normalization factor. Hence they are equal.

F1L1givenalgebra

1.2

In the shuffle formula of [F1], swapping the $k$ inputs destined for $α$ with the $ℓ$ inputs destined for $β$ contributes the sign of the block permutation, namely $(- 1)^{k ℓ}$ . Therefore every term of $α \land β$ matches the corresponding term of $(- 1)^{k ℓ} β \land α$ .

F1givenalgebra

2.1

Therefore the wedge product is associative and graded commutative.

step 1.1step 1.2∎

Depends on

Used by

Wedge products of the standard dual basis Example
A nonzero one-form need not have a nonzero square under the wedge product False statement
The wedge product is not commutative False statement
Differential forms form a graded commutative algebra Proposition

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)