TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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Exterior multiplication is well defined, graded, associative, unital, and graded-commutative

Statement

The wedge product of The graded exterior algebra $Λ V$ is well defined and bilinear in each homogeneous pair of degrees. It is associative, has unit $1 \in F = Λ^{0} V$ , is graded in the sense that $Λ^{k} V \land Λ^{ℓ} V \subseteq Λ^{k + ℓ} V$ , and is graded-commutative:

$α \land β = (- 1)^{k ℓ} β \land α (α \in Λ^{k} V, β \in Λ^{ℓ} V) .$

All of this holds over every field, including characteristic two.

Facts & Assumptions

Given: A vector space $V$ over a field $F$ and homogeneous elements of degrees $k, ℓ$ .

[L1]

The exterior algebra is the graded sum of the exterior powers, with the wedge product defined by concatenation on decomposables and extended bilinearly; the definition checks that a repeated pair persists under concatenation, so the product is well defined (The graded exterior algebra $Λ V$ ).

[L2]

The basic wedge map is multilinear and alternating (The basic wedge map $(v_{1}, \dots, v_{k}) \mapsto v_{1} \land \dots \land v_{k}$ is multilinear and alternating).

Proof

technique · direct

1.1L1

Well-definedness and bilinearity in each degree pair are recorded in [L1], and the graded containment is the degree count of the concatenation.

1.2L1

Unitality: for a decomposable $α$ , the convention $1 \land α = α$ is the definition in [L1]; bilinearity extends it.

1.3L1L2

Associativity: on decomposables, $(α \land β) \land γ$ and $α \land (β \land γ)$ are both the concatenation of the three lists by [L1]; multilinearity of [L2] extends the equality to all elements.

1.4L2algebra

For vectors $v, w$ , alternation of [L2] applied to $(v + w, v + w)$ gives $0 = v \land w + w \land v$ , so $v \land w = - w \land v$ ; no division by $2$ is used, so this holds in characteristic two as well, where it reads $v \land w = w \land v$ .

2.1step 1.3step 1.4L2

Block swap: for decomposable $α = v_{1} \land \dots \land v_{k}$ and $β = w_{1} \land \dots \land w_{ℓ}$ , move each of the $ℓ$ factors $w_{j}$ leftward past all $k$ factors $v_{i}$ using step 1.4, collecting $ℓ k$ factors of $- 1$ ; associativity of step 1.3 makes the block move legitimate, giving $α \land β = (- 1)^{k ℓ} β \land α$ , and multilinearity extends this to all homogeneous elements.

3.1step 1.1step 1.2step 1.3step 1.4step 2.1∎

Steps 1.1 through 2.1 prove the five asserted laws in every characteristic, because the only sign rule used is the alternation identity of step 1.4.

Depends on

Dependency tree · 2 levels

Sources

Keith Conrad, Exterior Powers