4.20.5 Tensor Alternating Exterior Algebra Preparation

Tensor Alternating Exterior Algebra Preparation is the body of facts about alternating multilinear maps that must be in place before the exterior algebra can be constructed and shown to have its expected properties: the universal property specific to alternating maps, the identification of the defining ideal to quotient by, and the graded multiplicative structure that assembles the individual exterior powers into a single algebra.

What Must Be Assembled First

The Alternating Universal Property

Before the exterior algebra is built, the correct universal property for alternating maps must be isolated: for every alternating multilinear map f: V × ... × V → W (n copies), there must exist a unique linear map f̃: ⋀ⁿV → W with f̃(v₁ ∧ ... ∧ vₙ) = f(v₁,...,vₙ). This is the alternating analogue of the tensor product's universal property, and establishing it is the preparatory step that guarantees ⋀ⁿV, however it is constructed, is the correct object to serve as the home for alternating n-linear maps.

The Ideal Generated by Repeated Factors

Constructing ⋀V = ⊕ₙ ⋀ⁿV as a quotient of the full tensor algebra T(V) = ⊕ₙ V^{⊗n} requires identifying the correct two-sided ideal I to quotient by: the ideal generated by all elements v ⊗ v for v ∈ V. Verifying that this specific ideal, rather than some other candidate, produces a quotient in which the repeated-argument result holds for the induced wedge product is a preparatory step resting entirely on the alternating pattern already established for multilinear maps.

Deriving the Anticommutativity of the Product From the Pattern

From Repeated Factors to Sign-Reversal

Working inside T(V)/I, the vanishing of v ⊗ v for every v, together with bilinearity of the tensor product, forces (v + w) ⊗ (v + w) ≡ 0, which expands to v ⊗ w ≡ -w ⊗ v modulo I. This is exactly the alternating slot-exchange rule, now derived at the level of the algebra's defining relations rather than assumed directly for a given multilinear map, confirming that the wedge product inherited from T(V)/I will automatically be anticommutative on degree-one elements.

Extending to Higher Degree Products

The anticommutativity established for single vectors extends, via the same repeated-argument reasoning applied to sums of longer wedge products, to the rule

more precisely α ∧ β = (-1)^{pq} β ∧ α for α ∈ ⋀ᵖV and β ∈ ⋀ᑫV, the graded-commutativity law that governs the full exterior algebra, obtained by tracking the number of pairwise transpositions needed to move each factor of α past each factor of β.

Assembling the Graded Algebra Structure

Direct Sum Across All Degrees

The exterior algebra ⋀V = ⊕_{n=0}^{d} ⋀ⁿV (with d = dim V in the finite-dimensional case, since ⋀ⁿV = 0 for n > d by the repeated-argument result applied to any n > d vectors) collects every exterior power into a single graded vector space, with the wedge product providing multiplication that respects the grading: ⋀ᵖV ∧ ⋀ᑫV ⊆ ⋀^{p+q}V.

Associativity Inherited From the Tensor Algebra

Associativity of the wedge product, (α ∧ β) ∧ γ = α ∧ (β ∧ γ), is inherited directly from the associativity of the tensor product in T(V), since the wedge product is defined as the tensor product followed by projection to the quotient T(V)/I, and this projection is a ring homomorphism; no separate verification of associativity specific to alternating structure is required once the ideal I has been shown to be two-sided.

Dimension Count as a Consistency Check

Binomial Coefficients Summing Correctly

The preparatory dimension count, dim(⋀ⁿV) = C(d, n) for V of dimension d, is confirmed using the basis of increasing-index wedge products established from the alternating basis component pattern; summing across all degrees gives

matching the dimension of the full exterior algebra as 2^d, a check that the exterior power construction, quotienting T(V) correctly, has produced the expected total space.

Confirming the Universal Property Once the Algebra Is Built

Verifying Existence of the Factorization

Once ⋀ⁿV is constructed as T^n(V)/I_n, existence of the factorization f̃ for a given alternating f follows the same reasoning used for the ordinary tensor product's universal property: f, being multilinear, first factors through V^{⊗n} as f̂, and since f is alternating, f̂ vanishes on the ideal I_n, so f̂ descends to a well-defined linear map f̃ on the quotient ⋀ⁿV, which is exactly the required factorization.

Verifying Uniqueness of the Factorization

Uniqueness follows because the wedge products v₁ ∧ ... ∧ vₙ span ⋀ⁿV, just as elementary tensors span the ordinary tensor product, so any two linear maps agreeing with f on these spanning wedge products must agree everywhere on ⋀ⁿV, completing the preparatory work needed to declare that ⋀ⁿV genuinely satisfies the universal property for alternating multilinear maps of arity n.

Why This Preparation Precedes the Exterior Algebra Proper

Separating the Combinatorial Pattern From the Algebraic Construction

Establishing the alternating pattern, repeated-argument vanishing, sign change under transposition, the basis component pattern, prior to constructing the exterior algebra keeps the purely combinatorial content of alternation separate from the specifically algebraic task of building a quotient ring with a well-defined graded multiplication; the exterior algebra construction then consists of transporting already-established facts about alternating multilinear maps into statements about a concrete associative graded algebra, rather than re-deriving them from first principles inside that algebra.