TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exterior powers are functorial

Statement

For linear maps $T : V \to W$ and $S : W \to U$ and every $k \geq 0$ , the induced maps of The induced map $Λ^{k} T$ on exterior powers satisfy

$Λ^{k} ({id}_{V}) = {id}_{Λ^{k} V}, Λ^{k} (S \circ T) = Λ^{k} S \circ Λ^{k} T .$

Thus $V \mapsto Λ^{k} V$ , $T \mapsto Λ^{k} T$ is a functor from $F$ -vector spaces to $F$ -vector spaces, for each fixed $k$ .

Facts & Assumptions

Given: Linear maps $T : V \to W$ and $S : W \to U$ and a degree $k \geq 0$ .

[L1]

The induced map is $Λ^{k} T (v_{1} \land \dots \land v_{k}) = T (v_{1}) \land \dots \land T (v_{k})$ (The induced map $Λ^{k} T$ on exterior powers).

[L2]

A linear map out of $Λ^{k} V$ is determined by its values on decomposable wedges, by the uniqueness clause of the universal property (Exterior powers represent alternating multilinear maps and are unique up to unique isomorphism).

Proof

technique · direct

1.1

By [L1], $Λ^{k} ({id}_{V}) (v_{1} \land \dots \land v_{k}) = v_{1} \land \dots \land v_{k}$ , so it agrees with ${id}_{Λ^{k} V}$ on decomposables; [L2] then gives equality everywhere.

L1L2

1.2

By [L1] applied to $S \circ T$ and to each factor, $Λ^{k} (S \circ T) (v_{1} \land \dots \land v_{k}) = S (T (v_{1})) \land \dots \land S (T (v_{k})) = Λ^{k} S (T (v_{1}) \land \dots \land T (v_{k})) = (Λ^{k} S \circ Λ^{k} T) (v_{1} \land \dots \land v_{k})$ ; [L2] extends the equality from decomposables to all of $Λ^{k} V$ .

L1L2

2.1

Steps 1.1 and 1.2 are the identity and composition laws of a functor, with the $k = 0$ and $k = 1$ cases the conventions of [L1].

step 1.1step 1.2L1∎

Depends on

Used by

Dependency tree · two levels

11 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

Keith Conrad, Exterior Powers (standard reference, not scraped)