LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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.

The coinvariant dual is the invariant-functional space

Statement

For $W = {span}_{C} {x \cdot m : x \in g_{o u t}, m \in M_{∙}}$ and $C_{X} (M_{∙}) = M_{∙} / W$ , pullback along the quotient map identifies $C_{X} (M_{∙})^{*}$ with ${λ \in M_{∙}^{*} : λ (x \cdot m) = 0 for every x \in g_{o u t}, m \in M_{∙}} .$ The dual is the algebraic linear dual; no topology or continuous-dual convention is imposed.

Statement references

Coinvariants and conformal blocks on a pointed curve

Facts & Assumptions

Given: The algebraic tensor product, outgoing Lie-algebra action, and quotient subspace in Coinvariants and conformal blocks on a pointed curve.

[F1]

Coinvariants and conformal blocks on a pointed curve: $C_{X} (M_{∙}) = M_{∙} / W$ , where $W = {span}_{C} {x \cdot m}$ , and $B_{X} (M_{∙}) = {Hom}_{C} (C_{X} (M_{∙}), C)$ .

Proof

1.1givenF1algebra

Let $q : M_{∙} \to M_{∙} / W$ be the quotient map. If $\overset{ˉ}{λ} \in (M_{∙} / W)^{*}$ , then $λ : = \overset{ˉ}{λ} \circ q$ vanishes on $W = \ker q$ , so $λ (x \cdot m) = 0$ for every $x, m$ .

2.1givenF1step 1.1construct∎

Conversely, if $λ \in M_{∙}^{*}$ vanishes on every $x \cdot m$ , linearity makes it vanish on their span $W$ . Define $\overset{ˉ}{λ} (q (m)) : = λ (m)$ . If $q (m) = q (m^{'})$ , then $m - m^{'} \in W$ , whence $λ (m) = λ (m^{'})$ ; thus $\overset{ˉ}{λ}$ is well-defined and linear. The two constructions are inverse by surjectivity of $q$ , proving the equality.

The coinvariant dual is the invariant-functional space

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources