ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (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.

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The same Verma in two ambient categories

Example

Assume the Axiom of Choice (The Axiom of Choice).

Let $g = {s l}_{2}$ and consider the weight $- 2$ . In the one-label truncation $Γ = {- 2}$ of the linkage class ${- 2, 0}$ , the weight $- 2$ is maximal in $Γ$ , so $Δ (- 2) = M (- 2)$ is projective in $O_{Γ}$ by A maximal-label Verma is projective in its truncation. In the full regular integral block ${- 2, 0}$ , the same Verma $Δ (- 2)$ is not projective, because its projective cover is the nonsplit extension $0 \to Δ (0) \to P (- 2) \to Δ (- 2) \to 0$ . This shows that the ambient truncation in the hypothesis of A maximal-label Verma is projective in its truncation cannot be dropped: projectivity of a maximal-label Verma is a statement about the chosen finite downward-closed ideal, and enlarging the ideal can destroy it.

Facts & Assumptions

Given: The Axiom of Choice, $g = {s l}_{2}$ , the linkage class ${- 2, 0}$ , the truncation $Γ = {- 2}$ , and the full block $C = {- 2, 0}$ .

[F1]

In the root order $- 2 \leq 0$ , and $Γ = {- 2}$ is a finite downward-closed ideal of the class ${- 2, 0}$ in which $- 2$ is maximal: the only element of the class below $- 2$ is $- 2$ itself (Truncation at a finite downward-closed ideal of a linkage class, The two projectives in the principal sl2 block).

[F2]

If $λ$ is maximal in a finite downward-closed ideal $Γ$ of a linkage class, then $Δ (λ)$ is projective in $O_{Γ}$ (A maximal-label Verma is projective in its truncation).

[F3]

In the full block, $P (- 2)$ fits into the nonsplit sequence $0 \to Δ (0) \to P (- 2) \to Δ (- 2) \to 0$ (The two projectives in the principal sl2 block).

Verification

technique · direct: apply the maximal-label lemma in the small truncation and use the nonsplit cover to refute projectivity in the large block

1.1F1F2given

By [F1] the set $Γ = {- 2}$ is a finite downward-closed ideal of the linkage class ${- 2, 0}$ in which $- 2$ is maximal, so [F2] makes $Δ (- 2)$ projective in $O_{Γ}$ .

1.2F3given

In the full block $C = {- 2, 0}$ , suppose $Δ (- 2)$ were projective. Since $Δ (- 2) = L (- 2)$ is the head of $P (- 2)$ and $P (- 2) ↠ Δ (- 2)$ is an epimorphism onto a projective object, it would split, contradicting the nonsplitness of the sequence in [F3]. Hence $Δ (- 2)$ is not projective in $O_{C}$ .

2.1step 1.1step 1.2∎

The same Verma $Δ (- 2)$ is thus projective in the one-label truncation $O_{Γ}$ but not in the full block $O_{C}$ : projectivity of a maximal-label Verma depends on the ambient finite downward-closed ideal, as claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Pavel Etingof, Representations of Lie Groups (18.757, Fall 2023), Proposition 16.4 and its proof applied to a support truncation (standard reference, not scraped)
Lin Chen, lecture notes (Spring 2024), Lecture 9, Warning 1.9 and Example 3.16 (standard reference, not scraped)