CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Finite weight spaces and bounded support do not replace finite generation

Statement refuted

False claim: local $n^{+}$ -finiteness, finite-dimensional weight spaces and support in finitely many downward cones suffice for membership in $O$ without finite generation.

For ${s l}_{2}$ , $X = ⨁_{n \geq 0} M (- 2 n)$ has these three local properties, with $supp X = - 2 Z_{\geq 0}$ and $\dim X_{- 2 k} = k + 1$ for $k \geq 0$ , but is not finitely generated.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement refuted.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

[F2]

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ . (Weights of a Verma module lie below lambda)

Counterexample

1.1

Each Verma has one basis vector at each weight $- 2 n - 2 j$ , $j \geq 0$ : the negative root algebra has the single generator $f$ , so the induced Verma basis is $f^{j} v_{- 2 n}$ . At weight $- 2 k$ the contributing summands are exactly $n = 0, \dots, k$ , proving the count $k + 1$ . The direct sum is a weight module supported in the single cone $0 - Q^{+}$ .

F2algebra

2.1

The operator $e$ raises the Verma weight, so it kills every vector after sufficiently many applications inside each summand. A vector of the direct sum has nonzero coordinates in finitely many summands; taking the maximum of their bounds proves local $C [e]$ -finiteness.

algebrastep 1.1

3.1

Any finite list of vectors is supported in a finite set of summands. Those summands form a $g$ -submodule, so the submodule generated by the list remains there. There is always a further nonzero Verma summand outside that finite set. Hence the list cannot generate $X$ , which fails the finite-generation axiom of $O$ . The empty list generates zero, not $X$ .

F1algebrastep 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Lecture 2 Definition 3.1; direct explicit hypothesis test (standard reference, not scraped)