CounterexampleConstruction: Literature-sourcedVerification: 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.

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

A quotient stack need not be a scheme

Statement refuted

False claim: every algebraic stack over a field that is presented as a quotient of a scheme by a finite group action is equivalent to the stack of a scheme.

Facts & Assumptions

Given: A field $k$ , a nontrivial finite group $G$ (for instance $Z / 2$ ), its constant group scheme $G_{k}$ over $k$ , the classifying stack $B G$ of right $G$ -torsors, the trivial action of $G$ on $Spec k$ , and the inherited AC.

[F1]

$B G$ is an algebraic stack over $k$ with presentation $Spec k \to B G$ given by the trivial torsor, and the automorphism sheaf of the trivial torsor $G_{T}$ is $G_{T}$ by left multiplication (The classifying stack of a finite group).

[F2]

The stack in setoids $S_{X}$ of a scheme $X$ has only trivial automorphism groups in its fibre categories, and the inertia projection of a stack whose fibres are setoids is an equivalence (The inertia of a stack in setoids is trivial, Descent data, prestacks and stacks in groupoids over the fppf site).

[F3]

The fppf quotient sheaf of the trivial action of $G$ on $Spec k$ is the representable sheaf $h_{Spec k}$ : the naive quotient presheaf takes every $T$ to the one-point quotient of ${Mor}_{k} (T, Spec k)$ (which is a one-point set for every $T$ over $k$ ), it is already an fppf sheaf, and it is represented by $Spec k$ ; this is the field-level quotient-sheaf convention of Quotient sheaves and representable quotients for pre-relations and group actions (Presheaves, covariantly and contravariantly representable functors, and representations).

Proof

1.1F1given

$B G$ has nontrivial inertia. By [F1] the trivial torsor $G_{k}$ over $Spec k$ has automorphism group $G (Spec k) = G$ acting by left multiplication, and $G$ is nontrivial by hypothesis. Hence the fibre category of $B G$ over $Spec k$ is not a setoid.

2.1F1F2step 1.1

$B G$ is not the stack of a scheme. Suppose $B G$ were equivalent to $S_{X}$ for some $k$ -scheme $X$ . Then by [F2] every fibre category of $B G$ would be a setoid, contradicting step 1.1. Hence no $k$ -scheme $X$ has $B G ≃ S_{X}$ , even though $B G$ is an algebraic stack by [F1].

3.1F1F3step 2.1∎

Separation of the sheaf and stack levels. For the trivial action of $G$ on $Spec k$ , the fppf quotient sheaf $(Spec k) / G$ is the representable sheaf $h_{Spec k}$ by [F3], so at the level of quotient sheaves the quotient is a scheme, namely $Spec k$ ; at the level of quotient stacks the same data present the classifying stack $B G$ , which is not a scheme by step 2.1. This witnesses that passing from quotient sheaves to quotient stacks genuinely enlarges the category, and the failed conclusion is exactly the identification $B G ≃ S_{X}$ for some scheme $X$ . The supplier definition Quotient sheaves and representable quotients for pre-relations and group actions is now authored, and this field-level use is reconciled directly by [F3]: the quotient presheaf is terminal on the big fppf site and therefore already a sheaf.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

39 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

The Stacks Project, Chapter 65 (Algebraic Spaces), Section 65.14, and Chapter 94 (Algebraic Stacks) (standard reference, not scraped)