TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Critical quiver CoHA is associative and unital, including the standard realization

Statement

Assume AC. For every finite complex quiver $Q$ , every finite potential $W$ and every coherent datum $K$ of Quiver potentials and realized monodromic coefficient data, the critical CoHA of Critical quiver CoHA and its standard rational realization is associative and unital. In particular its standard rational monodromic vanishing-cycle instance exists and is associative and unital, for arbitrary critical values, loops and multiple arrows. For $W = 0$ it recovers the ordinary intrinsic rational quiver CoHA with the same multiplication and grading.

Facts & Assumptions

Given: $Q, W, K$ and the critical algebraic direct sum with the subobject-first order.

[F1]

Quiver potentials and realized monodromic coefficient data specifies primary homological coefficients, the smooth identification $q^{*} (K_{a} ⊠ K_{b}) [2 r] ≃ K_{E}$ , the proper coefficient morphism $R p_{*} K_{E} \to K_{a + b}$ , their flag/external-product/base-change/monodromy coherence and the dimension-zero identity.

[F2]

Critical quiver CoHA and its standard rational realization defines the product using that datum and the intrinsic index, with shift $- 2 χ_{Q} (b, a)$ and value $z_{1} + z_{2}$ .

[F3]

A quiver potential trace is additive on exact sequences, The quiver exact-sequence correspondence is smooth and proper and Borel–Moore operations for quiver extension and flag quotient stacks give trace additivity, smooth/proper block geometry and the two Cartesian three-step-flag comparisons with coherent positive orientations.

[F4]

Standard rational quiver vanishing cycles realize the critical coefficient package ACTUALLY realizes every clause of [F1] for normalized rational vanishing cycles, including proper coefficient morphisms, the intrinsic shifts, all critical values and the $W = 0$ ordinary comparison. It does not presume an isomorphism for closed restriction.

[A1]

The Axiom of Choice is inherited through [F3]–[F4]; no additional choice is required by the finite flag arguments.

Proof

technique · direct

1.1F1F2F3givenconstruct

For each pair of dimension/value summands, [F1] and [F3] define the external product, smooth inverse image with its shift, and the proper coefficient pushforward. By [F2] the result has value $z_{1} + z_{2}$ and dimension $a + b$ . A finite sum of supported inputs therefore has a finite sum as output. This checks well-definedness on the stated algebraic direct sum.

2.1F1F3step 1.1

The two parenthesizations of a product of three inputs parametrize the same flag $0 \subset M_{a} \subset M_{a + b} \subset M_{a + b + c}$ . The proper/smooth Cartesian comparisons of [F3] express either iterated operation through this flag stack. Its trace function is $W_{a} ⊞ W_{b} ⊞ W_{c}$ by repeated trace additivity. The two coefficient identifications agree by the associative external-product/flag requirement of [F1], and the two proper coefficient maps agree by composition and Cartesian pasting in [F1]. Thus both parenthesizations induce exactly the same coefficient pull-push map on the common flag, preserving the same monodromy and total value.

2.2F1F2F3step 1.1algebra

The zero-subobject and whole-subobject correspondences are identity maps. Their dimension-zero coefficient is $Q$ , their coefficient morphisms are identities by [F1], and $χ_{Q} (0, d) = χ_{Q} (d, 0) = 0$ . The specified class is therefore a two-sided degree-zero unit.

3.1F2F3step 2.1algebra

The two degrees agree, since bilinearity gives $χ_{Q} (b, a) + χ_{Q} (c, a + b) = χ_{Q} (c, b) + χ_{Q} (b + c, a)$ . The proper comparisons have degree zero and the smooth orientations in [F3] concatenate with their even real ranks. Hence the equality of step 2.1 is an equality in the same intrinsic graded summand, proving associativity.

4.1A1F4step 1.1step 2.1step 3.1step 2.2∎

Apply the preceding argument to the STANDARD datum supplied by the actual realization [F4]. This proves the asserted nonconditional rational critical CoHA for every stated quiver and potential, retaining all its critical-value and monodromy summands. The $W = 0$ identification in [F4] identifies both its coefficient morphisms and smooth orientations with the ordinary intrinsic construction, so it identifies the algebras rather than merely their graded groups. AC is inherited exactly through [A1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

42 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

Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, §§2.3 and 7.6 (standard reference, not scraped)
Ben Davison, The critical CoHA of a quiver with potential, §§2.5–2.7 and 3.2 (standard reference, not scraped)