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Motivic Hall Algebras and Integration Maps: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Objective

These calculations test the A-page's quotient-first convention, stack automorphism weights, and Euler twist. The finite-field realizations are groupoid cardinalities of the stated rational-point groupoids, not unweighted counts of isomorphism classes.

Examples

The motivic Hall product for a one-vertex zero-arrow quiver computes the split extension fiber for a one-vertex zero-arrow quiver. The extension has no nonzero class, but its endpoint-fixing automorphisms contribute the factor $L^{- 1}$ .
An untwisted monoid map fails for the one-arrow quiver uses the one-arrow quiver over $F_{2}$ to show that omitting the Euler twist fails: the extension groupoid has cardinality $2$ , while the untwisted monomial product has coefficient $1$ .

Both items use the extension-stack formula and the finite-field groupoid-cardinality specialization proved on the A page. Neither example supplies a general theorem beyond its stated model and hypotheses.

Companion: motivic-hall-algebras-and-integration.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passOpen item page →

The motivic Hall product for a one-vertex zero-arrow quiver

Example

Let $Q$ have one vertex and no arrows.

\begin{tikzpicture}
\node[circle,draw,inner sep=1.5pt,label=below:{1}] (v) at (0,0) {};
\end{tikzpicture}

The vertex label is $1$ ; the absence of edges means a representation is only a finite-dimensional vector space. Let $S$ be its one-dimensional representation and let $δ_{S} = [B {GL}_{1} \to M_{1} (Q)]$ be the identity stack function. For every field $k$ , $I_{Q} (δ_{S} * δ_{S}) = L^{- 1} [B {GL}_{1}]^{2} x^{2} .$ On the finite-field groupoid-count specialization $k = F_{q}$ , this coefficient is $q^{- 1} (q - 1)^{- 2}$ .

Facts & Assumptions

Given: The one-vertex zero-arrow quiver, its representation stacks, and the quotient-first motivic Hall product.

[F1]

The Hall product is the stack of sequences $0 \to B \to E \to A \to 0$ with quotient first and subrepresentation second (The motivic Hall algebra of a finite quiver).

[F2]

Its extension-stack relative class is $L^{- χ_{Q} (d, e)}$ (The motive of a quiver extension stack).

[F3]

The integration map sends a stack function $[X \to M_{d}]$ to $[X] x^{d}$ and the target twist is the stated Euler factor (Quiver stack integration is multiplicative with the Euler twist).

[F4]

For a finite-field quotient stack in this construction, groupoid cardinality is the sum of inverse automorphism orders and respects the motivic stack relations (Finite-field groupoid cardinality for quiver stack specializations, Finite-field groupoid cardinality respects the motivic stack relations).

[A1]

The algebraic-stack convention inherits its represented-sheaf descent dependency on AC (Algebraic stacks and their inertia stacks, The Axiom of Choice); the calculation itself uses only finite-dimensional vector spaces and finite groupoids after specialization.

Verification

technique · direct

1.1F1F2F3A1

For this quiver, ${Rep}_{1} (Q)$ is a point and $G_{1} = {GL}_{1}$ , so $M_{1} (Q) = B {GL}_{1}$ . For the two degree-one factors, $d = e = 1$ and $χ_{Q} (1, 1) = 1$ . By [F2], the extension stack over the pair of endpoint stacks has relative class $L^{- 1}$ . Pushing to the middle and then integrating therefore gives $I_{Q} (δ_{S} * δ_{S}) = L^{- 1} [B {GL}_{1}]^{2} x^{2}$ .

2.1F1F2F4algebra∎

Over $F_{q}$ , the groupoid $B {GL}_{1} (F_{q})$ has one isomorphism class with automorphism group of order $q - 1$ , hence groupoid cardinality $(q - 1)^{- 1}$ . The extension fiber over the two fixed endpoints is $B G_{a} (F_{q})$ : the middle sequence splits, and its endpoint-fixing automorphisms are the linear maps $S \to S$ , a group of order $q$ . Its groupoid cardinality is $q^{- 1}$ by [F4]. Multiplying these three factors gives $q^{- 1} (q - 1)^{- 2}$ , as stated.

Remarks

Here $χ_{Q} (1, 1) = 1$ because there is one vertex and no arrows. This is the untwisted Hall stack coefficient with its automorphism group retained; it is not the finite subobject count $q + 1$ used by the separate untwisted Ringel product.

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passOpen item page →

An untwisted monoid map fails for the one-arrow quiver

Statement refuted

The claim that stack integration always maps quiver motivic Hall products multiplicatively to the untwisted dimension-vector monoid algebra is false. Here “untwisted” means the target rule $x^{d} x^{e} = x^{d + e}$ , while the source map is $I_{Q} ([X \to M_{d}]) = [X] x^{d}$ .

Let $Q$ have vertices $1, 2$ and one arrow $1 \to 2$ .

1 \arrow[r] & 2

For $k = F_{2}$ , let $S_{1}$ and $S_{2}$ be the representations supported at vertices $1$ and $2$ . Use $S_{1}$ as quotient and $S_{2}$ as subrepresentation. There are two exact sequences with these fixed endpoints, distinguished by the arrow map of the middle representation. The stack groupoid count of $δ_{S_{1}} * δ_{S_{2}}$ is $2$ , whereas the product of the two integrated classes in the untwisted target has coefficient $1$ .

Facts & Assumptions

Given: The one-arrow quiver over $F_{2}$ , its simple representations $S_{1}, S_{2}$ , and the quotient-first Hall convention.

[F1]

The extension stack with quotient dimension $d$ and subobject dimension $e$ has relative class $L^{- χ_{Q} (d, e)}$ (The motive of a quiver extension stack).

[F2]

The stack integration on a homogeneous class is its absolute stack class times the corresponding monomial, and the Euler-twisted target uses $L^{- χ_{Q} (d, e)}$ (Quiver stack integration is multiplicative with the Euler twist).

[F3]

An arrow of a quiver representation is a linear map from the space at its source vertex to the space at its target vertex; for the short exact sequence the off-diagonal arrow map belongs to $Hom (A_{1}, B_{2})$ (The motivic Hall algebra of a finite quiver).

[F4]

Finite-field groupoid cardinality is the sum of inverse automorphism orders and respects the stack relations (Finite-field groupoid cardinality for quiver stack specializations, Finite-field groupoid cardinality respects the motivic stack relations).

[A1]

The algebraic-stack convention inherits its represented-sheaf descent use of AC (Algebraic stacks and their inertia stacks, The Axiom of Choice). All vector-space and groupoid calculations in this finite example are explicit.

Counterexample

technique · direct

1.1A1F1F3F4givenalgebra

The quotient $S_{1}$ has dimension vector $d = (1, 0)$ and the subobject $S_{2}$ has dimension vector $e = (0, 1)$ . At each vertex the middle space is the direct sum of the endpoint spaces. The only possibly nonzero off-diagonal arrow entry is a scalar $ξ \in {Hom}_{F_{2}} (F_{2}, F_{2})$ , so there are exactly two extensions, $ξ = 0$ and $ξ = 1$ . Since ${GL}_{1} (F_{2})$ is the trivial group, neither endpoint automorphisms nor their identifications identify these two sequences, and each sequence has trivial automorphism group fixing the endpoints. Thus the extension groupoid has cardinality $2$ . The quotient-stack convention is the one in [A1]; the remaining calculation is a finite groupoid count.

2.1F1F2F4step 1.1

The Hall integration of this product therefore has coefficient $2$ at $x^{(1, 1)}$ in its finite-field point-count realization. Each endpoint stack $B {GL}_{1} (F_{2})$ has groupoid cardinality $1$ , so their integrated product in the untwisted target has coefficient $1$ . The two coefficients differ, refuting the claimed multiplicativity.

3.1F1F2F4step 1.1algebra∎

The Euler pairing is $χ_{Q} (d, e) = 0 - (1) (1) = - 1$ , so the correct target twist is $L^{- (- 1)} = L$ . It specializes to $2$ over $F_{2}$ and matches the source extension count from step 1.1. This also verifies the direction and sign of the twist.

Motivic Hall Algebras and Integration Maps: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Objective

Examples

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The motivic Hall product for a one-vertex zero-arrow quiver

Example

Facts & Assumptions

Verification

Remarks

An untwisted monoid map fails for the one-arrow quiver

Statement refuted

Facts & Assumptions

Counterexample

Sources