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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 motive of a quiver extension stack

Statement

Let $Q$ be a finite quiver over $k$ , and let $X \to M_{d} (Q)$ and $Y \to M_{e} (Q)$ be finite-type stack families with affine stabilizers of quotient representations $A$ and subrepresentations $B$ . Let $E_{X, Y}$ be the stack of exact sequences $0 \to B \to E \to A \to 0$ with the endpoint identifications supplied by $X$ and $Y$ . Define the bilinear Euler form $χ_{Q} (d, e) = \sum_{i \in Q_{0}} d_{i} e_{i} - \sum_{a \in Q_{1}} d_{s (a)} e_{t (a)} .$ Then the relative motivic class over $X \times Y$ is $[E_{X, Y} \to X \times Y] = L^{- χ_{Q} (d, e)} [X \times Y \to X \times Y]$ in the vector-bundle-stack coefficient theory of Relative motivic stack-function groups for quiver representation stacks. The statement concerns extension groupoids, so automorphisms of an extension that act trivially on its two endpoints are retained.

Facts & Assumptions

Given: A finite quiver $Q$ , families of representations $A, B$ of fixed dimension vectors $d, e$ , and the extension-stack class above.

[F1]

The relative coefficient group imposes $[V_{1} / V_{0} \to S] = L^{rk V_{1} - rk V_{0}} [S \to S]$ for a constant-rank linear map of vector bundles $V_{0} \to V_{1}$ (Relative motivic stack-function groups for quiver representation stacks).

[F2]

For each arrow $a : i \to j$ , an extension has an off-diagonal arrow map $A_{i} \to B_{j}$ after a local splitting of the vertexwise vector-bundle sequence; changing that splitting by $h_{i} : A_{i} \to B_{i}$ changes the off-diagonal map by $B_{a} h_{i} - h_{j} A_{a}$ (The motivic Hall algebra of a finite quiver).

[A1]

The algebraic-stack convention used here inherits the represented-sheaf descent use of AC stated in Algebraic stacks and their inertia stacks and The Axiom of Choice. The explicit finite quiver quotient presentations require no further choice.

Proof

technique · direct

1.1A1F2givenconstruct

Over the base $X \times Y$ , form the vector bundles $V_{0} = ⨁_{i \in Q_{0}} Hom (A_{i}, B_{i}), V_{1} = ⨁_{a \in Q_{1}} Hom (A_{s (a)}, B_{t (a)}) .$ Their ranks are $\sum_{i} d_{i} e_{i}$ and $\sum_{a : i \to j} d_{i} e_{j}$ , respectively. For any scheme $T \to X \times Y$ , the vertexwise vector-bundle sequences of an extension split Zariski locally on $T$ . Their simultaneous splittings form a torsor under $V_{0} ∣_{T}$ , and the off-diagonal arrow entries give an equivariant map from this torsor to $V_{1} ∣_{T}$ for the translation action $\partial (h)_{a} = B_{a} h_{s (a)} - h_{t (a)} A_{a}$ . Conversely, such a torsor with an equivariant map glues the split vertex bundles and block upper-triangular arrow maps into an extension. These constructions are inverse on objects and isomorphisms and commute with base change, giving a global equivalence $E_{X, Y} ≃ [V_{1} / V_{0}]$ over $X \times Y$ . No Zariski splitting over the stack base is required.

2.1F1F2step 1.1algebra

The rank of $\partial$ may vary, so stratify $X \times Y$ by its finitely many possible ranks. On each locally closed stratum the kernels and cokernels of $\partial$ are vector bundles, and the defining vector-stack relation in [F1] gives class $L^{rk V_{1} - rk V_{0}}$ times that stratum. This exponent is constant across strata because the two bundle ranks were fixed by $d, e$ . Additivity over the strata yields $[E_{X, Y} \to X \times Y] = L^{\sum_{a : i \to j} d_{i} e_{j} - \sum_{i} d_{i} e_{i}} [X \times Y \to X \times Y] .$

3.1F1step 2.1algebra∎

The exponent in step 2.1 is $- χ_{Q} (d, e)$ by the definition in the Statement. This proves the claimed relative class.

Depends on

Used by

An untwisted monoid map fails for the one-arrow quiver Counterexample
The motivic Hall product for a one-vertex zero-arrow quiver Example
Quiver stack integration is multiplicative with the Euler twist Theorem

Dependency tree · two levels

10 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

Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §4.1 and §7.1 (standard reference, not scraped)
Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v2, §§4–5 (standard reference, not scraped)