Finite-field stability data for an acyclic quiver

Let $Q=(Q_0,Q_1,s,t)$ be a finite quiver: $Q_0$ and $Q_1$ are finite sets, and $s,t:Q_1\to Q_0$ give the source and target of each arrow. Assume $Q$ has no directed cycles. For a field $K$, a finite-dimensional representation $M$ assigns a finite-dimensional $K$-vector space $M_i$ to each $i\in Q_0$ and a linear map $M_a:M_{s(a)}\to M_{t(a)}$ to each $a\in Q_1$. A morphism is a family of linear maps at the vertices commuting with all arrow maps. Write ${\mathcal{A}}_Q(K)$ for this category and $\mathbf{d}(M)=({\dim }_K{M}_i{)}_{i\in Q_0}\in {\mathbb{N}}^{Q_0}$ for its dimension vector. For the finite-field Hall branch, fix $k={\mathbb{F}}_q$ and write ${\mathcal{A}}_Q={\mathcal{A}}_Q(k)$. The next lemma proves the abelian and Euler properties over any field; over $k$ it constructs a small finitary abelian skeleton of ${\mathcal{A}}_Q$, so ${\mathcal{A}}_Q$ is essentially small.

For $\mathbf{d},\mathbf{e}\in {\mathbb{Z}}^{Q_0}$ define the bilinear form ${\chi }_Q(\mathbf{d},\mathbf{e})={\sum }_{i\in Q_0}{d}_i{e}_i-{\sum }_{a:i\to j}{d}_i{e}_j,$ and its skew-symmetrization $⟨\mathbf{d},\mathbf{e}⟩={\chi }_Q(\mathbf{d},\mathbf{e})-{\chi }_Q(\mathbf{e},\mathbf{d})$. The next lemma proves that ${\chi }_Q(\mathbf{d}(M),\mathbf{d}(N))$ is the Hom-minus-Yoneda-Ext${}^1$ dimension for this arrow convention.

Fix $z_i\in \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$ for each vertex and extend $Z(\mathbf{d})={\sum }_i{d}_i{z}_i$ additively. If $\mathbf{d}\in {\mathbb{N}}^{Q_0}$ is nonzero, then $\mathrm{Im}Z(\mathbf{d})={\sum }_i{d}_i\mathrm{Im}{z}_i>0$; thus its principal argument lies in $(0,\pi )$. For a nonzero representation $M$, put ${\varphi }_Z(M)=\arg Z(\mathbf{d}(M))\mathrm{/}\pi \in (0,1)$. Call $M$ $Z$-semistable if every nonzero subrepresentation $U\subseteq M$ satisfies ${\varphi }_Z(U)\le {\varphi }_Z(M)$, and $Z$-stable if the inequality is strict for every proper nonzero subrepresentation.

For the semistable groupoids and stacks below, include the zero representation as semistable without assigning it a phase. Stable objects and HN factors remain nonzero.

For a dimension vector $\mathbf{d}$, let ${\mathrm{Rep}}_{\mathbf{d}}(Q)={\prod }_{a:i\to j}{\mathrm{Hom}}_k(k^{d_i},k^{d_j}),G_{\mathbf{d}}={\prod }_{i\in Q_0}{\mathrm{GL}}_{d_i,k}\mathrm{.}$ The group $G_{\mathbf{d}}$ acts by change of bases. Define ${\mathrm{Rep}}_{\mathbf{d}}^{ss}(Q,Z)$ geometrically: its points over an algebraic closure $\bar{k}$ are representations for which every nonzero $\bar{k}$-subrepresentation satisfies the semistability inequality. The associated quotient stack is ${\mathfrak{M}}_{\mathbf{d}}^{ss}(Q,Z)=[{\mathrm{Rep}}_{\mathbf{d}}^{ss}(Q,Z)\mathrm{/}{G}_{\mathbf{d}}]$. For $\mathbf{d}=0$, set ${\mathrm{Rep}}_{\mathbf{0}}^{ss}={\mathrm{Rep}}_{\mathbf{0}}=\mathrm{Spec}k$. The group $G_{\mathbf{0}}$ is trivial, so its rational-point groupoid has one zero object, trivial automorphism group, and cardinality $1$. Here $[X\mathrm{/}G]$ denotes the groupoid-valued quotient: over a scheme $S$ its objects are $G$-torsors on $S$ with equivariant maps to $X$, and its arrows are equivariant isomorphisms over $S$. The openness lemma below proves that ${\mathrm{Rep}}_{\mathbf{d}}^{ss}$ is open.

For a finite groupoid $\mathcal{G}$, define its groupoid cardinality by $\mathrm{\mid }\mathcal{G}\mathrm{\mid }={\sum }_{[x]\in \mathrm{Iso}(\mathcal{G})}\frac{1}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{G}}(x)\mathrm{\mid }}\mathrm{.}$ The rational-point groupoid of ${\mathfrak{M}}_{\mathbf{d}}^{ss}$ is finite, and the finite-field descent lemma identifies it with the groupoid of finite-field semistable representations and their isomorphisms. No Calabi–Yau structure, potential, orientation data, vanishing cycles, or physical state-counting identification is part of this definition.