Finite-dimensional representations of an acyclic quiver form a hereditary finitary Hall category

Fix orderings of the finite sets $Q_0$, $Q_1$, and $k$. Kernels, images and cokernels of a morphism of representations are formed at each vertex, with the induced arrow maps; the vector-space kernel and cokernel identities imply the abelian-category axioms vertexwise. For each dimension vector, enumerate the finitely many arrow-matrix tuples on the fixed spaces $k^{d_i}$ and retain the lexicographically least tuple in each orbit under ${\prod }_i{\mathrm{GL}}_{d_i}(k)$. Their union ${\mathcal{R}}_Q$ is an explicit small skeleton; no representatives from a proper class are selected.

For a vertex $i$, let $P_i$ have at vertex $j$ the vector space with basis the directed paths $i\to j$, with arrows acting by path extension. Acyclicity and finiteness of $Q$ imply there are finitely many paths, so $P_i$ is finite-dimensional. Evaluation gives a natural isomorphism $\mathrm{Hom}(P_i,W)\cong W_i$; since evaluation is exact, $P_i$ is projective in ${\mathrm{Rep}}_k(Q)$. Its chosen representative in ${\mathcal{R}}_Q$ is projective as well, since the skeleton is equivalent to the full representation category.

The simple $S_i$ is $k$ at vertex $i$ and zero elsewhere. If a representation is simple, choose a sink $i$ in its nonzero support; a one-dimensional subspace at that vertex is a nonzero subrepresentation, so simplicity forces the representation to equal $S_i$. A composition series then shows $[V]={\sum }_i(\dim V_i)[S_i]$ in $K_0$ by [F3]. The dimension-vector map $K_0\to {\mathbb{Z}}^{Q_0}$ sends $[S_i]$ to the standard basis vector; it is inverse to the map sending that vector to $[S_i]$.

Every Hom set is a subset of the finite product ${\prod }_i{\mathrm{Hom}}_k(V_i,W_i)$ and is therefore finite. For fixed endpoints $V,W$, an extension has vertex dimensions $\dim V_i+\dim W_i$; there are only finitely many arrow-matrix collections and endpoint maps in those dimensions. Yoneda classes are a quotient of this finite set, so ${\mathrm{YExt}}^1(V,W)$ is finite.

For a representation $V$, define $P_0(V)={⨁}_i{P}_i{\otimes }_k{V}_i,P_1(V)={⨁}_{a:i\to j}{P}_j{\otimes }_k{V}_i\mathrm{.}$ The augmentation $P_0(V)\to V$ sends a path $p:i\to r$ tensored with $v$ to the image of $v$ under the composite map $V_p$. For $a:i\to j$ and a path $p:j\to r$, the differential sends $p\otimes v$ to $(pa)\otimes v-p\otimes V_a(v)$. These maps commute with every arrow.

In an extension $0\to S_j\to E\to S_i\to 0$, the vertex dimensions are fixed. If $i\mathrm{\ne }j$, all arrow maps vanish except the maps on arrows $i\to j$, each of which is a scalar in $k$. Endpoint-fixed equivalence leaves these scalars unchanged, and every scalar tuple defines an extension. Under Baer sum, the direct sum is pulled back along the diagonal and pushed out along the codiagonal; on each arrow coordinate these two operations add the two scalars. Pushing out along multiplication by $\lambda \in k$ multiplies each coordinate by $\lambda$. If $i=j$, looplessness leaves every arrow map zero and the extension splits. Thus ${\mathrm{YExt}}^1(S_i,S_j)\cong k^{c_{ij}}$ as $k$-spaces.

More generally, in an extension $0\to W\to E\to V\to 0$, choose a vector-space splitting at each vertex. Each arrow map on $E_i=W_i\oplus V_i$ then has block form $\left(\begin{array}{cc}{W}_a& {\xi }_a\\ 0& V_a\end{array}\right)$ with ${\xi }_a:V_i\to W_j$. Changing the vertex splittings by maps ${\theta }_i:V_i\to W_i$ changes the family $({\xi }_a)$ by the image of $\mathrm{\partial }(\theta {)}_a=W_a{\theta }_i-{\theta }_j{V}_a$. Conversely every family $({\xi }_a)$ defines such an extension, and an endpoint-fixed isomorphism changes it by exactly such a splitting change. Hence ${\mathrm{YExt}}^1(V,W)\cong \mathrm{coker}\mathrm{\partial }$, while $\mathrm{Hom}(V,W)=\ker \mathrm{\partial }$ because a morphism is precisely a family of vertex maps commuting with every arrow. Rank–nullity gives $\dim \mathrm{Hom}(V,W)-\dim {\mathrm{YExt}}^1(V,W)={\sum }_i\dim V_i\dim W_i-{\sum }_{a:i\to j}\dim V_i\dim W_j\mathrm{.}$ The right side is bilinear in the two dimension vectors.

At vertex $r$, every positive-length path ending at $r$ has a unique first arrow $a:i\to j$ followed by a path $p:j\to r$. The relation in step 2.2 reduces $(pa)\otimes v$ to $p\otimes V_a(v)$; induction on path length shows the augmentation is surjective and its kernel is generated by those relations. A linear combination of relations has a unique maximal path length; its terms of that length have distinct first-arrow/path/vector indices, so they cannot cancel unless their coefficients vanish. Descending on path length proves the differential is injective. Hence $0⟶P_1(V)⟶P_0(V)⟶V⟶0$ is a projective resolution of length one.

To see directly that every Yoneda two-extension of $V$ by $W$ is zero, write it as $0\to W\to E_1\to E_0\to V\to 0$. Projectivity of $P_0(V)$ gives a lift $s:P_0(V)\to E_0$ of its augmentation. The composite of $P_1(V)\to P_0(V)\stackrel{s}{\to }{E}_0$ lands in $J=\ker (E_0\to V)=\mathrm{im}(E_1\to E_0)$. Projectivity of $P_1(V)$ lifts it through $E_1\twoheadrightarrow J$ to $t:P_1(V)\to E_1$. The resulting map from the split two-extension $0\to W\to W\oplus P_1(V)\to P_0(V)\to V\to 0$ to the given extension is the identity on $W,V$ and commutes with all arrows. The split two-extension is zero in Yoneda ${\mathrm{Ext}}^2$, so the given extension is zero.

Step 1.1 constructs the small abelian model, step 2.1 verifies Hom and extension finiteness, and step 4.1 proves Yoneda ${\mathrm{Ext}}^2=0$. Steps 1.3, 2.3 and 2.4 give the asserted $K_0$, simple extension dimensions and Euler-form formula. Thus all clauses of [F2] hold for ${\mathcal{R}}_Q$, and ${\mathrm{Rep}}_k(Q)$ is essentially small with this finite-field hereditary Hall skeleton.