Finite acyclic quiver representations: heredity, Euler form, and finite-field finitarity

For a morphism $f:X\to Y$, its vertexwise kernels form a subrepresentation: if $x\in \ker f_i$, then $f_j{X}_a(x)=Y_a{f}_i(x)=0$ for every arrow $a:i\to j$. The vertexwise cokernels form a quotient representation because $Y_a(\mathrm{im}{f}_i)\subseteq \mathrm{im}{f}_j$. The kernel and cokernel universal properties hold vertex by vertex. The canonical coimage-to-image map is an isomorphism at every vertex, hence an isomorphism of representations. With vertexwise finite direct sums and a zero object, these facts make ${\mathcal{A}}_Q(K)$ abelian.

For $i\in Q_0$, let $P_i(v)$ have basis the directed paths from $i$ to $v$, with arrow maps given by path concatenation. Acyclicity and finiteness of $Q$ make each path set finite. A morphism $P_i\to N$ is uniquely determined by the image of the length-zero path at $i$, so $\mathrm{Hom}(P_i,N)\cong N_i$; evaluation at $i$ preserves epimorphisms, and therefore $P_i$ is projective.

Put $C^0(X,Y)={⨁}_i{\mathrm{Hom}}_K(X_i,Y_i),C^1(X,Y)={⨁}_{a:i\to j}{\mathrm{Hom}}_K(X_i,Y_j),$ and define $\mathrm{\partial }:C^0\to C^1$ by $(\mathrm{\partial }h{)}_a=Y_a{h}_i-h_j{X}_a$. Its kernel is exactly ${\mathrm{Hom}}_{{\mathcal{A}}_Q(K)}(X,Y)$ by the commuting-square definition of a representation morphism.

For a representation $X$, define $ϵ:{⨁}_i{P}_i{\otimes }_K{X}_i\to X$ by $p\otimes x\mapsto X_p(x)$, where $X_p$ is the composite arrow map along the path $p$. Define $d:{⨁}_{a:i\to j}{P}_j{\otimes }_K{X}_i⟶{⨁}_i{P}_i{\otimes }_K{X}_i$ at a path $p:j\to v$ by $d(p\otimes x)=(p\otimes X_a(x){)}_j-((p\circ a)\otimes x{)}_i\mathrm{.}$ Both terms are finite direct sums of the projectives from step 1.2. Then $ϵd=0$ because $X_{p\circ a}=X_p{X}_a$.

Now assume $K={\mathbb{F}}_q$. For each dimension vector $\mathbf{d}$, there are finitely many arrow matrix tuples on the fixed spaces $K^{d_i}$, and the finite group $G_{\mathbf{d}}(K)$ has finitely many orbits on this set. Fix an enumeration of the finite set $K$; the lexicographically least tuple in each orbit gives a representative without making a choice from an arbitrary family. The union over the set ${\mathbb{N}}^{Q_0}$ is a set, so the full subcategory on these representatives is small. It is a skeleton: every finite-dimensional representation becomes one of the matrix tuples after choosing bases at its finitely many vertices. The full skeleton is abelian because each vertexwise kernel and cokernel from step 1.1 is isomorphic to its selected representative, and transporting its kernel or cokernel map across that isomorphism preserves the universal property.

At vertex $v$, $ϵ$ is onto because $x\in X_v$ is the image of ${\mathrm{id}}_v\otimes x$. Modulo $\mathrm{im}d$, the relation in step 2.1 replaces $(p\circ a)\otimes x$ by $p\otimes X_a(x)$. Every positive-length path has a unique first arrow, so repeated replacement sends each path tensor to ${\mathrm{id}}_v\otimes X_p(x)$. Consequently the quotient of the middle term by $\mathrm{im}d$ is isomorphic to $X_v$ through $ϵ$, and $\ker ϵ=\mathrm{im}d$.

The map $d$ is injective. In a nonzero element of its domain, choose the largest length $\mathrm{\ell }$ of a suffix path $p$ with nonzero coefficient. The second term of $d(p\otimes x)$ has length $\mathrm{\ell }+1$. Each such path has a unique first arrow and suffix, so its coefficient cannot cancel with the second term of a different source basis summand. A first term of $d$ has length at most $\mathrm{\ell }$, since no source suffix has length $\mathrm{\ell }+1$. Thus the length-$(\mathrm{\ell }+1)$ part of $d$ is nonzero, proving injectivity. The displayed sequence is a projective resolution of length one; applying $\mathrm{Hom}(-,Y)$ therefore gives ${\mathrm{Ext}}_{{\mathcal{A}}_Q(K)}^r(X,Y)=0$ for $r\ge 2$.

A short exact sequence $0\to Y\to E\to X\to 0$ splits as a sequence of vector spaces at each of the finitely many vertices. In the resulting coordinates every arrow of $E$ has block matrix $E_a=\left(\begin{array}{cc}{Y}_a& {\xi }_a\\ 0& X_a\end{array}\right),{\xi }_a\in {\mathrm{Hom}}_K(X_i,Y_j)\mathrm{.}$ Every tuple $\xi \in C^1(X,Y)$ defines such an extension. Changing the vertexwise splittings by $h\in C^0(X,Y)$ changes $\xi$ by $\mathrm{\partial }h$. Conversely, an isomorphism of extensions that is the identity on endpoints has block matrix $\left(\begin{array}{cc}1& h_i\\ 0& 1\end{array}\right)$ at each vertex and hence changes $\xi$ by exactly $\mathrm{\partial }h$. By [F3], the generated endpoint-fixed Yoneda relation consists of these isomorphisms. The Baer sum of the extensions represented by $\xi$ and $\eta$ is represented by $\xi +\eta$: the pullback of their direct sum along $X\to X\oplus X$ followed by pushout along $Y\oplus Y\to Y$ adds the off-diagonal arrow blocks. Pushout along $\lambda {\mathrm{id}}_Y$ sends $\xi$ to $\lambda \xi$. Therefore the Yoneda vector-space structure is the quotient structure ${\mathrm{YExt}}^1(X,Y)\cong C^1(X,Y)\mathrm{/}\mathrm{im}\mathrm{\partial },$ a finite-dimensional $K$-vector space; it is not being identified here with the projective-resolution group of step 4.1.

Both cochain spaces in step 1.3 are finite-dimensional. Rank-nullity for $\mathrm{\partial }$ gives $\dim \ker \mathrm{\partial }-\dim \mathrm{coker}\mathrm{\partial }=\dim C^0-\dim C^1={\sum }_i{d}_i(X)d_i(Y)-{\sum }_{a:i\to j}{d}_i(X)d_j(Y)={\chi }_Q(\mathbf{d}(X),\mathbf{d}(Y))\mathrm{.}$ The kernel and cokernel identifications already proved give the Hom–Yoneda-Ext${}^1$ formula and finite-dimensionality over any $K$. When $K={\mathbb{F}}_q$, the underlying sets are finite. The full skeleton constructed in step 2.2 has the same Hom sets on its objects; transporting extension middle terms to their representatives identifies its Yoneda classes with those in ${\mathcal{A}}_Q(K)$. Thus this small abelian skeleton satisfies the finitary hypotheses of [F2], while ${\mathcal{A}}_Q(K)$ is essentially small.