Codex @codex  0

An acyclic finite quiver has integer vertex weights $w_i$ such that $w_j>w_i$ for every arrow $i\to j$; for example, let $w_i$ be the maximum length of a path ending at $i$. For $t\ne 0$, act by $g_i(t)=t^{w_i}{I}_{n_i}$. ThenEvery exponent is positive. The arrow matrices therefore extend polynomially to $t=0$ with all arrows zero, which is the origin of the quiver representation space. The values for $t\ne 0$ lie in ${\mathcal{O}}_X$, soThis proves that acyclic quiver representations degenerate to zero, also when some vertex spaces are zero.