Proposition 4

The mapping Orb($\alpha$ )$:$ Orb(G) $\rightarrow$ Orb(Coh($\varphi , \psi$ )) is surjective; its fibres are the connected components of the framework of the pair $(\varphi , \psi )$.

The morphism $\alpha$ is the composite of the morphisms $\alpha_1: G\rightarrow G_1$, $\theta_1: G_1\rightarrow$ Grp(G$_1$) and $\pi :$ Grp(G$_1$)$\rightarrow$ Coh($\varphi , \psi$ ). The mapping $\theta_1$ induces a bijection of the set of connected components of the quiver $G_1$ onto the set of orbits of Grp(G$_1$), and the mapping Orb($\pi$ )$:$ Orb(Grp(G$_1$))$\rightarrow$ Orb(Coh($\varphi , \psi$ )) is bijective (II, p. 170, remark 1). It is therefore enough to prove that the mapping of Orb(G) into $\pi_0(G_1)$ deduced from $\alpha_1$ is surjective and that its fibres are the connected components of the framework of the pair $(\varphi , \psi )$. The surjectivity results from the fact that the mapping Som($\alpha_1$) is the identity mapping. The equivalence relation in Som(G) defined by “$a$ and $b$ are in the same connected component of $G_1$” is generated by the relations “there exists an arrow of G joining $a$ to $b$” and “there exists a vertex $h$ of H such that $\varphi (h) =a$ and $\psi (h) =b$”. This equivalence relation is compatible with the mapping of Som(G) into Orb(G), and the relation deduced from it in Orb(G) is generated by the relation “there exists an orbit $\eta$ of H such that Orb($\varphi$ )$(\eta ) =\alpha$ and Orb($\psi$ )$(\eta ) =\beta$”. It is therefore the relation “$\alpha$ and $\beta$ are in the same connected component of the framework of the pair $(\alpha , \beta )$”.