Given a quiver $Q$ without oriented cycles and a dimension vector $α$, we can consider the representation space $\textRep(Q,α)$. On this space we have a canonical action of $ GL_α$. For a character $χ$ of $ GL_α$ (every character corresponds to an integer valued weight vector $θ$) we can consider the graded ring of semi-invariants of weights $nθ$ for all $n$ in $\BbbN$ and this gives us a projective variety $M^ ss_θ(Q,α)$: the moduli space of $θ$-semistable $α$-dimensional representations of $Q$. <P> In this paper the authors show that the étale local structure in a point of this moduli space is the same as the local structure of the zero in the quotient space $\textRep(Q_L,α_L)/\!\!/ GL_α_L$ of a new quiver $Q_L$ and dimension vector $α_L$. This quiver is called the local quiver. <P> To prove this statement they first use the technique of universal localizations. In this way they can cover $M^ ss_α(Q,θ)$ by a collection of affine open subsets which correspond to quotients of representation spaces of new algebras $\Bbb C Q_σ$ that are universal localizations of the path-algebra $\Bbb C Q$. Simple representations of the $\Bbb C Q_σ$ correspond to stable representations of $\Bbb C Q$. Using the Luna-slice machinery they conclude that the étale local structure of a point $p$ in such a quotient space is the same as $\textExt^1_\Bbb C Q_σ(V_p,V_p)/\!\!/\textStab V_p$ (where $V_p$ is the $\Bbb C Q_σ$-representation corresponding to $p$). Finally they show that the latter is in fact of the form $\textRep(Q_L,α_L)/\!\!/ GL_α_L$. <P> Since the étale isomorphism conserves simplicity, this technique also allows the authors to compute the dimension of simple representations of $\Bbb C Q_σ$ and hence stable representations of $Q$. The last section of the paper illustrates this by calculating the dimension vectors of stable representations for the fully bipartite quiver on $p+q$ vertices with weight $-1$ on the $p$ starting vertices and $1$ on the $q$ ending vertices. The result they obtain is that there exist stable representations if and only if the sum of the dimension of a starting vertex and an ending vertex never exceeds the total sum of the dimension vector.
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