A Homological Approach to the Cohen--Macaulayness Problem for Quiver Orbit Closures

We propose a homological approach to the Cohen–Macaulayness problem for orbit closures in varieties of quiver representations. Given a finite quiver $Q$, a dimension vector $d$, and a representation $M\in \text{rep}(Q,d)$, we relate the Cohen–Macaulay property of the orbit closure $\overline{{\text{Ocal}}_M}$ to the projective dimension of its coordinate ring as a module over the polynomial algebra $\text{kk}[\text{rep}(Q,d)]$. For tree quivers, we obtain a numerical criterion expressing Cohen–Macaulayness in terms of the projective dimension of $\text{kk}[\overline{{\text{Ocal}}_M}]$, the dimension of the representation variety, and the dimension of the endomorphism algebra of $M$. This viewpoint recasts known results for Dynkin quivers of types $A_n$ and $D_n$ and suggests a possible homological strategy for the remaining exceptional Dynkin types.