Schofield sequences in the Euclidean case

Abstract

Let $k$ be a field and consider the path algebra $kQ$ of the quiver $Q$. A pair of indecomposable $kQ$-modules $(Y,X)$ is called an orthogonal exceptional pair if the modules are exceptional and $\operatorname{Hom}(X,Y)=\operatorname{Hom}(Y,X)=\operatorname{Ext}^{1}(X,Y)=0$. Denote by $\mathcal{F}(X,Y)$ the full subcategory of objects having filtration with factors $X$ and $Y$. By the theorem of Schofield if $Z$ is exceptional but not simple, then $Z\in\mathcal{F}(X,Y)$ for some orthogonal exceptional pair $(Y,X)$, and $Z$ is not a simple object in $\mathcal{F}(X,Y)$. In fact, there are precisely $s(Z)-1$ such pairs, where $s(Z)$ is the support of $Z$ (i.e the number of nonzero components in ${\underline\dim}Z$). Whereas it is easy to construct $Z$ given $X$ and $Y$, there is no convenient procedure yet to determine the possible modules $X$ (called Schofield submodules of $Z$) and then $Y$ (called Schofield factors of $Z$), when $Z$ is given. We present such an explicit procedure in the tame case, i.e when $Q$ is Euclidean.