Irreducible components of varieties of representations I. The local case

Abstract

Let $\Lambda$ be a local truncated path algebra over an algebraically closed field $K$, i.e., $\Lambda$ is a quotient of a path algebra $KQ$ by the paths of length $L+1$, where $Q$ is the quiver with a single vertex and a finite number of loops and $L$ is a positive integer. For any $d>0$, we determine the irreducible components of the varieties that parametrize the $d$-dimensional representations of $\Lambda$, namely, the components of the classical affine variety ${\mathbf{Rep}}_d(\Lambda )$ and -- equivalently -- those of the projective parametrizing variety ${\mathrm{GRASS}}_d(\Lambda )$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from ${\mathbf{Rep}}_d(\Lambda )$ to a poset consisting of sequences of semisimple modules. An excerpt of the main result is as follows. Given a sequence $\mathbf{S}=({\mathbf{S}}_0,...,{\mathbf{S}}_L)$ of semisimple modules with $\dim {⨁}_{0\le l\le L}{\mathbf{S}}_l=d$, let $\mathbf{Rep}\mathbf{S}$ be the subvariety of ${\mathbf{Rep}}_d(\Lambda )$ consisting of the points that parametrize the modules with radical layering $\mathbf{S}$. (The radical layering of a $\Lambda$-module $M$ is the sequence $(J^{l}M/J^{l+1}M{)}_{0\le l\le L}$, where $J$ is the Jacobson radical of $\Lambda$.) Suppose the quiver $Q$ has $r\ge 2$ loops. If $d\le L+1$, the variety ${\mathbf{Rep}}_d(\Lambda )$ is irreducible. If, on the other hand, $d>L+1$, then the irreducible components of ${\mathbf{Rep}}_d(\Lambda )$ are the closures of the subvarieties $\mathbf{Rep}\mathbf{S}$ for those sequences $\mathbf{S}$ which satisfy the inequalities $\dim {\mathbf{S}}_l\le r\dim {\mathbf{S}}_{l+1}$ and $\dim {\mathbf{S}}_{l+1}\le r\dim {\mathbf{S}}_l$ for $0\le l<L$. As a byproduct, the main result provides generic information on the modules corresponding to the irreducible components of the parametrizing varieties.