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 ${\rm\bf{Rep}}_{d}(\Lambda)$ and -- equivalently -- those of the projective parametrizing variety ${\rm GRASS}_d(\Lambda)$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from ${\rm\bf{Rep}}_{d}(\Lambda)$ to a poset consisting of sequences of semisimple modules. An excerpt of the main result is as follows. Given a sequence ${\bf S} = ({\bf S}_0, ..., {\bf S}_L)$ of semisimple modules with $\dim \bigoplus_{0 \le l \le L} {\bf S}_l = d$, let ${\rm\bf{Rep}}\, {\bf S}$ be the subvariety of ${\rm\bf{Rep}}_{d}(\Lambda)$ consisting of the points that parametrize the modules with radical layering ${\bf S}$. (The radical layering of a $\Lambda$-module $M$ is the sequence $\bigl(J^l M / J^{l+1} M\bigr)_{0 \le l \le L}$, where $J$ is the Jacobson radical of $\Lambda$.) Suppose the quiver $Q$ has $r \ge 2$ loops. If $d \le L+1$, the variety ${\rm\bf{Rep}}_{d}(\Lambda)$ is irreducible. If, on the other hand, $d > L+1$, then the irreducible components of ${\rm\bf{Rep}}_{d}(\Lambda)$ are the closures of the subvarieties ${\rm\bf{Rep}}\, {\bf S}$ for those sequences ${\bf S}$ which satisfy the inequalities $\dim {\bf S}_l \le r \dim {\bf S}_{l+1}$ and $\dim {\bf S}_{l+1} \le r \dim {\bf S}_l$ 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.