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各种表示的不可约分量I。本地病例

Irreducible components of varieties of representations I. The local case

Birge Huisgen-Zimmermann

发表信息：
J. Algebra 464 (2016) 198-225
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摘要：

设$Λ$是代数闭域$K$上的局部截断路代数，即，$Λ$是路径代数$KQ$与长度为$L+1$的路径的商，其中$Q$是具有单个顶点和有限数量的环的商，并且$L$是正整数。对于任何$d>0$，我们确定的不可约组件的品种参数化的$d$维表示的$Λ$，即，组件的经典仿射品种${\rm\bf{Rep}}_{d}(Λ)$和-等价地-那些投影参数化品种${\rm GRASS}_d(Λ)$。我们的方法是通过一对上连续映射从${\rm\bf{Rep}}_{d}(Λ)$到由半单模序列组成的偏序集的方式来角化分量。主要结果摘录如下。给定具有$\dim \bigoplus_{0 \le l \le L} {\bf S}_l = d$的半单模序列${\bf S} = ({\bf S}_0, ..., {\bf S}_L)$，设${\rm\bf{Rep}}\, {\bf S}$是${\rm\bf{Rep}}_{d}(Λ)$的子簇，由用根分层${\bf S}$参数化模的点组成。(The $Λ$ -模块$M$的根分层是序列$\bigl(J^l M / J^{l+1} M\bigr)_{0 \le l \le L}$，其中$J$是$Λ$的Jacobson根。）假设$Q$有$r \ge 2$循环。若$d \le L+1$，则簇${\rm\bf{Rep}}_{d}(Λ)$不可约。另一方面，如果$d > L+1$，则${\rm\bf{Rep}}_{d}(Λ)$的不可约分量是满足$0 \le l < L$的不等式$\dim {\bf S}_l \le r \dim {\bf S}_{l+1}$和$\dim {\bf S}_{l+1} \le r \dim {\bf S}_l$的那些序列${\bf S}$的子簇${\rm\bf{Rep}}\, {\bf S}$的闭包。作为一个副产品，主要结果提供了通用的信息模块对应的参数化品种的不可约的组成部分。

英文摘要：

Let $Λ$ be a local truncated path algebra over an algebraically closed field $K$, i.e., $Λ$ 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 $Λ$, namely, the components of the classical affine variety ${\rm\bf{Rep}}_{d}(Λ)$ and -- equivalently -- those of the projective parametrizing variety ${\rm GRASS}_d(Λ)$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from ${\rm\bf{Rep}}_{d}(Λ)$ 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 ${\rm\bf{Rep}}_{d}(Λ)$ consisting of the points that parametrize the modules with radical layering ${\bf S}$. (The radical layering of a $Λ$-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 $Λ$.) Suppose the quiver $Q$ has $r \ge 2$ loops. If $d \le L+1$, the variety ${\rm\bf{Rep}}_{d}(Λ)$ is irreducible. If, on the other hand, $d > L+1$, then the irreducible components of ${\rm\bf{Rep}}_{d}(Λ)$ 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.

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