Irreducible components of varieties of representations: The local case

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• 作者：B. Huisgen-Zimmermann ^{birge@math.ucsb.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 16G10 ; secondary ; 16G20 ; 14M99 ; 14M15
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：464
• 期：Complete
• 页码：198-225
• 全文大小：551 K

文摘

For any positive integer d, we determine the irreducible components of the varieties that parametrize the d  -dimensional representations of a local truncated path algebra Λ. Here Λ is a quotient $KQ/〈\text{the paths of length}L+1〉$ of a path algebra KQ, where K is an algebraically closed field, L is a positive integer, and Q is the quiver with a single vertex and a finite number r   of loops. The components are determined in both the classical and the Grassmannian settings, Rep_d(Λ)${\mathbf{Rep}}_d(\mathrm{\Lambda })$ and GRASS_d(Λ)${\mathrm{GRASS}}_d(\mathrm{\Lambda })$. Our method is to corner the components by way of a twin pair of upper semicontinuous maps from Rep_d(Λ)${\mathbf{Rep}}_d(\mathrm{\Lambda })$ to a poset consisting of sequences of semisimple modules.

An excerpt of the main result is as follows. Given a sequence 46121ee55f07b45d529f6a9" title="Click to view the MathML source">S=(S₀,…,S_L)$\mathbb{S}=({\mathbb{S}}_0,\dots ,{\mathbb{S}}_L)$ of semisimple modules with dim⁡⨁_0≤l≤LS_l=d$\dim {⨁}_{0\le l\le L}{\mathbb{S}}_l=d$, let $\mathbf{Rep}\mathbb{S}$ be the subvariety of Rep_d(Λ)${\mathbf{Rep}}_d(\mathrm{\Lambda })$ consisting of the points that parametrize the modules with radical layering 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0021869316301570&_rdoc=1&_issn=00218693&md5=582e251daacf7774fd7ce5c8a8dcd453" title="Click to view the MathML source">S. (The radical layering of a Λ-module M   is the sequence 4677881cb031b7d67cd5c90e448a" title="Click to view the MathML source">(J^lM/J^l+1M)_0≤l≤L${(J^{l}M/J^{l+1}M)}_{0\le l\le L}$, where J is the Jacobson radical of Λ.) Suppose the quiver Q   has 4691528b1993e2f30" title="Click to view the MathML source">r≥2$r\ge 2$ loops. If d≤L+1$d\le L+1$, the variety Rep_d(Λ)${\mathbf{Rep}}_d(\mathrm{\Lambda })$ is irreducible and, generically, its modules are uniserial. If, on the other hand, d>L+1$d>L+1$, then the irreducible components of Rep_d(Λ)${\mathbf{Rep}}_d(\mathrm{\Lambda })$ are the closures of the subvarieties $\mathbf{Rep}\mathbb{S}$ for those sequences 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0021869316301570&_rdoc=1&_issn=00218693&md5=582e251daacf7774fd7ce5c8a8dcd453" title="Click to view the MathML source">S which satisfy the inequalities dim⁡S_l≤r⋅dim⁡S_l+1$\dim {\mathbb{S}}_l\le r\cdot \dim {\mathbb{S}}_{l+1}$ and dim⁡S_l+1≤r⋅dim⁡S_l$\dim {\mathbb{S}}_{l+1}\le r\cdot \dim {\mathbb{S}}_l$ for 0≤l<L$0\le l<L$; generically, the modules in any such component have socle layering (S_L,…,S₀)$({\mathbb{S}}_L,\dots ,{\mathbb{S}}_0)$. As a byproduct, the main result provides further installments of generic information on the modules corresponding to the irreducible components of the parametrizing varieties.