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• W2425986236 abstract "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/〈the paths of lengthL+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, Repd(Λ) and GRASSd(Λ). Our method is to corner the components by way of a twin pair of upper semicontinuous maps from Repd(Λ) to a poset consisting of sequences of semisimple modules. An excerpt of the main result is as follows. Given a sequence S=(S0,…,SL) of semisimple modules with dim⁡⨁0≤l≤LSl=d, let RepS be the subvariety of Repd(Λ) consisting of the points that parametrize the modules with radical layering S. (The radical layering of a Λ-module M is the sequence (JlM/Jl+1M)0≤l≤L, where J is the Jacobson radical of Λ.) Suppose the quiver Q has r≥2 loops. If d≤L+1, the variety Repd(Λ) is irreducible and, generically, its modules are uniserial. If, on the other hand, d>L+1, then the irreducible components of Repd(Λ) are the closures of the subvarieties RepS for those sequences S which satisfy the inequalities dim⁡Sl≤r⋅dim⁡Sl+1 and dim⁡Sl+1≤r⋅dim⁡Sl for 0≤l<L; generically, the modules in any such component have socle layering (SL,…,S0). As a byproduct, the main result provides further installments of generic information on the modules corresponding to the irreducible components of the parametrizing varieties." @default.
• W2425986236 created "2016-06-24" @default.
• W2425986236 creator A5055240273 @default.
• W2425986236 date "2016-10-01" @default.
• W2425986236 modified "2023-09-30" @default.
• W2425986236 title "Irreducible components of varieties of representations: The local case" @default.
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• W2425986236 doi "https://doi.org/10.1016/j.jalgebra.2016.05.025" @default.
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