Hereditary coalgebras and representations of species
Hereditary coalgebras and representations of species
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DOI：
10.1016/j.jalgebra.2005.04.019
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发表时间：
2005-11
影响因子：
0.9
通讯作者：
J. Kosakowska;D. Simson
中科院分区：
文献类型：
--
作者：
J. Kosakowska;D. Simson
设C是基本不可分解的遗传K-余代数，其中K是任意域。本文利用C的左值Gabriel算子、相应的Tits二次型和C的Ext-种的局部幂零表示，研究了C和左C-余模的一个技巧.本文的主要目的之一是证明以下结果。设[Formula：see text]是一个两两非同构的单左C-余模的完备集，给定i，j∈IC，我们设sij 1 = dimKExtC 1（S（i），S（j））.则每个左C-余模是有限维C-余模的直和当且仅当满足以下四个条件：在这种情况下，我们证明了C同构于与C的Ext-种相关联的（余）张量余代数TF□（M），其中F是除代数Fj=EndCS（j），[公式：见正文]和[公式：见正文]的直和。我们还刻划了Auslander-Reiten余模Γ（C-comod）和有限维左C-余模范畴C-comod的结构。
Let C be a basic indecomposable hereditary K-coalgebra, where K is an arbitrary field. We investigate a technique for studying C and left C-comodules by means of the left valued Gabriel quiver of C, an associated Tits quadratic form and locally nilpotent representations of the Ext-species of C. One of the main aims of the paper is to prove the following result. Let [Formula: see text] be a complete set of pairwise non-isomorphic simple left C-comodules and, given i,j∈IC, we set sij1=dimKExtC1(S(i),S(j)). Then every left C-comodule is a direct sum of finite dimensional C-comodules if and only if the following four conditions are satisfied: In this case, we show that C is isomorphic to the (co)tensor coalgebra TF□(M) associated to the Ext-species of C, where F is the direct sum of the division algebras Fj=EndCS(j), [Formula: see text] , and [Formula: see text] . We also describe the Auslander–Reiten quiver Γ(C-comod) and the structure of the category C-comod of left C-comodules of finite dimension.