Generalized canonical algebras and standard stable tubes-猫眼课题宝

广义正则代数和标准稳定管

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发表时间：

2001

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作者：

A. Skowroński

我们引入了一类新的有限维代数，它允许标准稳定管族（在Ringel [17]的意义下）。特别是，我们证明了有许多代数的任意非零（有限或无限）的全球范围内的维数Auslander-Reiten颤抖承认忠实的标准稳定管。导论.在整个文件K将表示一个固定的代数闭域。通过一个代数，我们指的是一个有限维的K-代数（结合的，有一个单位元），我们还假设它是基本的。一个代数A可以写为一个有界的路代数A <$= KQ/I，其中Q = QA是A的Gabriel代数，I是Q的路代数KQ中的一个容许理想。等价地，我们将A视为一个K-范畴，其对象类是QA的顶点集。对于代数A，我们用modA表示有限维（在K上）右A-模范畴，用rad（modA）表示modA的Jacobson根，用rad（modA）表示modA的无限根。回想一下，rad（modA）是由modA中不可分解对象之间的非同构生成的，rad（modA）是rad（modA）的所有有限幂rad（modA）的交集，i ≥ 1。A-模是指modA的对象。对于QA的每个顶点i，我们用SA（i）表示i处的单A-模，用PA（i）（分别为IA（i））表示SA（i）在modA中的投射覆盖（分别为内射包络）。此外，我们用D表示modA上的标准对偶HomK（−，K）。我们用ΓA表示A的Auslander-Reiten平移，用τA和τ A分别表示在ΓA中的Auslander-Reiten平移DTr和TrD。我们不区分不可分解的A-模和与之对应的ΓA的顶点，所谓ΓA的分支是指ΓA的连通分支。对于ΓA中的分支族C，我们用suppAC表示C的支集，用安娜表示C的零化子.回想一下，suppA C是由所有对象i给出的A的全子类，使得SA（i）是2000年数学主题分类：16 G10，16 G70，18 G 05。由波兰科学基金会资助。

We introduce a new wide class of finite-dimensional algebras which admit families of standard stable tubes (in the sense of Ringel [17]). In particular, we prove that there are many algebras of arbitrary nonzero (finite or infinite) global dimension whose Auslander–Reiten quivers admit faithful standard stable tubes. Introduction. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite-dimensional K-algebra (associative, with an identity), which we moreover assume to be basic. An algebra A can be written as a bound quiver algebra A ∼= KQ/I, where Q = QA is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q. Equivalently, we will consider A as a K-category whose class of objects is the set of vertices of QA. For an algebra A, we denote by modA the category of finite-dimensional (over K) right A-modules, by rad(modA) the Jacobson radical of modA and by rad(modA) the infinite radical of modA. Recall that rad(modA) is generated by nonisomorphisms between indecomposable objects in modA, and rad(modA) is the intersection of all finite powers rad(modA), i ≥ 1, of rad(modA). By anA-module we mean an object of modA. For each vertex i of QA, we denote by SA(i) the simple A-module at i, and by PA(i) (respectively, IA(i)) the projective cover (respectively, injective envelope) of SA(i) in modA. Moreover, we denote by D the standard duality HomK(−,K) on modA. We shall denote by ΓA the Auslander–Reiten quiver of A and by τA and τ A the Auslander–Reiten translations DTr and TrD in ΓA, respectively. We do not distinguish between an indecomposable A-module and the vertex of ΓA corresponding to it. By a component of ΓA we mean a connected component of ΓA. For a family C of components in ΓA, we denote by suppA C the support of C and by annA C the annihilator of C. Recall that suppA C is the full subcategory of A given by all objects i such that SA(i) is a 2000 Mathematics Subject Classification: 16G10, 16G70, 18G05. Supported by the Foundation for Polish Science.