Representation theory of graded Artin algebras
Representation theory of graded Artin algebras
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DOI：
10.1016/0021-8693(82)90241-1
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发表时间：
1982-05
影响因子：
0.9
通讯作者：
R. Gordon;E. Green
中科院分区：
文献类型：
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作者：
R. Gordon;E. Green
继[91]中开始了分级Artin代数A的研究之后，我们在这里开始对其表示理论的研究。我们的观点是研究分级/i-模块，我们认为分级/i-模块比未分级的模块更容易处理，以便获得有关所有/i-模块的信息。这一观点导致在第 1 节中介绍 mod/i 的完整子类别 mod,@!)，其中包含 mod/i 的可分级对象；即，支持分级的有限生成/i-模块。在这些方面，本文的主要结果之一断言 mod,@) 具有有限表示类型当且仅当 mod/i 具有有限表示类型。因此，在第 3 节中，我们引入了一个数字 G= G (4)，旨在测量 mod,(/i) 的大小。如果G= co，我们表明/i具有无限的表示类型。如果 G< co，我们证明存在某种指定形式的分级 Artin 代数 Q，使得当 0 具有无限表示类型时，i 恰好具有无限表示类型。 Artin 代数 fl 尤其具有理想的图解性质；我们将在其他地方利用这些。我们推测当 G 是有限的时，每个有限生成的/i-模块都是可分级的。现在，如果 n 具有有限表示类型，则显然 G 是有限的。事实上，我们证明，对于有限表示类型的给定 Artin 代数的每个分级，每个模块都是可分级的。这个结果和其他引用的结果主要是第 4 节中证明的结果的结果：如果分级 Artin 代数的 Auslander-Reiten 图的一个分量包含可分级模块，则该分量
Having initiated the study of graded Artin algebras A in [91, here we initiate the study of their representation theory. Our point of view is to study graded/i-modules, which we believe to be somewhat more tractable than ungraded ones, in order to obtain information about all/i-modules. This point of view leads to the introduction, in Section 1, of the full subcategory mod,@!) of mod/i consisting of the gradable objects of mod/i; that is, the finitely generated/i-modules which support a gradation. In these terms, one of the major results of the paper asserts that mod,@) has finite representation type if and only if mod/i has finite representation type. Thus, in Section 3 we introduce a number G= G (4) designed to measure the size of mod,(/i). If G= co, we show that/i has infinite representation type. If G< co, we show that there is a graded Artin algebra Q of a certain specified form such that/i has infinite representation type precisely when 0 has infinite representation type. The Artin algebra fl has, in particular, desirable diagrammatic properties; and these we will exploit elsewhere. We speculate that when G is finite, every finitely generated/i-module is gradable. Now, in case n has finite representation type, it is obvious that G is finite. We show, indeed, that for every gradation of a given Artin algebra of finite representation type, every module is gradable. This result, and the others cited, are chiefly consequences of a result proved in Section 4: If a component of the Auslander-Reiten graph of a graded Artin algebra contains a gradable module, then the component