On the Composition Factors of Periodic Modules
On the Composition Factors of Periodic Modules
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论周期模的构成因素
DOI：
10.1112/jlms/49.3.477
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发表时间：
1994
影响因子：
1.2
通讯作者：
A. Skowroński
中科院分区：
文献类型：
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作者：
A. Skowroński
在本文中，我们用周期模来表示Artin代数A上的一个周期生成的不可分解右模，它关于Auslander-Reiten平移DTr的作用是周期的。模群代数上的周期模已被同调方法广泛研究，我们对这类模的结构也有了很多了解（见[1，2，7，8，9，14]）。在这种情况下，DTr与合宿算子Q重合。2.对于非对称的artin代数，周期模的知识相对较少。在[11]中，Happel，Preiser和Ringel证明了A的Auslander-Reiten环TA的正则分支2 T包含周期模M当且仅当F是形式为ZA^/（xr）的稳定管，其中r，称为λ ^/（xr）的秩，是M的周期。此外，在[13]中，Hoshino证明了，如果A不是局部Nakayama代数，则每个DTr-不变不可分解A-模都位于秩为1的稳定管中。在文[23]中，我们引入了广义标准Auslander-Reiten分支的概念，除了关于这类分支的一般结果外，我们还给出了广义标准稳定管的秩的一些界（见第3节）。本文的主要目的是给出广义标准稳定管中周期模的组成因子的结果。我们的结果解释了一些重要的代数类的Auslander-Reiten箭图的稳定管的共同性质。作为结果的应用，我们得到了一些关于不含无限短圈的模范畴中不可分解模的结构的信息。设A是交换Artin环R上的Artin代数.所谓模，我们指的是由模生成的右A-模范畴mod A中的对象。给定一个模M，我们用[M]表示M在A的Grothendieck群K 0（A）中的象。因此[M]=[N]当且仅当模M和N具有相同的复合因子，包括重数。我们可能会问，当两个不可分解模M和N具有相同的合成因子时。特别是，找到一个不可分解模M唯一地由它的组成因子决定直到同构的充分条件是很有趣的。在[19]中证明了当M不位于非零非同构的短圈M-* X-> M上且A '是不可分解模时就是这种情况。不幸的是，这个准则不能应用于稳定管的周期模，因为这样的模总是位于短圈上。但是包含来自稳定管的模的短圈通常是有限的，也就是说，形成这些圈的态射不属于mod A的无限根rad 00（mod A）。我们将在第3节中证明，稳定管5”是广义标准的，当且仅当对于从5”开始的任何模M，rad（Af，M）= 0，即，所有环M-> M在
Throughout this paper, by a periodic module we mean a finitely generated indecomposable right module over an artin algebra A which is periodic with respect to the action of the Auslander-Reiten translation DTr. Periodic modules over modular group algebras have been extensively investigated by using homological techniques, and we know much about the structure of such modules (see [1, 2, 7, 8, 9, 14]). In this case, DTr coincides with the syzygy operator Q. 2. For artin algebras which are not symmetric, the knowledge of periodic modules is relatively poor. In [11], Happel, Preiser and Ringel proved that a regular component 2T of the Auslander-Reiten quiver TA of A contains a periodic module M if and only if F is a stable tube of the form ZA^/{xr) where r, called the rank of &~, is the period of M. Moreover, in [13] Hoshino proved that, if A is not a local Nakayama algebra, then every DTr-invariant indecomposable A-module lies in a stable tube of rank 1. In [23] we introduced the concept of a generalized standard Auslander-Reiten component and, besides general results on such components, we showed some bounds on the rank of generalized standard stable tubes (see Section 3). The main purpose of this paper is to present results on the composition factors of periodic modules lying in generalized standard stable tubes. Our results explain common properties of stable tubes of the Auslander-Reiten quivers of some important classes of algebras. As an application of our results we obtain some information on the structure of indecomposable modules in the module categories without infinite short cycles. Let A be an artin algebra over a commutative artin ring R. By a module we mean an object in the category mod A of finitely generated right A-modules. Given a module M, we denote by [M] the image of M in the Grothendieck group K0 (A) of A. Thus [M]=[N] if and only if the modules M and N have the same composition factors including the multiplicities. We may ask when two indecomposable modules M and N have the same composition factors. In particular, it would be interesting to find sufficient conditions for an indecomposable module M to be uniquely determined up to isomorphism by its composition factors. It was shown in [19] that this is the case when M does not lie on a short cycle M-* X-> M of non-zero non-isomorphisms with A'an indecomposable module. Unfortunately, this criterion cannot be applied to periodic modules from stable tubes, because such modules always lie on short cycles. But often the short cycles containing modules from stable tubes are finite, that is, the morphisms forming these cycles do not belong to the infinite radical rad00 (mod A) of mod A. We shall show in Section 3 that a stable tube 5" is generalized standard if and only if rad (Af, M)= 0 for any module M from 5", that is, all loops M-> M at