Applications of contravariantly finite subcategories
Applications of contravariantly finite subcategories
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DOI：
10.1016/0001-8708(91)90037-8
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发表时间：
1991-03
影响因子：
1.7
通讯作者：
M. Auslander;I. Reiten
中科院分区：
文献类型：
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作者：
M. Auslander;I. Reiten
在本文中，我们假设所有的模都是在一个artin代数L！上生成的。我们用mod/i表示所有n-生成的n-模的范畴，mod/i的反变和协变有限子范畴的概念（定义见第1节）是Auslander和Smals在[8，9]中为研究mod n的哪些子范畴有几乎可裂序列的问题而引入的。而模n中协变有限或逆变有限的子范畴也是本文的主要研究对象，但观点与[S]或[9]有很大不同。在这里，我们表明，这些概念是密切相关的问题（a）的子范畴PM（n）的mod/i的有限投射维数的模和（B）倾斜和cotilting理论组成。就ga（，4）而言，讨论了两个问题：Y '“（A）在mod/i中何时是逆变有限的，以及它是逆变有限的对θ =（/l）意味着什么？如果f i是有限表示型，则Y”（A）是逆变有限的，因为mod n的每个子范畴都是逆变有限的。事实上，n是有限表示型的当且仅当mod/i的每个子范畴在mod/i中是逆变（协变）有限的（见Prop. 1.2）。我们还表明，L？当n稳定等价于一个遗传代数时，（A）是逆变有限的。然而，由于Igusa et al. [13]证明了Y“（n）在模A中不总是逆变有限的。这一点不难看出。“（A）在mod/i中是逆变有限的当且仅当Yp”（T，（n））在mod T，（n）中是逆变有限的，其中T2（n）=（，”z）. Smalo在[181]中证明了一个更一般的结果。把这些观察结果结合起来，我们就得到了一个非常广泛的代数A
Throughout this paper we assume that all modules are finitely generated over an artin algebra L!. We denote by mod/i the category of all finitely generated/i-modules.The notions of contravariantly and covariantly finite subcategories of mod/i (see Sect. 1 for definitions) were introduced in [8, 9] by Auslander and Smals in connection with studying the problem of which subcategories of mod n have almost split sequences. While subcategories of mod n which are either covariantly or contravariantly finite in mod n are also the main objects of study in this paper, the point of view is quite different from [S] or [9]. Here we show that these notions are intimately related to questions about (a) the subcategory pm (n) of mod/i consisting of the modules of finite projective dimension and (b) tilting and cotilting theories. As far as ga (, 4) is concerned, two problems are discussed: when is Y’“(A) contravariantly finite in mod/i and what does its being contravariantly finite imply about 9=(/l)? If/i is of finite representation type, then Y”(A) is contravariantly finite since every subcategory of mod n is contravariantly finite. In fact, n is of finite representation type if and only if every subcategory of mod/i is contravariantly (covariantly) finite in mod/i (see Prop. 1.2). We also show that L?“(A) is contravariantly finite when n is stably equivalent to a hereditary algebra. However, an example due to Igusa et al.[13] shows that Y’“(n) is not always contravariantly finite in mod A. It is not hard to see that.!?“(A) is contravariantly finite in mod/i if and only if Yp”(T,(n)) is contravariantly finite in mod T,(n), where T2 (n)=(,” z). A more general result is proved by Smalo in [181. Combining these observations we get a very wide variety of algebras A