Extensions of homologically finite subcategories
Extensions of homologically finite subcategories
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同调有限子范畴的扩展
DOI：
10.1007/bf01236075
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发表时间：
1993
影响因子：
0.6
通讯作者：
S. Smalø
中科院分区：
文献类型：
--
作者：
S. A. Sikko;S. Smalø
介绍。同构有限子范畴的研究在许多情况下都是重要的，特别是在代数的表示理论中。本文研究了由有限生成模范畴的两个同调有限子范畴的模扩展得到的有限生成模范畴。在即将发表的一篇论文中，我们将回到其他类似结果成立的情况。设A是一个人工代数，用模A表示有限生成的左A模的范畴。通过路a的子范畴，我们总是理解一个关于直接和、直接和和和同构闭合的满子范畴。因此，一旦我们指定了子类别的对象，子类别就完全确定了。对于模A的子范畴的每一个有序对（d,~），我们用dg~表示子范畴，其对象是d中有子模A的模E与~中有E/A的模E的直接和。我们在本文中的目的是证明，如果d和~在模A中都是逆变有限的，那么~ c也是逆变有限的。这推广了Grecht， Vossieck, de la Pefia/Simson， Ringel和smallo先前的结果。我们首先回顾一些定义。设cg是模a的子范畴，在本文中，通过cg上的函子，我们总是理解为加性函子。如果在c~中存在模c和一个泛胚（,c）[4-~ F]，如果在cg中存在模C1和c2，并且cg上的函子有一个精确序列（,C1） l~—+(,Cz)] 4~ F~ 0，则我们说F是有限呈现的。对偶地，如果在cg中有一个协变函子（C,）[~ G]有C，则称协变函子G: cg~ db是有限生成的；如果在cg中有一个精确序列（C:,） I,~(C2,) l~ e~ G—>有C:， C ~ 2有C ~，则称G是有限呈现的。这里我们提醒读者，对于a中的模块X，对于cg中的每个对象Y，函子（~ X）[~]由Y~-~ Horn a （X， Y）给出，对于阿贝尔群Homa (X， Y)，我们简单地写成（X， Y）。如果YLY‘是cg中的态射，则（，X） l~(f)是阿贝尔群的态射Homa (Y’， X) Homa (f， X)) Homa （Y' X）。同样，模a中的态射M和N引出函子（，M）]~（, g) l%（， N）[~，其中（,g） l~(X)）的态射。
Introduction. The study of homologically finite subcategories has turned out to be important in several contexts, especially in the representation theory of artin algebras. In this paper we will study the category of finitely generated modules obtained by extensions of modules taken from two homologically finite subcategories of the category of finitely generated modules over an artin algebra. In a forthcoming paper we will come back to other contexts where analogous results hold. Let A be an artin algebra and denote by mod A the category of finitely generated left A-modules. By a subcategory of rood A we always understand a full subcategory closed with respect to direct sums, direct summands and isomorphisms. Thus a subcategory is fully determined once we have specified its objects. For each ordered pair (d,~) of subcategories of mod A we denote by dg~ the subcategory whose objects are the direct summands of the modules E having a submodule A in d with E/A in~. Our purpose in this article is to show that if d and~ are both contravariantly finite in mod A, then so is~ C. This generalizes earlier results by Grecht, Vossieck, de la Pefia/Simson, Ringel and Smalo.We start by recalling some definitions. Let cg be a subcategory of mod A. Throughout this paper, by a functor on cg we always understand an additive functor. A contravariant functor F from c~ to the category db of abelian groups is said to be finitely generated if there exists a module C in c~ and an epimorphism (, C)[4-~ F. If there are modules C1 and C 2 in cg and an exact sequence (, C1) l~--+(, Cz)] 4~ F~ 0 of functors on cg, we say that F is finitely presented. Dually, a covariant functor G: cg~ db is said to be finitely generated if there is an epimorphism (C,)[~ G with C in cg and G is said to be finitely presented if there is an exact sequence (C:,) I,~(C2,) l~ e~ G--> 0 with C: and C 2 in c~. Here we remind the reader that for a module X in rood A, the functor (~ X)[~ is given by Y~-~ Horn A (X, Y) for each object Y in cg and we simply write (X, Y) for the abelian group Homa (X, Y). If YLY'is a morphism in cg, then (, X) l~(f) is the morphism Homa (Y', X) Homa (f, X)) Homa (y'X) of abelian groups. Also a morphism M & N in modA induces the morphism of functors (, M)]~(, g) l%(, N)[~, where (, g) l~(X)