Relative homology and representation theory II: Relative Cotilting theory
Relative homology and representation theory II: Relative Cotilting theory
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DOI：
10.1080/00927879308824718
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发表时间：
1993
影响因子：
0.7
通讯作者：
M. Auslander;Ø. Solberg
中科院分区：
文献类型：
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作者：
M. Auslander;Ø. Solberg
本文是研究相对同调代数在artin代数表示论中的应用的三篇系列论文中的第二篇，致力于发展artin代数的相对余倾斜模的一般理论。读者可以参考系列[5]中的第一篇论文，了解本文中使用的有关相对同调代数的基本定义和结果以及符号。设mod Λ是Artin代数Λ上的n-生成左模范畴.设F是加法双函子ExtΛ（，）：（mod Λ）×mod Λ→ Ab的加法子函子。我们假设F有足够的投射词和内射词。文[5]讨论了F -正合列和函子ExtiF（，）对所有i ≥ 0的性质.与标准情形类似，相对F -余倾斜模定义如下.一个Λ-模T是F -余倾斜模，如果它满足（a）ExtF（T，T）= 0，（B）idFT 0，且满足（a）ExtF（T，T）= 0（i > 0），（b）IdFT 0，且满足（c）HomΓ（G，T）.这些结果是基于自然代数态射Λ→ EndΓ（ΓT）是同构这一事实。在标准的余倾斜理论中，我们知道所有的余倾斜Λ-模有相同数目的非同构不可分解的和项，即非同构的单Λ模的数目。我们证明了所有F -余倾斜Λ-模也有相同数目的非同构不可分解和项，即G的非同构不可分解和项的数目，其中addG = P（F）.由于Λ在P（F）中，因此F余倾斜模的非同构和项的数量大于或等于非同构简单Λ-模的数量。更
This paper, the second in a series of three papers, studying the use of relative homological algebra in the representation theory of artin algebras, is devoted to developing a general theory of relative cotilting modules for artin algebras. The reader is referred to the first paper in the series [5] for basic definitions and results, as well as notation, concerning the relative homological algebra used in this paper. Let mod Λ be the category of finitely generated left modules over an artin algebra Λ. Suppose F is an additive subfunctor of the additive bifunctor ExtΛ( , ): (mod Λ) ×mod Λ→ Ab. We assume that F has enough projectives and injectives. In [5] F -exact sequences and the functors ExtiF ( , ) for all i ≥ 0 were discussed. In analogy with the standard situation relative F -cotilting modules are defined as follows. A Λ-module T is an F -cotilting module if it satisfies (a) ExtF (T, T ) = 0 for all i > 0, (b) idF T 0 and ⊥ HomΓ(G, T ). Amongst other things these results are based on the fact that the natural algebra morphism Λ→ EndΓ(ΓT ) is an isomorphism. In standard cotilting theory, one knows that all cotilting Λ-modules have the same number of nonisomorphic indecomposable summands, namely the number of nonisomorphic simple Λmodules. We show that all F -cotilting Λ-modules also have the same number of nonisomorphic indecomposable summands, namely, the number of nonisomorphic indecomposable summands of G where addG = P(F ). Since Λ is in P(F ), the number of nonisomorphic summands of an F cotilting module is greater than or equal to the number of nonisomorphic simple Λ-modules. More