Stable equivalence of self-injective algebras
Stable equivalence of self-injective algebras
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DOI：
10.1016/0021-8693(76)90087-9
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发表时间：
1976-05
影响因子：
0.9
通讯作者：
I. Reiten
中科院分区：
文献类型：
--
作者：
I. Reiten
设A是一个artin代数，即一个artin环，它是其中心R上的一个n-生成模，R是一个artin环，设mod A表示n-生成（左）A-模的范畴。我们用mod A/P表示与mod A相关联的（投射）稳定范畴，其对象与mod A的对象相同，用M表示，其中M在mod A中。态射由Horn（M，N）= Hom，（M，N）/P（M，N）给出，其中P（M，N）表示Hom，（M，N）的子群，由通过投射A-模分解的映射f：JI-1组成.如果模A/P和模A '/P相等，则称A和A'稳定相等。文献[1 -6]研究了Artin代数的稳定等价性，或者更一般地说，研究了对偶R-簇的稳定等价性。我们知道两个阿廷代数A和A '在某些代数性质上可以有很大的不同，如同调维数、可交换与否，但仍然是稳定等价的。特别地，如果A的Loewy长度至多为2，即r2= 0，其中r表示A的根式，则A稳定等价于遗传代数A '，其Loewy长度也至多为2。本文研究了自内射代数的稳定等价。我们的主要定理是：如果A稳定等价于一个自内射代数，使得每个不可分解的直因子代数的Loewy长度大于2，则A也是自内射的。绿色关于群代数稳定等价的结果表明，两个稳定等价的自内射代数（不含半单直因子）不一定是Morita等价的。在第一节中，我们回顾了文献[2]中的一些结果，利用这些结果得到了Artin代数稳定等价于自内射代数的一些必要条件。在第二节中，我们回顾了[SJ]中关于广义三角矩阵环上模的一些结果，并利用这些结果证明了我们在第一节中描述的代数类恰好由自内射代数的乘积组成。
Let A be an artin algebra, ie, an artin ring that is a finitely generated module over its center R, which is an artin ring, and let mod A denote the category of finitely generated (left) A-modules. We denote by mod A/P the (projectively) stable category associated with mod A, whose objects are the same as those of mod A, denoted by M, for M in mod A. The morphisms are given by Horn& g)= Hom,(M, N)/P (M, N), where P (M,; V) denotes the subgroup of Hom,(M, N) consisting of the maps f: JI-1\’that factor through a projective A-module. A and A’are said to be stably equivalent if mod A/P and mod A’/P are equivalent. Stable equivalence of artin algebras, or more generally, of dualizing R-varieties, was studied in [l-6]. We know that two artin algebras A and A’can be very different with respect to certain algebraic properties, as homological dimension, being commutative or not, and still be stably equivalent. In particular, if the Loewy length of A is at most 2, ie, r2= 0, where r denotes the radical of A, then A is stably equivalent to an hereditary algebra A’, also of Loewy length at most 2. In this paper, we study stable equivalence of self-injective algebras. Our Main Theorem is that if A is stably equivalent to a self-injective algebra, such that each indecomposable direct factor algebra has Loewy length greater than 2, then A is also self-injective. Results of Green on stable equivalence of group algebras show that two stably equivalent self-injective algebras (with no semisimple direct factor) are not necessarily Morita equivalent.We have divided the paper in two sections. In Section 1, we recall some results from [2], which we use to get some necessary conditions on the artin algebras stably equivalent to self-injective algebras. In Section 2, we recall some results on modules over generalized triangular matrix rings from [SJ, and use these to show that the class of algebras we describe in Section 1 consists exactly of products of self-injective algebras 63