Dominant dimension and tilting modules
Dominant dimension and tilting modules
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DOI：
10.1007/s00209-018-2111-4
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发表时间：
2017-06
影响因子：
0.8
通讯作者：
Van C. Nguyen;I. Reiten;G. Todorov;Shijie Zhu
中科院分区：
文献类型：
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作者：
Van C. Nguyen;I. Reiten;G. Todorov;Shijie Zhu
我们研究哪些代数具有由射影-内射模生成和共同生成的倾斜模。 Crawley-Boevey 和 Sauter 已经证明 Auslander 代数具有这样的倾斜模；对于全局维度 2 的代数，Auslander 代数根据此类倾斜模的存在进行分类。在本文中，我们证明这种倾斜模块的存在相当于主维数至少为 2 的代数，与其全局维数无关。一般来说，这种倾斜模块不一定是共同倾斜的。在这里，我们证明具有由射影-内射模生成-共同生成的倾斜-共倾斜模的代数恰好是 1-最小 Auslander-Gorenstein 代数。当考虑这样一个倾斜模块时，在不假设它是共倾斜的情况下，我们研究其自同态代数的全局维数，并讨论与有限维数猜想的联系。此外，作为特殊情况，我们证明从 Auslander 代数和某些单射模获得的三角矩阵代数具有这样的倾斜模。我们还描述了哪些中山代数具有这样的倾斜模。
We study which algebras have tilting modules that are both generated and cogenerated by projective–injective modules. Crawley–Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting–cotilting module generated–cogenerated by projective–injective modules are precisely 1-minimal Auslander–Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.