Reflexive modules over the endomorphism algebras of reflexive trace ideals
Reflexive modules over the endomorphism algebras of reflexive trace ideals
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DOI：
10.1016/j.jpaa.2024.107662
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发表时间：
2023-01
影响因子：
0.8
通讯作者：
Naoki Endo;S. Goto
中科院分区：
文献类型：
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作者：
Naoki Endo;S. Goto
本文研究了一维Cohen-Macaulay局部环上自反轨迹理想自同态代数上的自反模。主定理推广了S. Goto， N. Matsuoka， T. T. Phuong（[20，定理5.1]）和T. Kobayashi（[30，定理1.3]）关于其最大理想的自同态代数的结果。我们还探讨了自反模的范畴何时是有限型的问题，即基环上只有有限多个不可分解自反模的同构类。我们证明，如果范畴是有限型的，环是解析非分枝的，并且只有有限个乌尔里希理想。因此，当归一化是局部环时，局部环只包含有限个Ulrich理想。
In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong ([20, Theorem 5.1]) and T. Kobayashi ([30, Theorem 1.3]) concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, Arf local rings contain only finitely many Ulrich ideals once the normalization is a local ring.