关于局部环上双周期模的一些注记
Some remarks on two-periodic modules over local rings
AI总结：
本文研究了交换 Noetherian 局部环上有限生成双周期模的性质，证明了特定条件下自然映射的同构性及 Auslander 深度公式成立，并将 Huneke-Wiegand 猜想推广至任意一维局部环上的双周期模。
摘要：
在本文中，我们研究了交换 Noetherian 局部环上有限生成双周期模的一些性质。我们证明，在对一对模 $(M,N)$（其中 $M$ 是双周期的）施加某些假设下，自然映射 $M \otimes_R N \to Hom_R(M^*,N)$ 是同构。作为推论，我们得到 Auslander 深度公式对这样的模对成立。Celikbas 等人最近证明了 Huneke-Wiegand 猜想在一维整环上对双周期模成立。我们将他们的结果推广到任意一维局部环上具有秩的双周期模的情形。更一般地，在对模施加某些假设下，我们证明一维局部环上的一对模具有非零挠当且仅当它们是 Tor-独立的。
英文摘要：
In this note, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules $\left(M,N \right)$ with $M$ two-periodic, the natural map $M \otimes_R N \to Hom_R(M^*,N)$ is an isomorphism. As a consequence, we have that the Auslander's depth formula holds for such a pair. Celikbas et al. recently showed the Huneke-Wiegand conjecture holds over one-dimensional domain for two-periodic modules. We generalize their result to the case of two-periodic module with rank over any one-dimensional local ring. More generally, under certain assumptions on the modules, we show that a pair of modules over an one-dimensional local ring has non-zero torsion if and only if they are Tor-independent.
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