Koszul对的进一步性质及应用
Further Properties and Applications of Koszul Pairs
摘要：
Koszul对在[arXiv：1011.4243]中被引入作为研究Koszul环的工具。在本文中，我们继续调查这些对，重点是第二个组成部分的描述，作为研究的后续[arXiv：1605.05458]。因此，我们引入Koszul核，并证明了他们的几个等价刻画。作为应用，在局部有限$R$ -环的情形下，我们证明了分次$R$ -环是Koszul当且仅当它的左（或右）分次对偶核是Koszul.最后，对有限分次偏序集，我们得到了相应的关联环是Koszul当且仅当关联核是Koszul。
英文摘要：
Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.
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