Koszul对及其应用
Koszul pairs and applications
摘要：
设$R$是半单环.一对$(A,C)$称为几乎-Koszul，如果$A$是连通分次$R$ -环，$C$是相容连通分次$R$ -核.对于几乎-Koszul对，人们将三个链复形和三个上链复形联系起来，使得其中一个是正合的，当且仅当其他的都是正合的。在这种情况下，$(A,C)$被称为Koszul。证明了一个连通的$R$ -环$A$是Koszul当且仅当存在一个连通的$R$ -核$C$使得$(A,C)$是Koszul.这一结果使我们能够研究Koszul环的Hochschild（上）同调。我们应用我们的方法证明了两个Koszul环的扭曲张量积是Koszul环。本文最后讨论了Koszul对的更多例子和应用，包括Fröberg定理的推广。
英文摘要：
Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem, are discussed in the last part of the paper.
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