Koszul pairs. Applications
Koszul pairs. Applications
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科祖尔对。
DOI：
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发表时间：
2010
期刊：
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通讯作者：
D. Ştefan
中科院分区：
文献类型：
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作者：
Pascual Jara Martínez;J. L. Peña;D. Ştefan
设$R$是一个半单环。如果$A$是连通的分级$R$-环，$C$是兼容的连通的分级$R$-环，则称一对$(A，C)$为近科祖尔。对于一个近科祖尔对，有三个链配合物和三个共链配合物，其中一个是精确的当且仅当其他是精确的。在这种情况下，$(A，C)$被称为Koszul。证明一个连通的R环a是Koszul当且仅当存在一个连通的R环C使得$(a，C)$是Koszul。这一结果允许我们研究kozul环的Hochschild （co）同源性。我们应用我们的方法证明了两个Koszul环的扭曲张量积是Koszul。本文的最后一部分讨论了更多的Koszul对的例子和应用，包括对Fr\ oberg定理的推广。
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 Fr\"oberg Theorem, are discussed in the last part of the paper.