Abstract To study s-homogeneous algebras, we introduce the category of quivers with s-homogeneous corelations and the category of s-homogeneous triples. We show that both of these categories are equivalent to the category of s-homogeneous algebras. We prove some properties of the elements of s-homogeneous triples and give some consequences for s-Koszul algebras. Then we discuss the relations between the s-Koszulity and the Hilbert series of s-homogeneous triples. We give some application of the obtained results to s-homogeneous algebras with simple zero component. We describe all s-Koszul algebras with one relation recovering the result of Berger and all s-Koszul algebras with one dimensional s-th component. We show that if the s-th Veronese ring of an s-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all s-homogeneous algebras with s-th Veronese rings k 〈 x , y 〉 / ( x y , y x ) and k 〈 x , y 〉 / ( x 2 , y 2 ) . In particular, we show that all of these algebras are not s-Koszul while their s-homogeneous duals are s-Koszul.

中文翻译：

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通过齐次三元组的齐次代数

摘要 为了研究s齐次代数，我们引入了s齐次相关的颤动范畴和s齐次三元组范畴。我们证明这两个类别都等价于 s 齐次代数的类别。我们证明了 s-齐次三元组元素的一些性质，并给出了 s-Koszul 代数的一些结论。然后讨论s-Koszulity和s-齐次三元组的希尔伯特级数之间的关系。我们将所得结果应用于具有简单零分量的 s 齐次代数。我们描述了所有 s-Koszul 代数具有一种恢复 Berger 结果的关系，以及所有 s-Koszul 代数具有一维 s-th 分量。我们证明如果 s-齐次代数的第 s 个 Veronese 环有两个生成元，那么它至少有两个关系。最后，我们用第 s 个 Veronese 环 k 〈 x , y 〉 / ( xy , yx ) 和 k 〈 x , y 〉 / ( x 2 , y 2 ) 对所有 s 齐次代数进行分类。特别是，我们证明所有这些代数都不是 s-Koszul，而它们的 s-齐次对偶是 s-Koszul。