Given a standard graded algebra over a field, we consider the relationship between G-quadraticity and the existence of a Koszul filtration. We show that having a quadratic Gröbner basis implies the existence of a Koszul filtration for toric algebras equipped with the degree reverse lexicographic term order and for algebras defined by binomial edge ideals. We also resolve a conjecture of Ene, Herzog, and Hibi by constructing an example where this implication fails. These results are underpinned by algorithms we develop for constructing Koszul filtrations. We also demonstrate the utility of these algorithms on the pinched Veronese algebra.

翻译：给定域上的标准分次代数，我们研究G-二次性与Koszul滤链存在性之间的关系。我们证明：对于配备度反字典序项序的环面代数，以及由二项式边理想定义的代数，拥有二次Gröbner基意味着存在Koszul滤链。通过构造一个反例，我们解决了Ene、Herzog和Hibi的一个猜想，该反例表明上述蕴含关系并不普遍成立。这些结果的基础是我们提出的构造Koszul滤链的算法。我们还通过捏挤Veronese代数的实例展示了这些算法的实用性。