标题： 波兰丘克和波西捷尔斯基关于二次代数的书中提出的两个问题 显示英文标题

标题： Two problems from the Polishchuk and Positselski book on Quadratic algebras

摘要： 在波兰奇克和波西采尔斯基所著的《二次代数》一书中[23]，考虑了生成元数量较少（n=2,3）的代数。 对于某些数量r的关系，可能的Hilbert级数被列出，并指明了哪些是Koszul代数的级数。 第一个无法做到的情况，即三个生成元n=3和六个关系r=6的情况被列为开放问题。 我们在这里对该问题给出了完整的答案，即对于二次代数满足dim A_1=dim A_2=3的情况，我们列出了所有可能的Hilbert级数，并确定了其中哪些可以来自Koszul代数，哪些不能。 作为这一分类的结果，我们发现了一个代数，它作为对同一本书[23]（第七章，第1节，猜想2）中另一个问题的反例，该问题指出Koszul代数具有有限的全局同调维数d时，有dim A_1 >= d。具体来说，由关系xx+yx=xz=zy=0生成的3个生成元代数A是Koszul代数，其Koszul对偶代数A^!的4次Hilbert级数为H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4，因此A具有4次全局同调维数。 显示英文摘要

摘要： In the book 'Quadratic algebras' by Polishchuk and Positselski [23] algebras with a small number of generators (n=2,3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and six relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with dim A_1=dim A_2=3, we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23] (Chapter 7, Sec. 1, Conjecture 2), saying that Koszul algebra of finite global homological dimension d has dim A_1 >= d. Namely, the 3-generated algebra A given by relations xx+yx=xz=zy=0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4, hence A has global homological dimension 4.