Two problems from the Polishchuk and Positselski book on quadratic algebras

Abstract

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 $\dim A_1\ge d$. 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.