Two problems from the Polishchuk and Positselski book on Quadratic algebras

TL;DR

This paper classifies quadratic algebras with three generators and six relations, resolving an open problem, and provides a counterexample to a conjecture relating Koszul algebra dimension and global homological dimension.

Contribution

It completes the classification of Hilbert series for certain quadratic algebras and identifies a counterexample to a conjecture about Koszul algebras and their dimensions.

Findings

Complete list of Hilbert series for quadratic algebras with 3 generators and 6 relations.

Identification of which Hilbert series correspond to Koszul algebras.

Counterexample disproving the conjecture relating global homological dimension and the dimension of A_1.

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 >= 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.