Some constructions of bi-Koszul algebras. (Q2859276)

A connected graded associative algebra \(A\) over a field \(k\) is bi-Koszul of type \(d\) if the trivial \(A\)-module \(k\) has a minimal free resolution \(P\) in which \(P_n\) is generated by elements of bi-degree \(\Delta(n)\) for some specific function \(\Delta\colon\mathbb N\to\mathbb N\times\mathbb N\). Suppose that \(D\) is a normal extension of \(A\) such that \(D/Dz=A\) and \(z\) has degree 1. Then \(D\) is a bi-Koszul algebra of type \(d\) if and only if \(A\) is a 3-Koszul algebra of global dimension at most 3 in the sense of \textit{R. Berger} [J. Algebra 239, No. 2, 705-734 (2001; Zbl 1035.16023)]. In this case \(d=2\) and \(\text{gl\,}D=\text{gl\,}A+1\). In particular let \(D=A[z;\sigma,\delta]\) be an Ore extension. Then \(D\) is bi-Koszul of type \(d\) if and only if \(A\) is 3-Koszul, \(\text{gl\,}A\leqslant 3\). Moreover \(\text{gl\,}D=\text{gl\,}A+1\) and \(d=2\).