Twisted Segre products

Abstract: We introduce the notion of the twisted Segre product $A\circ_\psi B$ of $\mathbb Z$-graded algebras $A$ and $B$ with respect to a twisting map $\psi$. It is proved that if $A$ and $B$ are noetherian Koszul Artin-Schelter regular algebras and $\psi$ is a twisting map such that the twisted Segre product $A\circ_\psi B$ is noetherian, then $A\circ_\psi B$ is a noncommutative graded isolated singularity. To prove this result, the notion of densely (bi-)graded algebras is introduced. Moreover, we show that the twisted Segre product $A\circ_\psi B$ of $A=k[u,v]$ and $B=k[x,y]$ with respect to a diagonal twisting map $\psi$ is a noncommutative quadric surface (so in particular it is noetherian), and we compute the stable category of graded maximal Cohen-Macaulay modules over it.