Twisted Segre products

We introduce the notion of the twisted Segre product $A\circ_ψB$ of $\mathbb Z$-graded algebras $A$ and $B$ with respect to a twisting map $ψ$. It is proved that if $A$ and $B$ are noetherian Koszul Artin-Schelter regular algebras and $ψ$ is a twisting map such that the twisted Segre product $A\circ_ψB$ is noetherian, then $A\circ_ψB$ 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 $A\circ_ψB$ of $A=k[u,v]$ and $B=k[x,y]$ with respect to a diagonal twisting map $ψ$ 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.

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