Twisted Segre products

TL;DR

This paper introduces the twisted Segre product of graded algebras, proves its properties under certain conditions, and explores its structure as a noncommutative geometric object, including explicit examples and module category analysis.

Contribution

It defines the twisted Segre product for graded algebras, establishes its properties as a noncommutative isolated singularity, and analyzes specific examples like noncommutative quadrics.

Findings

Twisted Segre product of certain algebras is a noncommutative graded isolated singularity.

Explicit example of a noncommutative quadric surface as a twisted Segre product.

Computed the stable category of graded maximal Cohen-Macaulay modules over the example.

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.