
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 of -graded algebras and with respect to a twisting map . It is proved that if and are noetherian Koszul Artin-Schelter regular algebras and is a twisting map such that the twisted Segre product is noetherian, then 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 of and with respect to a diagonal twisting map 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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