Artin-Schelter Regular Algebras, Subalgebras, and Pushouts

TL;DR

This paper explores how pushouts of certain regular quadratic algebras of dimension three can produce new regular algebras of dimension four, preserving some module structures.

Contribution

It demonstrates that pushouts of specific regular quadratic algebras of dimension three yield new regular algebras of dimension four, revealing structural inheritance.

Findings

Pushouts of certain regular algebras are regular of higher dimension.

Some point module structures are preserved in the pushout.

The construction provides new examples of regular algebras of dimension four.

Abstract

Take $A$ to be a regular quadratic algebra of global dimension three. We observe that there are examples of $A$ containing a dimension three regular cubic algebra $C$. If $B$ is another dimension three regular quadratic algebra, also containing $C$ as a subalgebra, then we can form the pushout algebra $D$ of the inclusions $i_1:C\hookrightarrow A$ and $i_2:C\hookrightarrow B$. We show that for a certain class of regular algebras $C\hookrightarrow A,B$, their pushouts $D$ are regular quadratic algebras of global dimension four. Furthermore, some of the point module structures of the dimension three algebras get passed on to the pushout algebra $D$.