On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings

TL;DR

This paper extends the concept of complete intersection from skew polynomial rings to a broader class of non-commutative algebras, including quantum and Clifford algebras, supporting their analogy to polynomial rings.

Contribution

It generalizes the definition of complete intersection to include regular graded skew Clifford algebras, quantum matrix rings, and homogenized universal enveloping algebras.

Findings

Extended the notion of complete intersection to new classes of non-commutative algebras.

Provided algebraic and geometric characterizations applicable beyond skew polynomial rings.

Supported the view of regular algebras as non-commutative analogues of polynomial rings.

Abstract

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of quantum matrices and homogenizations of universal enveloping algebras. Regular algebras are often considered to be non-commutative analogues of polynomial rings, so the results herein support that viewpoint.