The Eliahou-Kervaire resolution over a skew polynomial ring

TL;DR

This paper extends the Eliahou-Kervaire resolution from polynomial rings to skew polynomial rings, demonstrating that stable ideals in this non-commutative setting share similar homological properties and admit a compatible algebraic product.

Contribution

It generalizes the Eliahou-Kervaire resolution to skew polynomial rings and establishes analogous homological properties and algebraic structures for stable ideals in this broader context.

Findings

Homological properties of stable ideals extend to skew polynomial rings.

Constructed a minimal resolution with a compatible algebraic product.

Demonstrated the resolution's properties mirror those in the commutative case.

Abstract

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. Finally we show that the resolution we construct admits a product generalizing the one given by Eliahou and Kervaire in the commutative case.