A note on explicit homological invariants of graded double Ore extensions

TL;DR

This paper computes explicit homological invariants for a class of graded double Ore extensions, linking algebraic structure to homological properties.

Contribution

It provides the first explicit calculations of minimal graded free resolutions and Betti numbers for these extensions, connecting PBW structure to homological behavior.

Findings

Determined minimal graded free resolutions for the trivial module.

Computed graded Betti numbers for a family of algebras.

Established linear resolutions for natural cyclic modules.

Abstract

We compute explicit homological invariants of a trimmed graded double Ore extension of the quantum plane. For a pilot family of type (14641), we determine the minimal graded free resolution and graded Betti numbers of the trivial right module and also compute linear resolutions for two natural cyclic right modules. This provides a concrete link between the PBW structure of the algebra and the homological behavior of its natural quotients.