Shift modules, strongly stable ideals, and their dualities

TL;DR

This paper introduces shift modules to expand the framework of strongly stable ideals, providing a new duality perspective and a novel type of projective resolution within an enriched categorical setting.

Contribution

It develops shift modules as a new category for strongly stable ideals, formalizes their duality, and introduces a projective resolution concept in this context.

Findings

Duality on SSI's is made more effective and computational.

Shift modules encompass SSI's and facilitate new resolutions.

Study of SSI's in infinite dimensional polynomial rings.

Abstract

We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSI's. The recently introduced duality on SSI's is given an effective conceptual and computational setting. We study strongly stable ideals in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSI's is introduced. This is the projective resolution in the category of shift modules.