Shift modules, strongly stable ideals, and their dualities

Abstract: Polynomial rings over a field $k$ are the prime objects in algebra. Ideals in polynomial rings are the prime objects relating algebra and geometry via the zero set of the ideal.

To understand ideals in a polynomial ring, a common approach is to see what simpler ideals they degenerate to, for instance what monomial ideals. But what are the most degenerate ideals you can find? Those that cannot be degenerated any further? These are the so-called Borel-fixed ideals, or, when the field k has characteristic zero, the strongly stable ideals. This class is for instance the essential tool for understanding numerical invariants of ideals in polynomial rings.

We enrich the setting of strongly stable ideals by:

1. Extending them to a category of modules

2. Investigating the recently discovered duality on these ideals

3. Getting a new type of projective resolution of such ideals

4. Letting the ambient polynomial ring be infinite dimensional

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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