Codimension and Projective Dimension up to Symmetry

TL;DR

This paper studies the asymptotic behavior of codimension and projective dimension in chains of symmetric ideals in polynomial rings, extending known results and providing explicit growth rates and bounds.

Contribution

It extends linear growth results of codimension to arbitrary symmetric ideals and establishes lower bounds for projective dimensions, with applications to Cohen-Macaulayness.

Findings

Codimension grows linearly with explicit slope.

Established lower linear bounds for projective dimensions.

Provided obstructions to Cohen-Macaulayness in symmetric ideals.

Abstract

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen-Macaulayness obstructions.