Eventual linearity of regularity and projective dimension for Inc-invariant ideal chains

Establish that the Castelnuovo–Mumford regularity and projective dimension of every Inc-invariant chain of ideals are eventually linear functions of the index n.

Background

The paper places its study of Inc-invariant graph chains within the broader theory of Inc-invariant chains of ideals. Equivariant Noetherianity guarantees stabilization of such ideal chains up to increasing relabelings, but stabilization of the ideals does not by itself determine the asymptotic behavior of numerical invariants associated with them.

The authors identify eventual linearity of Castelnuovo–Mumford regularity and projective dimension as conjectured asymptotic behavior for arbitrary Inc-invariant ideal chains. They note that substantial progress is known, including results for chains of edge ideals, but that the general conjectures remain unresolved. The graph-theoretic results developed in the paper address related combinatorial invariants rather than settling these two algebraic conjectures in full generality.