Trace ideals of canonical modules over Schubert cycles and determinantal rings
Kaito Kimura
TL;DR
This work uniformly analyzes canonical traces of two fundamental combinatorial algebras—Schubert cycles and determinantal rings—by describing their trace ideals and non-Gorenstein loci through block/gap data. The authors derive explicit formulas for the canonical trace in both settings, connect these to Gorenstein criteria, and establish that the CTR property is preserved under base change, contingent on simple numerical gaps (κ−κ' and λ−λ' ≤ 1). By leveraging dehomogenization and ASL techniques, they extend field-case results to general base rings, enabling base-change stability results and a clear affine-descent description of singular loci. The results provide practical criteria to identify non-Gorenstein points and to verify CTR in families arising from Schubert and determinantal rings. Overall, the paper links combinatorial data to precise ring-theoretic invariants with implications for singularities and base-change behavior.
Abstract
In this paper, we study the canonical trace of Schubert cycles and determinantal rings. As an application, we give an explicit description of the non-Gorenstein locus and show that its structure is compatible with the known representations of the singular locus and the canonical module. Furthermore, for the CTR property recently introduced by Miyazaki, we establish its stability under base change and provide a characterization in the case of determinantal rings.