Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings

We study finitely generated modules of minimal multiplicity, a notion introduced by Puthenpurakal that extends the classical concept of minimal multiplicity from rings to modules. Our main result characterizes when trace ideals or reflexive ideals yield modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings. As a consequence, we show that a one-dimensional non-Gorenstein reduced local ring with a canonical module has minimal multiplicity if and only if its canonical module has minimal multiplicity as a module. We also construct several examples and compare them with Burch and Ulrich modules, highlighting cases where minimal multiplicity coincides with the Burch or Ulrich property.

💡 Research Summary

The paper investigates the notion of “modules of minimal multiplicity,” a concept introduced by Puthenpurakal that extends the classical idea of minimal multiplicity from rings to finitely generated torsion‑free modules. The setting throughout is a one‑dimensional Noetherian Cohen–Macaulay local ring (R) with maximal ideal (\mathfrak m) and residue field (k). For an (\mathfrak m)-primary ideal (I) and a non‑zero torsion‑free (R)-module (M), the module is said to have minimal multiplicity with respect to (I) if the Hilbert–Samuel multiplicity satisfies
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