Partial trace ideals and Berger's conjecture

For any finitely generated module M with non-zero rank over a commutative one-dimensional Noetherian local domain, we study numerical invariant h(M) based on partial trace ideal of M. We its properties and explore relations between this the colength conductor. Finally apply to universally finite differentials ΩR/k, where R is complete k-algebra k perfect field, long-standing conjecture due R. W. Berger.