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 a numerical invariant $\mathrm{h}(M)$ based on a partial trace ideal of $M$. We study its properties and explore relations between this invariant and the colength of the conductor. Finally we apply this to the universally finite module of differentials ${\mathrm{\Omega }}_{R\mathrm{/}k}$, where $R$ is a complete $k$-algebra with $k$ any perfect field, to study a long-standing conjecture due to R. W. Berger.