Partial trace ideals and Berger's conjecture

Abstract

For any finitely generated module M with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant 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 ΩR/k, where R is a complete k-algebra with k any perfect field, to study a long-standing conjecture due to R. W. Berger.