In his answer here Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."

Could somebody give a reference for or explain this?

I ask because there is a natural definition of conductor of a proper morphism to a normal scheme (for example as in Serre's "Algebraic groups and class fields"), and some relationship between the conductor and discriminant, but the definitions of discriminants I have seen always seem situation-specific and ad-hoc.

Thanks.