Suppose we have a morphism of schemes, f: X -> Y. In the 80's 
Faltings and most principally Deligne formulated, under suitable
conditions on f, conjectural properties of the linebundle det Rf_* E;
among other things Riemann-Roch like behavior. I establish these
properties in my thesis. When applied to the case of curves and abelian
schemes one recovers classical results; for example a new proof of T.
Saito's conductor-discriminant formula and a refined version of the
"formule clef"  of Moret-Bailly.