  This article is part of a series of works by the authors with the goal of
completing a far-reaching program propounded by Deligne, aiming to extend the
codimension one part of the Grothendieck-Riemann-Roch theorem from isomorphism
classes of line bundles to isomorphisms thereof. The paper develops a relative
functorial intersection theory with values in line bundles, together with a
formalism that generalizes previous constructions by Deligne and Elkik, related
to the right-hand side of the theorem.
