Deligne pairing for equidimensional morphisms

Let $S$ be a noetherian normal scheme, and let $X\to S$ be a surjective projective morphism of pure relative dimension $d$. We construct a symmetric multi-additive functor $\mathcal{P}\mathrm{i}\mathrm{c}(X{)}^{d+1}\to \mathcal{P}\mathrm{i}\mathrm{c}(S)$, and prove its functorial properties. Our construction uses Elkik’s and García’s ideas, as well as algebraic Hartogs’ theorem. Moreover, our results can be used to define arithmetic intersection theory of hermitian line bundles for equidimensional morphisms.