Intersection sheaves over normal schemes

Intersection sheaves are usually defined for a projective flat surjective morphism of Noetherian schemes of relative dimension d and for d+1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the case of the equi-dimensional projective surjective morphisms to normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on polarized endomorphisms are also given.