Differentially Homogeneous Schemes

Let S be a Noetherian scheme and π: X → S be a flat morphism of finite type. Then π is said to be differentially homogeneous when X × SX is normally flat along the diagonal. Examples of these morphisms are smooth morphisms and finite field extensions. We study here different characterizations and properties of differentially homogeneous morphisms, and finally obtain the local structure of these morphisms.