2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230574Let $u:A\to B$ be a morphism of noetherian local rings. We obtain smoothness criteria for algebras with differential bases, in the case of rings containing a field of characteristic $p>0.$ We also give smoothness criteria for reduced morphisms.6 pagesCommutative Algebra13N05; 13B10.text