Smoothness, flatness and deformations of subalgebras

The smoothness of a morphism between noetherian schemes can be recognized i n terms of the induced mappings between tangent and obstruction spaces. Thi s observation can be effectively applied in the study of schemes parametriz ing certain objects of interest in deformation theory. Strong versions of t he classical results on rigidity and stability of subalgebras in finite dim ensional Lie algebras are derived as an application. Some special cohomolog ical conditions ensuring rigidity or stability are obtained in case of a fi eld of positive characteristic. (C) 1999 Elsevier Science B.V. All rights r eserved.