Smoothness, flatness and deformations of subalgebras

AbstractThe smoothness of a morphism between noetherian schemes can be recognized in terms of the induced mappings between tangent and obstruction spaces. This observation can be effectively applied in the study of schemes parametrizing certain objects of interest in deformation theory. Strong versions of the classical results on rigidity and stability of subalgebras in finite dimensional Lie algebras are derived as an application. Some special cohomological conditions ensuring rigidity or stability are obtained in case of a field of positive characteristic