A Jacobian criterion for smoothness

AbstractA sufficient condition for an algebra to be smooth is given in this paper; namely, if the Jacobian matrix of relations is right-invertible, then the algebra is smooth. It is also shown that right-invertibility is not a necessary condition for being smooth. Moreover, an elementary proof of the known fact that “smoothness implies the projectivity of the differential module” is also given. The criterion obtained here can be applied to the Jacobian problem [9, Theorem 38]