Corollary 1

Let k be a field and A a noetherian local k-algebra such that the extension κ_A of k is separable. The following conditions are equivalent:

(i) the k-algebra A is formally smooth for the $m_A$-adic topology; (ii) the ring A is regular; (iii) the k-algebra Λ is absolutely regular (§ 6, No. 4, Def. 1) ; (iv) the k-algebra Ā is isomorphic to κ_A[[T_1, ..., T_n]], with n = dim A.