Theorem 2

Let $A$ be a complete noetherian local ring of dimension $d$ containing a field. Let $K$ be a coefficient field of $A$, and let $m$ be the dimension of the vector space $m_A/m_A^2$ over the field $K$. a) There exists an ideal $a$ of $K[[T_1, ..., T_m]]$ such that the $K$-algebra $A$ is isomorphic to $K[[T_1, ..., T_m]]/a$. b) There exists a sub-$K$-algebra $A'$ of $A$, isomorphic to $K[[T_1, ..., T_d]]$ and such that $A$ is a finite algebra over $A'$. c) Suppose that the noetherian local ring $A$ is regular, i.e. $d = m$. Then there exists a $K$-isomorphism of $A$ onto $K[[T_1, ..., T_d]]$. Let $t_1, ..., t_m$ be elements of $m_A$ whose classes modulo $m_A^2$ generate the vector space $m_A/m_A^2$ over $K$. By lemma 3 of § 2, No. 5, there exists a surjective $K$-homomorphism of $K[[T_1, ..., T_m]]$ into $A$, mapping $T_i$ to $t_i$ for $1 \leq i \leq m$. This proves $a$. Analogously, assertion $b$ follows from lemma 4 of loc. cit. and from the existence of a maximal secant sequence for $A$ (VIII, § 3, No. 2, th. 1). Finally, assertion $c$ is nothing but cor. 3 of th. 1 of VIII, § 5, No. 2.