Theorem 2

Let A be a separated complete local ring, k its residue field and $\pi$ the canonical homomorphism of A onto k. It is assumed that k is a perfect field of characteristic p.

a) There exists a unique ring homomorphism $u : W(k) \to A$ such that $\pi(u(a)) = a_0$ for $a = (a_n)_{n \in \mathbf{N}}$ in W(k).

b) The homomorphism u is continuous when W(k) is endowed with the $pW(k)$-adic topology, and the image of u is the unique Cohen subring of A.

By th. 1 of No. 2, there exists a unique Cohen subring of A; let us denote it by C. Let u be a homomorphism of W(k) into A such that $\pi(u(a)) = a_0$ for every $a = (a_n)_{n \in \mathbf{N}}$ in W(k); it is immediate that the image of u is a Cohen subring of A, hence equal to C. The existence and uniqueness of u then result from prop. 6. The $pC$-adic topology of C is induced by the $m_A$-adic topology of A (No. 2, th. 1, b)), whence the continuity of u.

For a direct construction of u, see p. 70, exerc. 6.