Exercise 3, § 4

Let $A$ be a local ring, $\mathfrak{m}$ its maximal ideal, $k = A/\mathfrak{m}$ its residue field and $f : A \to k$ the canonical homomorphism. Show that the two following conditions are equivalent:

(H) For every monic polynomial $P \in A[X]$ and every decomposition of $f(P) \in k[X]$ as a product $f(P) = Q.Q'$ of relatively prime monic polynomials, there exist two monic polynomials $Q, Q'$ in $A[X]$ such that $f(Q) = \overline{Q}$, $f(Q') = \overline{Q}'$ and $P = QQ'$.

(C) Every commutative algebra over $A$ which is a finitely generated $A$-module is a direct composition of $A$-algebras which are local rings.

(To prove that (H) implies (C), argue as in Proposition 8 of no. 6, using Exercise 2 (a). To see that (C) implies (H), show first that, for every commutative $A$-algebra $B$ which is a finitely generated $A$-module, every decomposition of $B/\mathfrak{m}B$ as a direct sum of two ideals is necessarily of the form $b/\mathfrak{m}b \oplus b'/\mathfrak{m}b'$, where $B = b \oplus b'$ is a decomposition of $B$ as a direct sum of two ideals; then use Exercise 2 (b).)

A local ring satisfying conditions (H) and (C) is called Henselian. Every complete Hausdorff local ring is Henselian. If $A$ is Henselian and $B$ is a commutative $A$-algebra which is a local ring and a finitely generated $A$-module, then $B$ is Henselian.