Exercise 4, § 4

(a) Let $(A_\alpha, \phi_{\alpha\beta})$ be a direct system of Henselian local rings, the homomorphisms $\phi_{\alpha\beta}$ being local. Show that the local ring $A = \varprojlim A_\alpha$ (Chapter II, § 3, Exercise 16) is Henselian (use criterion (H) of Exercise 3).

(b) Let $K$ be a commutative field and $L$ an algebraic extension of $K$. Deduce from (a) that the ring $L \otimes_K K[[X_1, \ldots, X_n]]$ is Henselian.

*(c) Let $A$ be a Henselian local ring and $B$ a commutative $A$-algebra which is integral over $A$ (Chapter V) and is a local ring. Show that $B$ is a Henselian ring (use (a)).*