Theorem 58.32.4. Let $(R, \mathfrak m, \kappa )$ be a local ring. The following are equivalent:

1. $R$ is henselian,

2. for any $f\in R[T]$ and any factorization $\bar f = g_0 h_0$ in $\kappa [T]$ with $\gcd (g_0, h_0)=1$, there exists a factorization $f = gh$ in $R[T]$ with $\bar g = g_0$ and $\bar h = h_0$,

3. any finite $R$-algebra $S$ is isomorphic to a finite product of local rings finite over $R$,

4. any finite type $R$-algebra $A$ is isomorphic to a product $A \cong A' \times C$ where $A' \cong A_1 \times \ldots \times A_ r$ is a product of finite local $R$-algebras and all the irreducible components of $C \otimes _ R \kappa$ have dimension at least 1,

5. if $A$ is an étale $R$-algebra and $\mathfrak n$ is a maximal ideal of $A$ lying over $\mathfrak m$ such that $\kappa \cong A/\mathfrak n$, then there exists an isomorphism $\varphi : A \cong R \times A'$ such that $\varphi (\mathfrak n) = \mathfrak m \times A' \subset R \times A'$.

Proof. This is just a subset of the results from Algebra, Lemma 10.153.3. Note that part (5) above corresponds to part (8) of Algebra, Lemma 10.153.3 but is formulated slightly differently. $\square$

Comment #1713 by Yogesh More on

In condition (3), "any finite R-algebra S is isomorphic to a finite product of finite local rings", should the third/last instance of "finite" be there? For example, take $R=\mathbb{C}[[t]]$, $S=R$, then $S$ is module finite over $R$ but $S$ is not finite.

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