# The Stacks Project

## Tag 03QH

Theorem 53.32.4. 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 10.148.3. Note that part (5) above corresponds to part (8) of Algebra, Lemma 10.148.3 but is formulated slightly differently. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 4121–4140 (see updates for more information).

\begin{theorem}
\label{theorem-henselian}
Let $(R, \mathfrak m, \kappa)$ be a local ring. The following are equivalent:
\begin{enumerate}
\item $R$ is henselian,
\item 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 $\bar h = h_0$,
\item any finite $R$-algebra $S$ is isomorphic to a finite product of
local rings finite over $R$,
\item 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,
\item if $A$ is an \'etale $R$-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
$\varphi(\mathfrak n) = \mathfrak m \times A' \subset R \times A'$.
\end{enumerate}
\end{theorem}

\begin{proof}
This is just a subset of the results from
Algebra, Lemma \ref{algebra-lemma-characterize-henselian}.
Note that part (5) above corresponds to part (8) of
Algebra, Lemma \ref{algebra-lemma-characterize-henselian}
but is formulated slightly differently.
\end{proof}

Comment #1713 by Yogesh More on December 1, 2015 a 11:39 pm UTC

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.

Comment #1755 by Johan (site) on December 15, 2015 a 7:04 pm UTC

Thanks, fixed here.

There are also 2 comments on Section 53.32: Étale Cohomology.

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