\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

Theorem 54.32.4. 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. $\square$

Comments (2)

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 , , then is module finite over but is not finite.

There are also:

  • 2 comment(s) on Section 54.32: Henselian rings

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