The Stacks Project


Tag 03QH

Chapter 53: Étale Cohomology > Section 53.32: Henselian rings

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}

    Comments (2)

    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.

    Add a comment on tag 03QH

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?