\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

[Chapter XI, Henselian] and [Proposition 1, Gabber-henselian]

Lemma 15.11.6. Let $(A, I)$ be a pair. The following are equivalent

  1. $(A, I)$ is a henselian pair,

  2. given an étale ring map $A \to A'$ and an $A$-algebra map $\sigma : A' \to A/I$, there exists an $A$-algebra map $A' \to A$ lifting $\sigma $,

  3. for any finite $A$-algebra $B$ the map $B \to B/IB$ induces a bijection on idempotents,

  4. for any integral $A$-algebra $B$ the map $B \to B/IB$ induces a bijection on idempotents, and

  5. (Gabber) $I$ is contained in the radical of $A$ and every monic polynomial $f(T) \in A[T]$ of the form

    \[ f(T) = T^ n(T - 1) + a_ n T^ n + \ldots + a_1 T + a_0 \]

    with $a_ n, \ldots , a_0 \in I$ and $n \ge 1$ has root $\alpha \in 1 + I$.

Moreover, in part (5) the root is unique.

Proof. Assume (2) holds. Then $I$ is contained in the Jacobson radical of $A$, since otherwise there would be a nonunit $f \in A$ not contained in $I$ and the map $A \to A_ f$ would contradict (2). Hence $IB \subset B$ is contained in the Jacobson radical of $B$ for $B$ integral over $A$ because $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is closed by Algebra, Lemmas 10.40.6 and 10.35.22. Thus the map from idempotents of $B$ to idempotents of $B/IB$ is injective by Lemma 15.10.2. On the other hand, since (2) holds, every idempotent of $B$ lifts to an idempotent of $B/IB$ by Lemma 15.9.10. In this way we see that (2) implies (4).

The implication (4) $\Rightarrow $ (3) is trivial.

Assume (3). Let $\mathfrak m$ be a maximal ideal and consider the finite map $A \to B = A/(I \cap \mathfrak m)$. The condition that $B \to B/IB$ induces a bijection on idempotents implies that $I \subset \mathfrak m$ (if not, then $B = A/I \times A/\mathfrak m$ and $B/IB = A/I$). Thus we see that $I$ is contained in the Jacobson radical of $A$. Let $f \in A[T]$ be monic and suppose given a factorization $\overline{f} = g_0h_0$ with $g_0, h_0 \in A/I[T]$ monic. Set $B = A[T]/(f)$. Let $\overline{e}$ be the nontrivial idempotent of $B/IB$ corresponding to the decomposition

\[ B/IB = A/I[T]/(g_0) \times A[T]/(h_0) \]

of $A$-algebras. Let $e \in B$ be an idempotent lifting $\overline{e}$ which exists as we assumed (3). This gives a product decomposition

\[ B = eB \times (1 - e)B \]

Note that $B$ is free of rank $\deg (f)$ as an $A$-module. Hence $eB$ and $(1 - e)B$ are finite locally free $A$-modules. However, since $eB$ and $(1 - e)B$ have constant rank $\deg (g_0)$ and $\deg (h_0)$ over $A/I$ we find that the same is true over $\mathop{\mathrm{Spec}}(A)$. We conclude that

\[ f = \det \nolimits _ A(T : B \to B) = \det \nolimits _ A(T : eB \to eB) \det \nolimits _ A(T : (1 - e)B \to (1 - e)B) \]

is a factorization into monic polynomials reducing to the given factorization modulo $I$1. Thus (3) implies (1).

Assume (1). Let $f$ be as in (5). The factorization of $f \bmod I$ as $T^ n$ times $T - 1$ lifts to a factorization $f = gh$ with $g$ and $h$ monic by Definition 15.11.1. Then $h$ has to have degree $1$ and we see that $f$ has a root reducing to $1$ modulo $1$. Thus (1) implies (5).

Before we give the proof of the last step, let us show that the root $\alpha $ in (5), if it exists, is unique. Namely, Due to the explicit shape of $f(T)$, we have $f'(\alpha ) \in 1 + I$ where $f'$ is the derivative of $f$ with respect to $T$. An elementary argument shows that

\[ f(T) = f(\alpha + T - \alpha ) = f(\alpha ) + f'(\alpha ) \cdot (T - \alpha ) \bmod (T - \alpha )^2 A[T] \]

This shows that any other root $\alpha ' \in 1 + I$ of $f(T)$ satisfies $0 = f(\alpha ') - f(\alpha ) = (\alpha ' - \alpha )(1 + i)$ for some $i \in I$, so that, since $1 + i$ is a unit in $A$, we have $\alpha = \alpha '$.

Assume (5). We will show that (2) holds, in other words, that for every étale map $A \to A'$, every section $\sigma : A' \to A/I$ modulo $I$ lifts to a section $A' \to A$. Since $A \to A'$ is étale, the section $\sigma $ determines a decomposition
\begin{equation} \label{more-algebra-equation-GCHP} A'/IA' \cong A/IA \times C \end{equation}

of $A/IA$-algebras. Namely, the surjective ring map $A'/IA' \to A/I$ is étale by Algebra, Lemma 10.141.8 and then we get the desired idempotent by Algebra, Lemma 10.141.9. We will show that this decomposition lifts to a decomposition
\begin{equation} \label{more-algebra-equation-GCHP-want} A' \cong A'_1 \times A'_2 \end{equation}

of $A$-algebras with $A'_1$ integral over $A$. Then $A \to A'_1$ is integral and étale and $A/I \to A'_1/IA'_1$ is an isomorphism, thus $A \to A'_1$ is an isomorphism by Lemma 15.10.3 (here we also use that an étale ring map is flat and of finite presentation, see Algebra, Lemma 10.141.3).

Let $B'$ be the integral closure of $A$ in $A'$. By Lemma 15.11.5 we may decompose
\begin{equation} \label{more-algebra-equation-dec-mod-I} B'/IB' \cong A/I \times C' \end{equation}

as $A/IA$-algebras compatibly with ( and we may find $b \in B'$ that lifts $(1, 0)$ such that $B'_ b \to A'_ b$ is an isomorphism. If the decomposition ( lifts to a decomposition
\begin{equation} \label{more-algebra-equation-want-2} B' \cong B'_1 \times B'_2 \end{equation}

of $A$-algebras, then the induced decomposition $A' = A'_1 \times A'_2$ will give the desired ( indeed, since $b$ is a unit in $B'_1$ (details omitted), we will have $B'_1 \cong A'_1$, so that $A'_1$ will be integral over $A$.

Choose a finite $A$-subalgebra $B'' \subset B'$ containing $b$ (observe that any finitely generated $A$-subalgebra of $B'$ is finite over $A$). After enlarging $B''$ we may assume $b$ maps to an idempotent in $B''/IB''$ producing
\begin{equation} \label{more-algebra-equation-again-dec-mod-I} B''/IB'' \cong C''_1 \times C''_2 \end{equation}

Since $B'_ b \cong A'_ b$ we see that $B'_ b$ is of finite type over $A$. Say $B'_ b$ is generated by $b_1/b^ n, \ldots , b_ t/b^ n$ over $A$ and enlarge $B''$ so that $b_1, \ldots , b_ t \in B''$. Then $B''_ b \to B'_ b$ is surjective as well as injective, hence an isomorphism. In particular, we see that $C''_1 = A/I$! Therefore $A/I \to C''_1$ is an isomorphism, in particular surjective. By Lemma 15.10.4 we can find an $f(T) \in A[T]$ of the form

\[ f(T) = T^ n(T - 1) + a_ n T^ n + \ldots + a_1 T + a_0 \]

with $a_ n, \ldots , a_0 \in I$ and $n \ge 1$ such that $f(b) = 0$. In particular, we find that $B'$ is a $A[T]/(f)$-algebra. By (5) we deduce there is a root $a \in 1 + I$ of $f$. This produces a product decomposition $A[T]/(f) = A[T]/(T - a) \times D$ compatible with the splitting ( of $B'/IB'$. The induced splitting of $B'$ is then a desired ( $\square$

[1] Here we use determinants of endomorphisms of finite locally free modules; if the module is free the determinant is defined as the determinant of the corresponding matrix and in general one uses Algebra, Lemma 10.23.2 to glue.

Comments (4)

Comment #2171 by JuanPablo on

There is a couple of points that could be clearer. I don't remember a definition of determinant for endomorphisms of locally finite modules of constant rank here, so maybe it could be said that the definition comes from lemma 10.22.1 (tag 00EJ).

"we see that satisfies a monic polynomial whose reduction modulo factors as where generate the unit ideal in ."

Here lemma 15.8.5 (tag 09XG) is used, but it seems in this case the lemma applies to an element corresponding to not so instead applying lemma 15.8.5 to .

"By construction we have and "

I did not understand why this is so. I have an argument, but maybe I'm missing something simpler. This is because the idempotent corresponding to in is where in , so reducing modulo you get modulo . Now using that is an idempotent modulo , you get modulo . And using that, modulo , , obtain from evaluating in and multiplying by that (modulo ). So the idempotent corresponding to is as required.

"It follows that is integral as well as \'etale, hence finite locally free. However, thus has rank as an -module along . Since is contained in the Jacobson radical of we conclude that has rank everywhere and it follows that is an isomorphism."

I don't understand why integral and étale implies finite locally free, or why locally finite of rank 1 implies that is an isomorphism. I could see that is an isomorphism using Nakayama's lemma 10.19.1 (tag 00DV) to see that is onto, then lemma 10.141.9 (tag 00U8) to see that is projection into a factor, and then lemma 15.8.4 (tag 09XF) to see from that the idempotent in associated to is (so is an isomorphism).

Comment #2200 by on

OK, this proof is a bit of a mess. I've fixed the error of applying Lemma 15.9.9 wrong and I've fixed the other arguments in a way a bit more in line with the original intent of the proof. I wanted to use the helper lemma 15.9.9, but maybe there is a more straightforward proof?

In any case, many thanks! Edits can be found here but should be online soon.

Comment #3259 by Kestutis Cesnavicius on

In the last paragraph of the proof "maps isomorphically to " should be "maps isomorphically to ".

There are also:

  • 2 comment(s) on Section 15.11: Henselian pairs

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