The Stacks project

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10.148 Henselian local rings

In this section we discuss a bit the notion of a henselian local ring. Let $(R, \mathfrak m, \kappa )$ be a local ring. For $a \in R$ we denote $\overline{a}$ the image of $a$ in $\kappa $. For a polynomial $f \in R[T]$ we often denote $\overline{f}$ the image of $f$ in $\kappa [T]$. Given a polynomial $f \in R[T]$ we denote $f'$ the derivative of $f$ with respect to $T$. Note that $\overline{f}' = \overline{f'}$.

Definition 10.148.1. Let $(R, \mathfrak m, \kappa )$ be a local ring.

  1. We say $R$ is henselian if for every monic $f \in R[T]$ and every root $a_0 \in \kappa $ of $\overline{f}$ such that $\overline{f'}(a_0) \not= 0$ there exists an $a \in R$ such that $f(a) = 0$ and $a_0 = \overline{a}$.

  2. We say $R$ is strictly henselian if $R$ is henselian and its residue field is separably algebraically closed.

Note that the condition $\overline{f'}(a_0) \not= 0$ is equivalent to the condition that $a_0$ is a simple root of the polynomial $\overline{f}$. In fact, it implies that the lift $a \in R$, if it exists, is unique.

Lemma 10.148.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $f \in R[T]$. Let $a, b \in R$ such that $f(a) = f(b) = 0$, $a = b \bmod \mathfrak m$, and $f'(a) \not\in \mathfrak m$. Then $a = b$.

Proof. Write $f(x + y) - f(x) = f'(x)y + g(x, y) y^2$ in $R[x, y]$ (this is possible as one sees by expanding $f(x + y)$; details omitted). Then we see that $0 = f(b) - f(a) = f(a + (b - a)) - f(a) = f'(a)(b - a) + c (b - a)^2$ for some $c \in R$. By assumption $f'(a)$ is a unit in $R$. Hence $(b - a)(1 + f'(a)^{-1}c(b - a)) = 0$. By assumption $b - a \in \mathfrak m$, hence $1 + f'(a)^{-1}c(b - a)$ is a unit in $R$. Hence $b - a = 0$ in $R$. $\square$

Here is the characterization of henselian local rings.

slogan

Lemma 10.148.3. Let $(R, \mathfrak m, \kappa )$ be a local ring. The following are equivalent

  1. $R$ is henselian,

  2. for every $f \in R[T]$ and every root $a_0 \in \kappa $ of $\overline{f}$ such that $\overline{f'}(a_0) \not= 0$ there exists an $a \in R$ such that $f(a) = 0$ and $a_0 = \overline{a}$,

  3. for any monic $f \in R[T]$ and any factorization $\overline{f} = g_0 h_0$ with $\gcd (g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \overline{g}$ and $h_0 = \overline{h}$,

  4. for any monic $f \in R[T]$ and any factorization $\overline{f} = g_0 h_0$ with $\gcd (g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \overline{g}$ and $h_0 = \overline{h}$ and moreover $\deg _ T(g) = \deg _ T(g_0)$,

  5. for any $f \in R[T]$ and any factorization $\overline{f} = g_0 h_0$ with $\gcd (g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \overline{g}$ and $h_0 = \overline{h}$,

  6. for any $f \in R[T]$ and any factorization $\overline{f} = g_0 h_0$ with $\gcd (g_0, h_0) = 1$ there exists a factorization $f = gh$ in $R[T]$ such that $g_0 = \overline{g}$ and $h_0 = \overline{h}$ and moreover $\deg _ T(g) = \deg _ T(g_0)$,

  7. for any étale ring map $R \to S$ and prime $\mathfrak q$ of $S$ lying over $\mathfrak m$ with $\kappa = \kappa (\mathfrak q)$ there exists a section $\tau : S \to R$ of $R \to S$,

  8. for any étale ring map $R \to S$ and prime $\mathfrak q$ of $S$ lying over $\mathfrak m$ with $\kappa = \kappa (\mathfrak q)$ there exists a section $\tau : S \to R$ of $R \to S$ with $\mathfrak q = \tau ^{-1}(\mathfrak m)$,

  9. any finite $R$-algebra is a product of local rings,

  10. any finite $R$-algebra is a finite product of local rings,

  11. any finite type $R$-algebra $S$ can be written as $A \times B$ with $R \to A$ finite and $R \to B$ not quasi-finite at any prime lying over $\mathfrak m$,

  12. any finite type $R$-algebra $S$ can be written as $A \times B$ with $R \to A$ finite such that each irreducible component of $\mathop{\mathrm{Spec}}(B \otimes _ R \kappa )$ has dimension $\geq 1$, and

  13. any quasi-finite $R$-algebra $S$ can be written as $S = A \times B$ with $R \to A$ finite such that $B \otimes _ R \kappa = 0$.

Proof. Here is a list of the easier implications:

  1. 2$\Rightarrow $1 because in (2) we consider all polynomials and in (1) only monic ones,

  2. 5$\Rightarrow $3 because in (5) we consider all polynomials and in (3) only monic ones,

  3. 6$\Rightarrow $4 because in (6) we consider all polynomials and in (4) only monic ones,

  4. 4$\Rightarrow $3 is obvious,

  5. 6$\Rightarrow $5 is obvious,

  6. 8$\Rightarrow $7 is obvious,

  7. 10$\Rightarrow $9 is obvious,

  8. 11$\Leftrightarrow $12 by definition of being quasi-finite at a prime,

  9. 11$\Rightarrow $13 by definition of being quasi-finite,

Proof of 1$\Rightarrow $8. Assume (1). Let $R \to S$ be étale, and let $\mathfrak q \subset S$ be a prime ideal such that $\kappa (\mathfrak q) \cong \kappa $. By Proposition 10.141.16 we can find a $g \in S$, $g \not\in \mathfrak q$ such that $R \to S_ g$ is standard étale. After replacing $S$ by $S_ g$ we may assume that $S = R[t]_ g/(f)$ is standard étale. Since the prime $\mathfrak q$ has residue field $\kappa $ it corresponds to a root $a_0$ of $\overline{f}$ which is not a root of $\overline{g}$. By definition of a standard étale algebra this also means that $\overline{f'}(a_0) \not= 0$. Since also $f$ is monic by definition of a standard étale algebra again we may use that $R$ is henselian to conclude that there exists an $a \in R$ with $a_0 = \overline{a}$ such that $f(a) = 0$. This implies that $g(a)$ is a unit of $R$ and we obtain the desired map $\tau : S = R[t]_ g/(f) \to R$ by the rule $t \mapsto a$. By construction $\tau ^{-1}(\mathfrak q) = \mathfrak m$. This proves (8) holds.

Proof of 7$\Rightarrow $8. (This is really unimportant and should be skipped.) Assume (7) holds and assume $R \to S$ is étale. Let $\mathfrak q_1, \ldots , \mathfrak q_ r$ be the other primes of $S$ lying over $\mathfrak m$. Then we can find a $g \in S$, $g \not\in \mathfrak q$ and $g \in \mathfrak q_ i$ for $i = 1, \ldots , r$. Namely, we can argue that $\bigcap _{i=1}^{r} \mathfrak {q}_{i} \not\subset \mathfrak {q}$ since otherwise $\mathfrak {q}_{i} \subset \mathfrak {q}$ for some $i$, but this cannot happen as the fiber of an étale morphism is discrete (use Lemma 10.141.4 for example). Apply (7) to the étale ring map $R \to S_ g$ and the prime $\mathfrak qS_ g$. This gives a section $\tau _ g : S_ g \to R$ such that the composition $\tau : S \to S_ g \to R$ has the property $\tau ^{-1}(\mathfrak q) = \mathfrak m$. Minor details omitted.

Proof of 8$\Rightarrow $11. Assume (8) and let $R \to S$ be a finite type ring map. Apply Lemma 10.141.22. We find an étale ring map $R \to R'$ and a prime $\mathfrak m' \subset R'$ lying over $\mathfrak m$ with $\kappa = \kappa (\mathfrak m')$ such that $R' \otimes _ R S = A' \times B'$ with $A'$ finite over $R'$ and $B'$ not quasi-finite over $R'$ at any prime lying over $\mathfrak m'$. Apply (8) to get a section $\tau : R' \to R$ with $\mathfrak m = \tau ^{-1}(\mathfrak m')$. Then use that

\[ S = (S \otimes _ R R') \otimes _{R', \tau } R = (A' \times B') \otimes _{R', \tau } R = (A' \otimes _{R', \tau } R) \times (B' \otimes _{R', \tau } R) \]

which gives a decomposition as in (11).

Proof of 8$\Rightarrow $10. Assume (8) and let $R \to S$ be a finite ring map. Apply Lemma 10.141.22. We find an étale ring map $R \to R'$ and a prime $\mathfrak m' \subset R'$ lying over $\mathfrak m$ with $\kappa = \kappa (\mathfrak m')$ such that $R' \otimes _ R S = A'_1 \times \ldots \times A'_ n \times B'$ with $A'_ i$ finite over $R'$ having exactly one prime over $\mathfrak m'$ and $B'$ not quasi-finite over $R'$ at any prime lying over $\mathfrak m'$. Apply (8) to get a section $\tau : R' \to R$ with $\mathfrak m = \tau ^{-1}(\mathfrak m')$. Then we obtain

\begin{align*} S & = (S \otimes _ R R') \otimes _{R', \tau } R \\ & = (A'_1 \times \ldots \times A'_ n \times B') \otimes _{R', \tau } R \\ & = (A'_1 \otimes _{R', \tau } R) \times \ldots \times (A'_1 \otimes _{R', \tau } R) \times (B' \otimes _{R', \tau } R) \\ & = A_1 \times \ldots \times A_ n \times B \end{align*}

The factor $B$ is finite over $R$ but $R \to B$ is not quasi-finite at any prime lying over $\mathfrak m$. Hence $B = 0$. The factors $A_ i$ are finite $R$-algebras having exactly one prime lying over $\mathfrak m$, hence they are local rings. This proves that $S$ is a finite product of local rings.

Proof of 9$\Rightarrow $10. This holds because if $S$ is finite over the local ring $R$, then it has at most finitely many maximal ideals. Namely, by going up for $R \to S$ the maximal ideals of $S$ all lie over $\mathfrak m$, and $S/\mathfrak mS$ is Artinian hence has finitely many primes.

Proof of 10$\Rightarrow $1. Assume (10). Let $f \in R[T]$ be a monic polynomial and $a_0 \in \kappa $ a simple root of $\overline{f}$. Then $S = R[T]/(f)$ is a finite $R$-algebra. Applying (10) we get $S = A_1 \times \ldots \times A_ r$ is a finite product of local $R$-algebras. In particular we see that $S/\mathfrak mS = \prod A_ i/\mathfrak mA_ i$ is the decomposition of $\kappa [T]/(\overline{f})$ as a product of local rings. This means that one of the factors, say $A_1/\mathfrak mA_1$ is the quotient $\kappa [T]/(\overline{f}) \to \kappa [T]/(T - a_0)$. Since $A_1$ is a summand of the finite free $R$-module $S$ it is a finite free $R$-module itself. As $A_1/\mathfrak mA_1$ is a $\kappa $-vector space of dimension 1 we see that $A_1 \cong R$ as an $R$-module. Clearly this means that $R \to A_1$ is an isomorphism. Let $a \in R$ be the image of $T$ under the map $R[T] \to S \to A_1 \to R$. Then $f(a) = 0$ and $\overline{a} = a_0$ as desired.

Proof of 13$\Rightarrow $1. Assume (13). Let $f \in R[T]$ be a monic polynomial and $a_0 \in \kappa $ a simple root of $\overline{f}$. Then $S_1 = R[T]/(f)$ is a finite $R$-algebra. Let $g \in R[T]$ be any element such that $\overline{g} = \overline{f}/(T - a_0)$. Then $S = (S_1)_ g$ is a quasi-finite $R$-algebra such that $S \otimes _ R \kappa \cong \kappa [T]_{\overline{g}}/(\overline{f}) \cong \kappa [T]/(T - a_0) \cong \kappa $. Applying (13) to $S$ we get $S = A \times B$ with $A$ finite over $R$ and $B \otimes _ R \kappa = 0$. In particular we see that $\kappa \cong S/\mathfrak mS = A/\mathfrak mA$. Since $A$ is a summand of the flat $R$-algebra $S$ we see that it is finite flat, hence free over $R$. As $A/\mathfrak mA$ is a $\kappa $-vector space of dimension 1 we see that $A \cong R$ as an $R$-module. Clearly this means that $R \to A$ is an isomorphism. Let $a \in R$ be the image of $T$ under the map $R[T] \to S \to A \to R$. Then $f(a) = 0$ and $\overline{a} = a_0$ as desired.

Proof of 8$\Rightarrow $2. Assume (8). Let $f \in R[T]$ be any polynomial and let $a_0 \in \kappa $ be a simple root. Then the algebra $S = R[T]_{f'}/(f)$ is étale over $R$. Let $\mathfrak q \subset S$ be the prime generated by $\mathfrak m$ and $T - b$ where $b \in R$ is any element such that $\overline{b} = a_0$. Apply (8) to $S$ and $\mathfrak q$ to get $\tau : S \to R$. Then the image $\tau (T) = a \in R$ works in (2).

At this point we see that (1), (2), (7), (8), (9), (10), (11), (12), (13) are all equivalent. The weakest assertion of (3), (4), (5) and (6) is (3) and the strongest is (6). Hence we still have to prove that (3) implies (1) and (1) implies (6).

Proof of 3$\Rightarrow $1. Assume (3). Let $f \in R[T]$ be monic and let $a_0 \in \kappa $ be a simple root of $\overline{f}$. This gives a factorization $\overline{f} = (T - a_0)h_0$ with $h_0(a_0) \not= 0$, so $\gcd (T - a_0, h_0) = 1$. Apply (3) to get a factorization $f = gh$ with $\overline{g} = T - a_0$ and $\overline{h} = h_0$. Set $S = R[T]/(f)$ which is a finite free $R$-algebra. We will write $g$, $h$ also for the images of $g$ and $h$ in $S$. Then $gS + hS = S$ by Nakayama's Lemma 10.19.1 as the equality holds modulo $\mathfrak m$. Since $gh = f = 0$ in $S$ this also implies that $gS \cap hS = 0$. Hence by the Chinese Remainder theorem we obtain $S = S/(g) \times S/(h)$. This implies that $A = S/(g)$ is a summand of a finite free $R$-module, hence finite free. Moreover, the rank of $A$ is $1$ as $A/\mathfrak mA = \kappa [T]/(T - a_0)$. Thus the map $R \to A$ is an isomorphism. Setting $a \in R$ equal to the image of $T$ under the maps $R[T] \to S \to A \to R$ gives an element of $R$ with $f(a) = 0$ and $\overline{a} = a_0$.

Proof of 1$\Rightarrow $6. Assume (1) or equivalently all of (1), (2), (7), (8), (9), (10), (11), (12), (13). Let $f \in R[T]$ be a polynomial. Suppose that $\overline{f} = g_0h_0$ is a factorization with $\gcd (g_0, h_0) = 1$. We may and do assume that $g_0$ is monic. Consider $S = R[T]/(f)$. Because we have the factorization we see that the coefficients of $f$ generate the unit ideal in $R$. This implies that $S$ has finite fibres over $R$, hence is quasi-finite over $R$. It also implies that $S$ is flat over $R$ by Lemma 10.127.5. Combining (13) and (10) we may write $S = A_1 \times \ldots \times A_ n \times B$ where each $A_ i$ is local and finite over $R$, and $B \otimes _ R \kappa = 0$. After reordering the factors $A_1, \ldots , A_ n$ we may assume that

\[ \kappa [T]/(g_0) = A_1/\mathfrak m A_1 \times \ldots \times A_ r/\mathfrak mA_ r, \ \kappa [T]/(h_0) = A_{r + 1}/\mathfrak mA_{r + 1} \times \ldots \times A_ n/\mathfrak mA_ n \]

as quotients of $\kappa [T]$. The finite flat $R$-algebra $A = A_1 \times \ldots \times A_ r$ is free as an $R$-module, see Lemma 10.77.4. Its rank is $\deg _ T(g_0)$. Let $g \in R[T]$ be the characteristic polynomial of the $R$-linear operator $T : A \to A$. Then $g$ is a monic polynomial of degree $\deg _ T(g) = \deg _ T(g_0)$ and moreover $\overline{g} = g_0$. By Cayley-Hamilton (Lemma 10.15.1) we see that $g(T_ A) = 0$ where $T_ A$ indicates the image of $T$ in $A$. Hence we obtain a well defined surjective map $R[T]/(g) \to A$ which is an isomorphism by Nakayama's Lemma 10.19.1. The map $R[T] \to A$ factors through $R[T]/(f)$ by construction hence we may write $f = gh$ for some $h$. This finishes the proof. $\square$

Lemma 10.148.4. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring.

  1. If $R \subset S$ is a finite ring extension then $S$ is a finite product of henselian local rings.

  2. If $R \subset S$ is a finite local homomorphism of local rings, then $S$ is a henselian local ring.

  3. If $R \to S$ is a finite type ring map, and $\mathfrak q$ is a prime of $S$ lying over $\mathfrak m$ at which $R \to S$ is quasi-finite, then $S_{\mathfrak q}$ is henselian.

  4. If $R \to S$ is quasi-finite then $S_{\mathfrak q}$ is henselian for every prime $\mathfrak q$ lying over $\mathfrak m$.

Proof. Part (2) implies part (1) since $S$ as in part (1) is a finite product of its localizations at the primes lying over $\mathfrak m$. Part (2) follows from Lemma 10.148.3 part (10) since any finite $S$-algebra is also a finite $R$-algebra. If $R \to S$ and $\mathfrak q$ are as in (3), then $S_{\mathfrak q}$ is a local ring of a finite $R$-algebra by Lemma 10.148.3 part (11). Hence (3) follows from (1). Part (4) follows from part (3). $\square$

Lemma 10.148.5. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring. Any finite type $R$-algebra $S$ can be written as $S = A_1 \times \ldots \times A_ n \times B$ with $A_ i$ local and finite over $R$ and $R \to B$ not quasi-finite at any prime of $B$ lying over $\mathfrak m$.

Proof. This is a combination of parts (11) and (10) of Lemma 10.148.3. $\square$

Lemma 10.148.6. Let $(R, \mathfrak m, \kappa )$ be a strictly henselian local ring. Any finite type $R$-algebra $S$ can be written as $S = A_1 \times \ldots \times A_ n \times B$ with $A_ i$ local and finite over $R$ and $\kappa \subset \kappa (\mathfrak m_{A_ i})$ finite purely inseparable and $R \to B$ not quasi-finite at any prime of $B$ lying over $\mathfrak m$.

Proof. First write $S = A_1 \times \ldots \times A_ n \times B$ as in Lemma 10.148.5. The field extension $\kappa \subset \kappa (\mathfrak m_{A_ i})$ is finite and $\kappa $ is separably algebraically closed, hence it is finite purely inseparable. $\square$

Lemma 10.148.7. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring. The category of finite étale ring extensions $R \to S$ is equivalent to the category of finite étale algebras $\kappa \to \overline{S}$ via the functor $S \mapsto S/\mathfrak mS$.

Proof. Denote $\mathcal{C} \to \mathcal{D}$ the functor of categories of the statement. Suppose that $R \to S$ is finite étale. Then we may write

\[ S = A_1 \times \ldots \times A_ n \]

with $A_ i$ local and finite étale over $S$, use either Lemma 10.148.5 or Lemma 10.148.3 part (10). In particular $A_ i/\mathfrak mA_ i$ is a finite separable field extension of $\kappa $, see Lemma 10.141.5. Thus we see that every object of $\mathcal{C}$ and $\mathcal{D}$ decomposes canonically into irreducible pieces which correspond via the given functor. Next, suppose that $S_1$, $S_2$ are finite étale over $R$ such that $\kappa _1 = S_1/\mathfrak mS_1$ and $\kappa _2 = S_2/\mathfrak mS_2$ are fields (finite separable over $\kappa $). Then $S_1 \otimes _ R S_2$ is finite étale over $R$ and we may write

\[ S_1 \otimes _ R S_2 = A_1 \times \ldots \times A_ n \]

as before. Then we see that $\mathop{\mathrm{Hom}}\nolimits _ R(S_1, S_2)$ is identified with the set of indices $i \in \{ 1, \ldots , n\} $ such that $S_2 \to A_ i$ is an isomorphism. To see this use that given any $R$-algebra map $\varphi : S_1 \to S_2$ the map $\varphi \times 1 : S_1 \otimes _ R S_2 \to S_2$ is surjective, and hence is equal to projection onto one of the factors $A_ i$. But in exactly the same way we see that $\mathop{\mathrm{Hom}}\nolimits _\kappa (\kappa _1, \kappa _2)$ is identified with the set of indices $i \in \{ 1, \ldots , n\} $ such that $\kappa _2 \to A_ i/\mathfrak mA_ i$ is an isomorphism. By the discussion above these sets of indices match, and we conclude that our functor is fully faithful. Finally, let $\kappa \subset \kappa '$ be a finite separable field extension. By Lemma 10.141.15 there exists an étale ring map $R \to S$ and a prime $\mathfrak q$ of $S$ lying over $\mathfrak m$ such that $\kappa \subset \kappa (\mathfrak q)$ is isomorphic to the given extension. By part (1) we may write $S = A_1 \times \ldots \times A_ n \times B$. Since $R \to S$ is quasi-finite we see that there exists no prime of $B$ over $\mathfrak m$. Hence $S_{\mathfrak q}$ is equal to $A_ i$ for some $i$. Hence $R \to A_ i$ is finite étale and produces the given residue field extension. Thus the functor is essentially surjective and we win. $\square$

Lemma 10.148.8. Let $(R, \mathfrak m, \kappa )$ be a strictly henselian local ring. Let $R \to S$ be an unramified ring map. Then

\[ S = A_1 \times \ldots \times A_ n \times B \]

with each $R \to A_ i$ surjective and no prime of $B$ lying over $\mathfrak m$.

Proof. First write $S = A_1 \times \ldots \times A_ n \times B$ as in Lemma 10.148.5. Now we see that $R \to A_ i$ is finite unramified and $A_ i$ local. Hence the maximal ideal of $A_ i$ is $\mathfrak mA_ i$ and its residue field $A_ i / \mathfrak m A_ i$ is a finite separable extension of $\kappa $, see Lemma 10.147.5. However, the condition that $R$ is strictly henselian means that $\kappa $ is separably algebraically closed, so $\kappa = A_ i / \mathfrak m A_ i$. By Nakayama's Lemma 10.19.1 we conclude that $R \to A_ i$ is surjective as desired. $\square$

Lemma 10.148.9. Let $(R, \mathfrak m, \kappa )$ be a complete local ring, see Definition 10.154.1. Then $R$ is henselian.

Proof. Let $f \in R[T]$ be monic. Denote $f_ n \in R/\mathfrak m^{n + 1}[T]$ the image. Denote $f'_ n$ the derivative of $f_ n$ with respect to $T$. Let $a_0 \in \kappa $ be a simple root of $f_0$. We lift this to a solution of $f$ over $R$ inductively as follows: Suppose given $a_ n \in R/\mathfrak m^{n + 1}$ such that $a_ n \bmod \mathfrak m = a_0$ and $f_ n(a_ n) = 0$. Pick any element $b \in R/\mathfrak m^{n + 2}$ such that $a_ n = b \bmod \mathfrak m^{n + 1}$. Then $f_{n + 1}(b) \in \mathfrak m^{n + 1}/\mathfrak m^{n + 2}$. Set

\[ a_{n + 1} = b - f_{n + 1}(b)/f'_{n + 1}(b) \]

(Newton's method). This makes sense as $f'_{n + 1}(b) \in R/\mathfrak m^{n + 1}$ is invertible by the condition on $a_0$. Then we compute $f_{n + 1}(a_{n + 1}) = f_{n + 1}(b) - f_{n + 1}(b) = 0$ in $R/\mathfrak m^{n + 2}$. Since the system of elements $a_ n \in R/\mathfrak m^{n + 1}$ so constructed is compatible we get an element $a \in \mathop{\mathrm{lim}}\nolimits R/\mathfrak m^ n = R$ (here we use that $R$ is complete). Moreover, $f(a) = 0$ since it maps to zero in each $R/\mathfrak m^ n$. Finally $\overline{a} = a_0$ and we win. $\square$

Proof. Let $R \to S$ be a finite ring map. By Lemma 10.148.3 it suffices to show that $S$ is a product of local rings. By Lemma 10.35.21 $S$ has finitely many primes $\mathfrak m_1, \ldots , \mathfrak m_ r$ which all lie over $\mathfrak m$. There are no inclusions among these primes, see Lemma 10.35.20, hence they are all maximal. Every element of $\mathfrak m_1 \cap \ldots \cap \mathfrak m_ r$ is nilpotent by Lemma 10.16.2. It follows $S$ is the product of the localizations of $S$ at the primes $\mathfrak m_ i$ by Lemma 10.52.5. $\square$

The following lemma will be the key to the uniqueness and functorial properties of henselization and strict henselization.

Lemma 10.148.11. Let $R \to S$ be a ring map with $S$ henselian local. Given

  1. an étale ring map $R \to A$,

  2. a prime $\mathfrak q$ of $A$ lying over $\mathfrak p = R \cap \mathfrak m_ S$,

  3. a $\kappa (\mathfrak p)$-algebra map $\tau : \kappa (\mathfrak q) \to S/\mathfrak m_ S$,

then there exists a unique homomorphism of $R$-algebras $f : A \to S$ such that $\mathfrak q = f^{-1}(\mathfrak m_ S)$ and $f \bmod \mathfrak q = \tau $.

Proof. Consider $A \otimes _ R S$. This is an étale algebra over $S$, see Lemma 10.141.3. Moreover, the kernel

\[ \mathfrak q' = \mathop{\mathrm{Ker}}(A \otimes _ R S \to \kappa (\mathfrak q) \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak m_ S) \to \kappa (\mathfrak m_ S)) \]

of the map using the map given in (3) is a prime ideal lying over $\mathfrak m_ S$ with residue field equal to the residue field of $S$. Hence by Lemma 10.148.3 there exists a unique splitting $\tau : A \otimes _ R S \to S$ with $\tau ^{-1}(\mathfrak m_ S) = \mathfrak q'$. Set $f$ equal to the composition $A \to A \otimes _ R S \to S$. $\square$

Lemma 10.148.12. Let $\varphi : R \to S$ be a local homomorphism of strictly henselian local rings. Let $P_1, \ldots , P_ n \in R[x_1, \ldots , x_ n]$ be polynomials such that $R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n)$ is étale over $R$. Then the map

\[ R^ n \longrightarrow S^ n, \quad (h_1, \ldots , h_ n) \longmapsto (\varphi (h_1), \ldots , \varphi (h_ n)) \]

induces a bijection between

\[ \{ (r_1, \ldots , r_ n) \in R^ n \mid P_ i(r_1, \ldots , r_ n) = 0, \ i = 1, \ldots , n \} \]

and

\[ \{ (s_1, \ldots , s_ n) \in S^ n \mid P'_ i(s_1, \ldots , s_ n) = 0, \ i = 1, \ldots , n \} \]

where $P'_ i \in S[x_1, \ldots , x_ n]$ are the images of the $P_ i$ under $\varphi $.

Proof. The first solution set is canonically isomorphic to the set

\[ \mathop{\mathrm{Hom}}\nolimits _ R(R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n), R). \]

As $R$ is henselian the map $R \to R/\mathfrak m_ R$ induces a bijection between this set and the set of solutions in the residue field $R/\mathfrak m_ R$, see Lemma 10.148.3. The same is true for $S$. Now since $R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n)$ is étale over $R$ and $R/\mathfrak m_ R$ is separably algebraically closed we see that $R/\mathfrak m_ R[x_1, \ldots , x_ n]/(\overline{P_1}, \ldots , \overline{P_ n})$ is a finite product of copies of $R/\mathfrak m_ R$. Hence the tensor product

\[ R/\mathfrak m_ R[x_1, \ldots , x_ n]/(\overline{P_1}, \ldots , \overline{P_ n}) \otimes _{R/\mathfrak m_ R} S/\mathfrak m_ S = S/\mathfrak m_ S[x_1, \ldots , x_ n]/(\overline{P_1'}, \ldots , \overline{P_ n'}) \]

is also a finite product of copies of $S/\mathfrak m_ S$ with the same index set. This proves the lemma. $\square$

Lemma 10.148.13. Let $R$ be a henselian local ring. Any countably generated Mittag-Leffler module over $R$ is a direct sum of finitely presented $R$-modules.

Proof. Let $M$ be a countably generated and Mittag-Leffler $R$-module. We claim that for any element $x \in M$ there exists a direct sum decomposition $M = N \oplus K$ with $x \in N$, the module $N$ finitely presented, and $K$ Mittag-Leffler.

Suppose the claim is true. Choose generators $x_1, x_2, x_3, \ldots $ of $M$. By the claim we can inductively find direct sum decompositions

\[ M = N_1 \oplus N_2 \oplus \ldots \oplus N_ n \oplus K_ n \]

with $N_ i$ finitely presented, $x_1, \ldots , x_ n \in N_1 \oplus \ldots \oplus N_ n$, and $K_ n$ Mittag-Leffler. Repeating ad infinitum we see that $M = \bigoplus N_ i$.

We still have to prove the claim. Let $x \in M$. By Lemma 10.91.2 there exists an endomorphism $\alpha : M \to M$ such that $\alpha $ factors through a finitely presented module, and $\alpha (x) = x$. Say $\alpha $ factors as

\[ \xymatrix{ M \ar[r]^\pi & P \ar[r]^ i & M } \]

Set $a = \pi \circ \alpha \circ i : P \to P$, so $i \circ a \circ \pi = \alpha ^3$. By Lemma 10.15.2 there exists a monic polynomial $P \in R[T]$ such that $P(a) = 0$. Note that this implies formally that $\alpha ^2 P(\alpha ) = 0$. Hence we may think of $M$ as a module over $R[T]/(T^2P)$. Assume that $x \not= 0$. Then $\alpha (x) = x$ implies that $0 = \alpha ^2P(\alpha )x = P(1)x$ hence $P(1) = 0$ in $R/I$ where $I = \{ r \in R \mid rx = 0\} $ is the annihilator of $x$. As $x \not= 0$ we see $I \subset \mathfrak m_ R$, hence $1$ is a root of $\overline{P} = P \bmod \mathfrak m_ R \in R/\mathfrak m_ R[T]$. As $R$ is henselian we can find a factorization

\[ T^2P = (T^2 Q_1) Q_2 \]

for some $Q_1, Q_2 \in R[T]$ with $Q_2 = (T - 1)^ e \bmod \mathfrak m_ R R[T]$ and $Q_1(1) \not= 0 \bmod \mathfrak m_ R$, see Lemma 10.148.3. Let $N = \mathop{\mathrm{Im}}(\alpha ^2Q_1(\alpha ) : M \to M)$ and $K = \mathop{\mathrm{Im}}(Q_2(\alpha ) : M \to M)$. As $T^2Q_1$ and $Q_2$ generate the unit ideal of $R[T]$ we get a direct sum decomposition $M = N \oplus K$. Moreover, $Q_2$ acts as zero on $N$ and $T^2Q_1$ acts as zero on $K$. Note that $N$ is a quotient of $P$ hence is finitely generated. Also $x \in N$ because $\alpha ^2Q_1(\alpha )x = Q_1(1)x$ and $Q_1(1)$ is a unit in $R$. By Lemma 10.88.10 the modules $N$ and $K$ are Mittag-Leffler. Finally, the finitely generated module $N$ is finitely presented as a finitely generated Mittag-Leffler module is finitely presented, see Example 10.90.1 part (1). $\square$


Comments (2)

Comment #341 by on

It should be nice to have the Multivariate Hensel Lemma and some ideas about related algorithms

e.g. if K is a field and A=K[X1,...,Xn]_m where m = (X1,...,Xn) then the elements of A=K[[X1,………,Xn]] which are algebraic over Quot(A)=K(X1,………,Xn) give a henselisation of A

(so there is an explicit isomorphism with the "standard" construction as colimit of étale algebras)

Best

Comment #342 by on

For some reason I ended up strongly disliking the material on Henselian local rings as I was writing it. Probably because I did not start of with the more general notion of a Henselian couple. So, yes, I think there needs to be a lot more material in this section. In fact, we could have a whole chapter dedicated to this kind of thing as well as Weierstrass preparation, etc, etc.


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