10.153 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 10.153.1. Let (R, \mathfrak m, \kappa ) be a local ring.
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}.
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 10.153.2. 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.
Lemma 10.153.3.slogan Let (R, \mathfrak m, \kappa ) be a local ring. The following are equivalent
R is henselian,
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},
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},
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),
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},
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),
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 retraction \tau : S \to R of R \to S,
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 unique retraction \tau : S \to R of R \to S such that \mathfrak q = \tau ^{-1}(\mathfrak m),
any finite R-algebra is a product of local rings,
any finite R-algebra is a finite product of local rings,
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,
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
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:
2\Rightarrow 1 because in (2) we consider all polynomials and in (1) only monic ones,
5\Rightarrow 3 because in (5) we consider all polynomials and in (3) only monic ones,
6\Rightarrow 4 because in (6) we consider all polynomials and in (4) only monic ones,
4\Rightarrow 3 is obvious,
6\Rightarrow 5 is obvious,
8\Rightarrow 7 is obvious,
10\Rightarrow 9 is obvious,
11\Leftrightarrow 12 by definition of being quasi-finite at a prime,
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 10.144.4 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 (details omitted). 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 m) = \mathfrak q. By Lemma 10.153.2 the map \tau is unique. 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 10.143.4 for example). Apply (7) to the étale ring map R \to S_ g and the prime \mathfrak qS_ g. This gives a retraction \tau _ g : S_ g \to R such that the composition \tau : S \to S_ g \to R has the property \tau ^{-1}(\mathfrak m) = \mathfrak q. Details omitted.
Proof of 8\Rightarrow 11. Assume (8) and let R \to S be a finite type ring map. Apply Lemma 10.145.3. 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 retraction \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 10.145.3. 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 retraction \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 10.20.1 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 10.128.5. 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 10.78.5. 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 10.16.1) 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 10.20.1. 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 10.153.4. Let (R, \mathfrak m, \kappa ) be a henselian local ring.
If R \to S is a finite ring map then S is a finite product of henselian local rings each finite over R.
If R \to S is a finite ring map and S is local, then S is a henselian local ring and R \to S is a (finite) local ring map.
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 and finite over R.
If R \to S is quasi-finite then S_{\mathfrak q} is henselian and finite over R 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 by Lemma 10.153.3 part (10). Part (2) also follows from Lemma 10.153.3 part (10) since any finite S-algebra is also a finite R-algebra (of course any finite ring map between local rings is local).
Let R \to S and \mathfrak q be as in (3). Write S = A \times B with A finite over R and B not quasi-finite over R at any prime lying over \mathfrak m, see Lemma 10.153.3 part (11). Hence S_\mathfrak q is a localization of A at a maximal ideal and we deduce (3) from (1). Part (4) follows from part (3).
\square
Lemma 10.153.5. 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 10.153.3.
\square
Lemma 10.153.6. 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 10.153.5. The field extension \kappa (\mathfrak m_{A_ i})/\kappa is finite and \kappa is separably algebraically closed, hence it is finite purely inseparable.
\square
Lemma 10.153.7. 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 10.153.5 or Lemma 10.153.3 part (10). In particular A_ i/\mathfrak mA_ i is a finite separable field extension of \kappa , see Lemma 10.143.5. 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 '/\kappa be a finite separable field extension. By Lemma 10.144.3 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 10.153.8. 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 10.153.5. 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 10.151.5. 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 10.20.1 we conclude that R \to A_ i is surjective as desired.
\square
Lemma 10.153.9.slogan Let (R, \mathfrak m, \kappa ) be a complete local ring, see Definition 10.160.1. 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 + 2} 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
Lemma 10.153.10.slogan Let (R, \mathfrak m) be a local ring of dimension 0. Then R is henselian.
Proof.
Let R \to S be a finite ring map. By Lemma 10.153.3 it suffices to show that S is a product of local rings. By Lemma 10.36.21 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 10.36.20, hence they are all maximal. Every element of \mathfrak m_1 \cap \ldots \cap \mathfrak m_ r is nilpotent by Lemma 10.17.2. It follows S is the product of the localizations of S at the primes \mathfrak m_ i by Lemma 10.53.5.
\square
The following lemma will be the key to the uniqueness and functorial properties of henselization and strict henselization.
Lemma 10.153.11. Let R \to S be a ring map with S henselian local. Given
an étale ring map R \to A,
a prime \mathfrak q of A lying over \mathfrak p = R \cap \mathfrak m_ S,
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 10.143.3. 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 10.153.3 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 10.153.12. 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^\varphi _ i(s_2, \ldots , s_ n) = 0, \ i = 1, \ldots , n \}
where P^\varphi _ 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 10.153.3. 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 where \overline{P}_ i is the image of P_ i in R/\mathfrak m_ R[x_1, \ldots , x_ n]. 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}^\varphi _1, \ldots , \overline{P}^\varphi _ n)
is also a finite product of copies of S/\mathfrak m_ S with the same index set. This proves the lemma.
\square
Lemma 10.153.13. 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 10.92.2 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 10.16.2 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 10.153.3. 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 10.89.10 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 10.91.1 part (1).
\square
Comments (6)
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