Trivial Notions Seminar:
How Implicit is the Implicit Function Theorem?
Kai-Wen Lan
∗Oct 31, 2003
Out of the slimy mud of words, out of the sleet and hail of verbal imprecisions, approximate thoughts and feelings, words that have taken the place of thoughts and feelings, there springs the perfect order of speech, and the beauty of incantation.
— T.S. Eliot
Abstract
We will study Artin’s theorem on algebraic approximation of struc- tures over complete local rings, using his formulation of implicit func- tion theorem, essentially Hensel’s lemma, which amounts to solve the problem locally for the ´etale topology.
We will study Artin’s theorem on algebraic approximation of structures over complete local rings, using his formulation of implicit function theorem, which is essentially Hensel’s lemma. We follow mainly Artin’s article Alge- braic Approximation of Structures over Complete Local Rings ([Art69]), in the sense that we will reproduce his proof, sometimes almost step by step, without claiming any originality in the ideas. Artin’s proof is a beautiful example of reducing abstract problems to concrete ones. We hope that our presentation will be as close as possible to his original one in spirit.
1 Introduction
The main question in our mind is the following:
∗Department of Mathematics, Harvard University. Email: [email protected]
Question 1.1. Let A be a ring, let F be a functor
F : (A − algebras) → (Sets), (1.2) let c be an integer, and let ¯ξ ∈ F ( ˆA), where ˆA is the m-adic completion of A with respect to an ideal m of A. Does there exist an element ξ ∈ F (A) such that
ξ ≡ ¯ξ (mod mc)?
It is natural to put some finiteness condition on the considered functor F , and a fundamental one is Grothendieck’s notion of being locally of finite presentation, i.e. the canonical map
lim−→F (Bi) → F (lim
−→Bi) (1.3)
is bijective for any filtering inductive system of A-algebras {Bi}.
This definition is justified in the sense that, if F is represented by an A-algebra R, then R is locally of finite presentation if and only if
lim−→HomA−alg(R, Bi) → HomA−alg(R, lim
−→Bi) (This is in Grothendieck’s EGA [GD66, IV, 8.14.2.2]).
Using (1.3), for any A-algebra B, we can always write B as the inductive limit of filtering system of A-algebras of finite presentation:
lim−→Bi → B.
Then using (1.3), an element ξ ∈ F (B) may be induced from ξi ∈ F (Bi) for some sufficiently large i. Choose a finite presentation
Bi = A[Y ]/(f (Y ))
where Y = (Y1, . . . , YN) and f = (f1, . . . , fm) ∈ A[Y ]. Then a homomor- phism of A-algebras Bi → C is given by a solution in C of the system of polynomial equations f (Y ) = 0. Given such a morphism, the element ξ ∈ F (Bi) induces an element of F (C). Thus we conclude the following:
Lemma 1.4. Let F be a functor (1.2) which is locally of finite presentation, let B be an A-algebra, and let ξ ∈ F (B). There exists:
(i) A finite system of polynomial equations
f (Y ) = 0, (1.5)
where Y = (Y1, . . . , YN) and f = (f1, . . . , fm) ∈ A[Y ].
(ii) A functorial rule associating to every solution of this system of equa- tions in an A-algebra C an element of F (C).
(iii) A solution of the system (1.5) in B, so that (ii) applied to this solution yields ξ.
Now, given ¯ξ ∈ F ( ˆA), put B = ˆA in Lemma 1.4. We obtain a system of equations (1.5) and a solution by (iii), say ¯y, of this system in ˆA which yields ¯ξ via the rule (ii). It is clear that to find ξ ∈ F (A) as in Question 1.1, it suffices to approximate the solution ¯y of (1.5) by a solution y in A modulo mc, and to apply the rule (ii) to this approximation.
Hence we have the following:
Corollary 1.6. Suppose that a pair (A, m) is given. Then to solve Question 1.1 for a functor (1.2) locally of finite presentation, it suffices to solve the following Question 1.7.
Question 1.7. Let Y = (Y1, . . . , Yn) be variables, let f = (f1, . . . , fm) ∈ A[Y ] be polynomials, and suppose given elements ¯y = (¯y1, . . . , ¯yN) in ˆA which solve the system of polynomial equations
f (Y ) = 0. (1.8)
Let c be an integer. Does there exist a solution y = (y1, . . . , yN) in A of (1.8) such that
yi ≡ ¯yi (mod mc)?
It is natural to ask for conditions under which Question 1.7 has an affir- mative answer. We shall proceed by assuming that A is a Henselian local ring, which amounts to study the questions locally for the ´etale topology. Re- call that a ring is Henselian if the following analogue of the implicit function theorem holds (cf. [GD67, IV, 18.5.11(b)]):
Condition 1.9. Let
f (Y ) = (f1(Y ), . . . , fN(Y )) ∈ A[Y ]
be polynomials, where Y = (Y1, . . . , YN). Let y◦ = (y1◦, . . . , yN◦) ∈ k = A/m be elements such that
f◦(y◦) = 0 and that
det
∂fi
∂Yj
◦
(y◦) 6= 0
with the symbol ◦ denoting residue modulo m. Then there are elements y = (y1, . . . , yN) in A, with yi ≡ yi◦ (mod m), such that
f (y) = 0.
Remark 1.10. Here m denotes the maximal ideal of A. But in fact, when A is a Henselian local ring, the maximal ideal may be replaced by any proper ideal.
The Henselian condition is still not sufficient for Question 1.7, and what we will suppose further is that A is an excellent discrete valuation ring.
Theorem 1.11. Let R be a field or an excellent discrete valuation ring, and let A be the Henselization ([GD67, IV, 18.6]) of an R-algebra of finite type at a prime ideal. Let m be a proper ideal of A. Given an arbitrary system of polynomial equations
f (Y ) = 0
where Y = (Y1, . . . , YN) with coefficients in A, a solution ¯y = (¯y1, . . . , ¯yN) in the m-adic completion ¯A of A, and an integer c, there exists a solution y = (y1, . . . , yN) in A with
yi ≡ ¯yi (mod mc).
By Corollary 1.6, we have the following theorem, which states our main result in this section.
Theorem 1.12. With assumptions of Theorem 1.11, let F be a functor (1.2) which is locally of finite presentation. Let c be any integer. Then, for any ξ ∈ F ( ˆ¯ A), there is a ξ ∈ F (A) such that
ξ ≡ ¯ξ (mod mc).
The proof of Theorem 1.11 will be given in Section 3.
To apply the approximation results, denote by S a scheme which is of finite type over a field or over an excellent Dedekind domain. Let s be a point of S. By an ´etale neighborhood of s in S we mean an ´etale map S0 → S together with a rational lifting of s to S0:
s = Spec(k(s)) //
&&M MM MM MM MM MM
M S0
S
Since the Henselization eOS,s of the local ring of S at s is the limit of the rings Γ(S0,OS0) as S0 runs over the filtering category of ´etale neighborhoods, Theorem 1.11 translates immediately as
Corollary 1.13. Let Y = (Y1, . . . , YN) be variables, and let f = (f1, . . . , fm) be polynomials in OS[Y ] whose coefficients are global sections of OS. Let
¯
y = (¯y1, . . . , ¯yN) be a solution of the system of equations f (Y ) = 0
in the complete local ring ˆOS,s, and let c be an integer. There exists an ´etale neighborhood S0 of s in S, and a solution y = (y1, . . . , yn) in Γ(S0,OS0) of the system of equations, such that
y ≡ ¯y (mod mcs).
Similarly, we have the following translation of Theorem 1.12.
Corollary 1.14. Let
F : (Sch /S)◦ → (Sets)
be a (contravariant) functor locally of finite presentation, and let ¯ξ ∈ F ( ˆS) ( ˆS = Spec( ˆOS,s)). Let c be an integer. Then there is an ´etale neighborhood S0 of s in S and an element ξ0 ∈ F (S0) such that
ξ0 ≡ ¯ξ (mod mcs).
2 N´ eron’s p-desingularization
In this section, we consider a pair Λ, Λ0 of discrete valuation rings such that Λ0 is unramified over Λ in the weak sense that a local parameter p of Λ is also a local parameter of Λ0. We make no other restriction on the pair at present.
Put T = Spec(Λ), T0 = Spec(Λ0). Let X be a T -scheme of finite type, and let s0 : T0 → X be a point of X with values in T0/T . Suppose that X/T is smooth at the generic point of s0. (Strictly speaking, we should say the generic point of s0(T0). The terminology should not cause confusion.) Then we define, following N´eron ([N´er64]), a measure l(s0) of singularity of X at s0 as follows:
Choose an affine open of X containing s0. Say that this affine is the locus of zeros of f1, . . . , fm ∈ Λ[y] in an affine space ANT. Let r be the relative dimension of X/T at the generic point of s0. Then for every minor M of rank N − r of the Jacobian matrix
J =∂fi
∂yj
,
evaluation on s0 yields a matrix with values in Λ0, and we define l(s0) = min
M (v0(det M (s0))) (2.1) as M runs over the minors of rank N − r, and where v0 denotes the valuation of Λ0.
It is immediately checked that this is independent of the choice of the affine neighborhood and of {fi}. Moreover, l(s0) is zero if and only if X/T is smooth on s0.
Next, we define N´eron blowing up as follows: Let X be a T -scheme of finite type, and let Y ⊂ X be a closed subscheme. Let S = OX ⊕I ⊕ S2I ⊕ . . . be the symmetric algebra on the sheaf of ideals I of Y in OX, and consider the (nonhomogeneous) ideal a of S generated by sections of the form
p[g] − g, g a section of I ,
where [g] denotes the corresponding element of S of degree one, and g the element of degree zero (in OX). Here p is as above the local parameter of Λ. Clearly this defines a quasi-coherent sheaf of ideals of S . Let A be the quasi-coherent sheaf of OX-algebras obtained by killing p-torsion in S /a.
N´eron’s blowing up of Y in X is defined as X =S pecOXA , which is a scheme affine over X.
Suppose X = Spec(A) is affine, and let g1, . . . , gr generate the ideal of Y . Then it follows from the above description that X = Spec(A), where A is the ring obtained by killing p-torsion in
A[z1, . . . , zr]/a, where a is the ideal generated by the relations
pzν− gν = 0, ν = 1, . . . , r. (2.2) Note that adding p to the set of generators for the ideal of Y does not affect the blowing up. For it amounts to add an extra variable zr+1 with relation
pzr+1 = p, whence, modulo p-torsion,
zr+1 = 1.
Thus N´eron’s blowing up of Y in X depends only on the closed fiber Y◦ of Y over T . Moreover, the map X → X is an isomorphism outside of the locus {p = 0}, as follows again from (2.2).
We revert to the notation of the beginning of this section. Let Y◦ be the closure in X of the closed point of s0, with its reduced structure. We work locally in a neighborhood of s0. With the above notation for X = Spec(A), we have
gν(s0) ≡ 0 (mod p),
by definition of Y◦. Hence gν(s0) is divisible by p in Λ0, and so we can find unique elements zν in Λ0 satisfying the equation (2.2). Since Λ0 is p-torsion free, the map A → Λ0 defining the section s0 extends to A, and thus the section s0 lifts to a section ¯s0:
T0 ¯s
0 //
s@0@@@@ @
@@ X
X
(2.3)
N´eron’s fundamental and beautiful observation is the following:
Theorem 2.4. With the above notation, suppose that Y◦ is generically smooth over the residue field k of Λ, i.e., that its function field k(Y◦) is a separable extension of k. Then
l(¯s0) ≤ l(s0),
with equality if and only if l(s0) = 0, i.e., X/T is smooth on s0. Since k(Y◦) is a subfield of the residue field of Λ0, it follows that
Corollary 2.5. Suppose that the residue field k0 of Λ0 is a separable extension of k, and let s0 : T0 → X be a T -map such that X is smooth at the generic point of s0. Consider the operation of replacing X by N´eron’s blowing up X and s0 by ¯s0, as in (2.3). A finite number of repetitions of this operation results in a situation where X/T is smooth on s0.
Proof. The proof of Theorem 2.4 consists in a reduction to the case that X is affine and that Y◦ is the origin {y1 = . . . = yN = p = 0} in affine y-space ANT, and an explicit calculation in that case. We begin with the calculation:
Let X be the locus of zeros of f1, . . . fm ∈ Λ[y]. Then by (2.2) the blowing up is given by the equations
pzi = yi i = 1, . . . , N,
and the extra equations needed to kill p-torsion. Thus we may eliminate the variables yi and view X as a locus in affine z-space.
Set
fi(y) = pai0+
N
X
j=1
aijyj + (degree ≥ 2).
The constant terms are multiples of p, because of our assumption on the form of the origin. Then
fi(pz) = pai0+X
j
paijzj+ p2(degree ≥ 2), whence
fi(pz) = pϕi(z) with
ϕi(z) = ai0+X
j
aijzj + p(degree ≥ 2), (2.6) and X is the locus of zeros of the polynomials ϕi(z) together with some extra equations needed to kill p-torsion.
We have
p∂ϕi
∂zj(z) = ∂
∂zjf (pz) = p∂fi
∂yj(pz), i.e.,
∂ϕi
∂zj(z) = ∂fi
∂yj(y) (pzi = yi). (2.7) This shows that l(¯s0) ≤ l(s0), since the Jacobian matrix of X is a submatrix of that of X if we use generators {fi}, {ϕi} for the respective ideals.
Denote by a symbol ◦ the residue modulo p. Then by (2.6), the polyno- mials ϕ◦i are linear, and the Jacobian matrix
J◦ = ∂ϕ◦i
∂zj
(z) = ∂fi◦
∂yj
(0)
is just the constant matrix (a◦ij), whence J (s0)◦ = (a◦ij). If we assume that X is not smooth on s0, then the rank of (a◦ij) is < N − r.
Suppose that the minimum in (2.1) is taken on for a minor of the form
∂fi
∂yj
where i, j run from 1 to N − r. Since the rank of (a◦ij) is less than N − r, we may make an invertible linear transformation of f1, . . . , fN −r with coefficients in Λ so that, say, a◦1j = 0 for all j. Then f1 has the form
f1(y) = pa10+X
j
pα1jyj + (degree ≥ 2).
Hence
f1(pz) = pa10+ p2(X
j
α1jyj + (degree ≥ 2)).
Since f1(pz) vanishes on the section ¯s0, it follows that a10≡ 0 (mod p),
whence
ϕ1(z) ≡ 0 (mod p).
Thus p−1ϕ1 is a polynomial vanishing on X, and if we replace ϕ1 by this polynomial in the Jacobian matrix (2.7), the value of the subdeterminants involving i = 1 is decreased by one, whence l(¯s0) ≤ l(s0) − 1, as required.
It remains to reduce the general case to the above one. The problem is local on X in a neighborhood of s0, hence we may assume that X = Spec(A) is affine. Let d = dim Y . Since Y is generically smooth over Spec(k), it is generically ´etale and finite over (y1, . . . , yd)-space AdT. Let eΛ be the local ring of AdT at the generic point of its closed fiber. This ring eΛ is a discrete valuation ring with local parameter p. Since s0 maps the closed point of T0 to the generic point of Y , the induced map T0 → AdT carries T0 to Spec(eΛ) = eT . We have a cartesian diagram
AN −dTe //
Te
ANT //AdT
Let eX, eY◦ be subschemes of AN −dTe induced by X, Y◦ respectively. Then s0 lifts to a eT -map es0 : T0 → eX, and the image of the closed point of T0 is Ye◦, which is a closed point of eX◦. With the above notation, the schemes eX, Ye◦ are defined respectively by the equations {fi = 0}, {gν = 0}, where these elements are viewed as polynomials in eΛ[yd+1, . . . , yN]. It therefore follows from equations (2.2) that N´eron’s blowing up X of Y◦ in X induces the blowing up of eY◦ in eX. Moreover, it is clear that l(s0) = l(es0), provided at least that the coordinates y1, . . . , yN are chosen generically, and that l(¯s0) ≤ l(¯es0) in any case. (Here the word generically means so that the minimum in (2.1) is taken on for a minor of J = (∂fi/∂yj) in which j runs over indices
> d.) Hence we may replace (T, X, s0) by ( eT , eX,es0), which reduces us to the case that Y◦ is a closed point of X◦, with residue field separable over k. Finally, the integer l(s0) does not change if Λ0 is replaced by any larger discrete valuation ring Λ01 having p as local parameter. Moreover, it is clear
that N´eron’s blowing up commutes with ´etale extensions Λ → Λ1, where Λ1 is a discrete valuation ring. An appropriate choice of Λ1, Λ01 followed by a suitable localization reduces us to the case that Y◦ is a rational point of X◦ over k, whence by translation to the case that Y◦ is the point {y = 0}.
3 Proof of Theorem 1.11
We will need the following lemmas.
Lemma 3.1. Let A be a ring, let B be an arbitrary A-algebra, and let G be a functor locally of finite presentation on B-algebras. Define a functor F on A-algebras by
F (A0) = G(A0⊗AB).
Then F is locally of finite presentation. An analogous assertion holds for contravariant functors.
Proof. This follows immediately from the fact that tensor products commutes with direct limits.
Lemma 3.2. Let A be a ring, and let f : F → G be a morphism of functors on A-algebras, with G locally of finite presentation. For an A-algebra A0 and an element ξ ∈ G(A0), denote by Fξ the functor on A0-algebras defined by
Fξ(B0) = {η ∈ F (B0) | fB0(η) = ξB0},
where fB0 : F (B0) → G(B0) is the map, and where ξB0 is induced from ξ. If Fξ is locally of finite presentation for every pair (A0, ξ), then F is locally of finite presentation. A similar assertion holds for contravariant functors on A-schemes.
Proof. Suppose Fξ is locally of finite presentation for all (A0, ξ). Let B = lim−→Bi, where {Bi} is a filtrating inductive system of A-algebras, and let η ∈ F (B). The element f (η) = ξ ∈ G(B) is induced by ξi ∈ G(Bi) for suitable i, and η is an element of Fξi(B). By assumption, this element is induced by an ξj ∈ Fξi(Bj) for suitable j. Thus the map
lim−→F (Bi) → F (B) (3.3)
is surjective.
If1ηi,2ηi ∈ F (Bi) have the same image in F (B), then f (1ηj) = f (2ηj) = ξj in G(Bj) for some j. Hence1ηj and 2ηj are in Fξj(Bj), and they represent the same element of Fξj(B). Thus they become equal in F (Bk) for some k.
This shows injectivity of the map (3.3).
Now we are going to prove Theorem 1.11. We begin with some preliminary reductions.
First of all, it is enough to treat the case that m is the maximal ideal of A. For, suppose that the theorem has been proved in that case, and let m be any ideal. Let m1, . . . , mr be generators for the ideal mc. Under the assumptions of Theorem 1.11, the elements ¯yi are in the m-adic completion A of A. Thus there are elements yi0 ∈ A such that
yi0 ≡ ¯y (mod mcA), hence
¯
yi = y0i+X
j
¯ aijmj
for suitable ¯aij in A. The elements {¯yi, ¯aij} ∈ A are thus a solution of the larger system of equations
fν(Y1, . . . , YN) = 0 ν = 1, . . . , m Yi− yi0−X
j
Aijmj = 0 i = 1, . . . , N, j = 1, . . . , r,
and we can view these elements as lying in the completion ˆA of A with respect to its maximal ideal. The theorem in the know case implies the existence of a solution {yi, aij} ∈ A, i.e.,
yi = yi0+X
j
aijmj,
and yi ≡ y0i ≡ ¯yi (mod mc), as required.
Next, we may assume that R is a discrete valuation ring and that the maximal ideal m of A lies over the closed point of Spec(R). For, if R is a discrete valuation ring but m lies over the generic point, then we replace R by its field of fractions, and if R is a field, then we replace it by the power series ring R[[t]], where t acts on A as zero. Since R[[t]] is excellent, this is permissible.
Moreover, we may assume that K = A/m is finite over the residue field k of R. For, since A is the Henselization of an R-algebra of finite type at a prime ideal, K is a field extension of k of finite type. Let d be its transcendence degree. Then we can find elements z1, . . . , zd ∈ A so that K is finite over k(z1◦, . . . , zd◦), where ◦ denotes the residue modulo m. Consider the map R[Z] → A sending Zi → zi. The inverse image of m is the prime ideal of R[Z] generated by the local parameter p of R. Since A is local, this
map factors through the localization R0 of R[Z] at this prime ideal, which is an excellent discrete valuation ring (cf. [GD65, IV, 7.8.6(i) and 7.8.3(ii)]).
Clearly, A is the Henselization of an R0-algebra of finite type. Thus we may replace R by the ring R0, whose residue field is k(Z).
Finally, say that A is the Henselization of the R-algebra of finite type A0 at a maximal ideal which we will denote also by m. Then we can make A0 into a finite algebra over a polynomial ring R[X] (X = (X1, . . . , Xn)) in such a way that m lies over the origin (p, X1, . . . , XN) of Spec(R[X]). This is clear: Write A0 as a quotient of some R[Z1, . . . , Zn]. Let g◦i(Zi) ∈ k[Zi] be a monic polynomial satisfied by the residue of Zi in A0/m. Choose a monic polynomial gi(Zi) ∈ R[Zi] representing gi◦, and set Xi = gi(Zi). Then the resulting map R[X] → A0 is as required.
Now let R[X]∼denote the Henselization of R[X] at the origin (p, X). The R[X]∼-algebra eA0 obtained from A0 by extension of scalars is a product of local rings (cf. [GD67, IV, 18.5.11(a)]) which are the Henselizations of A0 at the various points lying over the origin. Our ring A is among them, hence is a finite R[X]∼-algebra.
We claim that it is enough to prove Theorem 1.11 for the ring R[X]∼ itself. Indeed, suppose that the theorem has been proved in that case, let A be any finite local R[X]∼-algebra, and
f (Y ) = 0, Y = (Y1, . . . , YN), f = (f1, . . . , fm) (3.4) a system of polynomial equations with coefficients in A. Let F be the functor which to an R[X]∼-algebra B associates the set of solutions of (3.4) in the ring A ⊗ B (the tensor product being over the ring R[X]∼). This functor is locally of finite presentation, by Lemma 3.1. Thus we may apply Corollary 1.6. Since A is finite over R[X]∼, we have
A ∼ˆ= A ⊗ R[X]∧,
where the symbol ∧ denotes the completion of a local ring with respect to its maximal ideal. Hence a solution ¯y of (3.4) in ˆA yields an element ¯ξ of F (R[X]∧), which we may approximate (p, X)-adically to arbitrary order by a ξ ∈ F (R[X]∼), to obtain the required solution y in A.
We are therefore reduced to the case that A = R[X]∼, (X = (X1, . . . , Xn)), and that m = (p, X). Proceeding by induction on the number n ≥ 0 of variables Xi, we may fix n and assume the theorem true for fewer than n variables.
Lemma 3.5. If suffices to treat the case that the given polynomials f = (f1, . . . , fm) ∈ A[Y ] have coefficients in the polynomial ring R[X].
Proof. Suppose that the theorem has been proved for such polynomials, and let f be arbitrary. Consider the homomorphism
A[Y ] → R[X]∧ (3.6)
defined by the substitution of ¯y for Y . Since R[X]∧ = ˆR[[X]] is an integral domain, its kernel is a prime ideal p, and p contains (f1, . . . , fm). Clearly, it is permissible to add extra equations to the system so that (f1, . . . , fm) generate the whole ideal p. Let p0 = pT R[X, Y ]. Since A is the Henselization of R[X], the ring A[Y ] is a limit of ´etale extensions of R[X, Y ]. Therefore A[Y ]/p0A[Y ] is reduced ([Gro71, SGA I, Proposition 9.2]1), i.e., p0A[Y ] is an intersection of prime ideals. The ring A being algebraic over R[X], it is easily seen for reasons of dimensions that p is one of these prime ideals. If p = p0A[Y ], we are enough. Otherwise we have
p0A[Y ] = p\
q⊃ pq
for some ideal q not containing p. Since p is the kernel of the map (3.6), it follows that there is an element g ∈ q with g(¯y) 6= 0.
Let ϕ = (ϕ1, . . . , ϕl) in R[X, Y ] be generators for p0. Applying the theo- rem to the solution ¯y of the system of equations
ϕ(Y ) = 0,
which is possible by assumption, we can find y = (y1, . . . , yn) in A arbitrarily close to ¯y, m-adically, so that ϕ(y) = 0. Since g(¯y) 6= 0, it follows that g(y) 6= 0 if y is sufficiently near ¯y. But since pq ⊂ p0A[Y ], we have
fi(y)g(y) = 0, hence
fi(y) = 0 for all i. Thus y is the required solution of (3.4).
We now suppose that f = (f1, . . . , fm) is in R[X, Y ]. Let S = Spec(R[X]), S = Spec(R[X]e ∼) and ˆS = Spec(R[X]∧). It is permissible to assume that the elements fi generate the whole kernel of the map
R[X, Y ] → R[X]∧ (3.7)
1Let f : X → Y be an ´etale morphism. If Y is reduced, then so is X. The converse is true when f is surjective.
defined by the substitution of ¯y for Y , which is a prime ideal p of R[X, Y ].
Then the map
Spec(R[X, Y ]/(f )) = V → S
is generically smooth. For, since R is excellent, so is R[X] ([GD65, IV, 7.8.6(i)]). Therefore ([GD65, IV, 7.8.2(ii)]) the map ˆS → S is regular, and so fract(R[X]∧) is a separable extension of fract(R[X]).2 Since R[X, Y ]/(f ) is a subring of R[X]∧, fract(R[X, Y ]/(f )) is a separable extension of fract(R[X]), which proves the assertion.
Let Λ be the localization of R[X] at the prime ideal P generated by the local parameter p of R, and similarly let Λ0 be the localization of R[X]∧ at ˆP = p · R[X]∧. These rings Λ and Λ0 are discrete valuation rings, and p is a local parameter of them both. Moreover, the residue field of Λ0 is a separable extension of that of Λ. This is because these fields are fract(k[[X]]) and fract(k[X]) respectively, and k[X] is excellent ([GD65, IV, 7.8.6(i)]), k = R/p. Thus we may apply Corollary 2.5 to this pair.
Write T = Spec(Λ), T0 = Spec(Λ0), VT = V ×S T . The solution ¯y yields an S-map
σ : ˆS → V, which induces a map
s0 : T0 → VT making a commutative diagram
T0 //
s0
@ @
@@
@@
@@
Sˆ
σ
= =
==
==
==
VT //
~~}}}}}}}} V
T //S
We want to reduce ourselves to the case that VT is smooth over T on s0. Since T , T0 are obtained by localization from S, ˆS respectively, this is equivalent with the assertion that V be smooth over S at the point σ( ˆP ), where ˆP = p · R[X]∧.
To do this, let W◦ ⊂ V be the closure of σ( ˆP ) with its reduced structure.
Since VT is a localization of V , the scheme WT◦ = W◦×S T is the closure in VT of the closed point of s0. Thus we can induce N´eron’s blowing up of WT◦ in VT by blowing up W◦ in V : Let g1, . . . , gr in R[X, Y ] generate the ideal of
2We write fract(D) for the field of fractions of an integral domain D.
W◦. Then we blow up W◦ in V by killing p-torsion in the ring R[X, Y, Z]/a, Z = (Z1, . . . , Zr), where a is the ideal generated by the relations
pZν − gν = 0 ν = 1, . . . , r (3.8)
fi = 0 i = 1, . . . , m. (3.9)
Let the blowing up be the spectrum of this ring, say V . Clearly, VT = V ×ST is N´eron’s blowing up of WT◦ in VT. Moreover, since σ( ˆP ) lies in W◦, it follows that
gν(¯y) ≡ 0 (mod p · R[X]∧).
Hence we can find ¯zν ∈ R[X]∧ satisfying the equations p¯zν − gν(¯y) = 0,
whence since R[X]∧ is p-torsion free, σ lifts to an S-map Sˆ σ¯ //
σ
>
>>
>>
>>
> V
V
and ¯σ induces the lifting of s0 to ¯s0 : T0 → VT.
Now our problem is to approximate m-adically the map σ : ˆS → V given by the solution ¯y of (3.4) by a map eS → V , and it is clearly sufficient to approximate the map ¯σ instead, i.e., to solve the system of equations given by (3.8) together with the additional equations needed to kill p-torsion. Since V is reduced and irreducible because V is, we may replace V by V and σ by ¯σ.
since V and V are isomorphic outside of the locus {p = 0} (cf. Section 2), the elements (f1, . . . , fm) still generate the whole kernel of (3.7). By Corollary 2.5, a finite number of repetitions of this process results in a situation where V is smooth over S at σ( ˆP ). We have therefore proved:
Lemma 3.10. It suffices to treat the case that f = (f1, . . . , fm) have coor- dinates in R[X, Y ], and that V = Spec(R[X, Y ]/(f )) is smooth at the point σ( ˆP ), where
σ : ˆS → V
is the map defined by the substitution of ¯y for Y , and where ˆP = p · R[X]∧. Assume the conditions of Lemma 3.10 hold. Let r be the relative dimen- sion of V /S at σ( ˆP ). Then by the Jacobian criterion for smoothness, there is an (N − r)-rowed minor M of the Jacobian matrix (∂fi/∂Yj) such that
δ = det M ∈ R[X, Y ]
has the property
δ(X, ¯y) 6≡ 0 (mod p)
in R[X]∧. We may suppose that M is the minor 1 ≤ i, j ≤ N − r. Let V0 ⊂ Spec(R[X, Y ]) be the locus of zeros of f1, . . . , fN −r. Then V0 and V are equal locally at the point σ( ˆP ), hence we can write, set theoretically,
V0 = V [ W
for some closed set W ⊂ Spec(R[X, Y ]) which does not contain the image σ( ˆS). Thus there is an element g ∈ R[X, Y ] vanishing on W such that
g(X, ¯y) 6= 0.
Let y = (y1, . . . , yN) in R[X]∼ be any solution of the system of equations f1(Y ) = . . . = fN −r(Y ) = 0. (3.11) If y ≡ ¯y modulo a sufficiently high power of the maximal ideal of R[X]∼, then it follows that g(X, y) 6= 0, hence that the image of Spec(R[X]∼) in Spec(R[X, Y ]) under the map defined by the substitution of y for Y does not lie in W . Since Spec(R[X]∼) is reduce and irreducible, and since the image lies in V0, it follows that it lies in the subscheme V . Hence
fi(y) = 0
for all i = 1, . . . , m. Thus it suffices to treat the system of equations (3.11), whence we have the following:
Lemma 3.12. With the notations of Lemma 3.10, we may assume in addi- tion that m = N − r, where r is the relative dimension of V /S at σ( ˆP ), and that the determinant δ of the minor (∂fi/∂Yj), 1 ≤ i, j ≤ m, satisfies
δ(X, ¯y) 6≡ 0 (mod p).
We can now complete the proof as in the analytic case (cf. [Art68]).
Recall the following:
Lemma 3.13. Let A be a ring and a an ideal of A such that the pair (A, a) satisfies implicit function theorem (Condition 1.9) with m = a. For instance, A may be a Henselian local ring and a any proper ideal. Let f = (f1, . . . , fm) in A[Y ] be polynomials in the variables Y = (Y1, . . . , YN), let M be the matrix (∂fi/∂Yj), 1 ≤ i, j ≤ m, and let δ = det M . Suppose that we are given elements y◦ = (y◦1, . . . , yN◦ ) in A such that
f (y◦) ≡ 0 (mod δ2(y◦) · a).
Then there are elements y = (y1, . . . , yN) in A such that f (y) = 0
and that
y ≡ y◦ (mod δ(y◦) · a)
We shall give a proof of the following stronger assertion, due to Tougeron:
Lemma 3.14. Let A, a, f be as in Lemma 3.13. Let J be the Jacobian matrix (∂fi/∂Yj), i = 1, . . . , m, j = 1, . . . , N . Suppose that we are given elements y◦ = (y◦1, . . . , yN◦ ) in A such that
f (y◦) ≡ 0 (mod ∆2· a)
where ∆ is the annihilator ideal of the A-module C presented by the relation matrix J (y◦) (i.e., C is the cokernel of the homomorphism AN → Am whose matrix is J (y◦)). Then there is a solution y = (y1, . . . , yN) of the system of equations f (Y ) = 0 with
y ≡ y◦ (mod ∆ · a).
Proof. Let δ1, . . . , δr generate the annihilator ∆. This means that, writing J = J (y◦), there are N × m matrices Ni with
J Ni = δiI, where I is the m × m identity matrix. Write
f (y◦) =X
i,j
δiδjεij
for suitable vectors εij = (εij1, . . . , εijm) with εijν ∈ a. We try to solve the equations
f (y◦+
r
X
i=1
δiUi) = 0
for elements Ui = (Ui1, . . . , UiN) of AN. Expansion by Taylor’s formula in vector notation yields
0 = f (y◦) + J ·X
i
δiUi+X
i,j
δiδjQij
= J ·X
i
δiUi+X
i,j
δiδj(Qij + εij)
=X
i
δiJ · Ui+X
i
δiJ · (X
j
Nj · (Qij + εij)),
where Qij are vectors of polynomials in the Ui all of whose terms are of degree
≥ 2. Thus it suffices to solve the r vector equations 0 = Ui+X
Nj· (Qij + εij),
which give N r equations in the N r unknowns {Uiν}. The Jacobian of this system of equations is the identity matrix at U = 0. Thus the implicit function theorem implies the existence of a solution ui = (ui1, . . . , uiN) with uiν ≡ 0 (mod a), and y = y◦ +P δiui is the required solution of f (Y ) = 0.
If we apply Lemma 3.13 with a = mc to our situation, it follows that, in the notation of Lemma 3.13, it suffices to find y◦ = (y1◦, . . . , y◦N) in A = R[X]∼ such that
y◦ ≡ ¯y (mod mc) and that
f (y◦) ≡ 0 (mod δ2(X, y◦) · mc).
For then Lemma 3.13 implies the existence of the required solution y in A.
Now since ¯y in ˆA is a solution of (3.4), we have trivially f (¯y) ≡ 0 (mod δ2(X, ¯y) · mc).
Thus, setting g = δ2, we may apply the following lemma to complete the proof.
Lemma 3.15. Suppose that the theorem is proved for the ring R[X]∼ when there are fewer than n variables X, and let A = R[X1, . . . , Xn]∼ (n ≥ 0). Let g, f1, . . . , fm ∈ R[X, Y ] be polynomials and let ¯y = (¯y1, . . . , ¯yN) be elements of ˆA such that
g(X, ¯y) 6≡ 0 (mod p) and that
fi(X, ¯y) ≡ 0 (mod g(X, ¯y))
for i = 1, . . . , m. Then there are elements y = (y1, . . . , yN) in A such that fi(X, y) ≡ 0 (mod g(X, y))
for all i and that
y ≡ ¯y (mod mc).
Proof. If g(X, ¯y) is invertible, g(X, y) will be invertible for all y ≡ ¯y (mod m).
Then the desired divisibility is trivial. We may thus assume that g(X, y) is not invertible. This completes the proof in the case n = 0, since g(X, ¯y) 6≡ 0 (mod p) just means that it is an invertible element in that case.
Suppose that n > 0. Since g(X, ¯y) 6≡ 0 (mod p), we may adjust the coordinates X via an automorphism of R[X] of the type
Xi0 = Xi+ Xnei i = 1, . . . , n − 1 Xn0 = Xn
in such a way that g(X, ¯y) 6≡ 0 (mod (p, X1, . . . , Xn−1)).
Suppose we have
g(X, ¯y) ≡ (Xnr) · (unit) (mod (p, X1, . . . , Xn−1)). (3.16) Then ˆB = R[X]∧/(g(X, ¯y)) is a finite algebra over the ring
R[[Xˆ 1, . . . , Xn−1]] = R[X1, . . . , Xn−1]∧,
because it is a finite module (R/p)[[Xn]]/(Xnr) after modulo (p, X1, . . . , Xn−1), from which it follows that {1, Xn, . . . , Xnr−1} generates ˆB (as a module) over R[[Xˆ 1, . . . , Xn−1]] (cf. [ZS58, page 259, Corollary 2]).
Now write
Xnr = g(x, ¯y) · (unit) − (¯ar−1Xnr−1+ . . . + ¯a1Xn+ ¯a0),
then ¯a0, . . . , ¯ar−1 are non-units in R[X1, . . . , Xn−1]∧by (3.16). In other words, the Weierstraß Preparation Theorem holds for g(X, ¯y) with respect to the variables Xn, i.e., that
g(X, ¯y) = ¯a(Xn) · (unit) (3.17) where
¯
a(Xn) = Xnr+ ¯ar−1Xnr−1+ . . . + ¯a1Xn+ ¯a0
is a monic polynomial with coefficients ¯aν which are non-units in R[X1, . . . , Xn−1]∧ (what we have done can be seen in ([ZS58, page 261])). Therefore
B ∼ˆ = ˆR[[X1, . . . , Xn−1]][Xn]/(¯a(Xn)). (3.18) We make the substitution
Yν∗ =
r−1
X
j=0
YνjXnj ν = 1, . . . , N
for Yν into the polynomials g, fi, where Yνj are variables. Dividing by the variable polynomial
A(Xn) = Xnr+ Ar−1Xnr−1+ . . . + A1Xn+ A0 we obtain
g(X, Y∗) = A(Xn)Q +
r−1
X
i=0
GjXnj
fi(X, Y∗) = A(Xn)Q0i+
r−1
X
j=0
FijXnj,
(3.19)
where Q, Q0i, Gj, Fij are polynomials in the variables {Xν, Yνµ, Aν} with coefficients in R, and where Gj, Fij do not involve the variable Xn.
Next, divide ¯yν by ¯a(Xn) (which is possible by (3.18)):
¯
yν = ¯a(Xn)¯zν +
r−1
X
j=0
¯
yνjXnj (3.20)
with ¯zν in R[X]∧ and ¯yνj in R[X1, . . . , Xn−1]∧. Set
¯ y∗ν =
r−1
X
j=0
¯ yνjXnj.
Since
¯
y ≡ ¯y∗ (mod ¯a(Xn)), it follows by Taylor’s formula that
g(X, ¯y∗) ≡ g(X, ¯y) (mod ¯a(Xn)) and that
fi(X, ¯y∗) ≡ fi(X, ¯y) (mod ¯a(Xn)), whence by (3.17)
g(X, ¯y∗) ≡ 0
fi(X, ¯y∗) ≡ 0 (mod ¯a(Xn)) for i = 1, . . . , m. (3.21) Substitute ¯yνµ, ¯aν for Yνµ, Aν in (3.19). The congruence (3.21) shows that
Gj(X1, . . . , Xn−1, {¯yνµ}, {¯aν}) = 0 Fij(X1, . . . , Xn−1, {¯yνµ}, {¯aν}) = 0
for all relevant i, j. Thus {¯yνµ, ¯aν} is a solution in R[X1, . . . , Xn−1]∧ of the system of equations
Gi = 0
Fij = 0. (3.22)
By the induction hypothesis, there are elements {yνµ, aν} in R[X1, . . . , Xn−1]∼ solving the system (3.22), with
yνµ≡ ¯yνµ
aν ≡ ¯aν (mod (p, X1, . . . , Xn−1)c) for arbitrarily large c.
Choose
zν ≡ ¯zν (mod mc) where ¯z is as in (3.20), and set
a(Xn) = Xnr+ ar−1Xnr−1+ . . . + a1Xn+ a0 y∗ν =
r−1
X
j=0
yνjXnj yν = a(Xn)zν + yν∗, so that
yν ≡ ¯yν (mod mc). (3.23)
Then since {yνµ, aν} is a solution of (3.22), we have by (3.19) g(X, y∗) ≡ 0
fi(X, y∗) ≡ 0 (mod a(Xn)), whence by Taylor’s formula
g(X, y) ≡ 0
fi(X, y) ≡ 0 (mod a(Xn)), i = 1, . . . , m. (3.24) Now it is clear that if we write by Weierstraß
g(X, y) = b(Xn) · (unit)
where b(Xn) is a monic polynomial whose coefficients are non-units of R[X1, . . . , Xn−1]∧, then the degree of b will be r (cf. (3.17)), provided that c (cf. (3.23)) is chosen sufficiently large. Since a(Xn) divides b(Xn) by (3.24), and since these polynomials have the same degree, it follows that they are equal. Thus (3.24) implies that
fi(X, y) ≡ 0 (mod g(X, y))
for all i. This completes the proof of Lemma 3.15 and of Theorem 1.11.
A Henselian Rings
We summarize here several basic facts on Henselian rings from [Mil80, I, 4].
Throughout this section, A will be a Noetherian local ring with maximal ideal m and residue field k. The homomorphisms A → k and A[T ] → k[T ] will be written as (a → ¯a) and (f → ¯f ).
Definition A.1. A local ring A is called Henselian if for any monic polyno- mial f with coefficients in A such that ¯f factors as ¯f = g0h0 with g0 and h0 monic and coprime, then f itself factors as f = gh with g and h monic and such that ¯g = g0 and ¯h = h0.
Theorem A.2. Let x be the closed point of X = Spec(A). The following statements are equivalent:
1. A is Henselian.
2. Any finite A-algebra B is a direct product of local rings B =Q Bi (the Bi are then necessarily isomorphic to the rings Bmi, where the mi are the maximal ideals of B).
3. If f : Y → X is quasi-finite and separated, then Y = Y0 q Y1q . . . Yn where f (Y0) does not contain x and Yi is finite over X and is the spectrum of a local ring, i ≥ 1.
4. If f : Y → X is ´etale and there is a point y ∈ Y such that f (y) = x and k(y) = k(x), then f has a section s : X → Y .
5. Let f1, . . . , fn ∈ A[T1, . . . , Tn]. If there exists an a = (a1, . . . , an) ∈ kn such that ¯fi(a) = 0, i = 1, . . . , n, and det((∂ ¯fi/∂Tj)(a)) 6= 0, then there exists a b ∈ An such that ¯b = a and fi(b) = 0, i = 1, . . . , n.
6. Let f (T ) ∈ A[T ]. If ¯f factors as ¯f = g0h0 with g0 monic and g0 and h0 coprime, then f factors f = gh with g monic and ¯g = g0, ¯h = h0. Proof. See [Mil80, I, Theorem 4.2].
Proposition A.3. Any complete local ring A is Henselian.
Proof. See [Mil80, I, Proposition 4.5].
Definition A.4. Let A be a local ring. The smallest Henselian ring contain- ing A is called the Henselization of A, which we denote by eA.
Before proving the existence of eA, we introduce the notion of an ´etale neighborhood of a local ring A. It is a pair (B, q) where B is an ´etale A- algebra and q is a prime ideal of B lying over m such that the induced map k → k(q) is an isomorphism.
Lemma A.5. 1. If (B, q) and (B0, q0) are ´etale neighborhoods of A such that Spec(B0) is connected, then there is at most one A-homomorphism f : B → B0 such that f−1(q0) = q.
2. Let (B, q) and (B0, q0) be ´etale neighborhoods of A. There is an
´
etale neighborhood (B00, q00) of A with Spec(B00) connected and A- homomorphisms f : B → B00, f0 : B0 → B00 such that f−1(q00) = q, f0−1(q00) = q0.
Proof. See [Mil80, I, Lemma 4.8].
It follows from the lemma that the ´etale neighborhoods of A with con- nected spectra form a filtered inductive system. Define ( eA,m) to be itse inductive limit, ( eA,m) = lime
−→(B, q). It is easy to check that eA is a local A-algebra with maximal ideal m, ee A/me = k and eA is the Henselization of A. Also eA is obviously flat over A. Slightly less trivial is the fact that eA is Noetherian, that is, we have not passes outside our category. The proofs of these facts may be found in [Art62, III, 4].
The following propositions give two fundamental examples of Henseliza- tions.
Proposition A.6. Let A be normal. Let K be the field of fractions of A, and let Ks be a separable closure of K. The Galois group G of Ks over K acts on the integral closure B of A in Ks. Let n be a maximal ideal of B lying over m, and let D ⊂ G be the decomposition group of n, that is, D = {σ ∈ G | σ(n) = n}. Let eA be the localization at nD of the integral closure BD of A in KsD. Here
BD = {b ∈ B | σb = b for all σ ∈ D}
etc. Then eA is the Henselization of A.
Proof. See [Mil80, I, Example 4.10(a)].
Proposition A.7. Let k be a field, and let A be the localization of k[T1, . . . , Tn] at (T1, . . . , Tn). The Henselization of A is the set of power series P ∈ k[[T1, . . . , Tn]] that are algebraic over A.
Proof. See [Mil80, I, Example 4.10(b)]. (For a good discussion on why this should be so, see [Art71]. For a proof, see [Art73, II, 2.9].)
Every ring is a quotient of a normal ring, and so it would have sufficed to construct eA for A normal. This is the approach adopted by Nagata in [Nag62].
Let X be a scheme and let x ∈ X. An ´etale neighborhood of x is a pair (Y, y) where Y is an ´etale X-scheme and y is a point of Y mapping to x such that k(x) = k(y). The connected ´etale neighborhoods of x form a filtered system and clearly the limit lim
−→Γ(Y,OY) = eOX,x. This gives the definition of a Henselization of a scheme at a point, which serves as an analogue of the Zariski localization in the ´etale topology of schemes.
B Excellent Rings
We summarize here the definition and some basic properties concerning ex- cellent rings. The reader should consult [GD65, IV, 7.8] or [Mat70, Chapter 13] for a more complete treatment.
Definition B.1. A ring A is catenary if, for each pair of primes ideals p and q with p ⊂ q, the length of any maximal prime chain between p and q is finite and the same.
A ring A is universally catenary if A is Noetherian and every finite gen- erated A-algebra is catenary.
Definition B.2. Let A be a ring containing a field k. We say that A is geometrically regular over k if, for any finite extension k0 of k, the ring A ⊗kk0 is regular.
Definition B.3. Let A be a ring, and p a prime ideal of A. The formal fibers of Ap are the fibers of the canonical homomorphism Ap → ˆAp of Ap into its completion.
Definition B.4. A ring A is called excellent if it is Noetherian and satisfies the following conditions:
1. A is universally catenary. (Equivalently, for all prime ideal p of A, Ap is universally catenary.)
2. For any prime ideal p of A, the formal fibers of Ap are geometrically regular.
3. For any prime ideal p of A, and for any finite purely inseparable3 exten- sion K0 of the total quotient ring K(A/p) of A/p, there exists a finite A-algebra A0 whose quotient field is K0 such that A/p ⊂ A0 ⊂ K0 and the set of regular points of Spec(A0) contains a nontrivial open set.
Theorem B.5. (Several properties of excellent rings)
1. If A is a Noetherian local ring, then it is excellent if it satisfies the first two of the above conditions.
2. If A is excellent, then so are its localizations at prime ideals and every A-algebra of finite type.
3. A complete local ring (in particular a field) is excellent. A Dedekind domain whose field of fractions is of characteristic zero (in particular Z) is excellent.
Proof. See [GD65, IV, 7.8.3]. In fact, there is a much more complete list of properties of excellent rings.
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