Obtaining reduced fibres after normalized base change

an appendix to

Every rationally connected variety over the function field of a curve has a rational point

by A.J. de Jong, MIT and J. Starr, MIT

Introduction

In the original preprint version the authors gave an argument which proves Proposition 2.1 below without referring to the article by Bosch, Lütkebohmert and Raynaud. Since the authors are rather fond of this argument, they provide this web-based version of it for the enjoyment of its audience.

Acknowledgements. Part of this research was conducted by A.J. de Jong for the Clay Mathematics Institute. A.J. de Jong was partially supported by an NSF grant. The authors greatfully acknowledge the help of the referee in pointing out some inconsistencies in the earlier writeup of this argument.

Obtaining reduced fibres after normalized base change

Suppose that f : P → X is a flat morphism of a normal variety P to an irreducible smooth projective curve X over k. We will assume that the following equivalent conditions hold: (a) the geometric generic fibre is reduced, (b) the general fibre is reduced, and (c) the smooth locus of the morphism f is dense in P.

We note that for x ∈ X(k) the fibre P_{x} = f^{−1}(x) is a scheme satisfying S_{1} (every local ring of dimension ≥ 1 had depth ≥ 1). Thus P_{x} is reduced if and only if P_{x} is generically reduced, i.e., all local rings of P_{x} at maximal points are fields.

Let π : Y → X be a finite morphism of another irreducible smooth curve Y to X. Set

P_{Y} = (P ×_{X} Y)^{normalized}.

In this section we prove a technical result that is similar to Lemma 2.3 in [2]. Namely we prove there is at least one finite generically étale morphism Y → X such that P_{Y} → Y has reduced fibres, i.e., such that the smooth locus of P_{Y} → Y is dense in every fibre.

2.1. Proposition. With the assumptions and notations as above. There exists a π : Y → X such that all fibres of P_{Y} → Y are reduced and such that π is generically étale.

Proof. Let x_{1}, …, x_{r} ∈ X(k) be the points of X such that P_{x_{i}} = f^{−1}(x_{i}) is not reduced. Let R_{i} be the complete local ring of x_{i} on X. For each generic point ξ_{ij} of the fibre P_{x_{i}} let O_{ij} = O_{P,ξ_{ij}}. By [2, Lemma 2.3] there exist finite extensions of discrete valuation rings R_{i} ⊂ R′_{ij} such that the normalization O′_{ij} of O_{ij} ⊗_{R_{i}} R′_{ij} is formally smooth over R′_{ij}. By Lemma 2.3 below, there exist finite generically separable extensions R_{i} ⊂ R′′_{ij} such that the normalization

O′′_{ij} = (O_{ij} ⊗_{R_{i}} R′′_{ij})^{normalized}

is formally smooth over R′′_{ij}. For each i we can pick a finite generically étale extension of discrete valuation rings R_{i} ⊂ R′_{i} dominating the extensions R_{i} ⊂ R′′_{ij} for all j. These extensions then have the property that the normalization of P ×_{X} Spec R′_{i} has reduced special fibre. Clearly we may choose our R′_{i} such that the field extensions K_{i} = f.f.(R_{i}) ⊂ f.f.(R′_{i}) = L_{i} have a degree d independent of i.

Let K = k(X). We claim that there exists a separable field extension K ⊂ L of degree d such that L ⊗_{K} K_{i} is isomorphic to L_{i}. To see this we write L_{i} = K_{i}(α_{i}) and we let f_{i} ∈ K_{i}[X] be the minimal polynomial of α_{i} over K_{i}. We choose a polynomial f(X) ∈ K[X] which approximates f_{i} simultaneously for all i and we set L = K[X]/(f). By Krasner’s lemma we’ll have L ⊗_{K} K_{i} ≅ L_{i} if the approximation is good enough, see our Lemma 1.4 (b). □

In the arguments below the notation rad R indicates the intersection of the maximal ideals of the ring R.

2.2. Lemma. (a)Let R be an excellent Dedekind domain with a finite number of maximal ideals. Let K ⊂ K′ be a finite algebra extension of the field of fractions K = f.f.(R) so that K′ is a product of fields. Let R′ be the integral closure of R in K′. For any N ≥ 1 there exists a finite separable algebra K ⊂ K′′ such that, if R′′ denotes the integral closure of R in K′′, we have

R′ ⊗_{R} R/(rad R)^{N} ≅ R′′ ⊗_{R} R/(rad R)^{N}

as R-algebras.

(b) Assumptions as in (a) where in addition we suppose that K′ = K[α_{1}, …, α_{n}] with α_{i} ∈ R′ and that we are given for each t ∈ {1, …, n} an equation

f_{t}(X_{1}, …, X_{t}) = X_{t}^{d_{t}} + ∑_{i_{j}<d_{j}} a_{i_{1}···i_{t}}X_{1}^{i_{1}} ⋯ X_{t}^{i_{t}}

such that [K′ : K] = d_{1} … d_{n}, and a_{i_{1}···i_{t}} ∈ R. Then: Any sufficiently good approximation of the coefficients a_{i_{1}···i_{t}} by b_{i_{1}···i_{t}} in R such that

R[β_{1}, …, β_{n}] := R[X_{1}, …, X_{n}]/(X_{t}^{d_{t}} + ∑_{i_{j}<d_{j}} b_{i_{1}···i_{t}}X_{1}^{i_{1}} ⋯ X_{t}^{i_{t}})

is generically étale over R will lead to a solution of (a) by setting K′′ = K(β_{1}, …, β_{n}).

(c) In the situation of (b), for every M > 0, there exist choices of the b_{i_{1}···i_{t}} ∈ R such that K′′ is separable over K, and a_{i_{1}···i_{t}} − b_{i_{1}···i_{t}} ∈ 𝔪_{R}^{M}.

Proof. Clearly we may write K′ = ∏ K′_{i} as a product of finite field extensions, and it suffices to prove the lemma for each extension K ⊂ K′_{i}. Hence to prove part (a) we assume from now on that K′ is a field. Pick an element t ∈ rad R such that dim R/(t) = 0, i.e., rad R = \sqrt{(}t). Write K′ = K(α_{1}, …, α_{n}). Let d_{1} be the degree of α_{1} over K, let d_{2} be the degree of α_{2} over K(α_{1}), and so on. Thus [K′ : K] = d_{1} … d_{n}. We will let f_{1}(X_{1}) = X_{1}^{d_{1}} + ∑_{i<d_{1}} a_{i}X_{1}^{i} ∈ K[X_{1}] be the minimal polynomial of α_{1} over K, let f_{2}(X_{1}, X_{2}) = X_{2}^{d_{2}} +∑_{i_{1}<d_{1},i_{2}<d_{2}} a_{i_{1}i_{2}}X_{1}^{i_{1}}X_{2}^{i_{2}} ∈ K[X_{1}, X_{2}] be a polynomial such that f_{2}(α_{1}, X_{2}) is the minimal polynomial of α_{2} over K(α_{1}). Continuing like this, we obtain f_{1}, …, f_{n} with

f_{t}(X_{1}, …, X_{t}) = X_{t}^{d_{t}} + ∑_{i_{j}<d_{j}} a_{i_{1}···i_{t}}X_{1}^{i_{1}} ⋯ X_{t}^{i_{t}}

and a_{i_{1}···i_{t}} ∈ K such that f_{t}(α_{1}, …, α_{t−1}, X_{t}) is the minimal polynomial of α_{t} over K(α_{1}, …, α_{t−1}). Changing our choice of α_{i} into t^{c_{i}}α_{i} for suitable c_{i} we may assume that α_{i} ∈ R′ ∀i and a_{i_{1}···i_{t}} ∈ R for all t and all (i_{1}, …, i_{t}). Thus we see we have reduced part (a) to part (b).

Next, we let g_{t}(X_{1}, …, X_{t}) = X_{t}^{d_{t}} +∑_{i_{j}<d_{j}} b_{i_{1}···i_{t}}X_{1}^{i_{1}} ⋯ X_{t}^{i_{t}}, where the coefficients b_{i_{1}···i_{t}} are indeterminates for the moment. The algebra 𝔽_{p}[b_{i_{1}···i_{t}}, X_{1}, …, X_{n}]/(g_{1}, …, g_{n}) is finite flat over 𝔽_{p}[b_{i_{1}···i_{t}}]. The induced morphism on Spec is generically étale. Thus for any integer M there exist choices of b_{i_{1}···i_{t}} ∈ R so that a_{i_{1}···i_{t}} − b_{i_{1}···i_{t}} ∈ t^{M}R and such that

R[β_{1}, …, β_{n}] := R[X_{1}, …, X_{n}]/(g_{1}, …, g_{n})

is generically étale (follows by a simple computation). In other words K ⊂ K(β_{1}, …, β_{n}) =: K′′ is a separable algebra extension. (This proves (c).) By construction there exists an isomorphism

(*)  R[β_{1}, …, β_{n}] ⊗_{R} R/(t^{M}) ≅ R[α_{1}, …, α_{n}] ⊗_{R} R/(t^{M}).

Note that R[α_{1}, …, α_{n}] ⊂ R′. Pick α_{n+1}, …, α_{n+m} ∈ R′ which generate R′ over R[α_{1}, …, α_{n}]. There is an integer c such that t^{c}α_{ℓ} ∈ R[α_{1}, …, α_{n}] for all n + 1 ≤ ℓ ≤ n + m. Write, for n + 1 ≤ ℓ ≤ n + m, t^{c}α_{ℓ} = ∑_{i_{j}<d_{j}} A_{ℓ,i_{1}···i_{n}}α_{1}^{i_{1}} ⋯ α_{n}^{i_{n}} with A_{ℓ,i_{1}···i_{n}} ∈ R and set

β_{ℓ} := t^{−c}( ∑_{i_{j}<d_{j}} A_{ℓ,i_{1}···i_{n}}α_{1}^{i_{1}} ⋯ β_{n}^{i_{n}}) ∈ K′′.

Let P ∈ R[X_{1}, …, X_{n+m}] be a polynomial of total degree d such that P (α_{1}, …, α_{n+m}) = 0.

Claim.

P (β_{1}, …, β_{n+m}) ∈ t^{M−dc}R[β_{1}, …, β_{n}].

We postpone the proof of the claim for the moment.

Suppose the claim holds. As each α_{ℓ}, n + 1 ≤ ℓ ≤ n + m is integral over R, we see that there is a monic polynomial P_{ℓ} ∈ R[X] such that P_{ℓ}(α_{ℓ}) = 0. Say deg P_{ℓ} = d_{ℓ}. Then, if M ≥ d_{ℓ}c, we see that P_{ℓ}(β_{ℓ}) ∈ R[β_{1}, …, β_{n}]. Hence β_{ℓ} is integral over R[β_{1}, …, β_{n}]. So R[β_{1}, …, β_{n+m}] is finite over R if M ≥ max{d_{ℓ}c}. Also, if Q_{1}, …, Q_{s} ∈ R[X_{1}, …, X_{n+m}] generate the ideal ker (R[\underline{X}] → R[\underline{α}]), and if M ≥ max\{c deg Q_{j}+N\}, then Q_{j}(β_{1}, …, β_{n+m}) ∈ t^{N}R[β_{1}, …, β_{n+m}]. Therefore, if M ≥ max{d_{ℓ}c, c deg Q_{j} + N} the map

(**)  \begin{matrix}R[β_{1}, …, β_{n}] ⊗_{R} R/(t^{M}) & ⟶ & R[α_{1}, …, α_{n}] ⊗_{R} R/(t^{M}), \\ α_{i} & ⟼ & β_{i} \end{matrix}

is well defined and surjective; by considering lengths it is an isomorphism.

Let 𝔭 ⊂ R[β_{1}, …, β_{n+m}] be a maximal ideal. Since t ∈ 𝔭 we see that 𝔭 corresponds, via the isomorphism (∗∗), to a maximal ideal 𝔮 ⊂ R′ = R[α_{1}, …, α_{n+m}]. If N ≥ 2 (which we may assume) then the isomorphism (∗∗) induces an isomorphism 𝔭/𝔭^{2} ≅ 𝔮/𝔮^{2}. Therefore R[\underline{β}]_{𝔭} is a discrete valuation ring, and hence R[\underline{β}] = R′′ and we are done.

It rests to prove the claim. To see this we write P (X_{1}, …, X_{n+m}) = ∑_{I=(i_{j})} a_{I}X_{1}^{i_{1}} ⋯ X_{n+m}^{i_{n+m}}. Since R[β_{1}, …, β_{n}] is finite flat over R it suffices to prove that t^{cd}P (β_{1}, …, β_{n+m}) ∈ t^{M}R[β_{1}, …, β_{n}]. We compute

t^{cd}P (β_{1}, …, β_{n+m}) = ∑_{I=(i_{j})} t^{(d−i_{n+1}−...−i_{n+m})c}a_{I}β_{1}^{i_{1}} ⋯ β_{n}^{i_{n}}(t^{c}β_{n+1})^{i_{n+1}} ⋯ (t^{c}β_{n+m})^{i_{n+m}}.

Recall that t^{c}β_{ℓ} ∈ R[β_{1}, …, β_{n}], n + 1 ≤ ℓ ≤ n + m map to their counterparts under the map (∗). Thus the claim follows. □

2.3. Lemma. Let R ⊂ O be a local homomorphism of excellent discrete valuation rings. Assume that the extension of fraction fields K = f.f.(R) ⊂ L = f.f.(O) is separable. Suppose that R ⊂ R′ is an extension of discrete valuation rings with K ⊂ K′ = f.f.(R′) finite. Let O′ be the integral closure of O in L′ = L ⊗_{K} K′. Finally, assume that O′ is formally smooth over R′, i.e., the localizations O′_{i} of O′ at its maximal ideals are discrete valuation rings with ramification index e(O′_{i}/R′) = 1 and inducing separable residue field extensions R′/𝔪′ ⊂ O′_{i}/𝔪′_{i}. Then there exists another extension R ⊂ R′′ for which the conclusions hold, and such that in addition K ⊂ K′′ is separable.

Proof. Write K′ = K(α_{1}, …, α_{n}) and R′ = R[α_{1}, …, α_{n+m}] as in the proof of Lemma 2.2. Next, let R ⊂ R[β_{1}, …, β_{n}] be constructed as in Lemma 2.2 (b), with b_{i_{1}···i_{t}} − a_{i_{1}···i_{t}} ∈ 𝔪_{R}^{M}. As in the proof of Lemma 2.2 we can make our choices such that the algebra K′′ = K[β_{1}, …, β_{n}] is separable over K.

Note that L′ is a product of fields as K ⊂ L is separable. Note that L′ = L[α_{1}, …, α_{n}], where the equations for the α_{i} over L[α_{1}, …, α_{i−1}] are the same as the equations of α_{i} over K[α_{1}, …, α_{i−1}]. Hence, by part (b) of Lemma 2.2 when M is large enough, we see that the conclusion of Lemma 2.2 (a) applies to both R′′ (the integral closure of R in K′′) and to O′′ (the integral closure of O in Ł′′).We conclude that, if the integer M is chosen large enough, we have R′′/(𝔪′′)^{2} ≅ R′/(𝔪′)^{2} and O′′/(rad O′′)^{2} ≅ O′/(rad O′)^{2}. Thus all the desired properties of the extension R′′ ⊂ O′′ can be read off from the corresponding properties of R′ ⊂ O′. If R′′ is not a discrete valuation ring, then we replace R′′ by a localization. □

References

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