POINTS HAVING THE SAME RESIDUE FIELD AS THEIR IMAGE UNDER A MORPHISM

BJORN POONEN

1. Main result

Our result, loosely speaking, is that in a nontrivial family of varieties f : X → Y over a perfect field k, some fiber X_{t} = f^{−1}(t) has a point rational over the field of definition of t. The precise statement, which is slightly more general, is given in Theorem 1 below. Denote by \overline{f(X)} the scheme-theoretic image of a morphism f : X → Y between noetherian schemes, and by κ(x) the residue field of a point x of a scheme. Theorem 1. Let X and Y be schemes of finite type over a field k. Let f : X → Y be a k-morphism such that dim \overline{f(X)} ≥ 1. Then there exists a closed point x ∈ X such that the extension κ(x) of κ(f(x)) is purely inseparable.

Proof. We begin with several straightforward reductions. If we cover Y with finitely many open affine subsets V, one of them must satisfy dim (V ∩ \overline{f(X)}) ≥ 1. Similarly, some open

affine subset U of f^{−1}(V) satisfies dim (V ∩ \overline{f(U)}) ≥ 1. By considering f|_{U} : U → V instead of f, we reduce to the case X = Spec A and Y = Spec B. Let ϕ : B → A correspond to f.

If f′ is a composition X′ →^{α} X →^{f} Y →^{β} Y′ of morphisms of schemes of finite type over k with dim \overline{f′(X′)} ≥ 1, and if we find a closed point x′ ∈ X′ with κ(x′) purely inseparable over κ(f′(x′)), then x = α(x′) ∈ X will do for f. For instance, composing X_{red} → X with f does not affect the dimension of the image in Y, so we may assume X is reduced. Some irreducible component of X will have positive-dimensional image in Y; hence we may assume X is integral.

Replacing Y by \overline{f(X)}, or equivalently B by ϕ(B), we may assume that ϕ is injective and dim Y ≥ 1. Since dim B = dim Y ≥ 1, the polynomial ring k[t] injects into B. Composing f with the associated morphism Y → A^{1}, we reduce to the case Y = A^{1}, B = k[t].

Let S = k[t] \ {0}, and let 𝔪 be a maximal ideal of S^{−1}A. Let A′ be the image of A in L := (S^{−1}A)/𝔪. The composition B = k[t] → A → A′ is still injective, so we may reduce to the case A = A′.

Now S^{−1}A = Frac(A) = L is both a field and a finitely generated k(t)-algebra, so [L : k(t)] < ∞ by the Nullstellensatz. Write k(t) ⊆ L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{r} = L with L_{0} separable over k(t) and L_{i+1} = L_{i}(u_{i}) purely inseparable of degree p over L_{i}, where p is the characteristic of k. By the Primitive Element Theorem, we may write L_{0} = k(t)(z).

Date: February 15, 2000.

1991 Mathematics Subject Classification. Primary 14A15; Secondary 14D99.

Key words and phrases. residue field.

This research was supported by NSF grant DMS-9801104, a Sloan Fellowship, and a Packard Fellowship.

This article has appeared in J. Algebra 243 (2001), 224–227. \begin{matrix}c \\ ◯ \end{matrix}2001 by Academic Press.

Multiplying z by a nonzero element of k[t], we may assume that the characteristic polynomial P (T) of z in L_{0} over k(t) has coefficients in B = k[t]. Let A_{i} = B[z, u_{1}, …, u_{i}], so S^{−1}A_{i} = L_{i}. Pick q ∈ B nonzero such that u_{i+1}^{p} ∈ A_{i}[q^{−1}] for each i and A_{r}[q^{−1}] = A[q^{−1}].

We claim that for some b ∈ B, b − z ∉ A_{0}[q^{−1}]^{∗}. It suffices to find b ∈ B such that P (b) = Norm_{L_{0}/k(t)}(b − z) is not a unit in B[q^{−1}]. Let T_{n} be the set of polynomials in t of exact degree n with coefficients in {0, 1}, so #T_{n} = 2^{n}. Let d = deg P. Then { P (b) : b ∈ T_{n} } consists of at least 2^{n}/d distinct polynomials, each monic of degree nd if n is larger than the t-degree of the coefficients of P. On the other hand, factoring q over \overline{k} shows that the number of monic polynomials of degree nd in B[q^{−1}]^{∗} is less than O ((nd)^{deg q}) as n → ∞. By taking n large, we find b ∈ T_{n} such that P (b) ∉ B[q^{−1}]^{∗}, and hence b − z ∉ A_{0}[q^{−1}]^{∗}.

Choose a maximal ideal 𝔫_{0} of A_{0}[q^{−1}] containing b − z. Since A_{i+1}[q^{−1}] has the form A_{i}[q^{−1}][U]/(U^{p} − α_{i}) for some α_{i} ∈ A_{i}[q^{−1}], there is a unique maximal ideal 𝔫 of A_{r}[q^{−1}] = A[q^{−1}] above 𝔫_{0}. Let x_{0} ∈ Spec A_{0}[q^{−1}] and x ∈ Spec A[q^{−1}] ⊆ Spec A = X be the corresponding closed points, so κ(x) is purely inseparable over κ(x_{0}). It remains to show that the extension κ(x_{0}) of κ(f(x)) is trivial. Let t̄ and b̄ = z̄ denote the images of t, b, and z in κ(x_{0}). Then κ(x_{0}) = k(t̄, z̄) = k(t̄, b̄) = k(t̄) = κ(f(x)). □

Remark. In Theorem 1, if in addition some dense open subset of X is smooth over its image in Y, then we can find a closed point x ∈ X with κ(x) = κ(f(x)): by (IV, 17.16.3(ii)) of [2] one can reduce to the case where X is étale over its image, and then by (IV, 17.6.1(a,c′)) of [2] all residue field extensions are separable. (When we write κ(x) = κ(f(x)), we mean that the field homomorphism κ(f(x)) → κ(x) induced by f is an isomorphism.) Corollary 2. Let X and Y be schemes of finite type over a perfect field k. Let f : X → Y be a k-morphism such that dim \overline{f(X)} ≥ 1. Then there exists a closed point x ∈ X such that κ(x) = κ(f(x)).

Proof. Theorem 1 provides x. The extension κ(x) of κ(f(x)) is automatically separable, since both fields are finite extensions of the perfect field k. □

Remark . Corollary 2 can fail for nonperfect k. Here is a counterexample. Let k_{0} be a perfect field of characteristic p, let k = k_{0}(s, t) where s, t are indeterminates, and let L = k_{0}(s^{1/p}, t^{1/p}). Let f : X → Y be the morphism of affine k-schemes associated to the inclusion k[z] ↪ L[z]. Suppose that there exists a closed point x ∈ X with [κ(x) : κ(f(x))] = 1. Let α ∈ \overline{L} be a root of the polynomial in L[z] generating the prime x. Then [L(α) : k(α)] = 1, so L ⊆ k(α). We obtain the contradiction

p^{2} = [L : kL^{p}] ≤ [k(α) : k · k(α)^{p}] = [k(α) : k(α^{p})] ≤ p.

(The first inequality is the case F = k(α) of the inequality [L : kL^{p}] ≤ [F : kF^{p}] for finite extensions of fields k ⊆ L ⊆ F of characteristic p: this follows from [F : L] = [F^{p} : L^{p}] ≥ [kF^{p} : kL^{p}] and [F : L][L : kL^{p}] = [F : kF^{p}][kF^{p} : kL^{p}].)

2. Arithmetic analogues

Theorem 4 below is an arithmetic analogue of Corollary 2. Lemma 3 is a special case of Theorem 4, and will be used to prove it. Lemma 3. Let f : X → Spec Z be a dominant morphism of finite type. Then there exists x ∈ X such that κ(x) = κ(f(x)) (or equivalently, such that κ(x) has prime order).

Proof. Mimic the proof of Theorem 1 with Z playing the role of k[t]. Eventually we reduce to the statement that if q is a nonzero integer, and P ∈ Z[T] is a nonconstant polynomial, then there exists b ∈ Z such that P (b) ∉ Z[q^{−1}]^{∗}. This holds, by a counting argument again: if deg P = d, then {P (1), …, P (n)} is a set of at least n/d distinct integers of absolute value O(n^{d}), but the number of integers in Z[q^{−1}] up to this bound grows like a power of log n only. □

Remark. Alternatively, after reducing to the case dim X = 1, one could invoke the Chebotarev Density Theorem. Of course, this would make the proof less elementary. Theorem 4. Let X and Y be schemes of finite type over Spec Z, and let f : X → Y be a morphism such that dim \overline{f(X)} ≥ 1. Then there exists a closed point x ∈ X such that κ(x) = κ(f(x)).

Proof. If X dominates Spec Z, use the x given by Lemma 3. Otherwise there are finitely many nonzero primes p of Z for which the fiber X_{p} of X → Spec Z is nonempty, so dim \overline{f(X_{p})} ≥ 1 for some p. Apply Corollary 2 to the morphism of fibers f_{p} : X_{p} → Y_{p} over F_{p} to find x. □

3. Application to Shafarevich-Tate groups in a family

The paper [1] constructs a nonisotrivial smooth proper family X → U of genus 1 curves over an open subset U of P_{Q}^{1}, such that for each t ∈ U(Q), the fiber X_{t} violates the Hasse principle. It also constructs a nonisotrivial smooth proper family Y → U of torsors of abelian surfaces over an open subset U of P_{Q}^{1} such that for every t ∈ U of odd degree over Q, Y_{t} violates the Hasse principle over the number field κ(t). In other words, these fibers represent nonzero elements of the Shafarevich-Tate groups of the associated abelian varieties. Corollary 2 shows that such results cannot be extended to all closed fibers of a family: there will always be a closed point t ∈ U above which the fiber has a point rational over κ(t).

Acknowledgement

I thank Laurent Moret-Bailly for many comments. In particular, the suggestion to prove Theorem 1 for not necessarily perfect fields and the suggestion to prove arithmetic analogues are due to him.

References

[1] J.-L. Colliot-Thélène and B. Poonen, Algebraic families of nonzero elements of Shafarevich-Tate groups, J. Amer. Math. Soc. 13 (2000), 83–99.

[2] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. No. 32, 1967.

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: poonen@math.berkeley.edu