A NOTE ON LOCALLY NILPOTENT DERIVATIONS AND VARIABLES OF k[X, Y, Z]

DANIEL DAIGLE AND SHULIM KALIMAN

Abstract. We strengthen certain results concerning actions of (ℂ, +) on ℂ^{3} and embeddings of ℂ^{2} in ℂ^{3}, and show that these results are in fact valid over any field of characteristic zero.

Introduction

The Lefschetz principle [16] suggests that any result, which has been proven over the field ℂ of complex numbers and which involves a finite number of points and of varieties, remains valid over any universal domain (i.e. over an algebraically closed field with infinite transcendence degree over the prime field) of characteristic zero. In this form the principle was proven by Eklof in [4]. Furthermore, any statement of the first order predicate calculus true over ℂ is valid over any algebraically closed field of characteristic zero, without restriction on the transcendence degree (e.g., see [12]). However, the situation changes when one tries to extend results to all fields of characteristic zero, which is what the present paper does for the following results from [1], [6] and [5] respectively:

(1) Let B = ℂ[X, Y, Z], let D : B → B be a locally nilpotent derivation and let A = ker D. Then the map Spec B → Spec A, determined by the inclusion A ↪ B, is surjective.

(2) Every free algebraic ℂ_{+}-action on ℂ^{3} is a translation in a suitable polynomial coordinate system.

(3) Every polynomial f ∈ ℂ[X, Y, Z] whose general fiber is isomorphic to ℂ^{2} is a variable of ℂ[X, Y, Z].

Our ability to go beyond the Lefschetz principle in (1) and (3) is essentially based on Kambayashi’s theorem (see 1.2) in combination with other serious results. We also present slightly stronger versions of results (1) and (3) and we show (see 1.7, 2.8 and 2.11) that under some additional assumptions the analogue of Kambayashi’s theorem holds in dimension 3.

In the stronger version of (3) the assumption on the general fiber is replaced by the condition that there exists a Zariski-dense subset U of ℂ such that the fiber of f over any point of U is an affine plane. We shall verify in Section 2 that this weaker hypothesis implies the stronger one, but it is worth mentioning that this implication is also a corollary to the fact (proved in [3]) that for any morphism φ : X → Y of algebraic varieties, { y ∈ Y | φ^{−1}(y) ≅ 𝔸^{2} } is a constructible subset of Y. Actually, consider the following more general question: If φ : X → Y is a morphism of algebraic varieties and S is an affine algebraic surface, is { y ∈ Y′ | φ^{−1}(y) ≅ S } a constructible subset of Y′ (where Y′ is the set of closed points of Y)? Classification of surfaces S with this property will be given in our subsequent paper [3].

2000 Mathematics Subject Classification. Primary: 14R10. Secondary: 14R20, 14R25, 13N15.

Key words and phrases. Locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space.

Research supported by a grant from NSERC Canada.

Research supported by the NSA grant MDA904-03-1-0009.

1. Locally nilpotent derivations of k[X, Y, Z]

Definitions.

(1) If B is an algebra over a ring A, then the notation B = A^{[n]} means that B is A-isomorphic to a polynomial ring in n variables over A.

(2) Let B be a ring. A derivation D : B → B is locally nilpotent if for each b ∈ B there exists an integer r > 0 satisfying D^{r}(b) = 0.

(3) A subring A of an integral domain B is called an inert subring of B if the conditions x, y ∈ B \ {0} and xy ∈ A imply x, y ∈ A. It is well-known that if B is an integral domain of characteristic zero and D : B → B is a locally nilpotent derivation then ker(D) is an inert subring of B; it follows that ker(D) contains all units of B and, consequently, that if k is any field contained in B then D is a k-derivation.

We begin with the following simple observation:

1.1. Lemma. Let k′/k be a field extension, B a k-algebra and B′ = k′ ⊗_{k} B.

(1) The map Spec B′ → Spec B, corresponding to the natural map B ↪ B′, is surjective.

(2) Suppose that D′ : B′ → B′ is a k′-derivation, D : B → B is a k-derivation and D′ extends D. Then ker(D′) ≅ k′ ⊗_{k} ker(D). Consequently, Spec(ker D′) → Spec(ker D) is surjective.

Proof. Since k′ is free over k, B′ is a free, hence faithfully flat, B-module; assertion (1) follows (see for instance [8], p. 28). Applying the exact functor k′ ⊗_{k} to the exact sequence 0 → ker(D) → B ⟶^{D} B gives ker(D′) ≅ k′ ⊗_{k} ker(D). Then assertion (1) implies that Spec(ker D′) → Spec(ker D) is surjective. □

We need the following result of Kambayashi (see [7], or [13] for a different proof):

1.2. Let k′/k be a separable field extension and let A be a k-algebra. If k′ ⊗_{k} A = k′^{[2]} then A = k^{[2]}.

We also require the following result of Miyanishi:

1.3. Let k be a field of characteristic zero, B = k[X, Y, Z] = k^{[3]} and 0 ≠ D : B → B a locally nilpotent derivation. Then ker D = k^{[2]}.

Actually, Miyanishi proved the case k = ℂ in [9] and the general case 1.3 follows from Miyanishi’s result by a straightforward application of Kambayashi’s result 1.2. We include the proof for the sake of completeness.

Proof of 1.3. Choose a finite subset F of k which contains all coefficients of the polynomials DX, DY and DZ and define the field k_{0} = ℚ(F) and the algebra B_{0} = k_{0}[X, Y, Z]; then D restricts to a nonzero locally nilpotent derivation D_{0} : B_{0} → B_{0}. Since ker D = k ⊗_{k_{0}} ker D_{0} by 1.1, it suffices to show that ker D_{0} = k_{0}^{[2]}. Since k_{0} is isomorphic to a subfield of ℂ, it suffices to prove 1.3 in the case where k ⊆ ℂ.

Assume that k ⊆ ℂ, let B′ = ℂ[X, Y, Z] = ℂ^{[3]} and consider the extension D′ : B′ → B′ of D. By Miyanishi’s result we have ker(D′) = ℂ^{[2]}, so 1.1 gives ℂ ⊗_{k} ker(D) = ℂ^{[2]}. Then Kambayashi’s result 1.2 gives ker D = k^{[2]}. □

Next, we prove the following generalization of Bonnet’s result:

Theorem 1. Let k be a field of characteristic zero, B = k[X, Y, Z] = k^{[3]}, D : B → B a locally nilpotent derivation and A = ker D. Then:

(1) B is faithfully flat as an A-module.

(2) The ideal A ∩ D(B) of A is principal.

Remarks.

(1) Faithful flatness has the following standard consequences (see [8], pp. 28 and 46):

(a) The map π : Spec B → Spec A, determined by the inclusion A ↪ B, is submersive (i.e., π is surjective and a subset E of Spec A is closed iff π^{−1}(E) is closed).

(b) Each ideal I of A satisfies A ∩ IB = I.

(2) The fact that B is flat over A is well-known, for any field k of characteristic zero (if D ≠ 0, one can show that every nonempty fiber of π has dimension 1). Thus: π is also an open map.

(3) Bonnet also proved (but he did not publish this part) that the ideal A∩D(B) is principal when k = ℂ; our proof of this fact is a simplification of his argument.

Proof of Theorem 1

Consider the following condition on a field k:

(ℬ) Let B = k^{[3]}, then for every locally nilpotent derivation D : B → B the map

Spec B → Spec(ker D) is surjective. As mentioned in the introduction, P. Bonnet proved:

1.4. ℂ satisfies (ℬ).

We begin with:

1.5. If k′ is a field of characteristic zero which satisfies (ℬ), then every subfield of k′ satisfies (ℬ).

Proof. Let k be a subfield of k′, X, Y, Z indeterminates over k′ and

B = k[X, Y, Z] ⊆ B′ = k′[X, Y, Z].

Consider a locally nilpotent derivation D : B → B and let A = ker D. Note that D extends uniquely to a locally nilpotent derivation D′ : B′ → B′. Let A′ = ker D′. In the commutative diagram

Spec B ⟵^{β} Spec B′
π↓  _{π′}↓
Spec A ⟵^{α} Spec A′

the map π′ is surjective because k′ satisfies (ℬ) by assumption, and α is surjective by 1.1. It follows that π is surjective, as desired. □

1.6. Every field of characteristic zero satisfies (ℬ).

Proof. By 1.5, it suffices to prove that every algebraically closed field of characteristic zero satisfies (ℬ). This follows from 1.4 and Lefschetz Principle, but we include the proof for the sake of completeness (actually, verifying that the hypothesis of Eklof’s Theorem [4] is satisfied is roughly equivalent to the following argument).

Let k be an algebraically closed field of characteristic zero, B = k[X, Y, Z] = k^{[3]} and D : B → B a locally nilpotent derivation; let A = ker D. In order to show that π : Spec B → Spec A is surjective, we may assume that D ≠ 0; then A = k^{[2]} by 1.3. Let 𝔭 ∈ Spec A. If 𝔭 = 0 then clearly 𝔭 is in the image of π. If ht 𝔭 = 1 then 𝔭 = pA for some prime element p of A; since A is an inert subring of B, it follows that A∩pB = pA and that pB is a prime ideal of B, so again 𝔭 is in the image of π. So we may assume that 𝔭 is a maximal ideal of A. Since A = k^{[2]} and k is algebraically closed, we may choose f, g such that A = k[f, g] and 𝔭 = (f, g)A.

Consider a finite subset F of k which contains all coefficients of DX, DY, DZ, f and g; let k_{1} = ℚ(F) and B_{1} = k_{1}[X, Y, Z] ⊆ B. Then D restricts to a nonzero locally nilpotent derivation D_{1} : B_{1} → B_{1}; let A_{1} = ker D_{1}, then A_{1} = k_{1}^{[2]} by 1.3; actually we have A_{1} = k_{1}[f, g], because f, g ∈ B_{1}.

Spec B_{1} ⟵^{β} Spec B
π_{1}↓  ^{π}↓
Spec A_{1} ⟵^{α} Spec A

Note that β is surjective by part (1) of 1.1. Since k_{1} is a finitely generated field extension of ℚ, it is isomorphic to some subfield of ℂ. By 1.4 and 1.5, it follows that k_{1} satisfies (ℬ), so π_{1} is surjective. Consequently, α ∘ π is surjective. Let 𝔭_{1} = (f, g)A_{1} ∈ Spec(A_{1}), then the fibre of α over 𝔭_{1} is {𝔭} (because 𝔭_{1}A = 𝔭 is maximal). This and the fact that α ∘ π is surjective imply that 𝔭 belongs to the image of π, which shows that π is surjective and consequently that k satisfies (ℬ). □

We may now finish the proof of Theorem 1.

Let k, B, D and A be as in the statement of the theorem and let π : Spec B → Spec A be the map determined by A ↪ B. Since B is flat over A and (by 1.6) π is surjective, it follows that B is faithfully flat over A (e.g., [8] p. 28, Theorem 3). As mentioned in the remark following the statement of the theorem, it also follows that each ideal I of A satisfies A ∩ IB = I.

Next, we show that if f_{1}, f_{2} are nonzero elements of A ∩ D(B), then gcd(f_{1}, f_{2}) ∈ A ∩ D(B); of course, this implies that A ∩ D(B) is a principal ideal. (Remark: Given x, y ∈ A, gcd_{A}(x, y) = gcd_{B}(x, y) because A is an inert subring of B.)

For each i ∈ {1, 2}, choose s_{i} ∈ B such that D(s_{i}) = f_{i}. Define I = (f_{1}, f_{2})A then clearly s_{1}f_{2} − s_{2}f_{1} ∈ IB; since D(s_{1}f_{2} − s_{2}f_{1}) = 0, we have in fact s_{1}f_{2} − s_{2}f_{1} ∈ A ∩ IB = I, so there exist a_{1}, a_{2} ∈ A such that s_{1}f_{2} − s_{2}f_{1} = a_{1}f_{2} − a_{2}f_{1}. Thus (s_{1} − a_{1})f_{2} = (s_{2} − a_{2})f_{1}. Write f_{i} = f′_{i}d where d = gcd(f_{1}, f_{2}), then (s_{1} − a_{1})f′_{2} = (s_{2} − a_{2})f′_{1} and gcd(f′_{1}, f′_{2}) = 1. So if we define

s = \frac{s_{1} − a_{1}}{f′_{1}} = \frac{s_{2} − a_{2}}{f′_{2}}

then s ∈ B and clearly Ds = d ∈ A, whence d ∈ A ∩ D(B). This shows that A ∩ D(B) is a principal ideal of A.

The proof of Theorem 1 is complete. □

Free locally nilpotent derivations of k[X, Y, Z]

Definition. Let B be a ring. A locally nilpotent derivation D : B → B is free if the ideal of B generated by D(B) is equal to B.

Consider the case B = k^{[3]}, where k is a field of characteristic zero. There is a natural bijective correspondence between the set of locally nilpotent derivations of B and the set of G_{a}-actions on Spec(B) = 𝔸_{k}^{3} (e.g., see [11]). Consider an action α and the corresponding derivation D; the following facts are well-known:

(1) A maximal ideal 𝔪 of B is a fixed point of α if and only if D(B) ⊆ 𝔪; so D is free if and only if α is free of fixed points. As an example consider the locally nilpotent derivation D of ℚ[X, Y, Z] given by D(X) = D(Y) = 0 and D(Z) = X^{2} − 2; then D is not free and the corresponding action on 𝔸_{ℚ}^{3} has many fixed points, but no fixed point is in ℚ^{3}.

(2) The condition D(B) = B is equivalent to α being a translation in a suitable polynomial coordinate system of 𝔸_{k}^{3}.

Theorem 2. If k is a field of characteristic zero and D a free locally nilpotent derivation of B = k^{[3]}, then D(B) = B. (Equivalently, every fixed point free G_{a}-action on 𝔸_{k}^{3} is a translation in a suitable polynomial coordinate system.)

Proof. By [6] the case k = ℂ is true, so [4] implies that the theorem is true whenever k is a universal domain of characteristic zero. Suppose that k, B and D satisfy the hypothesis of the theorem. Consider a field extension k′/k where k′ is a universal domain. Then D extends uniquely to a locally nilpotent derivation D′ on B′ = k′ ⊗_{k} B = (k′)^{[3]}. As D is free, it is clear that D′ is free; since the theorem is true for k′, D′ : B′ → B′ is a surjective map. Note that D′ is obtained by applying the functor k′ ⊗_{k} to D; since k′ is a free (hence faithfully flat) k-module, D : B → B must be surjective. □

1.7. Corollary. Let B be an algebra over a field k of characteristic zero such that B ⊗_{k} k′ ≃ (k′)^{[3]} for some field extension k′/k. If D : B → B is a free locally nilpotent derivation then B = k^{[3]} and D(B) = B.

Proof. Extend D to a locally nilpotent derivation D′ of B′ = k′ ⊗_{k} B = (k′)^{[3]}. Then D′ is free, so D′ is surjective by Theorem 2 and D(B) = B by faithful flatness. By Proposition 2.1 of [17], the surjectivity of D implies that B = A^{[1]}, where A = ker(D). We have k′ ⊗_{k} A = ker(D′), and ker(D′) = (k′)^{[2]} by 1.3; so A = k^{[2]} by Kambayashi’s result 1.2 and this implies that B = k^{[3]}. □

2. Variables of k^{[3]}

Definition. Let k be a field, B = k^{[3]} and f ∈ B. We say that f is a variable of B if B = k[f, g, h] for some g, h.

2.1. Notation. Suppose that R is a subring of a ring B. If 𝔭 ∈ Spec R, we write κ(𝔭) = R_{𝔭}/𝔭R_{𝔭} for the residue field at 𝔭 and we consider the κ(𝔭)-algebra B ⊗_{R} κ(𝔭). Define

U(R, B) = { 𝔭 ∈ Spec(R) | B ⊗_{R} κ(𝔭) = κ(𝔭)^{[2]} }.

Suppose that f ∈ B = k^{[3]} is such that U(k[f], B) ≠ ∅, i.e., at least one fiber of f is an affine plane; then the Abhyankar-Sathaye Embedding Conjecture asserts that f is a variable of B. We shall prove a weaker statement, namely:

Theorem 3. Let B = k^{[3]}, where k is a field of characteristic 0. If f ∈ B is such that U(k[f], B) is a dense subset of Spec k[f], then f is a variable of B.

Let us first recall some known facts (2.2, 2.3 and 2.4):

2.2. Let f ∈ B = ℂ^{[3]} and suppose that one of the following conditions is satisfied:

(1) The zero ideal of ℂ[f] belongs to U(ℂ[f], B) (i.e., the generic fiber of f is an affine plane);

(2) U(ℂ[f], B) contains all closed points of Spec ℂ[f], except possibly a finite num-

ber of them (i.e., the general closed fiber of f is an affine plane). Then f is a variable of B.

2.3. Let φ : X → Y be a morphism of complex irreducible quasi-projective algebraic varieties. Then there exists a Zariski-dense open subset Y_{0} of Y so that the fibers of φ over any two closed points of Y_{0} are homeomorphic in the standard topology.

2.4. Let S be a smooth complex algebraic variety homeomorphic to ℝ^{4} in the standard topology. Then S is isomorphic to 𝔸_{ℂ}^{2} as an algebraic variety.

Case (2) of 2.2 is given in [5]; it follows that case (1) also holds, as it is easy to see that if the generic fiber of f is an affine plane then so is the general fiber. Statement 2.3 is a special case of the Varchenko equisingularity theorem (Theorem 5.2 in [15]) and 2.4 is a consequence of Ramanujam’s result [10]. Now 2.2–2.4 imply:

2.5. Theorem 3 is valid in the case k = ℂ.

So our task consists in showing that validity over ℂ implies validity over any field of characteristic zero. First note that Kambayashi’s result 1.2 has the following consequence:

2.6. Lemma. Let k′/k be an extension of fields of characteristic zero, let R ⊂ A be k-algebras and define R′ = k′ ⊗_{k} R and A′ = k′ ⊗_{k} A. Then

U(R′, A′) = π^{−1}(U(R, A)),

where π : Spec(R′) → Spec(R) is the surjective map determined by R ↪ R′.

Proof. Since R′ is a free R-module it is in particular faithfully flat and consequently π is surjective. Let 𝔮 ∈ Spec(R′); it suffices to show:

(1)  𝔮 ∈ U(R′, A′) ⟺ π(𝔮) ∈ U(R, A).

Let 𝔭 = π(𝔮) ∈ Spec(R) and write κ = κ(𝔭) = R_{𝔭}/𝔭R_{𝔭} and κ′ = κ(𝔮) = R′_{𝔮}/𝔮R′_{𝔮}; note that κ′ is a separable extension of κ. We have:

κ′ ⊗_{κ} (κ ⊗_{R} A) = κ′ ⊗_{R} A = κ′ ⊗_{R′} R′ ⊗_{R} A = κ′ ⊗_{R′} (k′ ⊗_{k} R ⊗_{R} A) = κ′ ⊗_{R′} (k′ ⊗_{k} A) = κ′ ⊗_{R′} A′.

Thus, by 1.2, κ′ ⊗_{R′} A′ = κ′^{[2]} if and only if κ⊗_{R} A = κ^{[2]}; in other words, (1) holds. □

We thank Sathaye for pointing out the following fact to us:

2.7 (Sathaye). Let B be a regular affine domain over a field k of characteristic zero. If f ∈ B and U(k[f], B) = Spec k[f], then B = k^{[3]} and f is a variable of B.

Proof. If 𝔪 is a maximal ideal of R = k[f] then the assumption implies that U(R_{𝔪}, B_{𝔪}) = Spec R_{𝔪}, so the main result of [14] gives B_{𝔪} = R_{𝔪}^{[2]}. Then [2] implies that B is the symmetric algebra of a finitely generated projective R-module P; but R = k^{[1]} implies that P is free and hence that B = R^{[2]}. □

Combining the results of Kambayashi and Sathaye, we obtain:

2.8. Proposition. Let k′/k be an extension of fields of characteristic zero, let B be a k-algebra and let f ∈ B. Suppose that k′ ⊗_{k} B = (k′)^{[3]} and that f is a variable of k′ ⊗_{k} B. Then B = k^{[3]} and f is a variable of B.

Proof. Let R = k[f], B′ = k′ ⊗_{k} B and R′ = k′[f] = k′ ⊗_{k} R. Since f is a variable of B′, we have B′ = R′^{[2]} so in particular U(R′, B′) = Spec(R′); then U(R, B) = Spec(R) by 2.6. Since the assumption k′ ⊗_{k} B = (k′)^{[3]} implies that B is a regular affine domain over k, the desired conclusion follows from Sathaye’s result 2.7. □

It immediately follows:

2.9. Corollary. Let k′/k be an extension of fields of characteristic zero, let X, Y, Z be indeterminates over k′ and let f ∈ k[X, Y, Z] ⊆ k′[X, Y, Z]. Then:

f is a variable of k[X, Y, Z] ⟺ f is a variable of k′[X, Y, Z].

We make the trivial observation:

2.10. Let k′/k be an extension of fields, let T be an indeterminate and let π : Spec k′[T] → Spec k[T] be the map defined by π(𝔮) = 𝔮 ∩ k[T]. For any subset E of Spec k[T],

E is dense in Spec k[T] ⟺ π^{−1}(E) is dense in Spec k′[T].

Proof of Theorem 3. Let k be a field of characteristic zero, let X, Y, Z be indeterminates and let f ∈ B = k[X, Y, Z] be such that U(k[f], B) is dense in Spec k[f]. Let F be a finite subset of k which contains all coefficients of f and define k_{0} = ℚ(F) and B_{0} = k_{0}[X, Y, Z]. Note that f ∈ B_{0} and that (by 2.6 and 2.10) U(k_{0}[f], B_{0}) is dense in Spec k_{0}[f]. Fix a homomorphism of fields k_{0} ↪ ℂ and regard f as an element of B′ = ℂ[X, Y, Z]. Using 2.6 and 2.10 again, we find that U(ℂ[f], B′) is dense in Spec ℂ[f]. Since Theorem 3 is known to be true over ℂ (see 2.5), f is a variable of B′. By 2.9, f is a variable of B_{0} and hence a variable of B. □

2.11. Corollary. Let B be an algebra over a field k of characteristic zero such that B ⊗_{k} k′ ≃ (k′)^{[3]} for some field extension k′/k. If f ∈ B is such that U(k[f], B) is a dense subset of Spec k[f], then B = k^{[3]} and f is a variable of B.

Proof. Let B′ = B ⊗_{k} k′ = (k′)^{[3]}. By 2.6 and 2.10, U(k′[f], B′) is dense in Spec k′[f], so Theorem 3 implies that f is a variable of B′. The desired conclusion follows from 2.8. □

Remark. Suppose for a moment that the Abhyankar-Sathaye Embedding Conjecture is true over ℂ (i.e., if f ∈ B = ℂ^{[3]} and U(ℂ[f], B) ≠ ∅ then f is a variable of B); then the conjecture holds over any field of characteristic zero and 2.11 remains true if the word “dense” is replaced by “nonempty”. To see this, simply replace “dense” by “nonempty” in 2.10 and in the proofs of Theorem 3 and of 2.11.

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Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5

  E-mail address: ddaigle@uottawa.ca

Department of Mathematics, University of Miami, Coral Gables, FL 33124 USA

  E-mail address: kaliman@math.miami.edu