ON KERNELS OF HOMOGENEOUS LOCALLY NILPOTENT DERIVATIONS OF k[X, Y, Z]

D. DAIGLE

Abstract. Let B = k[X_{0}, X_{1}, X_{2}] be the polynomial ring in three variables over a field k of characteristic zero, let ω_{0}, ω_{1}, ω_{2} be pairwise relatively prime positive integers and let B = ⊕_{i∈ℕ}B_{i} be the grading determined by B_{0} = k and X_{i} ∈ B_{ω_{i}}. Then the main result of the paper has the following corollary: For elements f, g of B which are homogeneous, geometrically irreducible and not associates, the following are equivalent:

(1) The homogeneous localization of B with respect to {1, fg, (fg)^{2}, … } is a polynomial ring in one variable over a subring.

(2) k[f, g] is the kernel of a homogeneous locally nilpotent derivation D : B → B. Moreover, if these equivalent conditions are satisfied then gcd(deg f, deg g) = 1.

Note that this relates (1) the affine rulings of the weighted projective plane Proj B to (2) the homogeneous locally nilpotent derivations of B.

Consider the case “n = 2” of our main result, Theorem 2.2: Corollary. Let B = k[X_{0}, X_{1}, X_{2}] be the polynomial ring in three variables over a field k of characteristic zero, let ω_{0}, ω_{1}, ω_{2} be pairwise relatively prime positive integers and let B = ⊕_{i∈ℕ}B_{i} be the grading determined by B_{0} = k and X_{i} ∈ B_{ω_{i}}. For elements f, g of B which are homogeneous, geometrically irreducible and not associates, the following are equivalent:

(1) B_{(fg)} is a polynomial ring in one variable over a subring.

(2) k[f, g] is the kernel of a homogeneous locally nilpotent derivation D : B → B. Moreover, if these equivalent conditions are satisfied then gcd(deg f, deg g) = 1.

Here, B_{(fg)} is the homogeneous localization of B with respect to {1, fg, (fg)^{2}, … }. By “geometrically irreducible”, we mean irreducible in k̄[X_{0}, X_{1}, X_{2}], where k̄ is an algebraic closure of k.

The reader should compare the above Corollary with 1.8. One notable difference is that the condition gcd(deg f, deg g) = 1, which is part of the assumption of 1.8, is in the conclusion of the present result; we are also replacing the assumption gcd(ω_{0}, ω_{1}, ω_{2}) = 1 of 1.8 by the stronger “ω_{0}, ω_{1}, ω_{2} are pairwise relatively prime”. The proof that gcd(deg f, deg g) = 1 is one of the crucial steps of this paper; it is achieved by Theorem 2.1, in the form gcd {i | A_{i} ≠ 0} = 1.

Date: August 30, 2001.

1991 Mathematics Subject Classification. Primary: 13B10. Secondary: 14E09, 14L30, 14J26.

Research supported by a grant from NSERC Canada.

The fact that condition (1) of the Corollary implies gcd(deg f, deg g) = 1 is needed in [4], which investigates the affine rulings of the weighted projective planes (see also the remark following 1.11). A proof of that implication is included in [4], but it relies on a considerable amount of machinery developed in [3, 4]; so we feel that it is appropriate to give a relatively self-contained proof, based on a different method.

Theorem 2.2 is also useful for establishing a precise correspondence between affine rulings and locally nilpotent derivations. That correspondence is used, in recent work, to relate the viewpoint of [3, 4] to that of [5].

1. Preliminaries

All rings are commutative and have a unity. For an A-algebra B, the notation B = A^{[n]} (where n is a positive integer) means that B is A-isomorphic to the polynomial ring in n variables over A.

Given a nonzero graded ring A = ⊕_{i∈ℤ}A_{i}, a homogeneous multiplicatively closed subset of A is a set S ⊆ ∪_{i∈ℤ}(A_{i} \ {0}) closed under multiplication and such that 1 ∈ S. Then A_{(S)} denotes the homogeneous localization of A with respect to S, i.e., the component of degree zero of the graded ring S^{−1}A. If a ∈ A_{i} \ {0} and S = {1, a, a^{2}, … }, we write A_{(a)} = A_{(S)}. By a homogeneous subring of A we mean a subring A′ of A satisfying A′ = ∑(A′ ∩ A_{i}).

Let R be a domain. A derivation Δ : R → R is locally nilpotent if for each r ∈ R we have Δ^{n}(r) = 0 for n sufficiently large; Δ is irreducible if the only principal ideal of R containing Δ(R) is R itself.

Facts 1.1–1.5 are needed in the proof of Theorem 2.1. The first one is due to W.V. Vasconcelos: 1.1 (Theorem 2.2 of [8]). Let B ⊇ R be an integral extension of domains containing ℚ. Suppose that Δ : R → R is a locally nilpotent derivation and that D : B → B is a derivation extending Δ. Then D is locally nilpotent.

The next statement is a well-known consequence of a result of David Wright (Proposition 2.1 of [9]): 1.2. Let D : B → B be a locally nilpotent derivation, where B is a domain containing ℚ, and let A = ker D. If b ∈ B satisfies Db ∈ A \ {0}, then B_{a} = A_{a}[b] = A_{a}^{[1]} where a = Db.

Statements 1.3 and 1.4 are well-known: 1.3. Let D : B → B be a nonzero derivation, where B is an integral domain satisfying the ascending chain condition on principal ideals. Then D = bD′, for some b ∈ B and some irreducible derivation D′ : B → B. 1.4. Let B be an integral domain of characteristic zero, D : B → B a nonzero derivation and b ∈ B \ {0}. The derivation bD : B → B is locally nilpotent if and only if D is locally nilpotent and b ∈ ker D.

Lemma 1.5. Let R be a ℤ-graded integral domain containing ℚ and Δ : R → R an irreducible, homogeneous locally nilpotent derivation. Suppose that A = ker Δ is a UFD and that each homogeneous prime element of A is a prime element of R. Then every derivation Δ′ : R → R satisfying ker Δ′ ⊇ A has the form Δ′ = ρΔ for some ρ ∈ R.

Proof. If A = R then Δ′ = 0 and the assertion is trivial. Assume that A ≠ R.

Choose a homogeneous t ∈ R such that Δ(t) ∈ A \ {0} and consider the multiplicatively closed set S = {1, α, α^{2}, … } ⊆ A where α = Δ(t). Then 1.2 gives S^{−1}R = (S^{−1}A)[t] = (S^{−1}A)^{[1]} and S^{−1}Δ and S^{−1}Δ′ are (S^{−1}A)-derivations going from (S^{−1}A)[t] to itself. Thus S^{−1}Δ = α · \frac{d}{dt} and S^{−1}Δ′ = Δ′(t) · \frac{d}{dt}, so

(1)  αΔ′ = Δ′(t) Δ.

Consider a factorization α = λ ∏_{i} p_{i}^{e_{i}} where λ ∈ A^{∗}, e_{i} ∈ ℕ and each p_{i} is a prime element of A. If some p_{i} divides Δ′(t) then we may cancel it both sides of equation (1); this yields

α′Δ′ = ρ Δ

where α′ | α in A, ρ ∈ R and no prime factor p_{i} of α′ divides ρ. In particular, α′ | ρ Δr in R, for every r ∈ R.

If α′ ∉ A^{∗} then p_{i} | α′ for some i. Since p_{i} is a homogeneous prime element of A, our assumption implies that p_{i} is a prime element of R. By irreducibility of Δ, we may choose r ∈ R such that p_{i} |̸ Δr; then p_{i} | ρ, a contradiction. Thus α′ ∈ A^{∗} and the lemma is proved. □

We now list the facts needed for the proof of Theorem 2.2. We begin with an “exercise” left to the reader: 1.6. Let k be a field, A = k^{[r]} (r ≥ 1) and let A = ⊕_{i∈ℕ}A_{i} be a grading such that A_{0} = k. If f_{1}, …, f_{n} are homogeneous elements of A satisfying k[f_{1}, …, f_{n}] = A, then there is a subset {g_{1}, …, g_{r}} of {f_{1}, …, f_{n}} satisfying A = k[g_{1}, …, g_{r}].

Part 1 of 1.7 is due to Miyanishi [7] when k is algebraically closed; then one uses [6] to deduce the general case. For part 2 of 1.7 (in particular for irreducibility of Δ_{(f,g)}), see Corollary 2.6 of [2]. 1.7. Let k be a field of characteristic zero and B = k[X_{0}, X_{1}, X_{2}] = k^{[3]}.

(1) If 0 ≠ D : B → B is a locally nilpotent derivation, then ker D = k^{[2]}.

(2) If f, g ∈ B are such that k[f, g] is the kernel of some locally nilpotent derivation of B, then the derivation Δ_{(f,g)} : B → B defined by the jacobian determinant

Δ_{(f,g)}(b) = |\frac{∂(f, g, b)}{∂(X_{0}, X_{1}, X_{2})}| (b ∈ B)

is locally nilpotent, irreducible and has kernel k[f, g].

For the next two facts, let k be a field of characteristic zero, B = k[X_{0}, X_{1}, X_{2}] = k^{[3]}, let ω_{0}, ω_{1}, ω_{2} be positive integers satisfying gcd(ω_{0}, ω_{1}, ω_{2}) = 1, and let B = ⊕_{i∈ℕ}B_{i} be the grading determined by B_{0} = k and X_{i} ∈ B_{ω_{i}}.

1.8 ([1], Theorem 3.5). Let f, g ∈ B be homogeneous and geometrically irreducible. If gcd(deg f, deg g) = 1, then the following are equivalent:

(1) B_{(fg)} is a polynomial ring in one variable over k[f, g]_{(fg)}.

(2) k[f, g] is the kernel of a homogeneous locally nilpotent derivation of B.

1.9. Assume that ω_{0}, ω_{1}, ω_{2} are pairwise relatively prime. If k[f, g] is the kernel of some locally nilpotent derivation D : B → B, where f, g ∈ B are homogeneous, then gcd(deg f, deg g) = 1.

Proof. Suppose that k[f, g] = ker D. Theorem 3.7 of [1] implies, in particular, that if gcd(deg f, deg g) > 1 then there exists a homogeneous coordinate system^{1} (X, Y, Z) of B satisfying gcd(deg X, deg Y) > 1. However, it is easy to see that if some homogeneous coordinate system of B has pairwise relatively prime degrees (which is the case here), then all homogeneous coordinate systems have that property. So we must have gcd(deg f, deg g) = 1. □

In 1.10, we gather some facts which can be found in [3];^{2} then we deduce 1.11 from 1.10. The proof of Theorem 2.2 requires 1.11.

1.10. Let k be an algebraically closed field of characteristic zero and X a projective algebraic surface over k; assume that X is normal, rational and affine-ruled, and that Pic(X_{s}) is a group of rank one, where X_{s} is the smooth locus of X; moreover, assume that all singularities of X are cyclic quotient (in [3], surfaces satisfying these conditions are said to “satisfy the condition (‡)”). Suppose that U ≠ ∅ is an open subset of X isomorphic to 𝔸^{1} × Γ, for some curve Γ. Since X is normal and rational, Γ must be an open subset of ℙ^{1}, so the projection U → Γ determines a rational map X → ℙ^{1}; let us consider the linear system^{3} Λ on X, without fixed components, determined by that rational map. The following facts are proved in [3]:

(i) Every member F of Λ has irreducible support, i.e., F = νC where ν ≥ 1 is an integer and C is an irreducible curve on X. If ν = 1 (resp. ν > 1) we call F a “reduced” (resp. “multiple”) member of Λ.

(ii) At most two members of Λ are multiple.

(iii) U = X \ supp(F_{1} + ⋯ + F_{n}) for some distinct members F_{1}, …, F_{n} of Λ (then define positive integers ν_{1}, …, ν_{n} by F_{i} = ν_{i}C_{i}, where C_{i} is an irreducible curve).

1We mean: X, Y, Z are homogeneous elements of B such that B = k[X, Y, Z].

2At the time of writing, the numbering of the results, in [3], is not available.

3We view Λ as a set of effective divisors; so a “member” of Λ is a divisor of X.

(iv) All multiple members of Λ belong to {F_{1}, …, F_{n}}.

(v) For a subset {F_{i}, F_{j}} of {F_{1}, …, F_{n}} (with i ≠ j), the following are equivalent: • {F_{i}, F_{j}} contains all multiple members of Λ;

• the isomorphism U ≅ 𝔸^{1}×Γ extends to an isomorphism X\supp(F_{i}+F_{j}) ≅

𝔸^{1} × (ℙ^{1} − two points). Moreover, if these conditions hold then Pic(X_{s}) = ℤ ⊕ ℤ/dℤ, with d = gcd(ν_{i}, ν_{j}).

In 1.11, given a homogeneous polynomial h ∈ B with prime factorization h = p_{1}^{e_{1}} ⋯ p_{r}^{e_{r}}, let div_{0}(h) denote the effective divisor ∑_{i} e_{i}V (p_{i}) of X, where V (p_{i}) ⊂ X is the zero set of p_{i}. Corollary 1.11. Let B = k[X_{0}, X_{1}, X_{2}] be the graded polynomial ring defined in the statement of Theorem 2.2, assume that k is algebraically closed and consider the weighted projective plane X = Proj B. Suppose that U ≠ ∅ is an open subset of X isomorphic to 𝔸^{1}×Γ, for some curve Γ, and consider the linear system Λ on X determined by the projection U → Γ, as in 1.10. Then:

(1) U = X \ (V (f_{1}) ∪ ⋯ ∪ V (f_{n})), for some homogeneous irreducible elements f_{1}, …, f_{n} of B (no two of which are associates).

(2) For each i = 1, …, n, there exists an integer ν_{i} ≥ 1 such that div_{0}(f_{i}^{ν_{i}}) ∈ Λ.

(3) If n ≥ 2 then there exist distinct elements i, j ∈ {1, …, n} satisfying: (a) X \ (V (f_{i}) ∪ V (f_{j})) ≅ 𝔸^{1} × (ℙ^{1} minus two points)

(b) Λ = {div_{0}(λf_{i}^{ν_{i}} + μf_{j}^{ν_{j}}) | (λ : μ) ∈ ℙ^{1}}

(c) For every (λ : μ) ∈ ℙ^{1} \ {(0 : 1), (1 : 0)}, λf_{i}^{ν_{i}} + μf_{j}^{ν_{j}} is irreducible in B.

(d) For every k ∈ {1, …, n} \ {i, j}, ν_{k} = 1 and f_{k} = λf_{i}^{ν_{i}} + μf_{j}^{ν_{j}} for some (λ : μ) ∈ ℙ^{1} \ {(0 : 1), (1 : 0)}.

(e) gcd(ν_{i}, ν_{j}) = 1.

Proof. The weighted projective plane X is normal and rational, the Picard group of its smooth locus is ℤ, and all its singularities are cyclic quotient, so we may apply 1.10. Assertions (1) and (2) follow immediately from parts (i) and (iii) of 1.10. Assume that n ≥ 2. By (ii) and (iv), there exists a subset {i, j} of {1, …, n} (with i ≠ j) satisfying

(2)  {div_{0}(f_{i}^{ν_{i}}), div_{0}(f_{j}^{ν_{j}})} contains all multiple members of Λ.

Then (v) gives (3a) and (3e); for (3b), simply note that Λ has (projective) dimension 1 and that div_{0}(f_{i}^{ν_{i}}) and div_{0}(f_{j}^{ν_{j}}) are distinct members of Λ. If (λ : μ) ∉ {(0 : 1), (1 : 0)} then, since {i, j} satisfies (2), div_{0}(λf_{i}^{ν_{i}} + μf_{j}^{ν_{j}}) is a reduced member of Λ; this gives (3c). For (3d), note that div_{0}(f_{k}^{ν_{k}}) ∈ Λ implies f_{k}^{ν_{k}} = λf_{i}^{ν_{i}} + μf_{j}^{ν_{j}}, and k ∉ {i, j} implies that div_{0}(f_{k}^{ν_{k}}) is reduced, so ν_{k} = 1. □

Remark. One consequence of this paper is that (ν_{i}, ν_{j}) = (deg f_{j}, deg f_{i}), in part (3) of 1.11. Indeed, we have (ν_{i}, ν_{j}) = \frac{1}{d}(deg f_{j}, deg f_{i}), where d = gcd(deg f_{i}, deg f_{j}), and the Corollary stated in the introduction implies that d = 1, because B_{(f_{i}f_{j})} is a polynomial ring in one variable over a subring.

2. The results

Theorem 2.1. Let B be an affine UFD over a field k of characteristic zero and let x_{1}, …, x_{n} (n ≥ 2) be prime elements of B no two of which are associates. Suppose that B = k[x_{1}, …, x_{n}] and that B = ⊕_{i∈ℤ}B_{i} is a ℤ-grading such that k ⊆ B_{0}, each x_{i} is homogeneous and

(i) gcd(deg(x_{1}), …, deg(x_{i−1}), deg(x_{i+1}), …, deg(x_{n})) = 1, for all i = 1, …, n. Suppose that A is a homogeneous subalgebra of B satisfying A ⊈ B_{0} and the following conditions:

(ii) A^{∗} = B^{∗}, A is a UFD and every homogeneous prime element of A is a prime

element of B. (iii) A = k[S] and B_{(S)} = A_{(S)}^{[1]}, for some homogeneous multiplicatively closed

subset S of A. Then gcd {i | A_{i} ≠ 0} = 1 and A is the kernel of a homogeneous locally nilpotent derivation D : B → B.

Proof. Let d = gcd {i | A_{i} ≠ 0} and let R = ⊕_{i∈ℤ}R_{i} be the homogeneous subring of B defined by R_{i} = B_{i} for all i ∈ dℤ and R_{i} = 0 otherwise. Note that A ⊆ R and that R is finitely generated as a k-algebra. Since A ⊈ B_{0}, we have d ≥ 1; in particular, B is integral over R. Also, observe that

(3)

If r ∈ R \ {0} is homogeneous, then deg r = deg s_{1} − deg s_{2} for some
s_{1}, s_{2} ∈ S.

To see this, note that the assumptions k ⊆ B_{0} and A = k[S] imply that the set E = {deg s | s ∈ S} is equal to {i | A_{i} ≠ 0}, so deg r belongs to the ideal (of ℤ) generated by E; since E is closed under addition, deg r = e_{1} − e_{2} for some e_{1}, e_{2} ∈ E.

We have B_{(S)} = A_{(S)}[\frac{h}{σ}], for some \frac{h}{σ} ∈ B_{(S)}, where h is a homogeneous element of B, σ ∈ S and deg h = deg σ (so h ∈ R). We claim that

(4)  S^{−1}R = (S^{−1}A)[h] = (S^{−1}A)^{[1]},

where S^{−1}R ⊇ (S^{−1}A)[h] is obvious. If r is any nonzero homogeneous element of R then, by (3), rs_{2}/s_{1} ∈ B_{(S)} for some s_{1}, s_{2} ∈ S. Thus rs_{2}/s_{1} ∈ A_{(S)}[\frac{h}{σ}] and it follows that r ∈ (S^{−1}A)[h]. This shows that R ⊆ (S^{−1}A)[h], so the equality S^{−1}R = (S^{−1}A)[h] holds. It remains to show that h is transcendental over S^{−1}A. If not, then h/σ is algebraic over S^{−1}A, hence algebraic over A, so there is a nonzero f(T) = ∑ a_{i}T^{i} ∈ A[T] satisfying f(h/σ) = 0. We may arrange that all nonzero a_{i} are homogeneous and of the same degree; then, by (3), we can find s_{1}, s_{2} ∈ S such that \frac{s_{2}}{s_{1}} f(T) ∈ A_{(S)}[T], which is absurd because h/σ is transcendental over A_{(S)}. So, (4) holds.

Next, we show:

(5)

If at least one of b, b′ ∈ B is homogeneous and bb′ ∈ A \ {0}, then b, b′ ∈ A.

For this, it’s enough to prove the case where both b and b′ are homogeneous. Consider a factorization bb′ = μ ∏_{i∈I} p_{i} where μ ∈ A^{∗} and each p_{i} is a prime (and homogeneous) element of A. By assumption (ii), each p_{i} is then a prime element of B so b = λ ∏_{j∈J} p_{j} where λ ∈ B^{∗} ⊂ A and J ⊆ I. So b ∈ A and, similarly, b′ ∈ A.

From (5), we easily deduce that

(6)  R ∩ S^{−1}A = A.

In fact, if r ∈ R ∩ S^{−1}A then r = a/s (a ∈ A, s ∈ S), so rs ∈ A; since s ≠ 0 is homogeneous, (5) implies that r ∈ A.

By (4), (6) and the fact that R is k-affine, we obtain

(7)

A = ker Δ, for some irreducible, homogeneous locally nipotent derivation Δ : R → R.

In fact, the “h-derivative” \frac{d}{dh} : (S^{−1}A)[h] → (S^{−1}A)[h] is a homogeneous locally nilpotent derivation with kernel S^{−1}A. Since R is finitely generated as an A-algebra, there exists s ∈ S such that the derivation s\frac{d}{dh} maps R into itself; the restriction Δ′ : R → R of s\frac{d}{dh} is a homogeneous derivation with kernel R ∩ S^{−1}A = A, and is locally nilpotent because s ∈ ker(\frac{d}{dh}) (see 1.4). By 1.3, we have Δ′ = ρ′Δ, where ρ′ ∈ R and Δ : R → R is an irreducible derivation; since Δ is homogeneous and locally nilpotent (1.4) and has the same kernel as Δ′, we proved (7).

Extend Δ to a derivation D′ : Frac B → Frac B and let m = (∏_{i=1}^{n} x_{i})^{d−1}; then mD′ maps B into itself. Indeed, for each i we have dx_{i}^{d−1}D′(x_{i}) = D′(x_{i}^{d}) = Δ(x_{i}^{d}) ∈ R, so mD′x_{i} ∈ B. Hence, the restriction D′′ : B → B of mD′ is a derivation and satisfies

D′′(r) = mΔ(r), for all r ∈ R.

Note that D′′ must be homogeneous, because its restriction to R is.

Using 1.3, write D′′ = βD where β is a homogeneous element of B and D : B → B is an irreducible, homogeneous derivation. Then

D(r) = \frac{m}{β} Δ(r), for all r ∈ R.

We claim that β divides m in B. To see this, consider the set M of all monomials M = x_{1}^{i_{1}} ⋯ x_{n}^{i_{n}} (i_{1}, …, i_{n} ∈ ℕ) satisfying deg(M) + deg(D) ∈ dℤ. Given any M ∈ M, the derivation MD : B → B maps R into itself, so we may consider the restriction Δ_{M} : R → R of MD.

Observe that Δ : R → R satisfies the hypothesis of Lemma 1.5 (if p is a homogeneous prime element of A then, by assumption (ii), p is a prime element of B, and it follows immediately that p is a prime element of R). Since ker Δ_{M} = ker Δ, Lemma 1.5 implies that Δ_{M} = ρ_{M} Δ for some ρ_{M} ∈ R. Note that Δ ≠ 0, choose r ∈ R such that Δr ≠ 0 and write

ρ_{M} Δr = M Dr = \frac{Mm}{β} Δr,

which implies that Mm/β = ρ_{M} ∈ R. In particular, β | Mm in B, and this holds for all M ∈ M. By assumption (i) we have gcd(M) = 1 in B, so β | m in B. Thus,

Dr = γ Δr, for all r ∈ R,

where γ = m/β = λ ∏_{i=1}^{n} x_{i}^{e_{i}}, λ ∈ B^{∗}, e_{i} ∈ ℕ.

Suppose that e_{1} > 0. By assumption (i), we may choose q_{2}, …, q_{n} ∈ ℕ such that deg(x_{1}) + q_{2} deg(x_{2}) + ⋯ + q_{n} deg(x_{n}) ∈ dℤ. Let N = x_{2}^{q_{2}} ⋯ x_{n}^{q_{n}}, then deg(x_{1}N) ∈ dℤ, so x_{1}N ∈ R and consequently

γ Δ(x_{1}N) = D(x_{1}N) = (Dx_{1})N + x_{1}DN ⟹ x_{1} | Dx_{1}.

Moreover, for each j ≠ 1 we have x_{j}^{d} ∈ R, so

γ Δ(x_{j}^{d}) = D(x_{j}^{d}) = dx_{j}^{d−1}Dx_{j} ⟹ x_{1} | Dx_{j},

which is absurd because D is irreducible. Hence, e_{1} = 0 and, by symmetry, e_{j} = 0 for all j. So γ ∈ B^{∗} and we proved:

(8)  Δ extends to a homogeneous derivation D : B → B.

Since B is integral over R, 1.1 gives

(9)  D : B → B is locally nilpotent.

Note that if α is a homogeneous element of ker D then α^{d} ∈ R ∩ ker D = ker Δ = A, so α ∈ A by (5). This implies that ker D ⊆ A, because ker D is a homogeneous subring of B. So

ker D = A.

Let a = Δh = Dh, where h ∈ R is as in (4). Then a ∈ A \ {0}; since D : B → B (resp. Δ : R → R) is locally nilpotent and has kernel A, 1.2 implies that

B_{a} = A_{a}[h] (resp. R_{a} = A_{a}[h])

so B_{a} = R_{a}. It follows that R = B, so d = 1. □

Theorem 2.2. Let B = k[X_{0}, X_{1}, X_{2}] = k^{[3]}, where k is a field of characteristic zero, let ω_{0}, ω_{1}, ω_{2} be pairwise relatively prime positive integers and let B = ⊕_{i∈ℕ}B_{i} be the grading determined by B_{0} = k and X_{i} ∈ B_{ω_{i}}. Consider elements f_{1}, …, f_{n} of B (n ≥ 2) which are homogeneous, geometrically irreducible and no two of which are associates. Then the following are equivalent:

(1) B_{(f_{1}···f_{n})} is a polynomial ring in one variable over a subring.

(2) k[f_{1}, …, f_{n}] is the kernel of a nonzero homogeneous locally nilpotent derivation

D : B → B. Moreover, if these equivalent conditions are satisfied then

(3) k[f_{1}, …, f_{n}] = k[f_{i}, f_{j}], for some distinct i, j ∈ {1, …, n}, and any such i, j satisfy gcd(deg f_{i}, deg f_{j}) = 1.

(4) B_{(f_{1}···f_{n})} = (k[f_{1}, …, f_{n}]_{(f_{1}···f_{n})})^{[1]}.

Proof.

Step 1. We show that, under the assumption that k is algebraically closed, (1) implies (2) and (3).

Assume that (1) holds and let A = k[f_{1}, …, f_{n}]. Consider the weighted projective plane X = Proj B; by (1), the open set U = X \ (V (f_{1}) ∪ ⋯ ∪ V (f_{n})) is isomorphic to the product of 𝔸^{1} with a curve. Consider distinct i, j ∈ {1, …, n} satisfying (3a–e) of 1.11. Then part (3d) gives that A = k[f_{i}, f_{j}], so A = k^{[2]} is a UFD; and it follows from part (3c) that every homogeneous prime element of A is prime in B. Now we claim:

(10)  B_{(f_{i}f_{j})} = A_{(f_{i}f_{j})}^{[1]}.

If this is the case then (2) and (3) follow immediately from Theorem 2.1, using S = {f_{i}^{k}f_{j}^{ℓ} | k, ℓ ∈ ℕ}.

By part (3a) of 1.11, we have B_{(f_{i}f_{j})} = R^{[1]} for a subring R of B_{(f_{i}f_{j})} satisfying R = k[ζ, ζ^{−1}] with ζ transcendental over k. Thus

(B_{(f_{i}f_{j})})^{∗} = R^{∗} = ⋃_{n∈ℤ}k^{∗}ζ^{n}.

On the other hand, if we define p′ = deg(f_{i}), q′ = deg(f_{j}), (p, q) = \frac{1}{gcd(p′,q′)}(p′, q′) and ξ = f_{i}^{q}/f_{j}^{p} then it is easy to see that

(B_{(f_{i}f_{j})})^{∗} = ⋃_{n∈ℤ}k^{∗}ξ^{n},

from which we obtain ζ = λξ^{±1} (λ ∈ k^{∗}). So R = k[ξ, ξ^{−1}] = A_{(f_{i}f_{j})}, (10) holds and Step 1 is complete.

Step 2. We show that (1) implies (2) and (3) (without assuming that k is algebraically closed).

Let k̄ be an algebraic closure of k and B̄ = k̄[X_{0}, X_{1}, X_{2}] = k̄^{[3]}. If (1) holds, it follows that B̄_{(f_{1}···f_{n})} is a polynomial ring in one variable over a subring. Since the f_{i} are irreducible in B̄ by assumption, Step 1 implies that k̄[f_{1}, …, f_{n}] = ker D̄ for some homogeneous locally nilpotent derivation 0 ≠ D̄ : B̄ → B̄, and that, for some i, j, k̄[f_{i}, f_{j}] = k̄[f_{1}, …, f_{n}] and gcd(deg f_{i}, deg f_{j}) = 1. By 1.7, the derivation D̄ = Δ_{(f_{i},f_{j})} : B̄ → B̄ satisfies the requirements. Since this D̄ maps the X_{i} to elements of B, it restricts to a derivation D : B → B (locally nilpotent and homogeneous). Since ker D = k̄[f_{i}, f_{j}] ∩ B = k[f_{i}, f_{j}] and k[f_{1}, …, f_{n}] ⊆ k̄[f_{i}, f_{j}] ∩ B = k[f_{i}, f_{j}], (2) and (3) hold and Step 2 is complete.

Step 3. We show that (2) implies (4).

Assume that (2) holds. Then 1.7 implies that k[f_{1}, …, f_{n}] = k^{[2]}, so, by 1.6, k[f_{1}, …, f_{n}] = k[f_{i}, f_{j}] for some i, j. Since (by 1.9) gcd(deg f_{i}, deg f_{j}) = 1, we may apply 1.8 and conclude that

(11)  B_{(f_{i}f_{j})} = (A_{(f_{i}f_{j})})^{[1]},

where A = k[f_{i}, f_{j}] = k[f_{1}, …, f_{n}]. Now (11) implies that B_{(f_{1}···f_{n})} = (A_{(f_{1}···f_{n})})^{[1]}, so (4) holds and the proof is complete. □

Remark. The Corollary stated in the introduction (hence, also Theorem 2.2) is no longer true if we replace the assumption “geometrically irreducible” by the weaker “irreducible”. Indeed, consider B = ℚ[X_{0}, X_{1}, X_{2}] with the standard total degree grading (deg(X_{i}) = 1), and let f = X_{0} and g = X_{0}^{2} + X_{1}^{2}. Then B_{(fg)} = (k[X_{0}, X_{1}]_{(fg)})[\frac{X_{2}}{X_{0}} ] = (k[X_{0}, X_{1}]_{(fg)})^{[1]} but k[f, g] = k[X_{0}, X_{1}^{2}] is not the kernel of a derivation of B.

References

[1] D. Daigle, Homogeneous locally nilpotent derivations of k[x, y, z], To appear in J. Pure Appl. Algebra.

[2] , On some properties of locally nilpotent derivations, J. Pure Appl. Algebra 114 (1997), 221–230.

[3] D. Daigle and P. Russell, Affine rulings of normal rational surfaces, To appear.

[4] , On weighted projective planes and their affine rulings, To appear.

[5] G. Freudenburg, Local slice constructions in K[X, Y, Z], Osaka J. Math. 34 (1997), 757–767.

[6] T. Kambayashi, On the absence of nontrivial separable forms of the affine plane, J. Algebra 35 (1975), 449–456.

[7] M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological Theories—to the memory of Dr. Takehiko MIYATA, Kinokuniya, 1985, pp. 37–51.

[8] Wolmer V. Vasconcelos, Derivations of Commutative Noetherian Rings, Math Z. 112 (1969), 229– 233.

[9] D. Wright, On the jacobian conjecture, Illinois J. of Math. 25 (1981), 423–440.

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

  E-mail address: daniel@zenon.mathstat.uottawa.ca