The Diffiety Institute Preprint Series

Preprint DIPS-6/96 October 1, 1996

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The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Field extensions and extensions of linear differential operators

by

Michael M. Vinogradov

FIELD EXTENSIONS AND EXTENSIONS OF

LINEAR DIFFERENTIAL OPERATORS

Michael M. Vinogradov

Diffiety Institute (Pereslavl-Zalessky, Russia), Institute of Economic Forecasting (Moscow, Russia)

Abstract. Let K ⊂ R ⊂ P be a tower of fields, N be a P -module, and Δ : R → N be a K-linear differential operator. The aim of this paper is to investigate whether the operator Δ has an extension to P, i.e. if these exists a differential operator Δ′ : P → N such that Δ′|_{R} = Δ. The results of this paper were published in Russian in 1981, see [5].

Introduction

Let K ⊂ R ⊂ P be a tower of fields, and Δ : R → P be a K-linear derivation from R to P. A K-linear derivation Δ′ : P → P is called a prolongation of Δ to the field P if Δ′|_{R} = Δ. Conditions for the prolongation existence of a given derivation are well known (see, for example, [1]). It depend on propeties of the extension R ⊂ P. The aim of this paper is to obtain a similar answer for the case of a differential operator of any order.

Note that the term “prolongation of a differential operator” in the theory of linear differantial operators is used in another situation; and the same is true for the case of homomorphisms (= differential operators of order 0). For this reason we use the term extension of a differential operator instead of the term “prolongation”.

0. Necessary notions and notations. Let K be a commutative ring with a unit, A be a commutative associative unitary K-algebra and M, N be an A-modules. We deal with the functors of differential calculus arising in the category M(A) of modules over A. The definition of differential operators over a commutative algebra and other necessary notions and notations may be found in the paper [2] of this issue. See also [3].

1. Field extensions generated by one element. The following proposition is needed for the sequel.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16S32, 16S80, 17B37, 16W30, 18D10, 35A99, 58A99, 58G05, 58G99, 81R50.

Key words and phrases. Differential operator, jets, field extensions, prolongations of derivations, extensions of linear differential operators.

Lemma 1. Let K be a commutative ring, A be a K-algebra, M, N be A-modules, and Δ ∈ Diff_{l}(M, N; M(A)). Fix an element x ∈ A and set Δ_{l−j} =^{def} δ_{x}^{j}Δ. Then

Δ_{i}(x^{n}m) = ∑_{j} \binom{n}{j}x^{n−j}Δ_{i−j}(m), i = 0,…, l  (1)

The set of DO {Δ_{i}}_{i=0,...,l−1} is said to be the x-set of Δ. Now assume that K ⊂ R ⊂ P is a tower of fields and P is generated over R by an element

x. Let N be a P -module and Δ ∈ Diff_{l}(N; M(R)) = Diff_{l}(R, N; M(R)). For simplicity we will write Diff_{l}(N; R) and Diff_{l}(N; P) instead of Diff_{l}(N; M(R)) and Diff_{l}(N; M(P)) respectively.

Theorem 1.

(1) Let x be transcendental over R. Then there exists a bijective correspondence between the set E_{P} (Δ) of all extensions of Δ to P and the set ⊕_{i=0}^{l−1} Diff_{i}(N; R). In this connection a set

(Δ_{(0)},…, Δ_{(l−1)}) ∈ ⨁_{i=0}^{l−1} Diff_{i}(N; R)

corresponding to an extension Δ′ is a restriction of the x-set of DO Δ′ on R, i.e. Δ′_{i}|_{R} = Δ_{(i)}.

(2) Let x be separable algebraic over R. Then there exists a unique extension Δ′ of the DO Δ to P.

(3) Let x be purely nonseparable and T^{p^{m}} − a be an irreducible polinomial over R

(char R = p), for which x is a root. Then the following conditions are equivalent: a. there exists an extension Δ′ of DO Δ to P; b. for any n ∈ ℤ, δ_{a}^{n}Δ ∈ Diff_{l−np^{m}}(N; R).

If the last conditions hold, then there exists a bijective correspondence between the set {Δ′} of all extensions of DO Δ on P and the set

⨁_{i=0}^{min(l,p^{m})−1} Diff_{i}^{or}(N; R).

Here by Diff_{i}^{or}(N; R) we denote the set of all DO of order i from R to N satisfying the condition b of order reduction formulated above.

◀ The proof is by induction over the order l of the DO Δ. For l = 1, there is nothing to prove.

(1.) The element x is transcendental over R. Any DO Δ : R[x] → N may be uniquely extended to P = R(x), i.e. there exists a

unique DO Δ′ : P = R(x) → N such that Δ′|_{R[x]} = Δ. It is clear that to extend Δ to P one needs to extend Δ to the algebra R[x] ⊂ P.

FIELD EXTENSIONS AND EXTENSIONS OF LINEAR DIFFERENTIAL OPERATORS 3

Consider a set Δ_{(0)},…, Δ_{(l−1)}, Δ_{(i)} ∈ Diff_{i}(N; R). Set Δ_{(l)} = Δ and define K-linear morphisms Δ′_{i} : R[x] → N, i = 0,…, l by

Δ′_{i}(ax^{n}) = ∑_{j} \binom{n}{j}x^{n−j}Δ_{(i−j)}(a), a ∈ R, k ≥ 0.  (2)

It can be easily checked that δ_{x}Δ′_{i} = Δ′_{i−1}. Consequently the set (Δ′_{0},…, Δ′_{i−1}) is an x-set of K-linear morphism Δ′_{i} . Now note that Δ′_{i}|_{R} = Δ_{(i)}. Therefore by the inductive hypothesis, Δ′_{i} is an extension of DO Δ_{(i)} to R[x] corresponding to

(Δ_{(0)},…, Δ_{(i−1)}) ∈ ⨁_{j=0}^{i−1} Diff_{j}(N; R)

Now let us prove that Δ′_{l} is a DO of order ≤ l, i.e. for any f ∈ R[x] one has δ_{f}Δ′_{l} ∈ Diff_{l−1}(N; R[x]). Without loss of generality it can be assumed that f = x or f = b ∈ R. By the previous statement, δ_{x}Δ′_{l} = Δ′_{l−1} ∈ Diff_{l−1}(N; R[x]). Furthermore, the operators δ_{x} and δ_{b} commute, hence δ_{b}Δ′_{l} is a DO of order ≤ l − 1. Indeed, δ_{b}Δ′_{l} is an extension of the DO δ_{b}Δ_{(l)} corresponding to

(δ_{b}Δ_{(1)},…, δ_{b}Δ_{(l−1)}) ∈ ⨁_{j=0}^{l−2} Diff_{j}(N; R).

This proves the first statement of the theorem.

(2.) The element x is separable algebraic over R.

Let f(T) = ∑_{i=0}^{n} a_{i}T^{i}, a_{n} = 1, be an irreducible polinomial over R, for which x is a root. Let us consider N as R[T]-module, then the statement in a consideration are equivalent to the following one:

there exists a unique DO Δ′ ∈ Diff_{l}(N; R[T]) such that

Δ′|_{R} = Δ  (3)

and

Δ′|_{{f(T)}} = 0.  (4)

First let us prove the uniqueness of the DO Δ′ . Let {Δ′_{0},…, Δ′_{l−1}} be a T -set of Δ′ . It is easy to prove that condition (4) is equivalent to the following one:

Δ′_{j}af(T) = 0 ∀a ∈ R, j = 0,…, l.  (5)

Let us consider the following relation.

Δ′_{j}(af(T)) = Δ′_{j}(a ∑_{i=0}^{n} a_{i}T^{i}) = ∑_{i=0}^{n}(δ_{a_{i}}Δ′_{j})(aT^{i}) + ∑_{i=0}^{n} a_{i}Δ′_{j}(aT^{i}).

The morphisms δ_{a_{i}}Δ′_{j} are elements of the T -set for the DO δ_{a_{i}}Δ′_{l} of order less than l. This implies that ∑_{i=0}^{n}(δ_{a_{i}}Δ′_{j})(aT^{i}) is uniquely determined. Set

β_{j}(a) =^{def} − ∑_{i=0}^{n}(δ_{a_{i}}Δ′_{j})(aT^{i}).

If we combine this with (5), we obtain

∑_{i=1}^{n} a_{i}Δ′_{j}(aT^{i}) = β_{j}(a) ∀a ∈ R, j = 1,…, l.

Using (1), we get

∑_{k=0}(∑_{i=1}^{n} \binom{i}{k}a_{i}T^{i−k})Δ′_{j−k}(a) = β_{j}(a), j = 1,…, l.  (6)

This system of linear equations with respect to Δ′_{j}(a) has triangular form and its determinant is equal to (\frac{df)}{dT})^{l}, Since \frac{df}{dT} ∉ \{f(t)\} it follows that (6) has a unique solution. This implies that the T -set for the DO Δ′_{l} is uniquely determined and consequently Δ′_{l} is uniquely determined too.

Let (u_{1}^{a},…, u_{l−1}^{a}) be a solution of system (6). Set u_{l}^{a} = Δ(a). It is readily seen that morphisms u_{i} : R → N, a ↦ u_{i}^{a}, are K-linear. It follows that K-linear homomorphisms

Δ′_{i}(aT^{k}) = ∑ \binom{k}{j}T^{k−j}u_{i−j}^{a}, i = 1,…, l,  (7)

are well defined and Δ′_{i}|_{{f(T)}} = 0.

Let us prove that Δ′_{i} is a DO of order ≤ i. To do this one needs to check that δ_{T} Δ′_{i} and δ_{b}Δ′_{i}, b ∈ R, are DO of order ≤ i − 1. The first of these equalities follows from our definitions, the second one is true by the inductive assumption. This completes the proof of the second statement of Theorem 1.

(3.) Let x be purely nonseparable and T^{p^{m}} − a be an irreducible polinomial over R (char R = p), for which x is a root. In this case, relation (5) has the form

Δ′_{i}(b(T^{p^{m}} − a)) = 0, ∀b ∈ R, i = 0,…, l

�ombining equalities (T^{p^{m}} − a)Δ(b) = 0 and δ_{T}^{p^{m}} = δ_{T^{p^{m}}}, we obtain

δ_{a}Δ′_{i}(b) = δ_{T^{p^{m}}} Δ′_{i}(b) = Δ′_{i−p^{m}}(b).

In other words, δ_{a}Δ_{i} = Δ_{i−p^{m}}, where Δ_{i} = Δ′_{i}|_{R}. It follows that

δ_{a}^{k}Δ_{i} = Δ_{i−kp^{m}}, ∀k ∈ ℤ  (8)

FIELD EXTENSIONS AND EXTENSIONS OF LINEAR DIFFERENTIAL OPERATORS 5

These formulas show that the condition of order reduction stated above must be true for all DO Δ′_{1},…, Δ′_{l} = Δ′.

Now suppose that a DO Δ satisfies this condition of order reduction. Consider a set of DO Δ_{l−i} ∈ Diff_{l−i}(N; R), i = 1,…, min(l, p^{m} − 1), each of which satisfies this condition too. Set

Δ_{l−i−kp^{m}} =^{def} δ_{a}^{k}Δ_{l−i}, k = 1,…, [\frac{l}{p}^{m}].

As above, we define the extension Δ′ of the DO Δ to R[T]. Relations (8) are true by the choice of the T -set. Therefore Δ′_{|}_{{f(T)}} = 0.

This completes the proof of Theorem 1. ▶

Corollary. Let K ⊂ R ⊂ P be a tower of fields and the extension R ⊂ P be finitely generated. Then the following conditions are equivalent:

(1) P is a separable algebraic extension of R.

(2) For any P -module N and any DO Δ ∈ Diff(N; R) there exists a unique extension Δ′ of the DO Δ to P.

(3) For any derivation Δ : P → P the relation Δ|_{R} = 0 implies Δ = 0.

◀ The implication (1) ⟹ (2) follows from Theorem 1 by the theorem on the primitive element (see [1], chapter V). The implication (2) ⟹ (3) is trivial. Finally the implication (3) ⟹ (1) was proved in the book [1]. ▶

2. Field extensions. General case. Let char K = p > 0, L be an extension of the field K, and M be a p-basis of L over K (see [1]). Now by induction we will construct a family of sets \overline{M}_{r} which are needed for the sequel.

Let \overline{M}_{1} = M. The field K(L^{p}) is generated over the field K(L^{p^{2}}) by elements of the form x^{p}, x ∈ M. Therefore one can choose a set \overline{M}_{2} ⊂ \overline{M}_{1} such that \overline{M}_{2}^{p} = \{x^{p}|x ∈ \overline{M}_{2}\} is a p-basis of K(L^{p}) over K. Arguing as above, we define the set \overline{M}_{r} as the subset of \overline{M}_{r−1}

such that \overline{M}_{r}^{p^{r−1}} is a p-basis of K(L^{pr−1}) over K. Let the function μ : M → ℤ_{+} ⋃\{∞\} be given by the rule

μ(x) = \{ \begin{matrix}min r | x ∉ \overline{M}_{r}, if x ∉ ⋂_{r} \overline{M}_{r} \\ ∞, if x ∈ ⋂_{r} \overline{M}_{r}. \end{matrix}

Let

M_{s} = {x_{1}^{k_{1}} ·. . . · x_{l}^{k_{l}} | x_{i} ∈ M, 0 ≤ ∑_{i=1}^{l} k_{i} ≤ s}.

Then \overline{M}_{p^{r}−1} is a basis of the linear space L over the field K(L^{p^{r}}). Indeed, the set

N_{p^{r−1}} = {x_{1}^{k_{1}} ·. . . · x_{l}^{k_{l}} | x_{i} ∈ M, 0 ≤ ∑_{i} k_{i} ≤ p^{r} − 1}.

generates L over K(L^{p^{r}}) and \overline{M}_{p^{r}−1} is a maximal linear independent subset over K(L^{p^{r}}) in N_{p^{r}−1}.

Proposition 1. Let N be an L-module. Any DO Δ ∈ Diff_{l}(N; L) is uniquely determined by its values on M_{l} and these values may be chosen in an arbitrary way.

◀ It is easily shown that for any set {m_{α}}_{α∈M_{l}} ∈ N there exists a unique DO Δ ∈ Diff_{l}(N; L) such that Δ(α) = m_{α}. Indeed, let r be an integer such that p^{r−1} ≤ l < p^{r}. By (1) any DO Δ ∈ Diff_{l}(N; L) is a K(L^{p^{r}})-linear mapping. The set \overline{M}_{p^{r}−1} is a basis of L over K(L^{p^{r}}); this implies that any DO Δ is determined by its values on \overline{M}_{p^{r}−1} .

Now consider a set {m_{α}}_{α∈M_{l}}. Let a map Δ′ : M_{l} → N be given by Δ′(α) = m_{α}. Using

(δ_{x_{0}} ∘ ⋯ ∘ δ_{x_{s}})(Δ)(1) = 0,  (9)

by induction on s from l to p^{r} − 1 the map Δ′ may be continued to a map \overline{M}_{p^{r}−1} → N. In other words, the map Δ′ may be continued to a K(L^{p^{r}})-linear map Δ̄ : L → N. It can easily be checked that Δ̄ is a DO of order ≤ l. Conditions (9) are true for any DO of order ≤ L; thus Δ̄ is a uniquely determined by Δ̄|_{M_{l}}. ▶

Proposition 2. Let κ be a transcendental basis of the field L over K such that the extension K(κ) ⊂ L is separable algebraic and

κ_{l} = {x_{1}^{i_{1}} ·. . . · x_{k}^{i_{k}} | x_{i} ∈ κ, 0 ≤ ∑_{j=1}^{k} i_{j} ≤ l}.

Then any DO Δ ∈ Diff_{l}(N; L) is uniquely determined by its values on κ_{l} and these values may be chosen in an arbitrary way.

◀ It is obvious that any DO Δ ∈ Diff(N; K(κ)) is uniquely determined by its values on κ_{l} and these values may be chosen in an arbitrary way. By statement 1 of Theorem 2, there exists a unique extension Δ to L. ▶

Theorem 2. Let K ⊂ R ⊂ P be a tower of fields. Then

(1) Suppose P is a separable algebraic extension of R; then for an arbitrary P -module N and any DO Δ ∈ Diff(N; R) there exists a unique extension to P.

(2) Suppose P is a separable transcendent extension of R; then for an arbitrary P - module N and any DO Δ ∈ Diff(N; R) there exist more than one extensions to P.

(3) Suppose that for any P -module N there exists a DO Δ ∈ Diff(N; R) that cannot be extended to P. Then the extension R ⊂ P is not separable.

(4) Suppose that the extension K ⊂ P is separable. Then the inverse statements to (1), (2) and (3) are true.

◀ (1). Let the extension R ⊂ P be separable algebraic, N be a P -module, and Δ ∈ Diff_{l}(N; R). By the corollary from Theorem 1, for any finite set H there exists a unique extension Δ_{H} of the DO Δ on R(H). At the same time, if H′ ⊂ H, then Δ_{H}|_{H′} = Δ_{H′}. This implies that for any element x ∈ P and any finite set H ∋ x, Δ_{H}(x) is independent of H. By definition, put Δ′(x) = Δ_{H}(x). It is easy to prove that Δ′ ∈ Diff_{l}(N; P).

FIELD EXTENSIONS AND EXTENSIONS OF LINEAR DIFFERENTIAL OPERATORS 7

(2). For the case p > 0 the statement follows from Proposition 1 and the McLane criterion. For the case p = 0 it follows from Proposition 2.

(3). This statement follows from statements (1) and (2).

(4). It is sufficient to prove the inverse statement to statement (3). Suppose an extension R ⊂ P is separable, and an extension R ⊂ P is nonseparable. Let us show the existence of a derivation Δ : R → P such that there are no extensions Δ to P. Let M be a p-basis R over K and M′ be a p-basis K over K^{p}. Then, by definition of a p-basis, we have R = K(R^{p})(M), K = K^{p}(M′). Hence R = R^{p}(M ⋃ M′). There exists a subset M′′ ⊂ M′ such that M ⋃ M′′ is a p-basis of R over R^{p}. By assumption, the extension R ⊂ P is nonseparable; thus by the McLane criterion the set M ⋃ M′′ is a p-dependent over P. There exists a p-dependent over P finite family

S = {x_{1},…, x_{r}, y_{1},…, y_{m}, } x_{i} ∈ M, y_{i} ∈ M′′.

From our assumptions it follows that m > 0. We may assume also that any proper subset of the set S is p-independent over P. Since the set S is p-dependent over P, there exists relations among elements of the form

x_{1}^{k_{1}} ·. . . · x_{r}^{k_{r}}y_{1}^{n_{1}} ·. . . · y_{m}^{n_{m}}, 0 ≤ k_{i} < p, 0 ≤ n_{i} < p.

Each of these relations may be represented in the form

∑_{i=1}^{k} γ_{i}x_{1}^{i} = 0, γ_{i} ∈ P^{p}(x_{2},…, x_{r}, y_{1},…, y_{m}).

Since the set x_{2},…, x_{r}, y_{1},…, y_{m} is p-independent over P, at least one of coefficients γ_{i} in each relation is not equal to zero. Choose a relation of the minimal degree n < p with response to x_{1}. Let this relation be ∑_{i=1}^{n} φ_{i}x_{1}^{i} = 0, where φ_{n} ≠ 0. By Proposition 2, the map Δ : M → P such that Δ(y_{i}) = 0 ∀i and

Δ(x_{i}) = \{ \begin{matrix}1, i = 1 \\ 0, i > 1 \end{matrix}

determines a derivation from R to P. Assume that there exists an extension Δ′ of Δ on P. Then Δ′(φ_{i}) = 0 ∀i and

Δ′(∑_{i=1}^{n} φ_{i}x_{1}^{i}) = ∑_{i=1}^{n} iφ_{i}x_{1}^{i−1} = 0

Since n < p, we have nφ_{n} ≠ 0. Therefore the degree of the relation ∑_{i=1}^{n} iφ_{i}x_{1}^{i−1} = 0 is less than n. This contradiction proves Theorem 2. ▶

Let us remark that without the assumption of the separability of the extension R ⊂ P statement (4) of Theorem 2 is not true. For example, let K be a quasifield, a ∈ K\K^{p}, and x be transcendental over K. Set R = K(x) and P = R(y), where y satisfies the relation T^{p^{m}} − a = 0. Then the extensions K ⊂ P and R ⊂ P are nonseparable. Nevertheless, for any P -module N and for any DO Δ ∈ Diff(N; R), there exists more than one extension of Δ to P. Indeed, this case satisfies the conditions of statement (3) of Theorem 1. Now let R = K and let P be any perfect field containing K. Then the extension R ⊂ P is nonseparable, but it is easily shown that for any DO Δ ∈ Diff(N; R) there exists a unique extension of Δ to P.

3. Maps of jets modules under extensions of fields. Let i : R ⊂ P be an extension of fields, let N be a P -module, Δ be an element of Diff_{l}(N; R), and Δ′ be an extension of Δ to P. For simplicity set

J^{l}(R; R) =^{def} J^{l}(R; M(R))

and J^{l}(P; P) =^{def} J^{l}(P; M(P)). Denote by i_{∗} the homomorphism from J^{l}(R; R) to J^{l}(P; P) j-associated to the natural composition of a DO j_{l;R} ∘i. In other words, i_{∗} ∘j_{l;R} = j_{l;P} ∘ i. Denote by φ : J^{l}(R; P) → N and φ′ : J^{l}(R; R) → N the homomorphisms j-associated to the DO Δ and Δ′ respectively. Define an R-homomorphism

α : J^{l}(R; R) → P ⊗ J^{l}(R; R)

by the rule α(θ) = 1 ⊗ θ, θ ∈ J^{l}(R; R). Let a P -homomorphism

β : P ⊗ J^{l}(R; R) → J^{l}(P; P)

be given by β(q ⊗ θ) = qi_{∗}(θ), q ∈ P, θ ∈ J^{l}(R; R). Consider the following commutative diagram:

[figure: \begin{aligned}R & & P \\ & N \\ J^{l}(R; R) & & J^{l}(P; P) \\ & P ⊗ J^{l}(R; R) \end{aligned}; i; Δ; j_{l;R}; Δ′; j_{l;P}; φ; i_{∗}; α; φ′; β]  (10)

Note that there exists an extension of DO Δ to P if and only if the R-homomorphism φ : J^{l}(R; R) → N j-associated to Δ may be represented as a composition

φ′ ∘ β ∘ α : J^{l}(R; R) → P ⊗ J^{l}(R; R) → J^{l}(P; P) → N

This implies the following

Proposition 3.

(1) Coker β = 0 ⟺ for any P -module N and for any DO Δ there exists no more than one extension of Δ to P.

(2) Ker β = 0 ⟺ for any P -module N and for any DO Δ there exists no less than one extension of Δ to P.

Theorem 3.

(1) Suppose P is a separable algebraic extension of R; then β is a homomorphism.

(2) Suppose P is a separable transcendental extension of R; then Coker β ≠ 0 and Ker β = 0.

(3) Suppose Ker β ≠ 0; then the extension R ⊂ P is not separable.

(4) Suppose the extension K ⊂ P is separable. Then the inverse statements to (1), (2) and (3) are true.

◀ The proof follows immediately from Theorem 2 and Proposition 3. ▶ This theorem generalizes results of S. Suzuki [4] obtained to the case of prime fields.

Acknowledgments. The author is grateful to A. M. Vinogradov for useful discussions and to A. B. Sosinskii and I. S. Krasil’shchik for their help in translating this paper in English.

References

1. Bourbaki N., Algèbre, livre II, Éléments de mathématique par N. Bourbaki. Premieŕe partie. Les structures fondamentales de l’analyse, Hermann & C^{ie}, Paris.

2. Krasil’shchik I. S., Calculus over commutative algebras: a concise user guide, Preprint DIPS 1/96.

3. Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1, Gordon and Breach, 1986.

4. Suzuki S., Differentials of commutative rings, Queen’s papers in Pure and Appl. Math., No 29, Queen’s university publ., Kingston, 1971.

5. Vinogradov M.M., Extensions of fields and extensions of linear differential operators, Mat. Zametky 30 (1981), no. 2, 237–248, (in Russian).

Institute of Economic Forecasting, 32, Krasikova str., Moscow 117418, Russia E-mail address: vin@ecfor.msk.su