The Diffiety Institute Preprint Series

Preprint DIPS–2/96 October 1, 1996

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The Diffiety Institute

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Characteristics of linear differential operators over commutative algebras

by

I.S.Krasil’shchik

Characteristics of linear differential operators over commutative algebras

I.S.Krasil’shchik Moscow Institute for Municipal Economy & the Diffiety Institute^{∗}

Abstract

Three definitions for characteristics of linear differential operators in the category of modules over a commutative unitary algebra are given. These definitions are compared with each other and some basic fact concerning their properties are proved. It is shown that for algebras without zero divisors the characteristic ideal is involutive and is the support of the symbolic module.

Key words and phrases: linear differential operators, characteristics, algebraic models, involutiveness, symbolic module.

1991 AMS Subject Classification: 35A30.

1 Introduction and notations

In this paper we construct a pure algebraic theory for characteristics of linear differential operator in the framework of [4]. We show that relations between the characteristic ideal and the symbolic module of a differential operator in the algebraic setting is the same as in the geometric one (cf. [2, 3]). It is also shown that junior symbols of differential operators can be introduced as module morphisms over appropriate characteristic ideals.

In what follows we consider a commutative unitary algebra A over a commutative ring R. We deal with the functors of differential calculus arising in the category M(A) of modules over A (see [4]). In this section we recall some essential definitions from [4] and [5].

∗For correspondence: 1st Tverskoy-Yamskoy per. 14, Apt.45, 125047 Moscow, Russia.

e.mail: josephk@glas.apc.org

Let P and Q be two A-modules and consider the algebra Diff_{∗}^{(+)}(P, Q) of linear differential operators acting from P to Q. Due to embeddings Diff_{k−1}^{(+)}(P, Q) ⊂ Diff_{k}^{(+)}(P, Q) one can define quotient modules

S_{k}(P, Q) = \frac{Diff_{k}^{(+)}(P, Q)}{Diff_{k−1}^{(+)}(P, Q)}.

Note that two different A-module structures (the left and the right ones) coincide in S_{k}(P, Q).

Definition 1 A-module

S(P, Q) = ∑_{k≥0} S_{k}(P, Q)

is called the module of symbols of differential operators acting from P to Q.

We also use the notations S(P) = ∑_{k≥0} S_{k}(P) for S(A, P) and S = ∑_{k≥0} S_{k} for S(A).

Let Diff_{k}^{(+)}(P, Q) → S_{k}(P, Q) be the natural projection and consider Δ ∈ Diff_{k}^{(+)}(P, Q). The image of Δ under this projection is denoted by |Δ|_{k} = |Δ| and is called the symbol of the operator Δ.

For any A-modules P, Q, R one has an A-module homomorphism

S_{k}(P, Q) ⊗_{A} S_{l}(Q, R) → S_{k+l}(P, R)

induced by the composition of differential operators:

|∇|_{l} · |Δ|_{k} = |∇ ∘ Δ|_{k+l}.

In particular, S(P, P) is an associative A-algebra.

Consider two operators Δ_{i} ∈ Diff_{k_{i}}^{(+)}(A, A). Evidently, their commutator [Δ_{1}, Δ_{2}] is an operator of the order ≤ k_{1} + k_{2} − 1 and consequently S(A) is a commutative algebra. Consider now two elements s = |Δ| ∈ S_{k} and t = |∇| ∈ S_{l} and define

{s, t} =^{def} |Δ ∘ ∇ − ∇ ∘ Δ|_{k+l−1}.

Proposition 1.1 For any elements s, t, r ∈ S one has (i) {s, t} + {t, s} = 0, (ii) {s, {t, r}} + {t, {r, s}} + {r, {s, t}} = 0, (iii) {s, tr} = {s, t}r + t{s, r}.

The operation {·, ·} : S ⊗_{R} S → S is called the Poisson bracket in S. Remark. For the case A = C^{∞}(M), where M is a smooth manifold, S plays the role of the algebra of functions on T^{∗}(M) polynomial along the fibers of π : T^{∗}(M) → m. This justifies the notation T^{∗}(A) for the space Spec_{gr}A of graded primitive ideal of A.

If Δ ∈ Diff_{i}^{(+)}(P, Q) is a differential operator, we define its symbolic map (or simply, the symbol of Δ)

s(Δ) = smbl (Δ) : S(P) → S(Q)

by s(Δ)(t) = |Δ ∘ ∇|_{k+i} for s = |∇|_{k} ∈ S_{k}(P), ∇ ∈ Diff_{k}^{(+)}(A, P).

In what follows we use standard concepts and notations from commutative algebra [1]. Namely, let A be a commutative algebra with a unit and P be an A-module. Then:

(i) Ann(P) ⊂ A denotes the annihilator of P in A,

(ii) Supp (P) ⊂ Spec (A) is the support of P,

(iii) if μ ⊂ A is a multiplicative set, then by P_{μ} corresponding localization is denoted,

(iv) in particular, if p ⊂ A is a primitive ideal, then by P_{p} localization with respect to A \ p is denoted,

(v) the quotient ring of A/p is denoted by ⟨A_{p}⟩.

2 Characteristic ideal

Let Δ ∈ Diff_{i}^{(+)}(P, Q) be a differential operator of the order i.

Definition 2 An element a ∈ A is called a characteristic of the operator Δ ∈ Diff_{i}^{(+)}(P, Q) if the kernel of the map δ_{a}^{i}(Δ) = \underbrace{[a, [a, … [a}_{i times}, Δ] …]] ∈ hom_{A}(P, Q)

is nontrivial.

The set of characteristics of the operator Δ is denoted by char (Δ) ⊂ A.

Proposition 2.1 If Δ_{1}, Δ_{2} ∈ Diff_{i}^{(+)}(P, Q) are differential operators such that |Δ_{1}| = |Δ_{2}|, then char (Δ_{1}) = char (Δ_{2}).

Hence, char (Δ) is determined by the symbol of Δ only and for any s ∈ S_{i}(P, Q) the set char (s) ⊂ A is well-defined.

Proposition 2.2 Let Δ_{1} ∈ Diff_{i}^{(+)}(P, Q), Δ_{2} ∈ Diff_{j}^{(+)}(Q, R) be two differential operators. Then one has the following embeddings

char (Δ_{1}) ⊂ char (Δ_{2} ∘ Δ_{1}) ⊂ char (Δ_{1}) ⋃ char (Δ_{2}).

Proof. It follows from the identity

δ_{a}^{i+j}(Δ_{2} ∘ Δ_{1}) = δ_{a}^{j}(Δ_{2}) ∘ δ_{a}^{i}(Δ_{1})

which is valid for any a ∈ A.

Corollary 2.3 For any two operators Δ_{1}, Δ_{2} ∈ Diff_{∗}^{(+)}(A, A) one has

char (Δ_{1} ∘ Δ_{2}) = char (Δ_{1}) ⋃ char (Δ_{2}).

Proof. Due to Proposition 2.2 one has

char (Δ_{1}) ⊂ char (Δ_{2} ∘ Δ_{1}) ⊂ char (Δ_{1}) ⋃ char (Δ_{2}).

On the other hand, since char (Δ) depends on |Δ| only, one has char (Δ_{1} ∘ Δ_{2}) = char (Δ_{2} ∘ Δ_{1}. Consequently

char (Δ_{2}) ⊂ char (Δ_{2} ∘ Δ_{1}) ⊂ char (Δ_{1}) ⋃ char (Δ_{2})

and

char (Δ_{1}) ⋃ char (Δ_{2}) ⊂ char (Δ_{2} ∘ Δ_{1}) ⊂ char (Δ_{1}) ⋃ char (Δ_{2}).

Corollary 2.4 For any operator Δ ∈ Diff_{∗}^{(+)}(A, A) and n ≥ 1 one has

char (Δ^{n}) = char (Δ).

¿From here till the end of Section 2 we suppose that A has no zero divisors. Note that in this case an element a ∈ A lies in char (Δ), Δ ∈ Diff_{i}^{(+)}(A, A) if and only if δ_{a}^{i}(Δ) = 0. Note also that the absence of zero divisors in A implies the same for property S(A).

Lemma 2.1 For any two operators Δ_{1}, Δ_{2} ∈ Diff_{i}^{(+)}(A, P) one has

char (Δ_{1}) ⋂ char (Δ_{2}) ⊂ char (Δ_{1} + Δ_{2}).

Proof. Let a ∈ char (Δ_{1}) ⋂ char (Δ_{2}). It means that there exist two nonzero elements a_{1}, a_{2} ∈ A such that

δ_{a}^{i}(Δ_{j})(a_{j}) = a_{j}p_{j} = 0, j = 1, 2,

where p_{j} = δ_{a}^{i}(Δ_{j})(1). Then a_{1}a_{2} ≠ 0 and consequently

δ_{a}^{i}(Δ_{1} + Δ_{2})(a_{1}a_{2}) = a_{1}a_{2}(p_{1} + p_{2}) = 0.

Consider an operator Δ ∈ Diff_{∗}^{(+)}(P, Q) and the sets J_{i} ⊂ S_{i} defined by

J_{i} = J_{i}(Δ) = {s ∈ S_{i} | char (Δ) ⊂ char (s)} .

¿From Lemma 2.1 it follows that these sets are additively closed, while from Proposition 2.2 one has that J_{Δ} = ∑_{i≥0} J_{i} is an ideal in S.

Definition 3 The ideal J_{Δ} is called the characteristic ideal of the operator Δ ∈ Diff_{∗}^{(+)}(P, Q).

Proposition 2.5 The characteristic ideal coincides with its radical, J_{Δ} = r(J_{Δ}).

Proof. Let s ∈ S and s = s_{1} + … + s_{k} be a decomposition of s in homogeneous elements satisfying deg (s_{1}) < deg (s_{2}) < … < deg (s_{k}). Suppose that s^{n} ∈ J_{Δ} for some n > 0 and prove by induction with respect to k that s ∈ J_{Δ}.

For k = 1 it follows from Corollary 2.4.

Let now k > 1 and suppose that the statement is valid for k − 1. Then in the decomposition

(s_{1} + … + s_{k})^{n} = ∑_{i_{1}+...+i_{k}=n} \frac{i_{1}! · . . . · i_{k}!}{n!}s_{1}^{i_{1}} … s_{k}^{i_{k}}

the summand s_{k}^{n} is of the maximal order (which equals to n · deg (s_{k})). Hence, s_{k}^{n} ∈ J_{Δ} which means that s_{k} lies in J_{Δ} as well. Consequently, (s_{1} +…+s_{k−1})^{n} ∈ J_{Δ}, from where it follows that s_{1} + … + s_{k−1} ∈ J_{Δ} by the induction hypothesis.

Proposition 2.6 The characteristic ideal is involutive, i.e. it is closed with respect to the Poisson bracket in S, {J_{Δ}, J_{Δ}} ⊂ J_{Δ}.

Proof. It is sufficient to prove the statement for homogeneous elements only. Let a ∈ char (Δ) and |Δ_{1}| ∈ J_{i_{1}}, |Δ_{2}| ∈ J_{i_{2}}. Then δ_{a}^{i_{1}}(Δ_{1}) = δ_{a}^{i_{2}}(Δ_{2}) = 0 and consequently

δ_{a}^{i_{1}+i_{2}−1}([Δ_{1}, Δ_{2}]) = \binom{i_{1}+i_{2}−1}{i_{1}}[δ_{a}^{i_{1}}(Δ_{1}), δ_{a}^{i_{2}−1}(Δ_{2})] +
+\binom{i_{1}+i_{2}−1}{i_{1}−1}[δ_{a}^{i_{1}−1}(Δ_{1}), δ_{a}^{i_{2}}(Δ_{2})] = 0.

But, by definition, {|Δ_{1}, Δ_{2}|} = |[Δ_{1}, Δ_{2}]_{i_{1}+i_{2}−1}|.

For any element a ∈ A define the sets

m_{i}(a) = {s ∈ S_{i} | a ∈ char (s)}

and

m(a) = ∑_{i≥0} m_{i}(a).

¿From Proposition 2.2 and Lemma 2.1 it follows that m(a) is an ideal in S.

Proposition 2.7 The ideal m(a) is primitive for any a ∈ A.

Proof. Consider elements s, t ∈ S and let s = s_{1} + … + s_{k}, t = t_{1} + … + t_{l} be their decompositions in homogeneous elements such that deg (s_{1}) < deg (s_{2}) < … < deg (s_{k}) and deg (t_{1}) < deg (t_{2}) < … < deg (t_{l}). Suppose that s · t ∈ m(a) and prove by induction with respect to k + l that either s ∈ m(a) or t ∈ m(a).

For k + l = 2 it follows from Corollary 2.3.

Let k + l > 2 and suppose that the statement is valid for k + l − 1. Then in the decomposition s · t = ∑_{i,j} s_{i}t_{j} the summand s_{k}t_{l} is of the maximal order (and equals to deg (s_{k}) + deg (t_{l})). Hence, s_{k}t_{l} belongs to m(a). From here it follows that either s_{k} or t_{l} belong to m(a). Let, say, s_{k} ∈ m(a). Then

(s_{1} + … + s_{k−1})(t_{1} + … + t_{l}) ∈ m(a)

and consequently either s_{1} +…+s_{k−1} or t_{1} +…+t_{l} lies in m(a) by the induction hypothesis. Remark. The absence of zero divisors in A is necessary for the proof of Lemma 2.1. This Lemma, in its turn, is used to prove that J_{i} and m_{i}(a) are closed with respect to summation. Multiplicative properties of these sets are independent of the existence of zero divisors in A. In particular, both J_{Δ} and m(a) are stable with respect to multiplication by any element of the symbolic ring, while S \m(a) is closed with respect to multiplication. Remark. Let a ∈ A. Consider the map δ_{a}^{∗} : S → A defined by

δ_{a}^{∗}(s) = δ_{a}^{i}(Δ), s ∈ S_{i}, |Δ| = s.

Obviously, δ_{a}^{∗} is an epimorphism. Denote its kernel by \overline{m}(a). If A possesses no zero divisors, one has \overline{m}(a) = m(a). The map δ_{a}^{∗} is an algebraic analog of the section of the cotangent bundle corresponding to the form da, while \overline{m}(a) correspond to a Lagrangian submanifold in T^{∗}(M) related to this form (cf. [5]). The ideal \overline{m}(a) is closed with respect to the Poisson bracket. It can be proved in the same way as it was done in Proposition 2.6. On the other hand, the ideal J_{Δ} plays the role of the Hamilton–Jacobi equation, and in the rest of this Section we deal with the algebraic viewpoint on the relations between characteristicity and solutions of the Hamilton–Jacobi equation corresponding to the operator Δ.

Proposition 2.8 For any a ∈ char (Δ) the ideal m(a) contains J_{Δ}. Moreover, one has the following equality

J_{Δ} = ⋂_{a∈char (Δ)} m(a).

Proof. It is sufficient to prove the statement for homogeneous components. Let s ∈ J_{Δ} be a homogeneous element and a ∈ char (Δ). Then a ∈ char (s), i.e. s ∈ m(a). Hence, J_{Δ} ⊂ ⋂_{a∈char (Δ)} m(A).

Conversely, if a ∉ J_{Δ}, then char (Δ) ⊄ char (s). Hence, there exists an element a ∈ char (Δ), such that a ∉ char (s). Therefore, s ∉ m(a).

Proposition 2.9 The following statements are equivalent:

(i) char (Δ) ⊂ ⋂_{s∈J_{Δ}} char (s) and

(ii) a ∈ char (Δ) if and only if J_{Δ} ⊂ m(a).

Remark. Condition (i) of the previous Proposition is holds, for example, in the following situation. Suppose that the operator Δ is such that the module

P′ = ⋂_{a∈char (Δ)} ker δ_{a}^{i}(Δ)

is nontrivial. Suppose further that for any nontrivial q ∈ Q there exists q^{∗} ∈ hom_{A}(Q, A) such that q^{∗}(q) ≠ 0. Note that the embedding

char (Δ) ⊂ ⋂_{s∈J_{Δ}} char (s)

is always valid. Let now a ∉ char (Δ). Then δ_{a}^{i}(Δ) : P → Q if a monomorphism. Consider an element p ∈ P′ and define a homomorphism φ_{p} : A → P by setting φ_{p}(1) = p. Choose an element q^{∗} ∈ hom_{A}(Q, A) such that (q^{∗} ∘ δ_{a}^{i}(Δ))(p) ≠ 0. Then for the operator ∂ = q ∘ Δ ∘ φ_{p} one has

δ_{a}^{i}(∂) = q^{∗} ∘ δ_{a}^{i}(Δ) ∘ φ_{p} ≠ 0,

i.e. a ∉ char (∂). On the other hand, if a′ ∈ char (Δ), then

(δ_{a′}^{i}∂)(1) = (q^{∗} ∘ δ_{a′}^{i}(Δ))(p) = 0,

i.e. ∂ ∈ J_{Δ}.

The result obtained allows us to generalize the definition of the characteristics of linear differential operators and to give the following

Definition 4 Let Δ ∈ Diff_{∗}^{(+)}(P, Q). A generalized characteristic of the operator Δ is a graded primitive ideal p of the symbolic ring S, such that J_{Δ} ⊂ p. A characteristic manifold of the operator Δ is the space

Spec_{gr} (S/J_{Δ}) ⊂ Spec_{gr}S = T^{∗}(A).

3 Symbolic module

Here we define the concept of symbolic module (cf. [2]) which makes it possible to get rid of the absence of zero divisors condition and to define the notion of characteristics in a general algebraic setting.

Consider an operator Δ ∈ Diff_{i}^{(+)}(P, Q). Then one can define an S-module homomorphism σ(Δ) : S(Q, A) → S(P, A) in the following way. For |∇| ∈ S_{k}(Q, A) we set

σ(Δ)(|∇|) = |∇ ∘ Δ|_{k+i} ∈ S_{k+i}(Q, A).

Co-kernel of this homomorphism is called the symbolic module of the operator Δ and is denoted by M_{Δ}: coker(σ(Δ)) = M_{Δ}.

Note that an element s = |∂|, deg (s) = r, lies in Ann(M_{Δ}) if and only if for any ∇ ∈ Diff_{i+k}^{(+)}(P, A) there exist ∇′ ∈ Diff_{r+k}^{(+)}(Q, A) and ε ∈ Diff_{i+r+k−1}^{(+)}(Q, A) such that

∂ ∘ ∇ = ∇′ ∘ Δ + ε. (1)

¿From (1) it follows that if s ∈ Ann(M_{Δ}), then

char (Δ) ⊂ char (s) ⋃ ( ⋂_{s′∈S(P,A)} char (s′)) .

Hence, if ⋂_{s′∈S(P,A)} char (s′) is empty, then one has the following

Proposition 3.1 The annihilator of M_{Δ} lies in J_{Δ}, Ann(M_{Δ}) ⊂ J_{Δ}.

Due to Proposition 2.5, one has also the following

Corollary 3.2 The radical of Ann(M_{Δ}) lies in J_{Δ}, r(Ann(M_{Δ})) ⊂ J_{Δ}.

In the sequel we restrict ourselves to the operators Δ for which the symbolic module M_{Δ} is of finite type. Then

Supp(M_{Δ}) = Spec (S/Ann(M_{Δ})) ⊂ T^{∗}(A)

and hence, due to Proposition 3.1, for any generalized characteristic p ∈ Spec_{gr}(A/J_{Δ}) localization (M_{Δ})_{p} is nontrivial. It means that the set of generalized characteristics of the operator Δ lies in Supp(M_{Δ}). These remarks motivate another

Definition 5 Let p be a primitive ideal of the algebra A. Denote by ⟨S_{p}⟩ the quotient field of the algebra S/p. A graded primitive ideal p is a generalized characteristic of the operator Δ ∈ Diff_{∗}^{(+)}(P, Q), if the map

s_{p}(Δ) : S(P)/(pS(P)) ⊗_{S/p} ⟨S_{p}⟩ → S(Q)/(pS(Q)) ⊗_{S/p} ⟨S_{p}⟩ ,

induced by the homomorphism s(Δ) : S(P) → S(Q), possesses a nontrivial kernel.

Denote by Λ^{1}(A) the A-module of 1-forms of the algebra A (see [4]).

Theorem 3.3 Suppose that the modules Λ^{1}(A), P, Q are projective and of finite type. Let the operator Δ ∈ Diff_{∗}^{(+)}(P, Q) be such that M_{Δ} is of finite type as well. Then a graded primitive ideal p ∈ Spec_{gr} (S) is a generalized characteristic of Δ in the sense of Definition 5 if and only if (M_{Δ})_{p} ≠ 0.

Proof. Consider a homomorphism η : S(P) → hom_{S}(S(P, A), S) defined by

η(|∇|)(|∂|) = |∇ ∘ ∂|, ∇ ∈ Diff_{∗}^{(+)}(A, P), ∂ ∈ Diff_{∗}^{(+)}(P, A).

Then η induces a homomorphism

η_{p} : S(P)/(pS(P)) ⊗_{S/p} ⟨S_{p}⟩ → hom_{S}(S(P, A), ⟨S_{p}⟩)

for any p ∈ Spec (S). Moreover, for any Δ ∈ Diff_{∗}^{(+)}(P, Q) one has a commutative diagram

[figure: \begin{matrix}S(P)/(pS(P)) ⊗_{S/p} ⟨S_{p}⟩ & S(Q)/(pS(Q)) ⊗_{S/p} ⟨S_{p}⟩ \\ hom_{S}(S(P, A), ⟨S_{p}⟩) & hom_{S}(S(Q, A), ⟨S_{p}⟩) \end{matrix}; η_{p}; s_{p}(Δ); η_{p}; σ^{∗}(Δ)]

where s_{p}(Δ) is induced by the symbolic map smbl(Δ) : S(P) → S(Q).

The functor hom_{S}(·, ⟨S_{p}⟩) is exact from the right. Hence, the kernel of s_{p}(Δ) is trivial if and only if hom_{S}(M_{Δ}, ⟨S_{p}⟩) = 0. But

hom_{S}(M_{Δ}, ⟨S_{p}⟩) ≃ hom_{⟨S_{p}⟩}(M_{Δ} ⊗_{S} ⟨S_{p}⟩ , ⟨S_{p}⟩).

Since ⟨S_{p}⟩ is a field, the map s_{p}(Δ) is a monomorphism if and only if M_{Δ}⊗_{S}⟨S_{p}⟩ = 0. On the other hand,

M_{Δ} ⊗_{S} ⟨S_{p}⟩ ≃ M_{Δ} ⊗_{S} S_{p} ⊗_{S_{p}} ⟨S_{p}⟩ ≃ (M_{Δ})_{p} ⊗_{S_{p}} ⟨S_{p}⟩ .

Let p_{0} be the maximal ideal of the local ring S_{p}. Then ⟨S_{p}⟩ ≃ S_{p}/p_{0} and

(M_{Δ})_{p} ⊗_{S_{p}} ⟨S_{p}⟩ ≃ \frac{(M_{Δ})_{p}}{p_{0}(M_{Δ})_{p}} .

Due to the Nakayama Lemma [1], it is equivalent to triviality of the module (M_{Δ})_{p}.

Corollary 3.4 If an ideal p ∈ Spec_{gr} (S) is a characteristic of the operator Δ in the sense of Definition 4, then p is a characteristic in the sense of Definition 5.

To prove the equivalence of Definitions 4 and 5, we suppose additionally that (a) the modules P, Q are free,

(b) S possesses the following property of divisibility: if s, s_{1}, …, s_{r} ∈ S and s ∈ ⋂_{i=1}^{r}(char (s_{i})), then there exist elements t_{1}, …, t_{r} ∈ S, such that

s = ∑_{i=1}^{r} t_{i}s_{i}, (2)

(c) the number of generators in Q is greater or equal than the one in P, i.e. Δ is not an underdetermined operator (naturally, it can always be done by embedding Q in some R with sufficient number of generator).

Choose free basises in P and Q and represent the operator Δ by the matrix ∥∂_{jk}∥, where ∂_{jk} ∈ Diff_{∗}^{(+)}(P, Q). Denote by ∇_{1}, …, ∇_{r} the leading minors of this matrix. Note that though these minors are not well-defined, their symbols are uniquely determined. Hence, there characteristics are well-defined.

Proposition 3.5 J_{Δ} = ∑_{l=1}^{r} J_{∇_{l}}

Proof. Let i be the order of Δ and n be the dimension of the leading minors. Let a ∈ A. Since ker δ_{a}^{i}(Δ) ≠ 0 if and only if δ_{a}^{ni}(∇_{l}) ≠ 0 for all l = 1, …, r, one has char (Δ) = ⋂_{l} char (∇_{l}). Our result follows now from the divisibility condition formulated above.

Under conditions (a), (b), (c) one also has the following

Proposition 3.6 The equality char (Δ) = ⋂_{s∈J_{Δ}} char (s) holds.

Thus in the case under consideration, an element a ∈ A is a characteristic of Δ if and only if J_{Δ} ⊂ m(a) (see Proposition 2.9).

We shall also need the following result valid in terms of conditions (a), (b),

(c) formulated above.

Lemma 3.1 If |∇| ∈ S, then J_{∇} is the principal ideal generated by |∇|: J_{∇} = (|∇|).

Consider again the localization s_{p}(Δ) of the symbol of the operator Δ. Its kernel is nontrivial if and only if all its leading minors (in some coordinate representation) vanish. In other words, the kernel is nontrivial if and only if for each ∇_{l} there exists an element t_{l} ∈ S \ p, such that t_{l}|∇_{l}| ∈ p. Since p is a primitive ideal, it is equivalent to the fact that |∇_{|}] ∈ p. This, together with Proposition 3.5 and Lemma 3.1, gives the following

Theorem 3.7 If the symbolic module of the operator Δ is of finite type and if the conditions (a), (b), (c) hold, then definitions 4 and 5 are equivalent.

Corollary 3.8 Under conditions of Theorem 3.7 one has

J_{Δ} = r(AnnM_{Δ}).

4 Characteristics of morphisms and junior // symbols

Let, as before, R be a commutative ring with a unit and B be a commutative unitary R-algebra. Consider B-modules P and Q.

Definition 6 A primitive ideal p ∈ Spec B is called a characteristic of a morphism φ ∈ hom_{B}(P, Q), if the kernel of the map

φ_{p} : P/(pP) ⊗_{B/p} ⟨B_{p}⟩ → Q/(pQ) ⊗_{B/p} ⟨B_{p}⟩

is nontrivial.

Denote the set of characteristics of φ by char (φ). We call

J_{φ} = ⋂_{p∈char (φ)} p

the characteristic ideal of φ. Example. Let V be a finite-dimensional space over a field k and L ∈ Endk(V). Consider a k[x]-module structure in V defined in the standard way: x · v = L(v), v ∈ V. Consider the trivial endomorphism 0 of this module. Then J0 = r(det(L − xE)). If k is algebraically closed, then char (0) = Spec (B/J0).

Consider relations between Spec (B/J_{φ}) and the set char (φ).

Definition 7 Let φ^{∗} : Q^{∗} → P^{∗} be a morphism dual to φ. The module M_{φ} = coker(φ^{∗}) is called the symbolic module of φ.

Proposition 4.1 If P and Q are finitely generated modules, then char (φ) = Supp(M_{φ}).

Proof. Consider the natural isomorphism

η_{p} : P/(pP) ⊗_{B/p} ⟨B_{p}⟩ → hom_{B}(P^{∗}, ⟨B_{p}⟩),

where p ∈ Spec (B). Then one has a commutative diagram

[figure: \begin{matrix}P/(pP) ⊗_{B/p} ⟨B_{p}⟩ & Q/(pQ) ⊗_{B/p} ⟨B_{p}⟩ \\ hom_{B}(P^{∗}, ⟨B_{p}⟩) & hom_{B}(Q^{∗}, ⟨B_{p}⟩) \end{matrix}; η_{p}; φ_{p}; η_{p}; φ_{p}^{∗∗}]

The rest of the proof literary repeats the proof of Theorem 3.3.

Corollary 4.2 J_{φ} = r(AnnM_{φ}).

Theorem 4.3 Let the conditions of Proposition 4.1 be valid. Then a primitive ideal p lies in char (φ) if and only if J_{φ} ⊂ p, i.e.

char (φ) = Spec (B/J_{φ}).

Proof. Let J_{φ} ⊂ p. Then Ann(M_{φ}) ⊂ p and consequently p ∈ char (φ).

Note that the result of previous subsections remain valid in the category of graded modules over a graded algebra B. Suppose additionally that the graded objects B, P, and Q are associated with filtered ones, B′, P′, Q′, i.e. B′ = ⋂_{i} B′_{i}, B′_{i−1} ⊂ B′_{i}, B_{i} = B′_{i}/B′_{i−1}, etc. Let φ = φ_{n} : P → Q be a morphism of the grading n ≥ 0 associated with a filtered morphism φ′ = φ′_{n} : P′ → Q′. Denote the characteristic ideal of φ by J_{φ}^{n} = J^{n}. Obviously, φ(J^{n}P) ⊂ J^{n}Q. Consider the natural homomorphism of graded B/J^{n}-modules

φ_{J^{n}} : P/J^{n}P → Q/J^{n}Q.

Let a ∈ P, deg (a) = i. Then the class aJ^{n}P lies in ker (φ_{J^{n}}) if and only if

φ′(a′) = b′ + b′′,

where |a′| = a, |b′| = b ∈ J^{n}Q, deg (b′′) = n + i − 1 (as before, by |s| we denote an element of the graded object corresponding to the element s from the filtered object). Since the element a′ is defined up to a sum of elements a′′, |a′′| ∈ J^{n}P, and a′′′, deg (a′′′) = i − 1, then one has a well-defined morphism

φ_{n−1} : ker (φ_{J^{n}}) → coker (φ_{J^{n}})

of the grading n − 1. Hence, one can define the set

char (φ_{n−1}) ⊂ Spec (B/J^{n}) ⊂ Spec (B)

and the ideal J_{φ_{n−1}}. Denote by J^{n−1} = J_{φ}^{n−1} the inverse image of J_{φ_{n−1}} under the natural projection B → B/J^{n}.

Proposition 4.4 For all j = n − 1, …, 0 one has well-defined morphisms

φ_{j} : ker (φ_{J^{j+1}}) → coker(φ_{J^{j+1}}),

where ideals J^{j+1} and morphisms

φ_{J^{j+1}} : \frac{ker(φ_{J^{j+2}})}{J_{φ_{j+1}} ker(φ_{J^{j+2}})} → \frac{coker(φ_{J^{j+2}})}{J_{φ_{j+1}}coker(φ_{J^{j+2}})}

are naturally constructed using the morphisms φ_{j+1}.

Proof. Use the inverse induction by j. For j = n − 1 the statement has been proved already. Let now j < n − 1 and suppose that φ_{j} has been constructed. Consider natural projections

π_{k} : coker (φ_{J^{k}}) → \frac{coker (φ_{J^{k}})}{J_{φ_{k−1}}coker (φ_{J^{k}})}

and

ρ_{k} : \frac{coker (φ_{J^{k}})}{J_{φ_{k−1}}coker (φ_{J^{k}})} → coker (φ_{J^{k−1}}),

where k = n + 1, …, j. Then it can be easily seen that an element a′ ∈ P′_{i} determines an element ā ∈ ker (φ_{J^{i}}) if and only if

φ′(a′) = b′ + φ′(a′′) + b′′,

where – b′ is of filtration i + n,

– (ρ_{j+2} ∘ π_{j+2} ∘ … ∘ ρ_{n+1} ∘ π_{n+1})(|b′|) ∈ J_{φ_{j}}coker (φ_{J^{j+1}}),

– a′′ is of filtration i − 1,

– b′′ is of filtration j − 1. Hence there exists a well-defined element

φ_{j−1}(ā) = (ρ_{j+1} ∘ π_{j+1} ∘ … ∘ ρ_{n+1} ∘ π_{n+1})(|b′′|).

Definition 8 The morphism

φ_{j} : ker (φ_{J^{j+1}}) → coker (φ_{J^{j+1}})

defined above is called the j-th symbol of the morphism φ, j = n − 1, …, 0.

Let J_{φ}^{j} = J^{j} be the inverse image of J_{φ_{j}} in B. Obviously, one has a sequence of embeddings

J^{n} ⊂ J^{n−1} ⊂ … ⊂ J^{1} ⊂ J^{0} ⊂ B.

Definition 9 The ideal J_{φ}^{j} is called the characteristic ideal of the rank j for the morphism φ. A primitive graded ideal

p ∈ Spec_{gr}(B/J^{j}) ⊂ Spec_{gr}(B)

is called a characteristic of the rank j for the morphism φ.

Denote the set of such characteristics by char_{j}(φ). Obviously, one has a sequence of embeddings

char (φ) = char_{0}(φ) ⊂ char_{1}(φ) ⊂ … ⊂ char_{n}(φ) ⊂ Spec_{gr}(B).

Theorem 4.5 Assume ker (φ_{J^{j+1}}) and coker (φ_{J^{j+1}}) to be projective B/J^{j+1}- modules of finite type. Then a primitive graded ideal p ∈ Spec_{gr}(B) belongs to char_{j}(φ) if and only iff the kernel of the map

(φ_{j})_{p} : \frac{ker (φ_{J^{j+1}})}{p ker (φ_{J^{j+1}})} ⊗_{B/p} ⟨B_{p}⟩ → \frac{coker (φ_{J^{j+1}})}{pcoker (φ_{J^{j+1}})} ⊗_{B/p} ⟨B_{p}⟩

is nontrivial.

Remark. Let B, C be unitary commutative algebras over a commutative ring R. Let f : B → C be a homomorphism due to which C can be considered as an B-algebra. If P is an B-module, then C ⊗_{B} P is a C-module. This construction is an algebraic analog of the induced bundle. In particular, if J is an ideal in B, then an B/J-module P/(JP) ≃ (B/J) ⊗_{B} P geometrically corresponds to the restriction onto a submanifold determined by the ideal J. Thus one can see that the (n − 1)-st symbol is defined on submanifold of degeneration of the senior symbol, the next one is defined on the submanifold of degeneration of the (n − 1)-st, etc. Remark. Junior symbols of a differential operator Δ : P → Q arise, when one considers S for B and an S-module homomorphism

φ = smbl(Δ) : S(P) → S(Q)

and applies the results of this section to φ.

References

[1] Atyah, M.F. and MacDonald, I.G.: Introduction to Commutative Algebra, Addisom Wesley Publ. Comp., Reading MA, 1969.

[2] Goldschmidt, H.: Existence theorems for analytic linear partial differential equations, Ann. of Math. 86 (1967), 246–270.

[3] Spencer, D.C.: Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179–239.

[4] Vinogradov, A.M.: The logic algebra of linear differential operators, Soviet Math. Dokl. 13(4) (1972), 1058–1062.

[5] Vinogradov, A.M. and Krasil’shchik, I.S.: What is the Hamiltonian formalism? London Math. Soc. Lecture Notes Ser. 60 (1981), 241–266.