Notes on Weyl Algebra and D-Modules

The present notes base on two short courses:

1. (1)

   Introduction to Weyl Algebras: five Twin Lessons, Thai Nguyen University of Science TNUS (Thai Nguyen, Vietnam), November 1–10, 2013.

2. (2)

   Weyl Algebras, Universal Gröbner Bases, Filtration Deformations and Characteristic Varieties of D-Modules: four Twin Lessons, Vietnam Institute for Advanced Study in Mathematics VIASM (Hanoi, Vietnam), November 12–26, 2013.

They were also the base for a third course:

1. (3)

   Introduction to Weyl Algebras and D-Modules: four Lessons and two Tutorial Sessions, “Workshop on Local Cohomology”, St. Joseph’s College Irinjalakuda, Kerala (India), June 20–July 2, 2016.

These notes aim to give an approach to Boldini’s stability and deformation results for characteristic varieties [11, 12] and to the bounding result [16] for the degrees of defining equations of characteristic varieties, including a self-contained introduction to the needed background on Weyl Algebras and D-modules. In particular, these notes should not be understood as an independent or complete introduction to the field of Weyl algebras and D-modules, which could replace one of the existing textbooks or monographs like [8, 9, 13, 24, 29, 37] or [38]: Too many core subjects are not treated at all or only marginally in these notes, as they are not needed on the way to our final results.

So, a few basic topics which are lacking in these notes—and which ought to be considered as indispensable in a complete introduction to the field—are:

• a systematic study of Bernstein’s Inequality and holonomic D-modules (we treat these subjects only briefly in Exercise and Remark 1.14.3),

• Bernstein’s result on singularities of generalized Γ-functions and Bernstein-Sato polynomials,

• weighted filtrations with negative weights,

• the sheaf theoretic and cohomological aspect,

• the analytic aspect.

Another subject which is not treated in our notes are Lyubeznik’s finiteness results for local cohomology modules of regular local rings in characteristic 0 (see [33] and also [34]), which brought a break-through in Commutative Algebra, as they base on the use of (holonomic) D-modules—and hence present a very important link between these two fields.

These notes are divided up in 14 sections:

1. 1.

   Introduction

2. 2.

   Filtered Algebras

3. 3.

   Associated Graded Rings

4. 4.

   Derivations

5. 5.

   Weyl Algebras

6. 6.

   Arithmetic in Weyl Algebras

7. 7.

   The Standard Basis

8. 8.

   Weighted Degrees and Filtrations

9. 9.

   Weighted Associated Graded Rings

10. 10.

    Filtered Modules

11. 11.

    D-Modules

12. 12.

    Gröbner Bases

13. 13.

    Weighted Orderings

14. 14.

    Standard Degree and Hilbert Polynomials

Sections 1.1–1.9 were the subject of the introductory course (1) at TNUS. In our course (2) at the VIASM we gave an account on all 14 sections, and discussed a few applications (to the Gelfand-Kirillow dimension of D-modules for example), which are not contained in these notes. In the course (3) at St. Joseph’s College we treated Sects. 1.1–1.9 and 1.14.

Our suggested basic reference is Coutinho’s introduction [24], although we do not follow that introduction and we partly use our own terminology and notations. We start in a slightly more general setting, than Coutinho, and so also we recommend the references [4, 10, 11, 32] and [35]. Files of [10] and [11] are available on request at the author. For readers who have already some background in the subject, we recommend as possible references [8, 9, 13, 29, 37], or the first part of the PhD thesis [11].

Finally, we aim to fix a few notations and make a few conventions which we shall keep throughout these notes. We do this on a fairly elementary level, according to the original intention of the short course (1): To give a first introduction to the subject to an audience having only some background in Linear Algebra and basic Abstract Algebra. Only in Sect. 1.14 we will need some background from Commutative Algebra, notably Hilbert functions and -polynomials, Local Cohomology and Castelnuovo-Mumford regularity. We shall give brief reminders on these more advanced preliminaries in Sect. 1.14.

Conventions, Reminders and Notations 1.1.1

(A) (General Notations) By \(\mathbb {Z}, \mathbb {Q}\) and \(\mathbb {R}\) we respectively denote the set of integers, of rationals and of real numbers. We also write

for the set of non-negative respectively of positive real numbers. Moreover, we use the following notations for the set of non-negative respectively the set of positive integers:

If \(\mathbb {S} \subset \mathbb {R}\) we form the supremum and infimum \(\mathrm {sup}(\mathbb {S})\) resp. \(\mathbb {\inf }(\mathbb {S})\) within the set \(\mathbb {R} \cup \{-\infty , \infty \},\) using the convention that sup(∅) = −∞ and inf(∅) = ∞.

Empty sums and empty products are respectively understood to be 0 or 1. We thus set

(B) (Rings) All rings R are understood to be associative, non-trivial and unital, so that they have a unit-element 1 = 1_R ∈ R ∖{0} and the following properties hold

1. (a)

   0x = x0 = 0 and 1x = x1 = x for all x ∈ R;

2. (b)

   x(yz) = (xy)z, x(y + z) = xy + xz and (x + y)z = xz + yz for all x, y, z ∈ R.

Rings need not be commutative.

If R is a ring, a subring of R is a subset R ₀ ⊆ R, such that 1_R ∈ R ₀ and x + y, xy ∈ R ₀ whenever x, y ∈ R ₀. If R ₀ ⊆ R is a subring and S ⊆ R is an arbitrary subset, we write R ₀[S] for the subring of R generated by R ₀ and S, hence for the smallest subring of R which contains R ₀ and S. Thus, R ₀[S] is the intersection of all subrings of R which contain R ₀ and S, and may be written in the form

If a ₁, a ₂, …, a _r is a finite collection of elements of R, we set

(C) (Homomorphisms of Rings) All homomorphisms of rings are understood to be unital, and hence are maps h : R→S, with R and S rings, such that

1. (a)

   h(x + y) = h(x) + h(y) and h(xy) = h(x)h(y) for all x, y ∈ R;

2. (b)

   h(1_R) = 1_S.

Clearly, the identity map Id_R : R→R is a homomorphism of rings, and the composition of homomorphisms of rings is again a homomorphism of rings. An isomorphism of rings is a homomorphism of rings admitting an inverse homomorphism. A homomorphism of rings is an isomorphism, if and only if it is bijective.

(D) (K-Algebras) All fields are considered as commutative. If K is a field, a K-algebra is understood to be a ring A together with a homomorphism of rings ε : K→A such that

As the ring A is non-trivial, the homomorphism ε : K→A is injective. So, we can and do always embed K into A by means of ε and thus identify c with ε(c) for all c ∈ K. Hence we have

Keep in mind, that a K-algebra A is a K-vector space in a natural way.

(E) (Homomorphisms of K-Algebras) Let K be a field. A homomorphism of K-algebras h : A→B is a map with K-algebras A and B such that:

1. (a)

   h : A→B is a homomorphism of rings;

2. (b)

   h(c) = c for all c ∈ K.

Observe, that a homomorphism of K-algebras is also a homomorphism of K-vector spaces.

(F) (Modules) We usually shall consider unital left-modules, hence modules M over a ring R, such that

for all x, y ∈ R and all m, n ∈ M. We shall refer to left-modules just as modules.

By a homomorphism of R-modules we mean a map h : M→N, with M and N both R-modules, such that

1. (a)

   h(m + n) = h(m) + h(n) for all m, n ∈ M.

2. (b)

   h(xm) = xh(m) for all x ∈ R and all m ∈ M.

A submodule of a R-module M is a subset N ⊆ M, such that m + n ∈ N and xm ∈ N whenever m, n ∈ N and x ∈ R. Clearly 0 := {0} and M are submodules of M.

If h : M→N is a homomorphism of R-modules, the kernel Ker(h) := {m ∈ M∣h(m) = 0} of h is a submodule of M and the image Im(h) := h(M) of h is a submodule of N.

A sequence of (homomorphisms of) R-modules

is said to be exact if Ker(h _i) = Im(h _i−1) for all i = 1, 2, …, r − 1. A short exact sequence of R-modules is an exact sequence of the form \(0 \longrightarrow M\stackrel {h}{\longrightarrow } N \stackrel {{l}}{\longrightarrow } P \longrightarrow 0,\) meaning that h is injective, l is surjective and Ker(l) = Im(h).

The annihilator of an R-module M is defined as the left ideal of R consisting of all elements which annihilate M, thus:

(G) (Noetherian Modules and Rings) Let R be a ring. A left R-module is said to be Noetherian, if it satisfies the following equivalent conditions

1. (i)

   Each left submodule N ⊆ M if finitely generated, and hence of the form \(N = \sum _{i=1}^r Rn_i\) with \(r \in \mathbb {N}_0\) and n ₁, n ₂, …, n _r ∈ N.

2. (ii)

   Each ascending sequence N ₀ ⊆ N ₁ ⊆⋯N _i ⊆ N _i+1 ⊆⋯ of left submodules N _i ⊆ M ultimately becomes stationary and thus satisfies \(N_{i_0} = N_{i_0 +1} = N_{i_0+2} = \ldots \) for some \(i_0 \in \mathbb {N}_0\).

We say that the ring R is left Noetherian if it is Noetherian as a left module.

Keep in mind the following facts:

1. (a)

   If 0→N→M→P→0 is an exact sequence of left R-modules then M is Noetherian if and only N and P are both Noetherian.

2. (b)

   If M and N are two Noetherian left R-modules, then their direct sum M ⊕ N is Noetherian, too.

3. (c)

   If R is left Noetherian, a left R-module M is Noetherian if and only if it is finitely generated.

(H) (Modules of Finite Presentation) Let R be a ring. A left R-module M is said to be of finite presentation if there is an exact sequence of left R-modules

In this situation, the above exact sequence is called a (finite) presentation of M and \(R^s \stackrel {h}{\longrightarrow } R^r\) is called a presenting homomorphism for M.

Keep in mind, that the presenting homomorphism is given by a matrix with entries in R, more precisely: There is a matrix

for all (x ₁, x ₂, …, x _s) ∈ R ^s. This matrix A is called a presentation matrix for M.

Note the following facts:

1. (a)

   A left R-module M of finite presentation is finitely generated.

2. (b)

   If R is left Noetherian, then each finitely generated left R-module is of finite presentation.

(I) (Graded Rings and Modules) A (positively) graded ring is a ring R together with a family \((R_i)_{i \in \mathbb {N}_0}\) of additive subgroups R _i ⊆ R such that

1. (1)

   \(R = \bigoplus _{i\in \mathbb {N}_0}R_i\);

2. (2)

   1 ∈ R ₀;

3. (3)

   for all \(i, j \in \mathbb {N}_0\) and all a ∈ R _i and all b ∈ R _j it holds ab ∈ R _i+j.

In this situation we also refer to \(R = \bigoplus _{i\in \mathbb {N}_0}R_i\) as (positively) graded R ₀ -algebra. If a ∈ R _i ∖{0}, we call a a homogeneous element of degree i.

Let \(R' = \bigoplus _{i \in \mathbb {N}_0} R^{\prime }_i\) be a second graded ring. A homomorphism of graded rings is a homomorphism f : R→R′ of rings which respects gradings, hence such that \(f(R_i) \subseteq R^{\prime }_i\) for all \(i \in \mathbb {N}_0.\) Clearly, the identity map Id_R : R→R of a graded ring as well as the composition of two homomorphisms of graded rings is a homomorphism of graded rings. An isomorphism of graded rings is a homomorphism of graded rings which admits an inverse which is a homomorphism of graded rings—or, equivalently—a bijective homomorphism of graded rings.

The (positively) graded ring \(R = \bigoplus _{i \in \mathbb {N}_0} R_i\) is called a homogeneous ring if it is generated over R ₀ by homogeneous elements of degree 1, hence if (in the notation introduced in part (B)) we have R = R ₀[R ₁].

A graded (left) module over the graded ring R is a left R-module together with a family \((M_j)_{j\in \mathbb {Z}}\) of additive subgroups M _j ⊆ M such that

1. (1)

   \(M = \bigoplus _{j \in \mathbb {Z}} M_j\);

2. (2)

   For all \(i \in \mathbb {N}_0\), all \(j \in \mathbb {Z}\), all a ∈ R _i and all m ∈ M _j it holds am ∈ M _i+j.

A homomorphism of graded (left) modules is a homomorphism h : M→N of R-modules, in which \(M = \bigoplus _{j\in \mathbb {Z}}M_j\) and \(N = \bigoplus _{j\in \mathbb {Z}}N_j\) are both graded and h(M _j) ⊆ N _j for all \(j \in \mathbb {Z}\). Clearly, the identity map of a graded R-module and the composition of two homomorphisms of graded R-modules are again homomorphisms of graded R-modules. An isomorphism of graded R-modules is a homomorphism of graded R-modules which admits an inverse which is a homomorphism of graded R-modules—or, equivalently—a bijective homomorphism of graded R-modules.

(K) (Prime Varieties) Let R be a commutative ring. We denote the prime spectrum of R, hence the set of all prime ideals in R, by Spec(R). If \(\mathfrak {a} \subseteq R\) is an ideal, we denote by \(\mathrm {Var}(\mathfrak {a})\) the prime variety of \(\mathfrak {a}\), thus

Let

denote the radical ideal of \(\mathfrak {a}\). Keep in mind the following facts:

1. (a)

   \(\mathrm {Var}(\mathfrak {a}) = \mathrm {Var}(\sqrt {\mathfrak {a}})\).

2. (b)

   If \(\mathfrak {a}, \mathfrak {b} \subseteq R\) are ideals, then \(\mathrm {Var}(\mathfrak {a}) = \mathrm {Var}(\mathfrak {b})\) if and only if \(\sqrt {\mathfrak {a}} = \sqrt {\mathfrak {b}}\).

(L) (Krull Dimension) Let R be as in part (K) and let M be a finitely generated R-module. Then, the (Krull) dimension dim_R(M) of M is defined as the supremum of the lengths of chains of prime ideals which can be found in the prime variety of the annihilator of M:

In particular, the (Krull) dimension dim(R) of R is the dimension of the R-module R:

Before giving a formal acknowledgement, the author likes very much to express his gratitude toward his Vietnamese fellow mathematicians, who gave him the chance to visit the Country so many times, to teach several invited short courses, to present talks and to discuss on Mathematics at various Universities since his first visit of Vietnam in 1996. He also looks back with pleasure to the many visits of Vietnamese mathematicians in Zürich as well as the numerous mathematical cooperations and the many personal friendships which resulted from them.

We begin with a few general preliminaries, which will pave our way to introduce and to treat Weyl algebras and D-modules. Our first preliminary theme are filtered algebras over a field. It will turn out later, that this concept is of basic significance for the theory of Weyl algebras.

Definition and Remark 1.2.1

(A) Let K be a field and let A be K-algebra (see Conventions, Reminders and Notations 1.1.1 (D)). By a filtration of A we mean a family

such that the following conditions hold:

1. (a)

   Each A _i is a K-vector subspace of A;

2. (b)

   A _i ⊆ A _i+1 for all \(i\in \mathbb {N}_0\);

3. (c)

   1 ∈ A ₀;

4. (d)

   \(A = \bigcup _{i\in \mathbb {N}_0}A_i\);

5. (e)

   A _i A _j ⊆ A _i+j for all \(i,j \in \mathbb {N}_0\).

In requirement (e) we have used the standard notation

which we shall use from now on without further mention. To simplify notation, we also often set

and then write our filtration in the form

If a filtration of A is given, we say that (A, A _•) or—by abuse of language—that A is a filtered K-algebra.

(B) Keep the notations and hypotheses of part (A) and let \(A_{\bullet } = (A_i)_{i \in \mathbb {Z}}\) be a filtered K-algebra. Observe that we have the following statements:

1. (a)

   A ₀ is a K-subalgebra of A.

2. (b)

   For all \(i \in \mathbb {Z}\) the K-vector space A _i is a left- and a right- A ₀-submodule of A.

Example 1.2.2 (The Degree Filtration of a Commutative Polynomial Ring)

Let \(n {\,\in \,} \mathbb {N}\) and let A = K[X ₁, X ₂, …, X _n] be the commutative polynomial algebra over the field K in the indeterminates X ₁, X ₂, …, X _n. Then clearly A is a K-space over its monomial basis:

where we have used use the standard notation

So, each f ∈ A can be written as

with a unique family

whose support

is finite. We also introduce the notation

Then, with the usual convention of Conventions, Reminders and Notations 1.1.1 (A) we may describe the degree of the polynomial f ∈ A by

Now, for each \(i \in \mathbb {N}_0\) we introduce the K-subspace A _i of A which is given by

With the usual convention that u + (−∞) = −∞ for all \(u \in \mathbb {Z} \cup \{-\infty \}\), we have the obvious relation

From this it follows easily:

is a filtration of A. This filtration is called the degree filtration of the polynomial algebra A = K[X ₁, X ₂, …, X _n].

Clearly filtrations also may occur in non-commutative algebras. The next example presents somehow the “generic occurrence” of this.

Example 1.2.3 (The Degree Filtration of a Free Associative Algebra)

Let \(n \in \mathbb {N}\), let K be a field and let A = K〈X ₁, X ₂, …, X _n〉 be the free associative algebra over K in the indeterminates X ₁, X ₂, …, X _n. We suppose in particular that (see Conventions, Reminders and Notations 1.1.1 (D))

and hence

Let \(i \in \mathbb {N}_0\). If

is a sequence of length i with values in the set {1, 2, …, n} we write

Then, with the usual convention that the product ∏_{j ∈∅} X _j of an empty family of factors equals 1 and using the notation

we can write A as a K-space over its monomial basis as follows:

Clearly, as in the case of a commutative polynomial ring, each f ∈ A may be written in the form

with a unique family

whose support

is finite. We also introduce the notion of length of a sequence \( \underline {\sigma } \in \mathbb {S}_n\) by setting

Now, we may define the degree of an element f ∈ A by

For each \(i \in \mathbb {N}_0\) we introduce a K-subspace A _i of A, by setting

We obviously have the relation

Moreover, it is easy to see:

is a filtration of A. This filtration is called the degree filtration of the free associative K-algebra A = K〈X ₁, X ₂, …, X _n〉.

Later, our basic filtered algebras will be Weyl algebras. These are non-commutative too, but they also admit the notion of degree and of degree filtration. From the point of view of filtrations, these algebras will turn out to be “close to commutative”, as we shall see later. To make this more precise, we will introduce the notion of associated graded ring with respect to a filtration in the next section.

Remark and Definition 1.3.1

(A) Let K be a field and let A = (A, A _•) be a filtered K-algebra. We consider the K-vector space

For all \(i \in \mathbb {N}_0\) we also use the notation

so that we may write

(B) Let \(i,j \in \mathbb {N}_0\), let f, f′∈ A _i and let g, g′∈ A _j such that

It follows that

So in \(A_{i+j}/A_{i+j-1} = \mathrm {Gr}_{A_{\bullet }}(A)_{i+j} \subset \mathrm {Gr}_{A_{\bullet }}(A)\) we get the relation

This allows to define a multiplication on the K-space \(\mathrm {Gr}_{A_{\bullet }}(A)\) which is induced by

With respect to this multiplication, the K-vector space \(\mathrm {Gr}_{A_\bullet }(A)\) acquires a structure of K-algebra.

Observe that, if \(r,s \in \mathbb {N}_0\) and

then

(C) Keep the above notations and hypotheses. Observe in particular, that \(\mathrm {Gr}_{A_{\bullet }}(A)_0\) is a K-subalgebra of \(\mathrm {Gr}_{A_{\bullet }}(A)\), and that there is an isomorphism of K-algebras

Moreover, with respect to our multiplication on \(\mathrm {Gr}_{A_{\bullet }}(A)\) we have the relations

So, the K-vector space \(\mathrm {Gr}_{A_{\bullet }}(A)\) is turned into a (positively) graded ring

by means of the above multiplication. We call this ring the associated graded ring of A with respect to the filtration A _•. From now on, we always furnish \(\mathrm {Gr}_{A_{\bullet }}(A)\) with this multiplication.

Example and Exercise 1.3.2

(A) Let \(n \in \mathbb {N}\), let K be a field and consider the commutative polynomial ring A = K[X ₁, X ₂, …, X _n]. Show that A has the following universal property within the category of all commutative K-algebras:

• If B is a commutative K-algebra and ϕ : {X ₁, X ₂, …, X _n}→B is a map, then there is a unique homomorphism of K-algebras \(\widetilde {\phi }:A \longrightarrow B\) such that \(\widetilde {\phi }(X_i) = \phi (X_i)\) for all i = 1, 2, …, n.

Show also, that A has the following relational universal property within the category of all associative K-algebras:

• If B is an associative K-algebra and ϕ : {X ₁, X ₂, …, X _n}→B is a map such that ϕ(X _i)ϕ(X _j) = ϕ(X _j)ϕ(X _i) for all i, j ∈{1, 2, …, n}, then there is a unique homomorphism of K-algebras \(\widetilde {\phi }:A \longrightarrow B\) such that \(\widetilde {\phi }(X_i) = \phi (X_i)\) for all i = 1, 2, …, n.

(B) Now, furnish A = K[X ₁, X ₂, …, X _n] with its degree filtration (see Example 1.2.2). Then, on use of the above universal property of A it is not hard to show that there is an isomorphism of K-algebras

We now introduce a class of filtrations, which will be of particular interest for our lectures.

Definition 1.3.3

Let K be a field and let A = (A, A _•) be a filtered K-algebra. The filtration A _• is said to be commutative if

It is equivalent to say that the associated graded ring \(\mathrm {Gr}_{A_{\bullet }}(A)\) is commutative. In this situation, we also say that (A, A _•) is a commutatively filtered K-algebra.

Later, in the case of Weyl algebras, we shall meet various interesting commutative filtrations—and precisely this makes these algebras to a subject which is intimately tied to Commutative Algebra. We now shall define three special types of commutative filtrations, which will play a particularly important rôle in Weyl algebras.

Definition and Remark 1.3.4

(A) Let (A, A _•) be a filtered K-algebra. The filtration A _• is said to be very good if it satisfies the following conditions:

1. (a)

   The filtration A _• is commutative;

2. (b)

   A ₀ = K;

3. (c)

   \(\operatorname {dim}_K(A_1) < \infty \);

4. (d)

   A _i = A ₁ A _i−1 for all \(i \in \mathbb {N}\).

Under these circumstances and on use of the notation introduced in Conventions, Reminders and Notations 1.1.1 (B) we clearly have

So, in this situation, the associated graded ring \(\mathrm {Gr}_{A_{\bullet }}(A)\) is a commutative homogeneous (thus standard graded) Noetherian K-algebra (see Conventions, Reminders and Notations 1.1.1 (I)). If A _• is a very good filtration of A, we say that (A, A _•)—or briefly A—is a very well-filtered K-algebra.

(B) Let (A, A _•) be a filtered K-algebra. The filtration A _• is said to be good if it satisfies the following conditions:

1. (a)

   The filtration A _• is commutative;

2. (b)

   A ₀ is a K-algebra of finite type;

3. (c)

   A ₁ is finitely generated as a (left-)module over A ₀;

4. (d)

   A _i = A ₁ A _i−1 for all \(i \in \mathbb {N}\).

Under these circumstances we clearly have

So, in this situation, the associated graded ring \(\mathrm {Gr}_{A_{\bullet }}(A)\) is a commutative homogeneous Noetherian A ₀-algebra (see Conventions, Reminders and Notations 1.1.1 (I)). If A _• is a good filtration of A, we say that (A, A _•)—or briefly A—is a well-filtered K-algebra.

Clearly, a very well-filtered K-algebra is also well-filtered. (C) Let (A, A _•) be a filtered K-algebra. The filtration A _• is said to be of finite type if it satisfies the following conditions:

1. (a)

   The filtration A _• is commutative;

2. (b)

   A ₀ is a K-algebra of finite type;

3. (c)

   There is an integer \(\delta \in \mathbb {N}\) such that A _j is finitely generated as a (left-)module over A ₀ for all j ≤ δ and

4. (d)

   \(A_i = \sum _{j=1}^{\delta } A_jA_{i-j}\) for all i > δ.

In this situation, we call the number δ a generating degree of the filtration A _•. Under these circumstances we clearly have

So, in this situation, the associated graded ring \(\mathrm {Gr}_{A_{\bullet }}(A)\) is a commutative Noetherian graded A ₀-algebra, which is generated by finitely many homogeneous elements of degree ≤ δ. If A _• is a filtration of A, which is of finite type, we say that (A, A _•) is a filtered algebra of finite type.

Clearly, a well-filtered K-algebra is also finitely filtered. Moreover, if A _• is of finite type and δ = 1, the filtration A _• is good.

Example and Exercise 1.3.5

(A) Let \(n \in \mathbb {N}\), let K be a field and consider the commutative polynomial ring A = K[X ₁, X ₂, …, X _n], furnished with its degree filtration. Then, it is easy to see, that A = K[X ₁, X ₂, …, X _n] is a very well filtered K-algebra.

(B) Let \(n \in \mathbb {N}\), let K be a field and consider the commutative polynomial ring A = K[X ₁, X ₂, …, X _n]. Let m ∈{0, 1, …, n − 1} and consider the subring B := K[X ₁, X ₂, …, X _m] ⊂ A, so that A = B[X _m+1, X _m+2, …, X _n]. For each polynomial \(f = \sum _{ \underline {\nu }} c^{(f)}_{ \underline {\nu }} \underline {X}^{ \underline {\nu }} \in A\) we denote by deg_B(f) the degree of f with respect to the indeterminates X _m+1, X _m+2, …, X _n, hence the degree of f considered as a polynomial in these indeterminates with coefficients in B. Thus we may write

where

Show, that by

a good filtration A _• on A is defined and that there is a canonical isomorphism of graded B-algebras

where A = B[X _m+1, X _m+2, …, X _n] is endowed with the standard grading with respect to the indeterminates X _m+1, …, X _n, hence with the grading given by deg(X _i) = 0 if 1 ≤ i ≤ m and deg(X _i) = 1 for m < i ≤ n.

(C) Let \(n \in \mathbb {N}\), with n > 1, let K be a field and consider the free associative K-algebra A = K〈X ₁, X ₂, …, X _n〉, furnished with its degree filtration A _•. For each i ∈{1, 2, …, n}, let

Show that

(D) Let the notations and hypotheses be as in part (C). Show that A = K〈X ₁, X ₂, …, X _n〉 has the following universal property in the category of K-algebras:

• If B is a K-algebra and ϕ : {X ₁, X ₂, …, X _n}→B is a map, there is a unique homomorphism of K-algebras \(\widetilde {\phi }: A \longrightarrow B\) such that \(\widetilde {\phi }(X_i) = \phi (X_i)\) for all i = 1, 2, …, n.

Use this to show, that there is a unique homomorphism of (graded) K-algebras (which must be in addition surjective)

(E) Let (A, A _•) be a filtered K-algebra, let \(r\in \mathbb {N}\) and let \(i_1,i_2,\ldots ,i_r \in \mathbb {N}_0\). We define inductively

In particular, if \(i \in \mathbb {N}_0\) we set

Assume now, that the filtration A _• is good and prove that

Assume that the filtration A _• is of finite type and has generating degree δ. Prove that

Filtered K-algebras and their associated graded rings are one basic ingredient of the theory of Weyl algebras. Another basic ingredient are derivations (or derivatives). The present section is devoted to this subject.

Definition and Remark 1.4.1

(A) Let K be a field, let A be a commutative K-algebra and let M be an A-module. A K-derivation (or K-derivative) on A with values in M is a map d : A→M such that:

1. (a)

   d is K-linear: d(αa + βb) = αd(a) + βd(b) for all α, β ∈ K and all a, b ∈ A.

2. (b)

   d satisfies the Leibniz Product Rule: d(ab) = ad(b) + bd(a) for all a, b ∈ A.

We denote the set of all K-derivations on A with values in M by Der_K(A, M), thus:

To simplify notations, we also write

(B) Keep in mind, that Hom_K(A, M) carries a natural structure of A-module, with scalar multiplication given by

It is easy to verify:

With our usual convention (suggested in Conventions, Reminders and Notations 1.1.1 (D)) that we identify c ∈ K with c1_A ∈ A, the rules (a) and (b) of part (A) imply d(c) = d(c1) = cd(1) and d(c1) = 1d(c) + cd(1), hence d(c) = d(c) + cd(1) = d(c) + d(c), thus

Next, we shall look at the arithmetic properties of derivations and gain an important embedding procedure for modules of derivations of K-algebras of finite type.

Exercise and Definition 1.4.2

(A) Let K be a field, let A be a commutative K-algebra and let M be an A-module. Let d ∈Der_K(A, M), let \(r \in \mathbb {N}\), let \(\nu _1,\nu _2,\ldots ,\nu _r \in \mathbb {N}\) and let a ₁, a ₂, …, a _r ∈ A. Use induction on r to prove the Generalized Product Rule

and the resulting Power Rule

(B) Let the notations and hypotheses be as in part (A). Assume in addition that A = K[a ₁, a ₂, …, a _r]. Let e ∈Der_K(A, M). Use what you have shown in part (A) together with the fact that e and d are K-linear to prove that the following uniqueness statement holds:

(C) Yet assume that A = K[a ₁, a ₂, …, a _r]. Prove that there is a monomorphism (thus an injective homomorphism) of A-modules

This monomorphism \(\varTheta ^M_{ \underline {a}}\) is called the embedding of Der_K(A, M) in M ^r with respect to \( \underline {a} := (a_1,a_2,\ldots ,a_r)\).

(D) Let the notations and hypotheses be as in part (C). Assume that M is finitely generated. Prove, that the A-module Der_K(A, M) is finitely generated.

Now, we turn to derivatives in polynomial algebras, a basic ingredient of Weyl algebras.

Exercise and Definition 1.4.3 (Partial Derivatives in Polynomial Rings)

(A) Let \(n \in \mathbb {N}\), let K be a field and consider the polynomial algebra K[X ₁, X ₂, …, X _n]. Fix i ∈{1, 2, …, n}. Then, using the monomial basis of K[X ₁, X ₂, …, X _n] we see that there is a unique K-linear map

such that for all \( \underline {\nu } = (\nu _1,\nu _2,\ldots ,\nu _n) \in \mathbb {N}_0^n\) we have

(B) Keep the notations and hypotheses of part (A). Let

and prove that

Use the K-linearity of ∂ _i to conclude that

The derivation \(\partial _i = \frac {\partial }{\partial X_i}\) is called the i-th partial derivative in K[X ₁, X ₂, …, X _n].

As we shall see in the proposition below, the embedding introduced in Exercise and Definition 1.4.2 (C) takes a particularly favorable shape in the case of polynomial algebras. The exercise to come is aimed to prepare the proof for this.

Exercise 1.4.4

(A) Let the notations and hypotheses be as in Exercise and Definition 1.4.3. For all \(i,j \in \mathbb {Z}\) let δ _i,j denote the Kronecker symbol, so that

Check that

(B) Keep the above notations and hypotheses. Show that

1. (a)

   For each i ∈{1, 2, …, n} it holds K[X ₁, X ₂, …, X _i−1, X _i+1, …, X _n] ⊆Ker(∂ _i) with equality if and only if Char(K) = 0.

2. (b)

   \(K \subseteq \bigcap _{i=1}^n\mathrm {Ker}(\partial _i)\) with equality if and only if Char(K) = 0.

Proposition 1.4.5 (The Canonical Basis for the Derivations of a Polynomial Ring)

Let \(n \in \mathbb {N}\) , let K be a field and consider the polynomial algebra K[X ₁, X ₂, …, X _n]. Then the canonical embedding of Der_K (K[X ₁, X ₂, …, X _n]) into K[X ₁, X ₂, …, X _n]ⁿ with respect to X ₁, X ₂, …, X _n (see Exercise and Definition 1.4.2 (C)) yields an isomorphism of K[X ₁, X ₂, …, X _n]-modules

given by

In particular, the n partial derivatives ∂ ₁, ∂ ₂, …, ∂ _n form a free basis of the K[X ₁, X ₂, …, X _n]-module Der_K (K[X ₁, X ₂, …, X _n]), hence

Proof

According to Exercise and Definition 1.4.2 (C), the map Θ is a monomorphism of K[X ₁, X ₂, …, X _n]-modules. By what we have seen in Exercise 1.4.4 (A) we have

for all i = 1, 2, …, n. As the n elements

form the canonical free basis of the K[X ₁, X ₂, …, X _n]-module K[X ₁, X ₂, …, X _n]ⁿ our claims follow immediately.

Now, we are ready to introduce Weyl algebras. We first remind a few facts on endomorphism rings of commutative K-algebras and relate these to modules of derivations.

Reminder and Remark 1.5.1

(A) Let K be a field and let A be a commutative K-algebra and let M be an A-module. Keep in mind, that the A-module

carries a natural structure of K-algebra, whose multiplication is given by composition of maps, thus:

The module End_K(M) endowed with this multiplication is called the K-endomorphism ring of M. Observe, that this endomorphism ring is not commutative in general.

(B) Keep the above notations and hypothesis. Then, we have a canonical homomorphism of rings

where id_M : M→M is the identity map on M, so that

It is immediate to verify that this canonical homomorphism is injective if M = A:

We therefore call the map ε _A : A→End_K(A) the canonical embedding of A into its K-endomorphism ring and we consider A as a subalgebra of End_K(A) by means of this canonical embedding. So, for all a ∈ A we identify a with ε _A(a).

Remark and Definition 1.5.2

(A) Let K be a field and let A be a commutative K-algebra. By the convention made in Reminder and Remark 1.5.1 we may consider A as a subalgebra of the endomorphism ring End_K(A). We obviously also have Der_K(A) ⊆End_K(A). So using the notation introduced in Conventions, Reminders and Notations 1.1.1 (B), we have may consider the K-subalgebra

of the K-endomorphism ring of A which is generated by A and all derivations on A with values in A. We call W _K(A) the Weyl algebra of the K-algebra A.

(B) Keep the hypotheses and notations of part (A). Assume in addition, that the commutative K-algebra A is of finite type, so that we find some \(r \in \mathbb {N}_0\) and elements a ₁, a ₂, …, a _r ∈ A such that

Then according to Exercise and Definition 1.4.2 (D), the A-module Der_K(A) is finitely generated. We thus find some \(s \in \mathbb {N}_0\) and derivations d ₁, d ₂, …, d _s ∈Der_K(A) such that

A straight forward computation now allows to see, that

In particular we may conclude, that the K-algebra W _K(A) is finitely generated.

(C) Keep the above notations and let \(n \in \mathbb {N}\). The n-th standard Weyl algebra \(\mathbb {W}(K,n)\) over the field K is defined as the Weyl algebra of the polynomial ring K[X ₁, X ₂, …, X _n], thus

Observe, that by Proposition 1.4.5 and according to the observations made in part (B) we may write

The elements of \(\mathbb {W}(K,n)\) are called polynomial differential operators in the indeterminates X ₁, X ₂, …, X _n over the field K. They are all K-linear combinations of products of indeterminates X _i and partial derivatives ∂ _j.

The differential operators of the form

with

are called elementary differential operators in the indeterminates X ₁, X ₂, …, X _n over the field K.

We now aim to study the structure of standard Weyl algebras. One of the main goals we are heading for is to find an appropriate ”monomial basis“ in each of these algebras. We namely shall see later that the previously introduced elementary differential operators form a K-basis of the standard Weyl algebra \(\mathbb {W}(K,n)\), provided K is of characteristic 0. To pave our way to this fundamental result, we first of all have to prove that in standard Weyl algebras certain commutation relations hold: the so-called Heisenberg relations. To establish these relations, we begin with the following preparations.

Remark and Exercise 1.5.3

(A) If K is a field and B is a K-algebra, we introduce the Poisson operation, that is the map

Show, that the Poisson operation has the following properties:

1. (a)

   [a, b] = −[b, a] for all a, b ∈ B.

2. (b)

   [[a, b], c] + [[b, c], a] + [[c, a], b] = 0 for all a, b, c ∈ B.

3. (c)

   [αa + α′a′, βb + β′b′] = αβ[a, b] + αβ′[a, b′] + α′β[a′, b] + α′β′[a′, b′]for all α, α′, β, β′∈ K and all a, a′, b, b′∈ B.

Observe in particular, that statement (a) says that the Poisson operation is anti-commutative, whereas statement (c) says that this operation is K-bilinear. We call [a, b] the commutator of a and b.

(B) Now, let K be a field, let A be a commutative K-algebra and consider the Weyl algebra W _K(A) := A[Der_K(A)]. Show that the following relations hold:

1. (a)

   [a, b] = 0 for all a, b ∈ A.

2. (b)

   [a, d] = −d(a) for all a ∈ A and all d ∈Der_K(A).

3. (c)

   [d, e] ∈Der_K(A) for all d, e ∈Der_K(A).

(C) Let the notations and hypotheses be as in part (B). Let d, e ∈Der_K(A), let \(r \in \mathbb {N}\), let \(\nu _1,\nu _2,\ldots ,\nu _r \in \mathbb {N}\) and let a ₁, a ₂, …, a _r ∈ A. Use statement (c) of part (B) and the Generalized Product Rule of Exercise and Definition 1.4.2 (A) to prove that

Proposition 1.5.4 (The Heisenberg Relations)

Let \(n\in \mathbb {N}\) , and let Kbe a field. Then, in the standard Weyl algebra

the following relations hold:

1. (a)

   [X _i, X _j] = 0, for all i, j ∈{1, 2, …, n};

2. (b)

   [X _i, ∂ _j] = −δ _i,j , for all i, j ∈{1, 2, …, n};

3. (c)

   [∂ _i, ∂ _j] = 0, for all i, j ∈{1, 2, …, n}.

Proof

1. (a)

   This is clear on application of Remark and Exercise 1.5.3 (B)(a) with a = X _i and b = X _j.

2. (b)

   If we apply Remark and Exercise 1.5.3 (B)(b) with a = X _i and d = ∂ _j, and observe that ∂ _j(X _i) = δ _j,i = δ _i,j we get our claim.

3. (c)

   Observe that for all i, k ∈{1, 2, …, n} we have ∂ _i(X _k) ∈{0, 1}⊆ K. So for all i, j, k ∈{1, 2, …, n} we obtain (see Definition and Remark 1.4.1 (B)):

   $$\displaystyle \begin{aligned}{}[\partial_i,\partial_j](X_k) = \partial_i\big(\partial_j(X_k)\big)-\partial_j\big(\partial_i(X_k)\big) \in \partial_i(K) + \partial_j(K) = \{0\}+ \{0\} = \{0\}.\end{aligned}$$

   Now, we get our claim by Exercise and Definition 1.4.2 (B) and Remark and Exercise 1.5.3 (B) (c) and (C).

The Heisenberg relations are of basic significance for the arithmetic in standard Weyl algebras. Before we show that the elementary differential operators provide a basis for a standard Weyl algebra we shall study the arithmetic of these algebras. In particular, in the next section we shall prove a product formula for elementary differential operators, which will be of basic significance. We shall do this in a slightly more general setting, namely just for K-algebras ”mimicking“ the Heisenberg relations. The next exercise is aimed to prepare this.

Exercise 1.5.5

(A) Let \(n \in \mathbb {N}\), let K be a field, let B be a K-algebra and let

be elements mimicking the Heisenberg relations, which means:

1. (1)

   [a _i, a _j] = 0, for all i, j ∈{1, 2, …, n};

2. (2)

   [a _i, d _j] = −δ _i,j, for all i, j ∈{1, 2, …, n};

3. (3)

   [d _i, d _j] = 0, for all i, j ∈{1, 2, …, n}.

Let \(\mu , \nu \in \mathbb {N}_0\). To simplify notations, we set

prove the following statements (using induction on μ and ν):

1. (a)

   \(a_i^{\mu }a_j^{\nu } = a_j^{\nu }a_i^{\mu }\);

2. (b)

   \(d_i^{\mu }d_j^{\nu } = d_j^{\nu }d_i^{\mu }\);

3. (c)

   \(d_i^{\mu } a_j^{\nu } = a_j^{\nu }d_i^{\mu }\) for all i, j ∈{1, 2, …, n} with i ≠ j.

4. (d)

   \(d_ia_i^{\nu } = a_i^{\nu }d_i + \nu a_i^{\nu -1}\) for all i ∈{1, 2, …, n}.

(B) Keep the notations and hypotheses of part (A). For all \((\lambda _1,\lambda _2,\ldots ,\lambda _n) \in \mathbb {N}_0^n\) and each sequence (b ₁, b ₂, …, b _n) ∈ B ⁿ we use again our earlier standard notation

Now, let

Prove that the following relations hold

1. (a)

   \( \underline {a}^{ \underline {\nu }} \underline {d}^{ \underline {\mu }} = \prod _{i=1}^na_i^{\nu _i}\prod _{j=1}^nd_j^{\mu _j} = \prod _{i=1}^n a_i^{\nu _i}d_i^{\mu _i}\).

2. (b)

   \(( \underline {a}^{ \underline {\nu }} \underline {d}^{ \underline {\mu }})( \underline {a}^{ \underline {\nu '}} \underline {d}^{ \underline {\mu '}}) = \big (\prod _{i=1}^na_i^{\nu _i}\prod _{j=1}^nd_j^{\mu _j}\big )\big (\prod _{i=1}^na_i^{\nu ^{\prime }_i}\prod _{j=1}^nd_j^{\mu ^{\prime }_j}\big ) = \prod _{i=1}^n a_i^{\nu _i}d_i^{\mu _i}a_i^{\nu ^{\prime }_i}d_i^{\mu ^{\prime }_i}\).

As announced above, we now aim to do some Arithmetic in standard Weyl algebras. This means in particular, that we make explicit a number of computations in the hope that readers who up to now were mainly faced with commutative rings, get fascinated by the complexity of the arithmetic in Weyl algebras.

The following arithmetical Lemma is formulated in a more general framework, namely in a situation, which “mimicks” the Heisenberg relation. If we specialize this Lemma to standard Weyl algebras, we get a most important formula, which expresses the product of two elementary differential operators as a \(\mathbb {Z}\)-linear combination of elementary differential operators. This will also give us an explicit presentation of the commutator [d, e] (see Remark and Exercise 5.3 (A)) of two elementary differential operators d and e. As a further application we will get the Reduction Principle for arbitrary products of elementary differential operators and thus pave our way to the standard basis presentation of Weyl algebras, which we shall introduce in the next section.

We prove the announced Lemma in a setting which is more general than just the framework of Weyl algebras, because in this form it will help us to prove the universal property of Weyl algebras formulated in Corollary 1.7.5. This property is an analogue of the (relational) universal property of commutative polynomial algebras (see Example and Exercise 1.3.2 (A)) or of free associative algebras (see Example and Exercise 1.3.5 (D)).

Lemma 1.6.1

Let \(n \in \mathbb {N}\) , let K be a field, let B be a K-algebra and let

such that:

1. (1)

   [a _i, a _j] = 0, for all i, j ∈{1, 2, …, n};

2. (2)

   [a _i, d _j] = −δ _i,j , for all i, j ∈{1, 2, …, n};

3. (3)

   [d _i, d _j] = 0, for all i, j ∈{1, 2, …, n}.

Then, the following statements hold:

1. (a)

   For all \(\mu , \nu \in \mathbb {N}_0\) and all i ∈{1, 2, …, n}we have

   $$\displaystyle \begin{aligned}d_i^{\mu}a_i^{\nu} = \sum_{k=0}^{\min\{\mu,\nu\}} \binom{\mu}{k}\prod_{p=0}^{k-1}(\nu-p)a_i^{\nu-k}d_i^{\mu-k}.\end{aligned} $$
2. (b)

   Let

   $$\displaystyle \begin{aligned}\underline{\mu} := (\mu_1,\mu_1,\ldots,\mu_n), \quad \underline{\nu} := (\nu_1,\nu_2,\ldots,\nu_n), \mathit{\mbox{ and }}\end{aligned}$$
   $$\displaystyle \begin{aligned} \underline{\mu'} := (\mu^{\prime}_1,\mu^{\prime}_1,\ldots,\mu^{\prime}_n), \quad \underline{\nu'} := (\nu^{\prime}_1,\nu^{\prime}_2,\ldots,\nu^{\prime}_n) \in \mathbb{N}_0^n.\end{aligned} $$

   Set

   $$\displaystyle \begin{aligned}\mathbb{I} := \{\underline{k} := (k_1,k_2,\ldots,k_n) \in \mathbb{N}_0^n \mid k_i \leq \min\{\mu_i,\nu^{\prime}_i\} \mathit{\mbox{ for }} i=1,2,\ldots,n\},\end{aligned} $$

   and let

   $$\displaystyle \begin{aligned}\lambda_{\underline{k}} := \huge[\prod_{i=1}^n\binom{\mu_i}{k_i}\huge] \times \huge[\prod_{i=1}^n\prod_{p=0}^{k_i-1}(\nu^{\prime}_i-p)\huge].\end{aligned}$$

   Then, we have the relation

   $$\displaystyle \begin{aligned} (\underline{a}^{\underline{\nu}}\underline{d}^{\underline{\mu}})(\underline{a}^{\underline{\nu'}}\underline{d}^{\underline{\mu'}}) &:= (\prod_{i=1}^na_i^{\nu_i}\prod_{j=1}^nd_j^{\mu_j})\prod_{i=1}^na_i^{\nu^{\prime}_j} \prod_{j=1}^nd_i^{\mu^{\prime}_j})\\ &=\prod_{i=1}^n a_i^{\nu_i+\nu^{\prime}_i}\prod_{i=1}^n d_i^{\mu_i+\mu^{\prime}_i} + \sum_{\underline{k} \in \mathbb{I} \setminus\{\underline{0}\}} \lambda_{\underline{k}} \prod_{i=1}^na_i^{\nu_i+\nu^{\prime}_i-k_i}\prod_{i=1}^n d_i^{\mu_i+\mu^{\prime}_i-k_i} \\ &= \underline{a}^{\underline{\nu}+\underline{\nu'}}\underline{d}^{\underline{\mu} + \underline{\mu'}} + \sum_{\underline{k} \in \mathbb{I}\setminus\{\underline{0}\}} \lambda_{\underline{k}}\underline{a}^{\underline{\nu} +\underline{\nu'}-\underline{k}}\underline{d}^{\underline{\mu}+\underline{\mu'} - \underline{k}}.\end{aligned} $$

Proof

(a) To simplify matters we use the notation

already introduced in the previous Exercise 1.5.5 (A). Then, it suffices to show that

We proceed by induction on μ. The case μ = 0 is obvious. The case μ = 1 is clear by Exercise 1.5.5 (A)(d). So, let μ > 1. By induction we have

It follows on use of Exercise 1.5.5 (A)(d) and the Pascal formulas for the sum of binomial coefficients, that

(b) According to Exercise 1.5.5 (B)(a),(b), the previous statement (a) and Exercise 1.5.5 (A)(a),(b) and (c) we may write

As an application we now get the announced product formula for elementary differential operators.

Proposition 1.6.2 (The Product Formula for Elementary Differential Operators)

Let \(n \in \mathbb {N}\) , let K be a field and consider the standard Weyl algebra

Moreover, let

Set

and let

Then, we have the equality

Proof

It suffices to apply Lemma 1.6.1 (b) with a _i := X _i and d _i := ∂ _i for i = 1, 2…, n.

Now, we can prove the main result of the present section. To formulate it, we introduce another notation and suggest a further exercise.

Notation and Remark 1.6.3

(A) Let \(n \in \mathbb {N}\) and let

We write

and

(B) Keep the notations of part (A). Observe that

and

(C) We now introduce a few notations, which we will have to use later very frequently. Namely, for

we set

and

Moreover, we write

Observe that

Exercise 1.6.4

(A) Let \(n \in \mathbb {N}\), let K be a field and consider the standard Weyl algebra

In addition, let

Moreover, let the sets

be defined according to Notation and Remark 1.6.3 (C). Prove that

and

(B) Let the notations be as in part (A) and let the set

be defined according to Notation and Remark 1.6.3 (C). Prove that

(C) To give a more precise statement than what was just said in part (B), keep the notations of Proposition 1.6.2 and set in addition

Use the product formula of Proposition 1.6.2 to show that

(D) Let i ∈{1, , 2, …, n} and consider the n-tuple \( \underline {\mathrm {e}}_i := (\delta _{i,j})_{j=1}^n = (0,\ldots ,0,1,0,\ldots ,0) \in \mathbb {N}_0^n.\) Use what you have shown in part (C) to prove the following statements

1. (a)

   \(\big [X_i,\underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\big ] = \begin {cases} - \mu _i \underline {X}^{ \underline {v}} \underline {\partial }^{ \underline {\mu } -\underline {\mathrm {e}}_i}&, \mbox{ if } \mu _i > 0; \\ 0&, \mbox{ if } \mu _i = 0. \end {cases}\)

2. (b)

   \(\big [\partial _i,\underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\big ] = \begin {cases}\nu _i \underline {X}^{ \underline {v}- \underline {\mathrm {e}}_i} \underline {\partial }^{ \underline {\mu }}&, \mbox{ if } \nu _i > 0; \\ 0&, \mbox{ if } \nu _i = 0. \end {cases}\)

Theorem 1.6.5 (The Reduction Principle)

Let \(n \in \mathbb {N}\) , let K be a field and consider the standard Weyl algebra

Let \(r \in \mathbb {N},\) let

and abbreviate

Moreover, let the set

be defined according to Notation and Remark 1.6.3 (C). Then, we have

Proof

We proceed by induction on r. The case r = 1 is obvious. The case r = 2 follows from Proposition 1.6.2, more precisely from its consequence proved in Exercise 1.6.4 (A) (see also Notation and Remark 1.6.3 (C)) . So, let r > 2. We set

By induction we have

By the case r = 2 we have (see once more Notation and Remark 1.6.3 (C) and Exercise 1.6.4 (A))

As

it remains to show that

Observe that

Observe also that

so that in the notation introduced in Notation and Remark 1.6.3 (C) we have

Hence, on application of Exercise 1.6.4 (A) it follows that

and this shows that indeed \(\varrho\underline {X}^{ \underline {\nu }^{(r)}} \underline {\partial }^{ \underline {\mu }^{(r)}} \in M\).

Now, in the next section, we can show that the elementary differential operators form a K-basis of the standard Weyl algebra \(\mathbb {W}(K,n)\), provided the field K has characteristic 0. To prepare this, we add an additional exercise.

Exercise 1.6.6

(A) Let \(n {\,\in \,} \mathbb {N}\) and consider the polynomial ring K[X ₁, X ₂, …, X _n]. Moreover, let

Fix i ∈{1, 2, …, n} and prove by induction on μ _i, that

(B) Let the notations and hypotheses be as in part (A) and use what you have shown there to prove that

Now, we are ready to prove the fact that over a base field of characteristic 0 the elementary differential operators form a vector space basis of the standard Weyl algebra.

Theorem 1.7.1 (The Standard Basis)

Let \(n \in \mathbb {N}\) and let K be a field of characteristic 0. Then, the elementary differential operators

form a K-vector space basis of the standard Weyl algebra

So, in particular we can say

1. (a)

   \(\mathbb {W}(K,n) = \bigoplus _{ \underline {\nu },\underline {\mu } \in \mathbb {N}_0^n} K \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }} = \bigoplus _{\mu _1,\mu _2,\ldots ,\mu _n,\nu _1,\nu _2,\ldots ,\nu _n \in \mathbb {N}_0} K\prod _{i=1}^n X_i^{\nu _i}\prod _{i=1}^n\partial _i^{\mu _i}\).

2. (b)

   Each differential operator \(d \in \mathbb {W}(K,n)\) can be written in the form

   $$\displaystyle \begin{aligned}d = \sum_{\underline{\nu}, \underline{\mu} \in \mathbb{N}_0^n} c^{(d)}_{\underline{\nu},\underline{\mu}} \underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}}\end{aligned}$$

   with a unique family

   $$\displaystyle \begin{aligned}\big(c^{(d)}_{\underline{\nu}, \underline{\mu}}\big)_{\underline{\nu}, \underline{\mu} \in \mathbb{N}_0^n} \in \prod_{\underline{\nu}, \underline{\mu} \in \mathbb{N}_0^n} K = K^{\mathbb{N}_0^n \times \mathbb{N}_0^n},\end{aligned}$$

   whose support

   $$\displaystyle \begin{aligned}\mathrm{supp}(d) = \mathrm{supp}\big((c^{(d)}_{\underline{\nu}, \underline{\mu}})_{\underline{\nu}, \underline{\mu} \in \mathbb{N}_0^n}\big) := \{(\underline{\nu}, \underline{\mu}) \in \mathbb{N}_0^n \times \mathbb{N}_0^n \mid c^{(d)}_{\underline{\nu}, \underline{\mu}} \neq 0\}\end{aligned}$$

   is a finite set. We thus can write

   $$\displaystyle \begin{aligned}d = \sum_{(\underline{\nu}, \underline{\mu}) \in \mathrm{supp}(d)}c^{(d)}_{\underline{\nu},\underline{\mu}}\underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}}\quad .\end{aligned}$$

Proof

We first show, that the elementary differential operators generate \(\mathbb {W}(K,n)\) as a K-vector space, hence that

Observe, that by definition each element d of \(\mathbb {W}(K,n)\) is a K-linear combination of products of elementary differential operators. But by the Reduction Principle of Theorem 1.6.5 each product of elementary differential operators is contained in the K-vector space M.

It remains to show, that the elementary differential operators are linearly independent among each other. Assume to the contrary, that there are linearly dependent elementary differential operators in \(\mathbb {W}(K,n)\). Then, we find a positive integer \(r \in \mathbb {N}\), families

with

and elements

such that

We may assume, that

and that for some s ∈{1, 2, …, r} we have

Then, it follows easily by what we have seen in Exercise 1.6.6 (B), that

So, we get

As Char(K) = 0, and as the monomials \( \underline {X}^{ \underline {\nu }^{(i)}}\) are pairwise different for i = s, s + 1, …, r, the last sum does not vanish, and we have a contradiction.

Definition and Remark 1.7.2

(A) Let the notations and hypotheses be as in Theorem 1.6.5. We call the basis of \(\mathbb {W}(K,n)\) which consists of all elementary differential operators the standard basis. If we present a differential operator \(d \in \mathbb {W}(K,n)\) with respect to the standard basis and write

as in statement (b) of Theorem 1.6.5, we say that d is written in standard form. The support of a differential operator d in \(\mathbb {W}(K,n)\) is always defined with respect to the standard form as in statement (b) of Theorem 1.7.1. We therefore call the support of d also the standard support of d.

(B) Keep the above notations and hypotheses. It is a fundamental task, to write an arbitrarily given differential operator \(d \in \mathbb {W}(K,n)\) in standard form. This task actually is reduced by the Reduction Principle of Theorem 1.6.5 to make explicit the coefficients of the differences

This task can be solved by a repeated application of the Product Formula of Proposition 1.6.2 or—directly—by a repeated application of the Heisenberg relations. Clearly, this is a task which usually is performed by means of Computer Algebra systems.

We now prove the following application, a result on supports, which will turn out to be useful in the next section.

Proposition 1.7.3 (Behavior of Supports)

Let \(n \in \mathbb {N}\) , let K be a field of characteristic 0 and consider the differential operators

For all \(( \underline {\alpha }, \underline {\beta }) \in \mathbb {N}_0^n \times \mathbb {N}_{0}^n\) , let the sets

be defined according to Notation and Remark 1.6.3 (C). Then, we have

1. (a)

   (supp(d) ∪supp(e)) ∖(supp(d) ∩supp(e)) ⊆supp(d + e) ⊆supp(d) ∪supp(e).

2. (b)

   supp(cd) = supp(d) for all c ∈ K ∖{0}.

3. (c)

   \(\mathrm {supp}(de) \subseteq \bigcup _{( \underline {\nu }, \underline {\mu })\in \mathrm {supp}(d), ( \underline {\nu }', \underline {\mu }')\in \mathrm {supp}(e)}\overline {\mathbb {M}}( \underline {\nu }+ \underline {\nu }', \underline {\mu }+ \underline {\mu }')\).

4. (d)

   \(\mathrm {supp}\big ([d,e]\big ) \subseteq \bigcup _{( \underline {\nu }, \underline {\mu })\in \mathrm {supp}(d), ( \underline {\nu }', \underline {\mu }')\in \mathrm {supp}(e)}\mathbb {M}( \underline {\nu }+ \underline {\nu }', \underline {\mu }+ \underline {\mu }')\).

Proof

(a), (b) These statements follow in a straight forward way from our definition of support, and we leave it as an exercise to perform their proof.

(c) In the notations of Theorem 1.7.1 we write

it follows that

But according to Exercise 1.6.4 (A) we have

Now, our claim follows easily on repeated application of statements (a) and (b).

(d) As in the proof of statement (c) we can write

and, similarly

It follows that

By Exercise 1.6.4 (B) we have

Now, statement (d) follows easily on repeated application of statements (a) and (b).

Exercise 1.7.4

1. (A)

   Let \(n \in \mathbb {N}\), let K be a field of characteristic 0 and consider the standard Weyl algebra

   $$\displaystyle \begin{aligned}\mathbb{W} = \mathbb{W}(K,n) = K[X_1,X_2,\ldots,X_n,\partial_1,\partial_2,\ldots,\partial_n].\end{aligned}$$

   Prove in detail statements (a) and (b) of Proposition 1.7.3.

2. (B)

   Let the notations and hypotheses be as in part (A). Present in standard form the following differential operators:

   $$\displaystyle \begin{aligned}\partial_1^2X_1^2 - X_1\partial_1X_1 - 1, \quad \partial_1^2X_1^2\partial_1^2 - \partial_1X_1^2, \quad \partial_2X_1X_2\partial_1 + \partial_1X_1X_2 \in \mathbb{W}(K,n).\end{aligned}$$
3. (C)

   Keep the notations of part (A), but assume that n = 1 and Char(K) = 2. Compute \(\partial _1(X_1^{\nu })\) for all \(\nu \in \mathbb {N}_0\) and comment your findings in view of the Standard Basis Theorem.

4. (D)

   Keep the notations of part (A), let

   $$\displaystyle \begin{aligned}d = \sum_{(\underline{\nu}, \underline{\mu}) \in \mathrm{supp}(d)}c^{(d)}_{\underline{\nu},\underline{\mu}}\underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}} \in \mathbb{W}, \quad \big(c^{(d)}_{\underline{\nu},\underline{\mu}} \in K \setminus \{0\}, \forall (\underline{\nu}, \underline{\mu}) \in \mathrm{supp}(d) \big)\end{aligned}$$

   (see Theorem 1.7.1) and let i ∈{1, 2, …, n}. Use Exercise 1.6.4 (D) to prove the following equalities:

   1. (a)

      \(\big [X_i,d \big ] = -\sum _{( \underline {\nu },\underline {\mu }) \in \mathrm {supp}(d):\mu _i > 0} \mu _ic^{(d)}_{ \underline {\nu }, \underline {\mu }} \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }- \underline {\mathrm {e}}_i}.\)

   2. (b)

      \(\big [\partial _i,d \big ] = \sum _{( \underline {\nu },\underline {\mu }) \in \mathrm {supp}(d):\nu _i > 0} \nu _ic^{(d)}_{ \underline {\nu }, \underline {\mu }} \underline {X}^{ \underline {\nu }- \underline {\mathrm {e}}_i} \underline {\partial }^{ \underline {\mu }}.\)

   Conclude that

   1. (c)

      d = 0 ⇔∀i ∈{1, 2, …, n} : [X _i, d] = [∂ _i, d] = 0.

As another application of the Standard Basis Theorem we now can prove

Corollary 1.7.5 (The Universal Property of Weyl Algebras)

Let n ≥ 2 and let the notations and hypotheses be as in Theorem 1.7.1 . Let B be a K-algebra and let

be a map ”which respects the Heisenberg relations“ and hence satisfies the requirements

1. (1)

   [ϕ(X _i), ϕ(X _j)] = 0, for all i, j ∈{1, 2, …, n};

2. (2)

   [ϕ(X _i), ϕ(∂ _j)] = −δ _i,j , for all i, j ∈{1, 2, …, n};

3. (3)

   [ϕ(∂ _i), ϕ(∂ _j)] = 0, for all i, j ∈{1, 2, …, n}.

Then, there is a unique homomorphism of K-algebras

such that

Proof

According to Theorem 1.7.1 there is a K-linear map

Next, we show, that the previously defined K-linear map \(\widetilde {\phi }\) is multiplicative, and hence satisfies the condition that

As the multiplication maps

are both K-bilinear, it suffices to verify the above multiplicativity condition in the special case where

with

But this can be done by a straight forward computation, on use of the Product Formula of Proposition 1.6.2 and on application of Lemma 1.6.1 with

It remains to show, that \(\widetilde {\phi }:\mathbb {W}(K,n) \longrightarrow B\) is the only homomorphism of K-algebras which satisfies the requirement that

But indeed, if a map \(\widetilde {\phi }\) satisfies this requirement and is multiplicative, it must be defined on the elementary differential operators as suggested above. This proves the requested uniqueness.

Exercise 1.7.6

1. (A)

   Let \(n \in \mathbb {N}\), let K be a field of characteristic 0. Show, that there is a unique automorphism of K-algebras

   $$\displaystyle \begin{aligned}\alpha: \mathbb{W}(K,n) \stackrel{\cong}{\longrightarrow}\mathbb{W}(K,n) \mbox{ with } \alpha(X_i) = \partial_i \mbox{ and } \alpha(\partial_i) = -X_i \mbox{ for all } i=1,2,\ldots,n.\end{aligned}$$
2. (B)

   Keep the notations and hypotheses of part (A). Present in standard form all elements \(\alpha (X_i^{\nu }\partial _i^{\mu }) \in \mathbb {W}(K,n)\) with \(\mu ,\nu \in \mathbb {N}_0\).

In this section we introduce and investigate a particularly nice class of filtrations of the standard Weyl algebras, the so-called weighted filtrations. To do so, we first will introduce the related notion of weighted degree of a differential operator.

Convention 1.8.1

Throughout this section we fix a positive integer n, a field K of characteristic 0 and we consider the standard Weyl algebra

Definition and Remark 1.8.2

(A) By a weight we mean a pair

such that

For

we frequently shall use the scalar product

(B) Fix a weight \(( \underline {v}, \underline {w}) \in \mathbb {N}_0^n \times \mathbb {N}_0^n\). We define the degree associated to the weight \(( \underline {v}, \underline {w})\) (or just the weighted degree) of a differential form \(d \in \mathbb {W}\) by

with the usual convention that sup(∅) = −∞.

Observe that by our definition of weight, for all \(d \in \mathbb {W}\) and all \( \underline {\mu },\underline {\nu } \in \mathbb {N}_0\)—and using the notations of Notation and Remark 1.6.3 (C)– we can say:

1. (a)

   \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) \in \mathbb {N}_0 \cup \{-\infty \}\) with \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) = -\infty \) if and only if d = 0.

2. (b)

   If \( \underline {\lambda } \leq\underline {\nu }\) and \( \underline {\kappa } \leq\underline {\mu }\) for all \(( \underline {\lambda }, \underline {\kappa }) \in \mathrm {supp}(d)\), then

   $$\displaystyle \begin{aligned}\mathrm{deg}^{\underline{v}\underline{w}}(d) \leq \underline{v}\cdot\underline{\nu}+\underline{w}\cdot\underline{\mu}.\end{aligned} $$
3. (c)

   If \(\mathrm {supp}(d) \subseteq \mathbb {M}_{\leq }( \underline {\nu }, \underline {\mu })\), then

   $$\displaystyle \begin{aligned}\mathrm{deg}^{\underline{v}\underline{w}}(d) < \underline{v}\cdot\underline{\nu}+\underline{w}\cdot\underline{\mu}.\end{aligned} $$

(C) Keep the notations and hypotheses of part (B). We fix some non-negative integer \(i \in \mathbb {N}_0\) and set

Observe, that we also may write

Lemma 1.8.3

Let \(( \underline {v}, \underline {w}) \in \mathbb {N}_0^n \times \mathbb {N}_0^n\) be a weight and let \(d,e \in \mathbb {W}\) . Then we have

1. (a)

   \(\mathrm {deg}^{ \underline {v} \underline {w}}(d+e) \leq \mathrm {max}\{\mathrm {deg}^{ \underline {v} \underline {w}}(d),\mathrm {deg}^{ \underline {v} \underline {w}}(e)\}\) , with equality if \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) \neq \mathrm {deg}^{ \underline {v} \underline {w}}(e)\) ;

2. (b)

   \(\mathrm {deg}^{ \underline {v} \underline {w}}(cd) = \mathrm {deg}^{ \underline {v} \underline {w}}(d)\) for all c ∈ K ∖{0}.

3. (c)

   \(\mathrm {deg}^{ \underline {v} \underline {w}}(de) \leq \mathrm {deg}^{ \underline {v} \underline {w}}(d) + \mathrm {deg}^{ \underline {v} \underline {w}}(e)\) ;

4. (d)

   \(\mathrm {deg}^{ \underline {v} \underline {w}}\big ([d,e]\big ) < \mathrm {deg}^{ \underline {v} \underline {w}}(d) + \mathrm {deg}^{ \underline {v} \underline {w}}(e)\).

Notice: In statement (c) actually equality holds. We shall prove this later (see Corollary 1.9.5).

Proof

(a) The stated inequality is clear by the second inclusion of the following relation (see Proposition 1.7.3 (a)):

It remains to establish the stated equality if \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) \neq \mathrm {deg}^{ \underline {v} \underline {w}}(e)\). It suffices to treat the case in which \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) < \mathrm {deg}^{ \underline {v} \underline {w}}(e)\). In this case, there is some

By the first of the previous inclusions we have \(( \underline {\nu }, \underline {\mu }) \in \mathrm {supp}(d+e)\) and hence

By the already proved inequality \(\mathrm {deg}^{ \underline {v} \underline {w}}(d+e) \leq \mathrm {max}\{\mathrm {deg}^{ \underline {v} \underline {w}}(d),\mathrm {deg}^{ \underline {v} \underline {w}}(e)\}\) it follows that \(\mathrm {deg}^{ \underline {v} \underline {w}}(d+e) = \mathrm {deg}^{ \underline {v} \underline {w}}(e)\).

(b) This is obvious.

(c) This follows easily by Proposition 1.7.3 (c) and Definition and Remark 1.8.2 (B) (b).

(d) This follows in a straight forward manner by Proposition 1.7.3 (d) and Definition and Remark 1.8.2 (B) (c).

Theorem 1.8.4 (Weighted Filtrations)

Let

be a weight. Then, the family

is a commutative filtration of the K-algebra \(\mathbb {W} = \mathbb {W}(K,n)\).

Moreover, the following statements hold.

1. (a)

   \(\mathbb {W}^{ \underline {v} \underline {w}}_0= K[X_i,\partial _j \mid v_i=0, w_j = 0],\) so that \(\mathbb {W}^{ \underline {v} \underline {w}}_0\) is a commutative polynomial algebra in the variables X _i and ∂ _j for which either v _i = 0 or else w _j = 0.

2. (b)

   Let \(\delta = \delta ( \underline {v} \underline {w}) =\mathrm {max}\{v_1,v_2,\ldots ,v_n,w_1,w_2,\ldots ,w_n\}\) . Then, for all i > δ it holds

   $$\displaystyle \begin{aligned}\mathbb{W}^{\underline{v}\underline{w}}_i = \sum_{j=1}^{\delta}\mathbb{W}^{\underline{v}\underline{w}}_j\mathbb{W}^{\underline{v}\underline{w}}_{i-j}.\end{aligned}$$
3. (c)

   The filtration \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }=\big (\mathbb {W}^{ \underline {v} \underline {w}}_i\big )_{i\in \mathbb {N}_0}\) is of finite type.

Proof

It is clear from our definitions, that

On use of Lemma 1.8.3 (c) it follows immediately that

So the family \(\big (\mathbb {W}^{ \underline {v} \underline {w}}_i := \{d \in \mathbb {W} \mid \mathrm {deg}^{ \underline {v} \underline {w}}(d)\leq i\}\big )_{i\in \mathbb {N}_0}\) constitutes indeed a filtration on the K-algebra \(\mathbb {W}\).

Now, let \(i,j \in \mathbb {N}_0\), let \(d \in \mathbb {W}^{ \underline {v} \underline {w}}_i\) and let \(e \in \mathbb {W}^{ \underline {v} \underline {w}}_j\). Then by Lemma 1.8.3 (d) we have

so that

This proves, that our filtration is commutative (see Definition 1.3.3).

(a) Set

Let \( \underline {\nu },\underline {\mu } \in \mathbb {N}_0^n\). Then

But this means that

It remains to show, that this latter ring is a commutative polynomial algebra in all the variables X _i and ∂ _j for which either v _i = 0 or else w _j = 0. In view of Theorem 1.7.1 it suffices to show that X _i ∂ _j = ∂ _j X _i for all i, j with v _i = v _j = 0. But as (v _k, w _k) ≠ (0, 0) for all k = 1, 2, …, n (see Definition and Remark 1.8.2 (A)), this is clear by the Heisenberg relations (see Proposition 1.5.4 (b)).

(b) Let i > δ. Let

We aim to show that

If σ ≤ 0 this is clear as i > 0 implies i ≥ 1, so that

So, let σ > 0. Then either

1. (1)

   there is some p ∈{1, 2, …, n} with v _p > 0 and ν _p > 0, or else,

2. (2)

   there is some q ∈{1, 2, …, n} with w _q > 0 and μ _q > 0.

In the above case (1) we can write

As \(\mathrm {deg}^{ \underline {v} \underline {w}}(X_p) = v_p \leq \delta \) and \(\mathrm {deg}^{ \underline {v} \underline {w}}(d) = \sigma - v_p\) it follows that

In the above case (2) we may first assume, that we are not in the case (1). This means in particular that either v _q = 0 or ν _q = 0, hence v _q ν _q = 0, so that

Now, in view of the Heisenberg relations, we may write

As v _q ν _q = 0, we have \(\mathrm {deg}^{ \underline {v} \underline {w}}(e) = \sigma - w_q\), and it follows that

But this shows, what we were aiming for, hence that

But this means that

and hence proves statement (b).

(c) This is an immediate consequence of statements (a) and (b) (see Definition and Remark 1.3.4 (C)).

Definition 1.8.5

Let the notations and hypotheses be as in Theorem 1.8.4. In particular, let

be a weight. Then, the filtration

is called the filtration induced by the weight \(( \underline {v}, \underline {w})\). Generally, we call weighted filtrations all filtrations which are induced in this way by a weight.

Definition and Remark 1.8.6

(A) We consider the strings

and a differential form \(d \in \mathbb {W}\). We define the standard degree or just the degree deg(d) of d as the weighted degree with respect to the weight \(( \underline {1}, \underline {1}) \in \mathbb {N}_0^n \times \mathbb {N}_0^n\), hence

Observe that

The corresponding induced weighted filtration

is called the standard degree filtration or just the degree filtration of \(\mathbb {W}\).

(B) Keep the notations and hypotheses of part (A). The order of the differential operator d is defined by

Observe that

The corresponding induced weighted filtration

is called the order filtration of \(\mathbb {W}\).

Now, as an immediate application of Theorem 1.8.4 we obtain:

Corollary 1.8.7

Let the notations be as in Convention 1.8.1 . Then it holds

1. (a)

   The degree filtration \(\mathbb {W}^{\mathrm {deg}}_{\bullet }\) is very good.

2. (b)

   The order filtration \(\mathbb {W}^{\mathrm {ord}}_{\bullet }\) is good and \(\mathbb {W}^{\mathrm {ord}}_0 = K[X_1,X_2,\ldots ,X_n]\).

Proof

In the notations of Theorem 1.8.4 (b) we have

Moreover, by Theorem 1.8.4 (a) we have

This proves our claim (see Definition and Remark 1.3.4 (C)).

Exercise 1.8.8

1. (A)

   Show that the degree filtration is the only very good filtration on \(\mathbb {W}\).

2. (B)

   Write down all weights \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0 \times \mathbb {N}_0^n\) for which the induced filtration \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\) is good.

This section is devoted to the study of the associated graded rings of weighted filtrations of standard Weyl algebras. We shall see, that these are all naturally isomorphic to polynomial rings.

Convention 1.9.1

Again, throughout this section we fix a positive integer n, a field K of characteristic 0 and consider the standard Weyl algebra

In addition, we introduce the polynomial ring

in the indeterminates Y ₁, Y ₂, …, Y _n, Z ₁, Z ₂, …, Z _n with coefficients in the field K.

Definition and Remark 1.9.2

(A) Fix a weight \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\) and consider the induced weighted filtration \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\). To write down the corresponding associated graded ring, we introduce the following notation:

(B) Keep the above notations and hypotheses. For each \(j \in \mathbb {Z}\) we introduce the notations:

Fix some \(i \in \mathbb {N}_0\). Observe that

As a consequence, we get an isomorphism of K-vector spaces

such that

In particular we can say:

We call this basis the standard basis of \(\mathbb {G}^{ \underline {v} \underline {w}}_i\). Its elements are called standard basis elements of the associated graded ring \(\mathbb {G}^{ \underline {v} \underline {w}}\).

(C) Keep the previously introduced notation. We add a few more useful observations on standard basis elements. First, observe that we may write

1. (a)

   \(( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }})^* \in \mathbb {G}^{ \underline {v} \underline {w}}_ { \underline {v}\cdot\underline {\nu } +\underline {w}\cdot\underline {\mu }}\) for all \(( \underline {\nu }, \underline {\mu })\in \mathbb {N}_0^n\times \mathbb {N}_0^n\).

2. (b)

   \(X_i^* \in \mathbb {G}^{ \underline {v} \underline {w}}_{v_i}\) and \(\partial _j^* \in \mathbb {G}^{ \underline {v} \underline {w}}_{w_j}\) for all i, j ∈{1, 2, …, n}.

Moreover, by the observations made in part (B) we also can say that all standard basis elements form a K-basis of the whole associated graded ring, thus:

1. (c)

   The family \(\big (( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }})^*\big )_ {( \underline {\nu }, \underline {\mu })\in \mathbb {N}_0^n\times \mathbb {N}_0^n}\) is a K-basis of \(\mathbb {G}^{ \underline {v} \underline {w}}\).

Finally, as the associated graded ring is commutative, and keeping in mind how the multiplication in this ring is defined (see Remark and Definition 1.3.1 (B)) we get the following product formula

1. (d)

   \(( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }})^* = \big (\prod _{i=1}^nX_i^{\nu _i}\prod _{j=1}^n\partial ^{\mu _j}\big )^*= \prod _{i=1}^n\big (X_i^*\big )^{\nu _i} \prod _{j=1}^n\big (\partial _j^*\big )^{\mu _j} =: \big ( \underline {X}^*\big )^{ \underline {\nu }} \big ( \underline {\partial }^*\big )^{ \underline {\mu }}\).

Exercise and Definition 1.9.3

(A) We fix a weight \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\). As in Definition and Remark 1.9.2 (A) we use again the notation

and consider the K-subspace

of our polynomial ring \(\mathbb {P} = K[Y_1,Y_2,\ldots ,Y_n,Z_1,Z_2,\ldots ,Z_n]\). Prove the following statements:

1. (a)

   \(K \subseteq \mathbb {P}^{ \underline {v} \underline {w}}_0\);

2. (b)

   \(\mathbb {P}^{ \underline {v} \underline {w}}_i \mathbb {P}^{ \underline {v} \underline {w}}_j \subseteq \mathbb {P}^{ \underline {v} \underline {w}}_{i+j}\) for all \(i,j \in \mathbb {N}_0\).

3. (c)

   \(\mathbb {P} = \bigoplus _{i \in \mathbb {N}_0} \mathbb {P}^{ \underline {v} \underline {w}}_i\).

(B) Let the hypotheses and notations be as in part (A). Conclude that

We call this grading the grading induced by the weight \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\). If we endow our polynomial ring with this grading we write it as \(\mathbb {P}^{ \underline {v} \underline {w}}\), thus

Theorem 1.9.4 (Structure of Weighted Associated Graded Rings)

Let \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\) be a weight. Then there exists an isomorphism of K-algebras, which preserves gradings (see Convention, Reminders and Notations 1.1.1 (I)).

given by

Proof

According to the universal property of the polynomial ring \(\mathbb {P}\) there is a unique homomorphism of K-algebras

such that

In view of the product formula of Definition and Remark 1.9.2 (C) we obtain

In particular \(\eta ^{ \underline {v} \underline {w}}\) yields a bijection between the monomial basis of the polynomial ring \(\mathbb {P}\) and the standard basis of the associated graded ring \(\mathbb {G}^{ \underline {v} \underline {w}}\). So, \(\eta ^{ \underline {v} \underline {w}}\) is indeed an isomorphism. But moreover, for each \(i \in \mathbb {N}_0\) it also follows that \(\eta ^{ \underline {v} \underline {w}}\) yields an bijection between the monomial basis of the subspace \(\mathbb {P}^{ \underline {v} \underline {w}}_i \subseteq \mathbb {P}\) and the standard basis of \(\mathbb {G}^{ \underline {v} \underline {w}}_i\). But this means, that \(\eta ^{ \underline {v} \underline {w}}\) preserves the gradings.

In Lemma 1.8.3 (c) we have seen that weighted degrees are sub-additive, which means that \(\mathrm {deg}^{ \underline {v} \underline {w}}(de) \leq \mathrm {deg}^{ \underline {v} \underline {w}}(d) +\mathrm {deg}^{ \underline {v} \underline {w}}(e) \mbox{ for all } d,e \in \mathbb {W}.\) As an application of Theorem 1.9.4 we now shall improve on this and show, that weighted degrees are indeed additive, which means that the above inequality is in fact always an equality.

Corollary 1.9.5 (Additivity of Weighted Degrees)

Let \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\) be a weight and let \(d,e \in \mathbb {W}\) . Then

Proof

If d = 0 or e = 0 our claim is clear. So let d, e ≠ 0. We have

We use again the notation

and set

We then may write

We thus have

By what we know already about degrees we have \(\mathrm {deg}^{ \underline {v} \underline {w}}(as+rb+rs) < i+j\) (see Lemma 1.8.3 (a), (c)). So, in view of Lemma 1.8.3 (a) it suffices to show that

To do so, we write

It follows that

By Exercise 1.6.4 (A) and in the notation of Notation and Remark 1.6.3 (C), it follows that

for all \(( \underline {\nu }, \underline {\mu }) \in \mathrm {supp}(a)\) and all \(( \underline {\nu }', \underline {\mu }') \in \mathrm {supp}(b)\). Observe that

So, by Definition and Remark 1.8.2 (B)(c) it follows that

for all \(( \underline {\nu }, \underline {\mu }) \in \mathrm {supp}(a)\) and all \(( \underline {\nu }', \underline {\mu }') \in \mathrm {supp}(b)\). If we set

and on repeated use of Lemma 1.8.3 (a) and (b) we thus get

So, we may write

By Lemma 1.8.3 (a) it thus suffices to show that \(\mathrm {deg}^{ \underline {v} \underline {w}}(h) = i+j\). As

It suffices to show that h ≠ 0. To do so, we consider the two polynomials

As supp(a) and supp(b) are non-empty, and all coefficients of f and g are non-zero, we have f ≠ 0 and g ≠ 0. As \(\mathbb {P}\) is an integral domain. it follows that fg ≠ 0. We set

so that

Applying the isomorphism

of Theorem 1.9.4, we now get

But this clearly implies that h ≠ 0.

Corollary 1.9.6 (Integrity of Standard Weyl Algebras)

The standard Weyl algebra \(\mathbb {W}\) is an integral domain:

Proof

Apply Theorem 1.9.4 and keep in mind that an element of \(\mathbb {W}\) vanishes if and only if its degree (with respect to any weight) equals −∞.

Exercise 1.9.7

1. (A)

   We fix a weight \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\) and set

   $$\displaystyle \begin{aligned}\varGamma^{\underline{v},\underline{w}} := \{\underline{v}\cdot\underline{\nu}+\underline{w}\cdot\underline{\mu} \mid \underline{\nu},\underline{\mu} \in \mathbb{N}_0^n\}.\end{aligned}$$

   Prove the following statements

   1. (a)

      \(0 \in \varGamma ^{ \underline {v} \underline {w}} \subseteq \mathbb {N}_0.\)

   2. (b)

      If \(i,j \in \varGamma ^{ \underline {v} \underline {w}}\), then \(i+j \in \varGamma ^{ \underline {v} \underline {w}}\).

   3. (c)

      \(\mathbb {G}^{ \underline {v} \underline {w}}_i \neq 0 \Leftrightarrow \mathbb {P}^{ \underline {v} \underline {w}}_i \neq 0 \Leftrightarrow i \in \varGamma ^{ \underline {v} \underline {w}}\).

   \(\varGamma ^{ \underline {v}, \underline {w}}\) is called the degree semigroup associated to the weight \(( \underline {v}, \underline {w})\).

2. (B)

   Let n = 1, \( \underline {v} = (p)\) and \( \underline {w} = (q)\), where \(p,q \in \mathbb {N}\) are two distinct prime numbers. Determine \(\varGamma ^{ \underline {v}, \underline {w}}\) and the standard bases of all K-vector spaces

   $$\displaystyle \begin{aligned}\mathbb{P}^{\underline{v}\underline{w}}_i \mbox{ and } \mathbb{G}^{\underline{v}\underline{w}}_i \mbox{ for } i \in \varGamma^{\underline{v}\underline{w}},\end{aligned}$$

   at least for some specified pairs like (p, q) = (2, 3), (2, 5), (5, 7), …

3. (C)

   Show, that the ring End_K(K[X ₁, X ₂, …, X _n]) is not an integral domain.

Now, we aim to consider finitely generated left-modules over standard Weyl algebras: the so-called D-modules. Our basic aim is to endow such modules with appropriate filtrations, which are compatible with a given weighted filtration of the underlying Weyl algebra. This will allow us to define associated graded modules over the corresponding associated graded ring of the Weyl algebra—hence over a weight graded polynomial ring. We approach the subject in a more general setting.

Definition and Remark 1.10.1

(A) Let K be a field and let A = (A, A _•) be a filtered K-algebra. Let U be a left-module over A. By a filtration of U compatible with A _• or just an A _• -filtration of U we mean a family

such that the following conditions hold:

1. (a)

   Each U _i is a K-vector subspace of U;

2. (b)

   U _i ⊆ U _i+1 for all \(i\in \mathbb {Z}\);

3. (c)

   \(U = \bigcup _{i\in \mathbb {Z}}U_i\);

4. (d)

   A _i U _j ⊆ U _i+j for all \(i\in \mathbb {N}_0\) and all \(j\in \mathbb {Z}\).

In requirement (d) we have used the standard notation

which we shall use from now on without further mention. If an A _•-filtration U _• of U is given, we say that (U, U _•) or—by abuse of language—that U is a A _• filtered A-module or just that U is a filtered A-module.

(B) Keep the notations and hypotheses of part (A) and let \(U_{\bullet } = (U_i)_{i \in \mathbb {Z}}\) be a filtered A-module. Observe that

(C) We say that two A _•-filtrations \(U^{(1)}_{\bullet }\), \(U^{(2)}_{\bullet }\) are equivalent if there is some \(r \in \mathbb {N}_0\) such that

1. (a)

   \(U^{(1)}_{i-r} \subseteq U^{(2)}_i \subseteq U^{(1)}_{i+r}\mbox{ for all } i \in \mathbb {Z}.\)

Later, we shall use the following observation.

Assume that the above condition (a) holds, let \(i \in \mathbb {N}\) and let a ∈ A _i. Then we have

1. (b)

   \(aU^{(1)}_j \subseteq U^{(1)}_{j+i-1} \mbox{ for all } j {\,\in \,} \mathbb {Z} \quad \Rightarrow \quad a^kU^{(1)}_j \subseteq U^{(1)}_{j+k(i-1)}\) for all \(j {\,\in \,} \mathbb {Z}\) and all \(k \in \mathbb {N}_0\).

2. (c)

   \(aU^{(1)}_j \subseteq U^{(1)}_{j+i-1} \mbox{ for all } j {\,\in \,} \mathbb {Z} \quad \Rightarrow \quad a^{2r+1}U^{(2)}_j \subseteq U^{(2)}_{j+(2r+1)i-1}\) for all \(j {\,\in \,} \mathbb {Z}\).

To prove statement (b), we assume that \(aU^{(1)}_j \subseteq U^{(1)}_{j+i-1}\) for all \(j \in \mathbb {Z}\) and proceed by induction on k. If k = 0 our claim is obvious. If k > 0, we may assume by induction that \(a^{k-1}U^{(1)}_j \subseteq U^{(1)}_{j + (k-1)(i-1)}\) for all \(j \in \mathbb {Z}\), so that indeed

and this proves statement (b). If we apply statement (b) with k = 2r + 1 and observe condition (a), we get

and this proves statement (c).

Remark and Definition 1.10.2

(A) Let K be a field and let A = (A, A _•) be a filtered K-algebra and let U = (U, U _•) be an A _•-filtered A-module. We consider the corresponding associated graded ring

and the K-vector space

For all \(i \in \mathbb {Z}\) we also use the notation

so that we may write

(B) Let \(i \in \mathbb {N}_0\), let \(j \in \mathbb {Z}\) let f, f′∈ A _i and let g, g′∈ U _j such that

It follows that

So in \(U_{i+j}/U_{i+j-1} = \mathrm {Gr}_{U_{\bullet }}(U)_{i+j} \subset \mathrm {Gr}_{U_{\bullet }}(U)\) we get the relation

This allows to define a \(\mathrm {Gr}_{A_{\bullet }}(A)\) -scalar multiplication on the K-space \(\mathrm {Gr}_{U_{\bullet }}(U)\) which is induced by

for all \(i\in \mathbb {N}_0\), all \(j \in \mathbb {Z}\), all f ∈ A _i g ∈ U _j. More generally, if \(r,s \in \mathbb {N}_0\), \(t \in \mathbb {Z}\),

then

(C) Keep the above notations and hypotheses. With respect to our scalar multiplication on \(\mathrm {Gr}_{U_{\bullet }}(U)\) we have the relations

So, the K-vector space \(\mathrm {Gr}_{U_{\bullet }}(U)\) is turned into a graded \(\mathrm {Gr}_{A_\bullet }(A)\)-module

by means of the above multiplication. We call this \(\mathrm {Gr}_{A_\bullet }(A)\)-module \(\mathrm {Gr}_{U_\bullet }(U)\) the associated graded module of U with respect to the filtration U _•. From now on, we always furnish \(\mathrm {Gr}_{U_{\bullet }}(U)\) with this structure of graded \(\mathrm {Gr}_{A_\bullet }(A)\)-module.

Definition 1.10.3

Let K be a field and let A = (A, A _•) be a filtered K-algebra. Assume that the filtration A _• is commutative, so that the corresponding associated graded ring

is commutative.

Moreover, let U = (U, U _•) be an A _•-filtered A-module and consider the corresponding associated graded module

in addition, consider the annihilator ideal

of the \(\mathrm {Gr}_{A_{\bullet }}(A)\)-module \(\mathrm {Gr}_{U_{\bullet }}(U)\). We define the characteristic variety \(\mathbb {V}_{U_{\bullet }}(U)\) of the A _•-filtered A-module U = (U, U _•) as the prime variety of the annihilator ideal of \(\mathrm {Gr}_{U_{\bullet }}(U)\), hence

We also call this variety the characteristic variety of the left A-module U with respect to the A _• -filtration U _• or just the characteristic variety of U with respect to U _•.

Proposition 1.10.4 (Equality of Characteristic Varieties for Equivalent Filtrations)

Let K be a field and let A = (A, A _•) be a filtered K-algebra. Assume that the filtration A _• is commutative (see Definition 1.3.3). Let U be an A-module which is endowed with two equivalent A _• -filtrations \(U^{(1)}_{\bullet }\) and \(U^{(2)}_{\bullet }\) . Then

Proof

We have to show that

By symmetry, it suffices to show that

In view of the fact that the formation of radicals of ideals is idempotent, it suffices even to show that

As \(\mathrm {Gr}_{U^{(1)}_{\bullet }}(U)\) is a graded \(\mathrm {Gr}_{A_{\bullet }}(A)\)-module, its annihilator is a graded ideal of \(\mathrm {Gr}_{A_{\bullet }}(A)\). So, it finally is enough to show, that

So, fix some \(i \in \mathbb {N}_0\) and some

We chose some a ∈ A _i with \(\overline {a} = a+A_{i-1} \in A_i/A_{i-1}.\). For all \(j \in \mathbb {Z}\) we have in \(\mathrm {Gr}_{U_{\bullet }^{(1)}}(U)\) the relation

and hence

According to our hypotheses we find some \(r \in \mathbb {N}_0\) such that \(U^{(1)}_{k-r} \subseteq U^{(2)}_k \subseteq U^{(1)}_{k+r}\) for all \(k \in \mathbb {Z}\). By Definition and Remark 1.10.1 (C)(c) we therefore have

So, for all \(j \in \mathbb {Z}\) we get in \(U^{(2)}_{j+(2r+1)i}/U^{(2)}_{j+(2r+1)i-1} = \mathrm {Gr}_{U_{\bullet }^{(2)}}(U)_{j+(2r+1)i}\) the relation:

This shows that \(\overline {a}^{2r+1} \in \mathrm {Ann}_{\mathrm {Gr}_{A_{\bullet }}(A)}\big (\mathrm {Gr}_{U^{(2)}_{\bullet }}(U)\big )\) and hence that indeed

So, provided (A, A _•) is a commutatively filtered K-algebra (see Definition 1.3.3), the characteristic variety of an A _•-graded A-module (U, U _•) depends only on the equivalence class of the filtration U _•. This allows us to define in an intrinsic way the notion of characteristic variety of a finitely generated (left-) module over the filtered ring A. We work this out in the following combined exercise and definition.

Exercise and Definition 1.10.5

(A) Let (A, A _•) be a filtered K-algebra and let U be a (left) module over A.

Prove the following claims:

1. (a)

   A _i V = 0 for all i < 0.

2. (b)

   The family \(A_\bullet V := \big (A_iV\big )_{i \in \mathbb {Z}}\) is an A _•-filtration of U.

The above filtration A _• V  is called the A _•-filtration of U induced by the subspace V .

(B) Let the notations and hypotheses be as in part (A). Assume in addition that

Prove that

1. (a)

   U is finitely generated as an A-module;

2. (b)

   A _i V  is a finitely generated (left-) module over A ₀.

3. (c)

   The graded \(\mathrm {Gr}_{A_{\bullet }}(A)\)-module \(\mathrm {Gr}_{A_{\bullet }V}(U)\) is generated by finitely many elements \(\overline {g}_1,\overline {g}_2,\ldots ,\overline {g}_s \in \mathrm {Gr}_{A_{\bullet }V}(U)_0\).

Keep in mind that we can always find a vector space V ⊆ U of finite dimension with AV = U if the A-module U is finitely generated.

(C) Let the notations and hypotheses be as above. Let V ⁽¹⁾, V ⁽²⁾ ⊆ U be two K-subspaces such that

Prove that

1. (a)

   The two induced A _•-filtrations A _• V ⁽¹⁾ and A _• V ⁽²⁾ are equivalent.

2. (b)

   If the filtration A _• is commutative, it holds

   $$\displaystyle \begin{aligned}\mathbb{V}_{A_\bullet V^{(1)}}(U) = \mathbb{V}_{A_\bullet V^{(2)}}(U).\end{aligned}$$

(D) Keep the above notations and hypotheses. Assume that the filtration A _• is commutative and that the (left) A-module U is finitely generated. By what we have learned by the previous considerations, we find a K-subspace V ⊆ U of finite dimension such that AV = U, and the characteristic variety \(\mathbb {V}_{A_\bullet V}(U)\) of U with respect to the induced filtration A _• V  is independent of the choice of V . So, we may just write

and we call \(\mathbb {V}_{A_\bullet }(U)\) the characteristic variety of U with respect to the (commutative !) filtration A _• of A. This is the announced notion of intrinsic characteristic variety.

(E) Keep the above notations. Assume that the filtration A _• is of finite type (see Definition and Remark 1.3.4 (C)) and that the (left) A-module U is finitely generated. The A _• filtration U _• of U is said to be of finite type if

1. (a)

   There is some \(j_0 \in \mathbb {Z}\) such that U _j = 0 for all j ≤ j ₀;

2. (b)

   There is an integer σ such that:

   1. (1)

      U _j is finitely generated as a (left) A ₀-module for all j ≤ σ and

   2. (2)

      U _i =∑_j≤σ A _j U _i−j for all i > σ.

In this situation σ is again called a generating degree of the A _•-filtration U _• (compare Definition and Remark 1.3.4 (C)). Prove that in this situation, we have

As U _σ is a finitely generated A ₀-module, we may chose a K-subspace V ⊆ U such that

Prove that for this choice of V  we have:

As a consequence it follows by Proposition 1.10.4 and the observations made in part (D), that

Convention 1.11.1

(A) As in Sect. 1.9, we fix a positive integer n, a field K of characteristic 0 and consider the standard Weyl algebra

In addition, we consider the polynomial ring

in the indeterminates Y ₁, Y ₂, …, Y _n, Z ₁, Z ₂, …, Z _n with coefficients in the field K.

(B) Let \(( \underline {v}, \underline {w}) \in \mathbb {N}^n_0\times \mathbb {N}_0^n\) be a weight. We consider the induced weighted filtration \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\) and also the corresponding associated graded ring.

(see Definition and Remark 1.9.2 (A)).

(C) Moreover, we shall consider the polynomial ring

furnished with the grading induced by our given weight \(( \underline {v}, \underline {w})\) (see Exercise and Definition 1.9.3 (B)), as well as the canonical isomorphism of graded rings (see Theorem 1.9.4):

Definition and Remark 1.11.2

(A) By a D-module we mean a finitely generated left module over the standard Weyl algebra \(\mathbb {W}\).

(B) Let U be a D-module. If U _• is a \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\)-filtration of U, we may again introduce the corresponding associated graded module of U with respect to the filtration U _• (see Definition 1.10.3):

which is indeed a graded module over the associated graded ring \(\mathbb {G}^{ \underline {v} \underline {w}}\). But, in fact, we prefer to consider \(\mathrm {Gr}_{U_{\bullet }}(U)\) as a graded \(\mathbb {P}^{ \underline {v} \underline {w}}\)-module by means of the canonical isomorphism \(\eta ^{ \underline {v} \underline {w}}:\mathbb {P} = \mathbb {P}^{ \underline {v} \underline {w}} \stackrel {\cong }{\longrightarrow } \mathbb {G}^{ \underline {v} \underline {w}}\).

(C) Keep the notations and hypotheses of part (B). Then, we may again consider the characteristic variety of U with respect to the filtration U _•, but under the previous view, that \(\mathrm {Gr}_{U_{\bullet }}(U)\) is a graded module over the graded polynomial ring \(\mathbb {P} = \mathbb {P}^{ \underline {v} \underline {w}}\). So, we define this characteristic variety by

Observe in particular, that the ideal

is graded.

(D) Finally, as U is finitely generated, we may again chose a finite dimensional K-subspace V ⊆ U such that \(\mathbb {W}V = U\), and then consider the induced filtration \(\mathbb {W}^{ \underline {v} \underline {w}}_\bullet V\) of U and the corresponding intrinsic characteristic variety (see Exercise and Definition 1.10.5 (D)) of U with respect to the weight \(( \underline {v}, \underline {w})\), hence:

Example 1.11.3

(A) Keep the above notations and let

with \(c^{(d)}_{ \underline {\nu } \underline {\mu }} \in K \setminus \{0\}\) for all \(( \underline {\nu }, \underline {\mu }) \in \mathrm {supp}(d)\). We also consider the so-called leading differential form of d with respect to the weight \(( \underline {v}, \underline {w})\), which is given by

Moreover, we introduce the polynomial

Now, consider the cyclic left \(\mathbb {W}\)-module

Endow U with the \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\)-filtration (see Exercise and Definition 1.10.5 (A)):

(B) Keep the above notations and hypotheses. Observe first, that for all \(i \in \mathbb {Z}\) we may write

By the additivity of weighted degrees (see Corollary 1.9.5) we have

So, we obtain

Consequently, there is a surjective homomorphism of graded \(\mathbb {G}^{ \underline {v} \underline {w}}\)-modules

If we set

it follows that

Consequently we get

As \(\eta ^{ \underline {v} \underline {w}}(f^{ \underline {v} \underline {w}}) = \overline {h}^{ \underline {v} \underline {w}}\) and if we consider \(\mathrm {Gr}_{U_\bullet }(U)\) as a graded \(\mathbb {P}^{ \underline {v} \underline {w}}\)-module by means of \(\eta ^{ \underline {v} \underline {w}}\), we thus may write

In particular we obtain:

Exercise 1.11.4

1. (A)

   Let n = 1, \(K = \mathbb {R}\) and let \(d := X_1^4 + \partial _1^2 - X_1^2\partial _1^2\). Determine the two characteristic varieties

   $$\displaystyle \begin{aligned}\mathbb{V}^{\underline{v}\underline{w}}(\mathbb{W}/\mathbb{W}d) \mbox{ for } (\underline{v},\underline{w}) = (1,1) \mbox{ and } (\underline{v},\underline{w}) = (0,1).\end{aligned}$$
2. (B)

   To make more apparent what you have done in part (A), determine and sketch the real traces

   $$\displaystyle \begin{aligned}\mathbb{V}^{\underline{v}\underline{w}}_{\mathbb{R}}(\mathbb{W}/\mathbb{W}d) := \{(y,z) \in \mathbb{R}^2 \mid (Y_1-y,Z_1-z)K[Y_1,Z_1] \in \mathbb{V}^{\underline{v}\underline{w}}(\mathbb{W}/\mathbb{W}d)\}\end{aligned}$$

   for \(( \underline {v}, \underline {w}) = (1,1)\) and \(( \underline {v}, \underline {w}) = (0,1)\). Comment your findings.

Now, we shall establish the fact that D-modules are finitely presentable. To do so we first will show that standard Weyl algebras are left Noetherian (see Conventions, Reminders and Notations 1.1.1 (G) and (H)). We begin with the following preparation.

Definition and Remark 1.11.5

(A) Let \(I\subseteq \mathbb {W}\) be a left ideal. We consider the following K-subspace of \(\mathbb {G}^{ \underline {v} \underline {w}}\):

It is immediate to see, that \(\mathbb {G}^{ \underline {v} \underline {w}}(I) \subseteq \mathbb {G}^{ \underline {v} \underline {w}}\) is graded ideal. We call this ideal the graded ideal induced by I in \(\mathbb {G}^{ \underline {v} \underline {w}}\).

(B) Let the notations and hypotheses as in part (A). It is straight forward to see, that the family

is a filtration of the (left) \(\mathbb {W}\)-module I, which we call the filtration induced by \(\mathbb {W}^{ \underline {v} \underline {w}}_{\bullet }\). Observe, that for all \(i \in \mathbb {Z}\) we have a canonical isomorphism of K-vector spaces

It is easy to see, that these isomorphisms of K-vector spaces actually give rise to a canonical isomorphism of graded \(\mathbb {G}^{ \underline {v} \underline {w}}\)-modules

So, by means of this canonical isomorphism we may identify

Lemma 1.11.6

Let \(I, J \subseteq \mathbb {W}\) be two left ideals with I ⊆ J. Then we can say:

1. (a)

   There is an inclusion of graded ideals \(\mathbb {G}^{ \underline {v} \underline {w}}(I) \subseteq \mathbb {G}^{ \underline {v} \underline {w}}(J)\) in the graded ring \(\mathbb {G}^{ \underline {v} \underline {w}}.\)

2. (b)

   If \(\mathbb {G}^{ \underline {v} \underline {w}}(I) = \mathbb {G}^{ \underline {v} \underline {w}}(J)\) , then I = J.

Proof

1. (a):

   This is immediate by Definition and Remark 1.11.5 (A).

2. (b):

   Assume that \(I \subsetneq J\). Then, there is a least integer \(i \in \mathbb {N}_0\) such that

   $$\displaystyle \begin{aligned}I^{\underline{v}\underline{w}}_i = I\cap \mathbb{W}^{\underline{v}\underline{w}}_i \subsetneq J^{\underline{v}\underline{w}}_i = J\cap \mathbb{W}^{\underline{v}\underline{w}}_i.\end{aligned}$$

   As \(I^{ \underline {v} \underline {w}}_{i-1} = J^{ \underline {v} \underline {w}}_{i-1}\) it follows that

   $$\displaystyle \begin{aligned}\mathbb{G}^{\underline{v}\underline{w}}(I)_i \cong I^{\underline{v}\underline{w}}_i/I^{\underline{v}\underline{w}}_{i-1} \mbox{ is not isomorphic to } I^{\underline{v}\underline{w}}_i/I^{\underline{v}\underline{w}}_{i-1} \cong \mathbb{G}^{\underline{v}\underline{w}}(J)_i,\end{aligned}$$

   so that indeed

   $$\displaystyle \begin{aligned}\mathbb{G}^{\underline{v}\underline{w}}(I) \neq \mathbb{G}^{\underline{v}\underline{w}}(J).\end{aligned}$$

Theorem 1.11.7 (Noetherianness of Weyl Algebras)

The Weyl algebra \(\mathbb {W}\) is left Noetherian.

Proof

Otherwise \(\mathbb {W}\) would contain an infinite strictly ascending chain of left ideals \(I^{(1)} \subsetneq I^{(2)} \subsetneq I^{(3)} \subsetneq \cdots .\) But then, by Lemma 1.11.6 we would have an infinite strictly ascending chain \(\mathbb {G}^{ \underline {v} \underline {w}}(I^{(1)}) \subsetneq \mathbb {G}^{ \underline {v} \underline {w}}(I^{(2)}) \subsetneq \mathbb {G}^{ \underline {v} \underline {w}}(I^{(3)}) \subsetneq \cdots \) of ideals in the Noetherian ring \(\mathbb {G}^{ \underline {v} \underline {w}} \cong \mathbb {P}^{ \underline {v} \underline {w}} = \mathbb {P},\) a contradiction.

Corollary 1.11.8 (Finite Presentability of D-Modules)

Each D-module U admits a finite presentation

Proof

This follows immediately by Theorem 1.11.7 and the observations made in Conventions, Reminders and Notations 1.1.1 (H).

Example 1.11.9

(A) Consider the polynomial ring U := K[X ₁, X ₂, …, X _n]. As

this polynomial ring can be viewed in a canonical way as a left module over \(\mathbb {W}\), the scalar being multiplication given by

As f ⋅ 1 = f for all f ∈ U it follows that

So, the \(\mathbb {W}\)-module U := K[X ₁, X ₂, …, X _n] is generated by a single element, and hence in particular a D-module.

(B) Keep the previous notations and hypotheses. Observe that

and hence

We thus have an exact sequence of K-vector spaces

in which \(\mathbb {W} \stackrel {\pi }{\longrightarrow } U\) is the canonical projection map given by

Our aim is to show:

To do so, it suffices to show that for all \( \underline {\nu },\underline {\mu },\underline {\nu }',\underline {\mu }' \in \mathbb {N}_0^n\) it holds

If \( \underline {\mu } =\underline {\mu }' =\underline {0}\), we have

If \( \underline {\mu } = 0\) and \( \underline {\mu }' \neq 0\) we have

So, let \( \underline {\mu } \neq\underline {0}\). By the Product Formula of Proposition 1.6.2 we have

with

and

Assume first, that \( \underline {\mu }' \neq 0\). Then we have

It thus follows, that

So, finally let \( \underline {\mu }' = 0\). Then \(dd' =\underline {X}^{ \underline {\nu }+ \underline {\nu }'} \underline {\partial }^{ \underline {\mu }} + s\), and

So, by what we have learned in Exercise 1.6.6 (B), we have

As s is a K-multiple of a monomial in the X _i’s we have π(s) = s. It thus follows

This proves, that π is indeed a homomorphism of left \(\mathbb {W}\)-modules.

(C) Keep the previous notations and hypotheses. Then, according the above observations, we have an exact sequence of left \(\mathbb {W}\)-modules

in which h is given by

This sequence clearly constitutes a presentation of the left \(\mathbb {W}\)-module U (see Conventions, Reminders and Notations 1.1.1 (H)) and the corresponding presentation matrix for U is the row

Exercise 1.11.10

(A) We consider the polynomial ring U = K[X ₁, X ₂, …, X _n] canonically as a D-module, as done in Example 1.11.9. Fix a weight \(( \underline {v}, \underline {w}) \in \mathbb {N}_0^n \times \mathbb {N}_0^n\). Consider the K-subspace K ⊂ U, observe that \(\mathbb {W}K = U\) and endow U with the induced filtration

Show, that there is an isomorphism of graded \(\mathbb {P}\)-modules

where

is the polynomial ring U endowed with the grading associated to the weight \( \underline {v} \in \mathbb {N}_0^n\). Determine the characteristic variety

(B) Keep the notations and hypotheses of part (A). Show, the left \(\mathbb {W}\)-module U is simple: If \(V \subsetneq U\) is a proper left \(\mathbb {W}\)-submodule, then V = 0. (Hint: Let f ∈ U ∖{0} be of degree r and assume that \( \underline {\nu } = (\nu _1,\nu _2,\ldots ,\nu _n) \in \mathrm {supp}(f)\) with \(\sum _{i=1}^n \nu _i = r\) and show that \( \underline {\partial }^{ \underline {\nu }} \in K \setminus \{0\}.\) Conclude that \(\mathbb {W}f = U.\))

Remark and Definition 1.11.11

(A) We furnish the polynomial ring K[X ₁, X ₂, …, X _n] with its canonical structure of D-module (see Example 1.11.9). We now consider a ring \(\mathscr {A}\) with the following properties

1. (1)

   \(\mathscr {A}\) is commutative;

2. (2)

   \(\mathscr {A}\) is a left \(\mathbb {W}\)-module;

3. (3)

   \(K[X_1,X_2,\ldots ,X_n] \subseteq \mathscr {A}\) is a left submodule.

In this situation, we call \(\mathscr {A}\) a ring of good functions in X ₁, X ₂, …, X _n over K.

The idea covered by this concept is that for all \(d \in \mathbb {W}\) and all \(f \in \mathscr {a}\) the product \(df \in \mathscr {A}\) should be viewed as the result of the application of the differential operator d to the function f. Therefore, one often writes

(B) Let the notations and hypotheses be as in part (A). By a system of polynomial differential equations in \(\mathscr {A}\) we mean a system of equations

with \(r,s \in \mathbb {N}\) such that

The above system of differential equations can be understood as a linear system of equations over the ring \(\mathscr {A}\). We namely may consider the matrix

Then, the above system may be written in matrix form as

We call \(\mathscr {D}\) the matrix of differential operators associated to our system of linear differential equations. So, systems of differential equations correspond to matrices with entries in a standard Weyl algebra.

(C) Keep the previous notations and hypotheses, then the matrix of differential operators \(\mathscr {D} \in \mathbb {W}^{s \times r}\) gives rise to an exact sequence of left \(\mathbb {W}\)-modules

In particular \(U_{\mathscr {D}}\) is a D-module and the previous sequence is a finite presentation of \(U_{\mathscr {D}}\). We call this presentation the presentation induced by the matrix \(\mathscr {D}\) and we call \(U_{\mathscr {D}}\) the D-module defined by the matrix \(\mathscr {D}\)—or the D-module associated with our system of differential equations. So, each system of differential equations defines a D-module. Obviously, one is particularly interested in the solution space of our system of differential equations, hence in the K-vector space

Observe, that \(\mathbb {S}_{\mathscr {D}}(\mathscr {A})\) is a K-subspace of \(\mathscr {A}^r\).

Proposition 1.11.12

Let \(r,s \in \mathbb {N}\) , let

be a matrix of differential operators, consider the induced presentation

and the corresponding solution space \(\mathbb {S}_{\mathscr {D}}(\mathscr {A})\).

For all i = 1, 2, …, r let \(\mathrm {e}_i := (\delta _{i,j})_{j=1}^r \in \mathbb {W}^r\) be the i-th canonical basis element. Then, there is an isomorphism of K-vector spaces

given by

Proof

Observe, that there is indeed a K-linear map

given by

If ε(m) = 0, then m(π(e_i)) = 0 for all i = 1, 2, …, r. As π is surjective, the elements π(e_i) (i = 1, 2, …, r) generate the left \(\mathbb {W}\)-module \(U = U_{\mathscr {D}}\). So, it follows that m = 0 and this proves, that the map ε is injective.

It remains to show that

To do so, let

be the canonical basis elements of \(\mathbb {W}^s.\)

First, let \(m \in \mathrm {Hom}_{\mathbb {W}}\big (U_{\mathscr {D}},\mathscr {A}\big )\). We aim to show, that \(\varepsilon (m) \in \mathbb {S}_{\mathscr {D}}(\mathscr {A})\). We have to show, that the column

vanishes. For each i = 1, 2, …, s we can write \(\sum _{j=1}^rd_{ij}\mathrm {e}_j = \mathrm {b}_i\mathscr {D} = h(\mathrm {b}_i)\), and hence get indeed

Conversely, let \((f_1,f_2,\ldots ,f_r) \in \mathbb {S}_{\mathscr {D}}(\mathscr {A})\), so that \(\sum _{j=1}^r d_{ij}f_j = 0\). We aim to show that \((f_1,f_2,\ldots ,f_r) \in \varepsilon \big (\mathrm {Hom}_{\mathbb {W}}(U,\mathscr {A})\big )\).

To this end, we consider the homomorphism of left \(\mathbb {W}\)-modules

Observe that

It follows that k ∘ h = 0. Therefore k induces a homomorphism of left \(\mathbb {W}\)-modules

It follows that m(π(e_j)) = k(e_j) = f _j for all j = 1, 2, …, r. But this means that \((f_1,f_2,\ldots ,f_r) = \varepsilon (m) \in \varepsilon \big (\mathrm {Hom}_{\mathbb {W}}(U,\mathscr {A})\big )\).

Exercise 1.11.13

(A) Let n = 1, \(K=\mathbb {R}\) and let \(\mathscr {A} := \mathscr {C}^{\infty }(\mathbb {R})\) be set of smooth functions on \(\mathbb {R}\). Fix \(d \in \mathbb {W} = \mathbb {W}(\mathbb {R},1) = \mathbb {R}[X,\partial ]\) and consider the matrix \(\mathscr {D} = (d) \in \mathbb {W}^{1\times 1}\). Determine

for all weights \(( \underline {v}, \underline {w}) = (v,w) \in \mathbb {N}_0\times \mathbb {N}_0 \setminus \{(0,0)\}\) and for

(B) Let \(n,m \in \mathbb {N}\), \(\mathscr {A} := K[X_1,X_2,\ldots ,X_n]\) and consider the matrix

Determine

In this section, we introduce and treat Gröbner bases of left ideals in standard Weyl algebras with respect to so-called admissible orderings of the set of elementary differential operators. What we get is a theory very similar to the theory of Gröbner bases of ideals in polynomial rings. A theory many readers may be familiar with already. Indeed a great deal of what we shall present in the sequel could also be deduced from the theory of Gröbner in polynomial rings. Nevertheless, we prefer to introduce the subject in a self contained way so that readers who are not familiar with Gröbner in polynomial rings can follow our approach without further prerequisites. As for Gröbner bases in (commutative) polynomial rings and their applications, there are indeed many introductory and advanced textbooks and monograph. So, we mention only a sample of possible references for this subject, namely [1, 6, 19, 25, 26, 30, 36] and [42].

In general, Gröbner bases are intimately related to Division Theorems, which generalize Euclid’s Division Theorem for univariate polynomial rings over a field. Gröbner bases and Division Theorems for rings of linear differential operators were introduced by Briançon and Maisonobe [14] in the univariate case and by Castro-Jiménez [21] in the multivariate case. Two more recent basic references in the field of are the textbook of Bueso,Gómez-Torricellas and Verschoren [20] and the PhD thesis [31] of Levandovskyy.

The main goal of the present section is to prove that left ideals in Weyl algebras admit so-called universal Gröbner bases. This existence result can actually be proved in the more general setting of admissible algebras. Readers, who are interested in this, should consult for example Boldini’s thesis [10] or else [38], [41] or [43].

Convention 1.12.1

(A) As previously, we fix a positive integer n, a field K of characteristic 0 and consider the standard Weyl algebra

Moreover, we consider the polynomial ring

in the indeterminates Y ₁, Y ₂, …, Y _n, Z ₁, Z ₂, …, Z _n with coefficients in the field K.

(B) In addition, we fix the isomorphism of K-vector spaces

Moreover we respectively consider the set \(\mathbb {E}\) of all elementary differential operators in \(\mathbb {W}\) and the set \(\mathbb {M}\) of all monomials in \(\mathbb {P}\), thus:

In a first step we now introduce some basic notions of our subject, namely: admissible orderings (of the set \(\mathbb {E}\) of elementary differential operators, leading (elementary) differential operators and (in the polynomial ring \(\mathbb {P}\)) leading monomials and leading terms. Mainly for those readers who have not met these concepts in the framework of polynomial rings, we shall add below a number of examples and exercises on these new notions.

Definition, Reminder and Exercise 1.12.2

(A) (Total Orderings) Let S be any set. A total ordering of S is a binary relation ≤⊆ S × S such that for all a, b, c ∈ S the following requirements are satisfied:

1. (a)

   (Reflexivity) a ≤ a.

2. (b)

   (Antisymmetry) If a ≤ b and b ≤ a, then a = b.

3. (c)

   (Transitivity) If a ≤ b and b ≤ c, then a ≤ c.

4. (b)

   (Totality) Either a ≤ b or b ≤ a.

We write TO(S) for the set of total orderings on S.

If ≤ ∈TO(S) and a, b ∈ S, we write

(B) (Well Orderings) Keep the above notations and hypotheses. A total ordering ≤ ∈TO(S) is said to be a well ordering of S, if it satisfies the following additional requirement:

1. (e)

   (Existence of Least Elements) For each non-empty subset T ⊆ S there is an element t ∈ T such that t ≤ t′ for all t′∈ T.

Inthe situation mentioned in statement (e), the element t ∈ T—if it exists at all—is uniquely determined by T and called the least element or the minimum of T with respect to ≤ and denoted by min_≤(T), thus

We write WO(S) for the set of all well orderings of S.

(C) (Admissible Orderings) A total ordering \(\leq \in \mathrm {TO}(\mathbb {E})\) of the set of all elementary differential operators is called an admissible ordering of \(\mathbb {E}\) if it satisfies the following requirements:

1. (a)

   (Foundedness) \(1 \leq\underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\) for all \( \underline {\nu }, \underline {\mu } \in \mathbb {N}_0^n\)

2. (b)

   (Compatibility) For all \( \underline {\lambda },\underline {\lambda }',\underline {\kappa },\underline {\kappa }',\underline {\nu }, \underline {\mu } \in \mathbb {N}_0^n\) we have the implication:

   $$\displaystyle \begin{aligned}\mbox{If } \underline{X}^{\underline{\lambda}}\underline{\partial}^{\underline{\kappa}} \leq \underline{X}^{\underline{\lambda}'}\underline{\partial}^{\underline{\kappa}'}, \mbox{ then } \underline{X}^{\underline{\lambda} + \underline{\nu}}\underline{\partial}^{\underline{\kappa}+\underline{\mu}} \leq \underline{X}^{\underline{\lambda}' + \underline{\nu}}\underline{\partial}^{\underline{\kappa}'+\underline{\mu}}.\end{aligned} $$

We write \(\mathrm {AO}(\mathbb {E})\) for the set of all admissible orderings of \(\mathbb {E}\).

Prove the following facts:

1. (c)

   If \( \underline {\nu }, \underline {\nu }', \underline {\mu }, \underline {\mu }', \underline {\lambda },\underline {\lambda }',\underline {\kappa },\underline {\kappa }', \in \mathbb {N}_0^n\) with \( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }} \leq\underline {X}^{ \underline {\nu }'} \underline {\partial }^{ \underline {\mu }'}\) and \( \underline {X}^{ \underline {\lambda }} \underline {\partial }^{ \underline {\kappa }} <\underline {X}^{ \underline {\lambda }'} \underline {\partial }^{ \underline {\kappa }'}\), then

   $$\displaystyle \begin{aligned}\underline{X}^{\underline{\lambda} + \underline{\nu}}\underline{\partial}^{\underline{\kappa}+\underline{\mu}} < \underline{X}^{\underline{\lambda}' + \underline{\nu}'}\underline{\partial}^{\underline{\kappa}'+\underline{\mu}'}.\end{aligned} $$
2. (d)

   \(\mathrm {AO}(\mathbb {E}) \subseteq \mathrm {WO}(\mathbb {E})\).

(D) (Leading Elementary Differential Operators and Related Concepts) From now on, for all \(d \in \mathbb {W},\) we use the notation

Keep the above notations and hypotheses. If \(\leq \in \mathrm {AO}(\mathbb {E})\) and \(d \in \mathbb {W}\setminus \{0\}\), we define the leading elementary differential operator of d with respect to ≤ by:

so that

Moreover, we define the leading coefficient LC_≤(d) of d with respect to ≤ as the coefficient of d with respect to LE_≤(d), and the leading differential operator LD_≤(d) of d with respect to ≤ as the product of the leading elementary differential operator with the leading coefficient, so that:

1. (a)

   LC_≤(d) ∈ K ∖{0} with LE_≤(d −LC_≤(d)LE_≤(d)) < LE_≤(d).

2. (b)

   LD_≤(d) = LC_≤(d)LE_≤(d).

3. (c)

   LE_≤(d −LD_≤(d)) < LE_≤(d).

Finally, we define the leading monomial and the leading term of d with respect to ≤ respectively by

Prove the following statements:

1. (d)

   If \(d,e \in \mathbb {W}\setminus \{0\}\), with d ≠ − e, then LE_≤(d + e) ≤max_≤{LE_≤(d), LE_≤(e)}, with equality if and only if LD_≤(d) ≠ −LD_≤(e).

The previously introduced notions are of basic significance for this and the next section. So, we hope to illuminate their meaning in the following series of examples and exercises, which were already announced prior to the definition of these concepts.

Examples and Exercises 1.12.3

(A) (Well Orderings) Keep the above notations and hypotheses. Prove the following statements:

1. (a)

   Let \(\varphi :\mathbb {N}_0 \longrightarrow \mathbb {N}^n_0\times \mathbb {N}^n_0\) be a bijective map. Show that the binary relation \(\leq _{\varphi } \subseteq \mathbb {E}\times \mathbb {E}\) defined by

   $$\displaystyle \begin{aligned}\underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}} \leq_{\varphi} \underline{X}^{\underline{\nu}'}\underline{\partial}^{\underline{\mu}'} \Leftrightarrow \varphi^{-1}(\underline{\nu},\underline{\mu}) \leq \varphi^{-1}(\underline{\nu},\underline{\mu})\end{aligned}$$

   for all \( \underline {\nu }, \underline {\mu }, \underline {\nu }', \underline {\mu }' \in \mathbb {N}^n_0\) is a well ordering of \(\mathbb {E}\).

2. (b)

   Show that in the notations of exercise (a) the well ordering ≤_φ is discrete, which means that the set \(\{e \in \mathbb {E} \mid e \leq _{\varphi } d\}\) is finite for all \(d \in \mathbb {E}\).

3. (c)

   Show, that there uncountably many discrete well orderings of \(\mathbb {E}\).

4. (d)

   Let n = 1, set X ₁ =: X, ∂ ₁ =: ∂ and define the binary relation ≤ on the set of elementary differential operators \(\mathbb {E} = \{X^{\nu }\partial ^{\mu } \mid \nu ,\mu \in \mathbb {N}_0\}\) by

   $$\displaystyle \begin{aligned}X^{\nu}\partial^{\mu} \leq X^{\nu'}\partial^{\mu'} \mbox{ if either } \begin{cases} \nu < \nu' \mbox{ or else }\\ \nu = \nu' \mbox{ and } \mu < \mu'\end{cases}\end{aligned}$$

   for all \(\nu , \mu \in \mathbb {N}_0\). Show, that ≤ is a non-discrete well ordering of \(\mathbb {E}\).

(B) (Admissible Orderings) Keep the above notations and hypotheses.

1. (a)

   We define the binary relation \(\leq _{\mathrm {lex}} \subseteq \mathbb {E} \times \mathbb {E}\) by setting (again for all \( \underline {\nu }, \underline {\mu }, \underline {\nu }', \underline {\mu }' \in \mathbb {N}^n_0\)):

   $$\displaystyle \begin{aligned}\underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}} \leq_{\mathrm{lex}} \underline{X}^{\underline{\nu}'}\underline{\partial}^{\underline{\mu}'} \mbox{ if either }\end{aligned}$$
   1. (1)

      \( \underline {\nu }= \underline {\nu }'\) and \( \underline {\mu }= \underline {\mu }'\), or

   2. (2)

      \( \underline {\nu }= \underline {\nu }'\) and \(\exists j \in \{1,2,\ldots ,n\}:\big [\mu _j < \mu ^{\prime }_j \mbox{ and } \mu _k = \mu ^{\prime }_k, \forall k < j\big ]\), or else

   3. (3)

      \(\exists i \in \{1,2,\ldots ,n\}:\big [ \nu _i < \nu ^{\prime }_i \mbox{ and } \nu _k = \nu ^{\prime }_k, \forall k < i\big ]\).

   Prove that ≤_lex ∈AO(E). The admissible ordering ≤_lex is called the lexicographic ordering of the set of elementary differential operators.

2. (b)

   Set n = 1, X ₁ =: X, ∂ ₁ =: ∂ and write down the first 20 elementary differential operators \(d \in \mathbb {E} = \{X^{\nu }\partial ^{\mu } \mid \nu ,\mu \in \mathbb {N}_0\}\) with respect to the ordering ≤_lex.

3. (c)

   Solve the similar task as in exercise (b), but with n = 2 instead of n = 1 and with 30 instead of 20.

4. (d)

   We define another binary relation \(\leq _{\mathrm {deglex}} \subseteq \mathbb {E} \times \mathbb {E}\) by setting

   $$\displaystyle \begin{aligned}d \leq_{\mathrm{deglex}} e \mbox{ if either } \begin{cases} \mathrm{deg}(d) < \mathrm{deg}(e) \mbox{ or else }\\ \mathrm{deg}(d) = \mathrm{deg}(e) \mbox{ and } d \leq_{\mathrm{lex}} e.\end{cases}\end{aligned}$$

   Show, that \(\leq _{\mathrm {deglex}} \in \mathrm {AO}(\mathbb {E})\). This admissible ordering is called the degree-lexicographic ordering of the set of elementary differential operators.

5. (e)

   Solve the previous exercises (b) and (c) but this time with the ordering ≤_deglex.

6. (f)

   We introduce a further binary relation \(\leq _{\mathrm {degrevlex}} \subseteq \mathbb {E} \times \mathbb {E}\) by setting (again for all \( \underline {\nu }, \underline {\mu }, \underline {\nu }', \underline {\mu }' \in \mathbb {N}^n_0\)):

   $$\displaystyle \begin{aligned}\underline{X}^{\underline{\nu}}\underline{\partial}^{\underline{\mu}} \leq_{\mathrm{degrevlex}} \underline{X}^{\underline{\nu}'}\underline{\partial}^{\underline{\mu}'} \mbox{ if either }\end{aligned}$$
   1. (1)

      \(\mathrm {deg}\big ( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\big ) < \mathrm {deg}\big ( \underline {X}^{ \underline {\nu }'} \underline {\partial }^{ \underline {\mu }'}\big )\), or else

   2. (2)

      \(\mathrm {deg}\big ( \underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\big ) = \mathrm {deg}\big ( \underline {X}^{ \underline {\nu }'} \underline {\partial }^{ \underline {\mu }'}\big )\) and either

      1. (i)

         \( \underline {\nu }= \underline {\nu }'\) and \( \underline {\mu }= \underline {\mu }'\), or

      2. (ii)

         \( \underline {\mu } =\underline {\mu }'\) and \(\exists i \in \{1,2,\ldots ,n\}:\big [ \nu _i > \nu ^{\prime }_i \mbox{ and } \nu _k = \nu ^{\prime }_k, \forall k > i\big ]\), or else

      3. (iii)

         \(\exists j \in \{1,2,\ldots ,n\}:\big [\mu _j > \mu ^{\prime }_j \mbox{ and } \mu _k = \mu ^{\prime }_k, \forall k > j\big ]\).

   Prove, that \(\leq _{\mathrm {degrevlex}} \in \mathrm {AO}(\mathbb {E})\). This admissible ordering is called the degree-reverse-lexicographic ordering of the set of elementary differential operators.

7. (g)

   Solve the previous exercise (e) but with ≤_degrevlex instead of ≤_deglex.

8. (h)

   An admissible ordering of the set \(\mathbb {M} = \{ \underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \mid\underline {\nu },\underline {\mu } \in \mathbb {N}^n_0\}\) of all monomials in \(\mathbb {P}\) is a total ordering of \(\mathbb {M}\) which satisfies the requirements

   1. (1)

      (Foundedness) 1 ≤ m for all \(m \in \mathbb {M}\).

   2. (2)

      (Compatibility) For all m, m′ and \(t\ \in \mathbb {M}\) we have the implication:

      $$\displaystyle \begin{aligned}\mbox{If } m \leq m', \mbox{ then } mt \leq m't.\end{aligned}$$

   For any \(\leq \in \mathrm {AO}(\mathbb {E})\) we define the binary relation \(\leq _{\varPhi } \subseteq \mathbb {M} \times \mathbb {M}\) by setting

   $$\displaystyle \begin{aligned}m \leq_{\varPhi} m' \Leftrightarrow \varPhi^{-1}(m) \leq \varPhi^{-1}(m') \mbox{ for all } m,m' \in \mathbb{M}.\end{aligned}$$

   Prove, that \(\leq _{\varPhi } \in \mathrm {AO}(\mathbb {M})\) and that there is indeed a bijection

   $$\displaystyle \begin{aligned}\bullet_{\varPhi}: \mathrm{AO}(\mathbb{E}) \stackrel{\cong}{\longrightarrow}\mathrm{AO}(\mathbb{M}), \mbox{ given by } \leq \mapsto \leq_{\varPhi}.\end{aligned}$$

   The names given in the previous exercises (a), (d) and (f) to the three admissible orderings of \(\mathbb {E}\) introduced in these exercises are ”inherited“ from the ”classical“ designations used in polynomial rings, via the above bijection.

9. (i)

   Prove, that ≤_deglex and ≤_degrevlex are both discrete in the sense of exercise (A) (b), where as ≤_lex is not.

(C) (Leading Elementary Differential Operators and Related Concepts) Keep the previous notations and hypotheses.

1. (a)

   Let n = 1, set X ₁ =: X, ∂ ₁ =: ∂, Y ₁ =: Y  and Z ₁ =: Z. Write down the leading elementary differential operator, the leading differential operator, the leading coefficient, the leading monomial and the leading term of each of the following differential operators, with respect to each of the admissible orderings ≤_lex, ≤_deglex and ≤_degrevlex:

   1. (1)

      5X ⁶ + 4X ⁴ ∂ − 2X ² ∂ ³ + X∂ ⁴ − 3∂ ⁶.

   2. (2)

      ∂ ⁴ − 4X∂ ³ + 6X ² ∂ ² − 4X∂ + X ⁴.

   3. (3)

      ∂ ¹² − X ⁵ ∂ ⁷ + X ⁷ ∂ ⁵ − X ⁹ ∂ ³ + X ¹².

2. (b)

   Let n = 2 solve the task corresponding to exercise (a) above for the differential operators

   1. (1)

      \(X_1^3X_2^2 + 2\partial _1^3\partial _2^2\).

   2. (2)

      \(X_1^2X_2^3\partial _1^2\partial _2^3 - \partial _1^4\partial _2^6.\)

   3. (3)

      \(X_1^k + X_2^k + \partial _1^k + \partial _2^k\) with \(k \in \mathbb {N}\).

The next proposition will play a crucial role for our further considerations. it tells us essentially, that ”leading differential operators behave as leading terms of polynomials“. It is precisely this property, which will allow us to introduce a fertile notion of Gröbner bases for left ideals in Weyl algebras.

Proposition 1.12.4 (Multiplicativity of Leading Terms)

Let \(\leq \in \mathrm {AO}(\mathbb {E})\) and let \(d,e \in \mathbb {W} \setminus \{0\}\) . Then it holds

1. (a)

   LT_≤(de) = LT_≤(d)LT_≤(e).

2. (b)

   LM_≤(de) = LM_≤(d)LM_≤(e).

Proof

The product formula for elementary differential operators of Proposition 1.6.2 yields that

We may write

with \(c^{(d)}_{ \underline {\nu } \underline {\mu }}, c^{(e)}_{ \underline {\nu }' \underline {\mu }'} \in K \setminus \{0\}\) for all \(( \underline {\nu }, \underline {\mu })\in \mathrm {supp}(d)\) and all \(( \underline {\nu }', \underline {\mu }')\in \mathrm {supp}(e)\). With appropriate pairs \(( \underline {\nu }^{(0)}, \underline {\mu }^{(0)})\in \mathrm {supp}(d)\) and \(( \underline {\nu }^{\prime (0)}, \underline {\mu }^{\prime (0)})\in \mathrm {supp}(e)\) we also may write

Now, bearing in mind the previous observation on leading elementary differential operators we may write

with \(r_{ \underline {\nu } \underline {\nu }' \underline {\mu } \underline {\mu }'} \in \mathbb {W}\), such that for all \(( \underline {\nu }, \underline {\mu })\in \mathrm {supp}(d)\) and all \(( \underline {\nu }', \underline {\mu }')\in \mathrm {supp}(e)\) it holds

By Definition, Reminder and Exercise 1.12.2 (C)(c) we have

By Definition, Reminder and Exercise 1.12.2 (D)(d) it now follows easily that

We thus obtain

But now it follows

The next result may be understood as an extension of the classical division algorithms of Euclid for univariate polynomials to the case of differential operators. It was first proved in 1984 by Briançon-Maisonobe in the univariate case and by Castro-Jiménez in the multivariate case.

Those readers, who are familiar with the Buchberger algorithm in multivariate polynomial rings will realize that our result corresponds to the division algorithm in multi-variate polynomial rings. Observe in particular that—as in the case of multi-variate polynomials—we will divide “by a family of denominators” and that the presented division procedure depends on an admissible ordering.

Proposition 1.12.5 (The Division Property, Briançon-Maisonobe [14] and Castro-Jiménez [21])

Let \(\leq \in \mathrm {AO}(\mathbb {E})\) , let \(d \in \mathbb {W}\) and let \(F \subset \mathbb {W}\) be a finite set. Then, there is an element \(r \in \mathbb {W}\) and a family \((q_f)_{f \in F} \in \mathbb {W}^F\) such that (in the notations of Convention 1.12.1 (B) and Definition, Reminder and Exercise 1.12.2 (D))

1. (a)

   d =∑_{f ∈ F} q _f f + r;

2. (b)

   \(\varPhi (s) \notin \mathbb {P}\mathrm {LM}_\leq (f)\) for all f ∈ F ∖{0} and all s ∈Supp(r).

3. (c)

   LE_≤(q _f f) ≤LE_≤(d) for all f ∈ F with q _f f ≠ 0.

Proof

We clearly may assume that \(F \subset \mathbb {W}\setminus \{0\}\). If d = 0, we choose r = 0 and q _f = 0 for all f ∈ F. Assume, that our claim is wrong, and let \(U \subsetneq \mathbb {W}\) be the non-empty set of all differential operators \(d \in \mathbb {W}\) which do not admit a presentation of the requested form. As \(\leq \in \mathrm {WO}(\mathbb {E})\) and \(U \subset \mathbb {W}\setminus \{0\}\), we find some d ∈ U such that

We distinguish the following two cases:

1. (1)

   There is some f ∈ F such that \(\mathrm {LM}_\leq (d) \in \mathbb {P} \mathrm {LM}_\leq (f)\).

2. (2)

   \(\mathrm {LM}_\leq (d) \notin \bigcup _{f\in F}\mathbb {P} \mathrm {LM}_\leq (f)\).

In the case (1) we find some \(e \in \mathbb {E}\) such that LM_≤(d) = Φ(e)LM_≤(f) and so we can introduce the element

If d′ = 0, we set

But then

is a presentation of d with the requested properties.

So, let d′≠ 0. Observe, that by Proposition 1.12.4 (a) we can write

If follows that \(\mathrm {LD}_\leq \big (\frac {\mathrm {LC}_\leq (d)}{\mathrm {LC}_\leq (f)}ef\big ) = \mathrm {LD}_\leq (d)\), and hence by Definition, Reminder and Exercise 1.12.2 (D)(d) we obtain that

Therefore, d′∉U and so we find an element \(r' \in \mathbb {W}\) and a family \((q^{\prime }_{f'})_{f'\in F} \in \mathbb {W}^F\) such that

1. (a)′

   \(d' = \sum _{f' \in F} q^{\prime }_{f'} f' + r'\);

2. (b)′

   \(\varPhi (s') \notin \mathbb {P}\mathrm {LM}_\leq (f')\) for all f′∈ F and all s′∈Supp(r′).

3. (c)′

   \(\mathrm {LE}_\leq (q^{\prime }_{f'}f') \leq \mathrm {LE}_\leq (d')\) for all f′∈ F with \(q^{\prime }_{f'} \neq 0\).

Now, we set

As

we get

Now, it follows easily, that the requirements (a),(b) and (c) of our proposition are satisfied in the case (1).

So, let us assume that we are in the case (2). We set

If d′ = 0 we have d′ = LD_≤(d) and it suffices to choose q _f := 0 for all f ∈ F and r = d.

So, let d′≠ 0. Then, we have LE_≤(d′) < LE_≤(d) (see Definition, Reminder and Exercise 1.12.2 (D)(c)), so that again d′∉U. But this means once more, that we get elements r′ and \(q^{\prime }_{f'} \in \mathbb {W}\) (for all f′∈ F) such that the above conditions (a)′, (b)′ and (c)′ are satisfied. Now, we set

As supp(r) ⊆supp(r′) ∪{LE_≤(d)} and LE_≤(q _f f) ≤LE(d′) ≤LE_≤(d) for all f ∈ F with q _f ≠ 0 the requirements (a),(b) and (c) are again satisfied for the suggested choice.

Now, we are ready to introduce the basic notion of this section: the concept of Gröbner basis.

Definition, Reminder and Exercise 1.12.6

(A) (Monomial Ideals) An ideal \(I \subseteq \mathbb {P}\) is called a monomial ideal if there is a set \(S \subset \mathbb {M} =\{ \underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \mid\underline {\nu },\underline {\mu } \in \mathbb {N}^n_0\}\) such that

Show that in this situation for all \(m \in \mathbb {M} \setminus \{0\}\) we have

1. (a)

   If \(m = \sum _{i=1}^t f_is_i\) with s ₁, s ₂, …, s _t ∈ S and \(f_1,f_2,\ldots ,f_t \in \mathbb {P}\), then there is some i ∈{1, 2, …, t} and some n _i ∈supp(f _i) such that m = n _i s _i.

2. (b)

   m ∈ I if and only if there are \(n\in \mathbb {M}\) and some s ∈ S such that m = ns.

(B) (Leading Monomial Ideals) Let \(\leq \in \mathrm {AO}(\mathbb {E})\) and \(T \subset \mathbb {W}\). Then, the ideal

is called the leading monomial ideal of T with respect to ≤.

Prove that for all \(m \in \mathbb {M}\), we have the following statements.

1. (a)

   If \(m = \sum _{i=1}^s f_i\mathrm {LM}_\leq (t_i)\) with t ₁, t ₂, …, t _s ∈ T and \(f_1,f_2,\ldots ,f_s \in \mathbb {P}\), then there is some i ∈{1, 2, …, s} and some n _i ∈supp(f _i) such that t _i ≠ 0 and m = n _iLM_≤(t _i).

2. (b)

   m ∈LMI_≤(T) if and only if there are elements \(u \in \mathbb {E}\) and t ∈ T such that m = LM_≤(u)LM_≤(t).

(C) (Gröbner Bases) Let \(\leq \in \mathrm {AO}(\mathbb {E})\) and let \(L \subseteq \mathbb {W}\) be a left ideal. A Gröbner basis of L with respect to ≤ (or a ≤-Gröbner basis of L) is a subset G ⊆ L such that

Prove the following facts:

1. (a)

   If G is a ≤-Gröbner basis of L and G ⊆ H ⊆ L with #H < ∞, then H is a ≤-Gröbner basis of L.

2. (b)

   If G is a ≤-Gröbner basis of L, then for each d ∈ L ∖{0} there is some \(u \in \mathbb {E}\) and some g ∈ G ∖{0} such that

   $$\displaystyle \begin{aligned}\mathrm{LM}_\leq(d) = \mathrm{LM}_\leq (u) \mathrm{LM}_\leq(g) = \mathrm{LM}_\leq(ug).\end{aligned}$$
3. (c)

   If G is a ≤-Gröbner basis of L, then for each d ∈ L ∖{0} there is some monomial \(m =\underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \in \mathbb {P}\) and some g ∈ G ∖{0} such that

   $$\displaystyle \begin{aligned}\mathrm{LM}_\leq(d) = m\mathrm{LM}_\leq(g).\end{aligned}$$

Now, we prove that Gröbner bases always exist, and that they deserve the name of ”basis“, as they generate the involved left ideal. Clearly, these statements correspond precisely to well known facts in multi-variate polynomial rings. After having established the announced existence and generating property of Gröbner bases, we shall add a few examples and exercises on the subject.

Proposition 1.12.7 (Existence and Generating Property of Gröbner Bases)

Let ≤ ∈AO(E) and let \(L \subseteq \mathbb {W}\) be a left ideal. Then the following statements hold.

1. (a)

   L admits a ≤-Gröbner basis.

2. (b)

   If G is any ≤-Gröbner basis of L, then \(L = \sum _{g\in G} \mathbb {W}g\).

Proof

1. (a):

   This is clear as the ideal LMI_≤(L) is generated by finitely many elements of the form LM_≤(g) with g ∈ L.

2. (b):

   Let G ⊆ L be a ≤-Gröbner basis of L and assume that \(\sum _{g\in G}\mathbb {W}g \subsetneq L\). As \(\leq \in \mathrm {WO}(\mathbb {E})\), we find some \(e \in L \setminus \sum _{g\in G}\mathbb {W}g\) such that

   $$\displaystyle \begin{aligned}\mathrm{LE}(e) = \min_\leq\{\mathrm{LE}_\leq(d) \mid d \in L \setminus \sum_{g\in G}\mathbb{W}g\}.\end{aligned}$$

   By Definition, Reminder and Exercise 1.12.6 (C)(b) we find some \(u \in \mathbb {E}\) and some g ∈ G such that

   $$\displaystyle \begin{aligned}\mathrm{LM}_\leq(e) = \mathrm{LM}_\leq(u)\mathrm{LM}_\leq(g).\end{aligned}$$

   Setting

   $$\displaystyle \begin{aligned}v := -\frac{\mathrm{LC}_\leq(e)}{\mathrm{LC}_\leq(g)}u\end{aligned}$$

   we now get on use of Proposition 1.12.4 (a) that

   $$\displaystyle \begin{aligned} \mathrm{LT}_\leq(e) &= \mathrm{LC}_\leq(e)\mathrm{LM}_\leq(e) = \mathrm{LC}_\leq(e)\mathrm{LM}_\leq(u)\mathrm{LM}_\leq(g) \\ &= \mathrm{LC}_\leq(e)\mathrm{LT}_\leq(u)\frac{1}{\mathrm{LC}_\leq(g)}\mathrm{LT}_\leq(g) =\frac{\mathrm{LC}_\leq(e)}{\mathrm{LC}_\leq(g)}\mathrm{LT}_\leq(u)\mathrm{LT}_\leq(g) \\ &=-\mathrm{LT}_\leq(v)\mathrm{LT}_\leq(g) = -\mathrm{LT}_\leq(vg). \end{aligned} $$

   As \(e \notin \sum _{g\in G}\mathbb {W}g\) and g ∈ G, we have

   $$\displaystyle \begin{aligned}e+vg \in L \setminus \sum_{g\in G}\mathbb{W}g.\end{aligned}$$

   In particular e + vg ≠ 0. So by Definition, Reminder and Exercise 1.12.2 (D)(d) it follows that

   $$\displaystyle \begin{aligned}\mathrm{LE}_\leq(e+vg) < \mathrm{LE}_\leq(e) = \min_\leq\{\mathrm{LE}_\leq(d) \mid d \in L \setminus \sum_{g\in G}\mathbb{W}g\}.\end{aligned}$$

   But this is a contradiction.

Now, we add the previously announced examples and exercises.

Examples and Exercises 1.12.8

1. (A)

   (Leading Monomial Ideals) Keep the above notations and hypotheses. Prove the following statements:

   1. (a)

      Let \(d \in \mathbb {W}\setminus \{0\}\) and \(\leq \in \mathrm {AO}(\mathbb {E})\). Prove that \(\mathrm {LMI}_{\leq }(\mathbb {W}d)\) is a principal ideal.

   2. (b)

      Let n = 1, X ₁ =: X and ∂ ₁ =: ∂. Set \(L := \mathbb {W}(X^2 - \partial ) + \mathbb {W}(X\partial )\) and determine LMI_≤(L) for ≤:=≤_lex, ≤_deglex and ≤:=≤_degrevlex.

2. (B)

   (Gröbner Bases) Keep the above notations and hypotheses. Prove the following statements:

   1. (a)

      Let the notations be as in exercise (a) of part (A) and prove that {cd} is a ≤-Gröbner basis of \(\mathbb {W}d\) for all c ∈ K ∖{0}, and that any singleton ≤-Gröbner bases of \(\mathbb {W}d\) is of the above form.

   2. (b)

      Let the notations and hypotheses be as in exercise (b) of part (A) and compute a ≤-Gröbner basis for ≤:=≤_lex, ≤_deglex and ≤:=≤_degrevlex

We now head for another basic result on Gröbner bases, which says that these bases enjoy a certain restriction property. This will be an important ingredient in our treatment of Universal Gröbner bases. We begin with the following preparations.

Notation 1.12.9

1. (A)

   For any set \(S \subseteq \mathbb {W}\) we write (see also Definition, Reminder and Exercise 1.12.2 (D)):

   $$\displaystyle \begin{aligned}\mathrm{supp}(S) := \bigcup_{s \in S} \mathrm{supp}(s) \mbox{ and } \mathrm{Supp}(S) := \bigcup_{s \in S} \mathrm{Supp}(s).\end{aligned} $$
2. (B)

   Let \(\leq \in \mathrm {TO}(\mathbb {E})\) (see Definition, Reminder and Exercise 1.12.2 (A)) and let \(T \subset \mathbb {E}\). We write ≤↾_T for the restriction of ≤ to T, thus—if we interpret binary relations on a set S as subsets of S × S:

   $$\displaystyle \begin{aligned}\leq \upharpoonright_T \quad := \quad \leq \cap (T \times T), \mbox{ so that }: d \leq \upharpoonright_T e \Leftrightarrow d \leq e \mbox{ for all } d,e \in T.\end{aligned} $$

Proposition 1.12.10 (The Restriction Property of Gröbner Bases)

Let \(L \subseteq \mathbb {W}\) be a left ideal. Let \(\leq , \leq ' \in \mathrm {AO}(\mathbb {E})\) and let G be a ≤-Gröbner basis of L. Assume that

Then G is also a ≤′-Gröbner basis of L.

Proof

Let d ∈ L ∖{0}. We have to show that \(\mathrm {LM}_{\leq '}(d) \in \mathrm {LMI}_{\leq '}(G)\). We may assume that 0∉G. If we apply Proposition 1.12.5 to the ordering ≤′, we find an element r and a family \((q_g)_{g \in G} \in \mathbb {W}^G\) such that

1. (1)

   d =∑_{g ∈ G} q _g g + r;

2. (2)

   \(\varPhi (s) \notin \mathbb {P}\mathrm {LM}_{\leq '}(g)\) for all g ∈ G and all s ∈Supp(r).

3. (3)

   \(\mathrm {LE}_{\leq '}(q_gg) \leq ' \mathrm {LE}_{\leq '}(d)\) for all g ∈ G with q _g ≠ 0.

Our immediate aim is to show that r = 0. Assume to the contrary that r ≠ 0. As r ∈ L and G is a ≤-Gröbner basis of L, we get LM_≤(r) ∈LMI_≤(G). So, there is some g ∈ G such that LM_≤(r) = mLM_≤(g) for some \(m \in \mathbb {M}\) (see Definition, Reminder and Exercise 1.12.6 (C)(c)). As ≤↾_Supp(G)  = ≤′↾_Supp(G) it follows that

As LT_≤(r) ∈Supp(r), this contradicts the above condition (2). Therefore r = 0.

But now, we may write

By the above condition (3) we have \(\mathrm {LE}_{\leq '}(q_gg) \leq ' \mathrm {LE}_{\leq '}(d)\) for all g ∈ G ^∗. So, there is some g ∈ G ^∗ such that \(\mathrm {LE}_{\leq '}(d) = \mathrm {LE}_{\leq '}(q_g g)\) (see Definition, Reminder and Exercise 1.12.2 (D)(d)), and hence \(\mathrm {LM}_{\leq '}(d) = \mathrm {LM}_{\leq '}(q_g g)\). Thus, on use of Proposition 1.12.4 (b) we get indeed

Now, we shall introduce the central concept of this section.

Definition 1.12.11 (Universal Gröbner Bases)

Let \(L \subseteq \mathbb {W}\) be a left ideal. A universal Gröbner basis of L is a (finite) subset \(G \subset \mathbb {W}\) which is a ≤-Gröbner basis for all \(\leq \in \mathrm {AO}(\mathbb {E})\).

Universal Gröbner bases have been studied by Sturmfels [41] in the polynomial ring K[X ₁, X ₂, …, X _n]—and indeed this notion can be immediately extended to the Weyl algebra \(\mathbb {W}.\) Gröbner bases for left ideals in the Weyl algebra were introduced by Assi, Castro-Jiménez and Granger [3] and also by Saito et al. [38].

Clearly, our next aim should be to show, that universal Gröbner bases always exist. There are indeed various possible ways to prove this. Here, we shall do this by a topological approach which relies on an idea of Sikora [40], and which can be found in greater generality in Boldini’s thesis [11]. We approach the subject by first introducing a natural metric on the set of total orderings of all elementary differential operators. Then, we make the reader prove in a series of exercises, that we get a complete metric space in this way.

Definition, Exercise and Convention 1.12.12

(A) (The Natural Metric on the Set \(\mathrm {TO}(\mathbb {E})\)) For all \(i \in \mathbb {Z}\) we introduce the notation

We define a map

Prove that

1. (a)

   For all \(\leq , \leq ' \in \mathrm {TO}(\mathbb {E})\) and all \(r \in \mathbb {N}_0\) we have

   $$\displaystyle \begin{aligned}\mathrm{dist}(\leq,\leq') < \frac{1}{2^r} \mbox{ if and only if } \leq\upharpoonright_{\mathbb{E}_{r+1}} \quad = \quad \leq'\upharpoonright_{\mathbb{E}_{r+1}}.\end{aligned}$$
2. (b)

   The map \(\mathrm {dist}: \mathrm {TO}(\mathbb {E}) \times \mathrm {TO}(\mathbb {E}) \longrightarrow \mathbb {R}\) is a metric on \(\mathrm {TO}(\mathbb {E})\).

From now on, we always endow \(\mathrm {TO}(\mathbb {E})\) with this metric and the induced Hausdorff topology.

(B) (Completeness of the Metric Space \(\mathrm {TO}(\mathbb {E})\) ) Let \((\leq _i)_{i \in \mathbb {N}_0}\) be a Cauchy sequence in \(\mathrm {TO}(\mathbb {E})\). This means:

We introduce the binary relation \(\leq \subseteq \mathbb {E} \times \mathbb {E}\) given for all \(d,e \in \mathbb {E}\) by

Prove the following statements:

1. (a)

   If \(r \in \mathbb {N}_0\), \(d,e \in \mathbb {E}_{r+1}\), and i, j ≥ n(r), then d ≤_i e if and only if d ≤_j e.

2. (b)

   If \(r \in \mathbb {N}_0\), \(d,e \in \mathbb {E}_{r+1}\), and i ≥ n(r), then d ≤_i e if and only if d ≤ e.

3. (c)

   \(\leq \in \mathrm {TO}(\mathbb {E})\).

4. (d)

   If \(r \in \mathbb {N}_0\), and i ≥ n(r), then \(\mathrm {dist}(\leq _i,\leq ) \leq \frac {1}{2^r}\).

5. (e)

   lim_i→∞≤_i =≤.

6. (f)

   \(\mathrm {TO}(\mathbb {E})\) is a complete metric space.

Now, we are ready to prove the basic ingredient of our existence proof for universal Gröbner bases.

Proposition 1.12.13 (Compactness of the Space of Total Orderings)

The space \(\mathrm {TO}(\mathbb {E})\) is compact.

Proof

Let \((\leq _i)_{i \in \mathbb {N}_0}\) be a sequence in \(\mathrm {TO}(\mathbb {E})\). It suffices to show, that \((\leq _i)_{i \in \mathbb {N}_0}\) has a convergent subsequence. Bearing in mind Definition, Exercise and Convention 1.12.12 (B)(f) (or (e)), it suffices to find a subsequence of \((\leq _i)_{i \in \mathbb {N}_0}\) which is a Cauchy sequence. Observe that all the sets \(\mathbb {E}_r\) are finite. We want to construct a sequence \((\mathbb {S}_r)_{r\in \mathbb {N}_0}\) of infinite subsets \(\mathbb {S}_r \subseteq \mathbb {N}_0\) such that for all \(s \in \mathbb {N}_0\) we have

1. (1)

   \(\mathbb {S}_{s+1} \subseteq \mathbb {S}_s\).

2. (2)

   \(\leq _j\upharpoonright _{\mathbb {E}_{s+1}}\quad = \quad \leq _k\upharpoonright _{\mathbb {E}_{s+1}}\) for all \(j,k \in \mathbb {S}_s\).

We construct the members \(\mathbb {S}_r\) of the sequence \((\mathbb {S}_r)_{r\in \mathbb {N}_0}\) by induction r. As \(\mathbb {E}_1\) is finite, we can find an infinite set \(\mathbb {S}_0 \subseteq \mathbb {N}_0\) such that requirement (2) is satisfied with s = 0. Now, let r > 0 and assume that the sets \(\mathbb {S}_0, \mathbb {S}_1,\ldots ,\mathbb {S}_r\) are already defined such that requirement (1) holds for all s < r and requirement (2) holds for all s ≤ r.

As \(\mathbb {E}_{r+2}\) is finite, we find an infinite subset \(\mathbb {S}_{r+1} \subseteq \mathbb {S}_r\) (which hence satisfies requirement (1) for s = r) such that requirement (2) is also satisfied with s = r + 1. This completes the step of induction and hence proves that a sequence \((\mathbb {S}_r)_{r\in \mathbb {N}_0}\) with the requested properties exists.

Now, we may choose a sequence \((i_k)_{k\in \mathbb {N}_0}\) in \(\mathbb {N}_0\), such that

In particular it follows that

and hence (see Definition, Exercise and Convention 1.12.12 (A)(a))

So, the constructed subsequence \((\leq _{i_k})_{k \in \mathbb {N}_0}\) of our original sequence \((\leq _i)_{i \in \mathbb {N}_0}\) is indeed a Cauchy sequence.

What we need indeed to prove our main result, is the compactness of subspace of admissible orderings in the topological space of total orderings.

Proposition 1.12.14 (Compactness of the Space of Admissible Orderings)

The set \(\mathrm {AO}(\mathbb {E})\) is a closed subset of \(\mathrm {TO}(\mathbb {E})\) and hence compact.

Proof

Let \((\leq _i)_{i \in \mathbb {N}_0}\) be sequence in \(\mathrm {AO}(\mathbb {E})\), which converges in \(\mathrm {TO}(\mathbb {E})\) and let

We aim to show, that \(\leq \in \mathrm {AO}(\mathbb {E})\). According to Definition, Reminder and Exercise 1.12.2 (C), we must show, that for all \( \underline {\lambda },\underline {\lambda }',\underline {\kappa },\underline {\kappa }',\underline {\nu }, \underline {\mu } \in \mathbb {N}_0^n\) the following statements hold.

1. (1)

   \(1 \leq\underline {X}^{ \underline {\nu }} \underline {\partial }^{ \underline {\mu }}\).

2. (2)

   If \( \underline {X}^{ \underline {\lambda }} \underline {\partial }^{ \underline {\kappa }} \leq\underline {X}^{ \underline {\lambda }'} \underline {\partial }^{ \underline {\kappa }'}\) then \( \underline {X}^{ \underline {\lambda } +\underline {\nu }} \underline {\partial }^{ \underline {\kappa }+ \underline {\mu }} \leq\underline {X}^{ \underline {\lambda }' +\underline {\nu }} \underline {\partial }^{ \underline {\kappa }'+ \underline {\mu }}\).

So, fix \( \underline {\lambda },\underline {\lambda }',\underline {\kappa },\underline {\kappa }',\underline {\nu }, \underline {\mu } \in \mathbb {N}_0^n\). Then we find some \(r \in \mathbb {N}_0\) such that all the elementary differential operators which occur in (1) and (2) belong to \(\mathbb {E}_{r+1}\). Now, we find some \(i \in \mathbb {N}_0\) such that \(\mathrm {dist}(\leq _i,\leq ) < \frac {1}{2^r}\), hence such that \(\leq \upharpoonright _{\mathbb {E}_{r+1}} \quad = \quad \leq _i\upharpoonright _{\mathbb {E}_{r+1}}\). As \(\leq _i \in \mathrm {AO}(\mathbb {E})\) the required inequalities hold for ≤_i. But then, by the coincidence of ≤ and ≤_i on \(\mathbb {E}_{r+1}\), they hold also for ≤.

Now, after having established the following auxiliary result, we are ready to prove the announced main result.

Lemma 1.12.15

Let \(L \subset \mathbb {W}\) be a left ideal and let G ⊆ L be a finite subset. Then, the set

is open in \(\mathrm {AO}(\mathbb {E})\).

Proof

We may assume that \(\mathbb {U}_L(G)\) is not empty and choose \(\leq \in \mathbb {U}_L(G)\). We find some \(r \in \mathbb {N}_0\) with \(\mathrm {supp}(G) \subseteq \mathbb {E}_{r+1}\). Let \(\leq ' \in \mathrm {AO}(\mathbb {E})\) such that \(\mathrm {dist}(\leq ,\leq ') < \frac {1}{2^r}\). So, we obtain that \(\leq \upharpoonright _{\mathbb {E}_{r+1}} \quad = \quad \leq '\upharpoonright _{\mathbb {E}_{r+1}}\) and hence in particular that ≤↾_Supp(G)  = ≤′↾_Supp(G). By Proposition 1.12.10 it follows that G is a ≤′-Gröbner basis of L and hence that \(\leq ' \in \mathbb {U}_L(G)\). But this means, that the open neighborhood

of ≤ belongs to \(\mathbb {U}_L(G)\).

Theorem 1.12.16 (Existence of Universal Gröbner Bases)

Each left ideal L of \(\mathbb {W}\) admits a universal Gröbner basis.

Proof

Let \(L \subseteq \mathbb {W}\) be a left ideal. For each ≤ ∈AO(E) we choose a ≤-Gröbner basis G _≤ of L. In the notations of Lemma 1.12.15 we have \(\leq \in \mathbb {U}_L(G_\leq )\). So, by this same Lemma the family

is an open covering of \(\mathrm {AO}(\mathbb {E})\). By Proposition 1.12.14 we thus find finitely many elements

such that

Let \(\leq \in \mathrm {AO}(\mathbb {E})\). Then \(\leq \in \mathbb {U}_L(G_{\leq _i})\) for some i ∈{1, 2, …, r}. Therefore \(G_{\leq _i}\) is a ≤-Gröbner basis of L. So \(\bigcup _{i=1}^rG_{\leq _i}\) is a Gröbner basis of L for all \(\leq \in \mathrm {AO}(\mathbb {E}).\)

As a first application of the previous existence result we get the following finiteness result.

Corollary 1.12.17 (Finiteness of the Set of Leading Monomial Ideals)

Let \(L \subseteq \mathbb {W}\) be a left ideal. Then the set

of all leading monomial ideals of L with respect to admissible orderings of \(\mathbb {E}\) is finite.

Proof

Let G ⊆ L be a universal Gröbner basis of L. Then we have

Therefore

This section is devoted to the study of admissible orderings which are compatible with a given weight and the related notion of weighted (admissible) ordering. Such weighted orderings were first studied by Assi, Castro-Jiménez and Granger [3] and by Saito et al. [38].

In relation to these weighted orderings, we shall introduce the fundamental notion of symbol of a differential operator with respect to a given weight. We will see, that these symbols, which are indeed polynomials, behave again multiplicatively. Moreover, we shall see that the symbols of all members of a Gröbner basis of a given left ideal generate the so-called induced ideal of the given left ideal. Our ultimate goal is to prove, that the number of characteristic varieties of given D-module with respect to all weights is finite. Moreover, we shall prove a certain stability result for characteristic varieties found in Boldini’s thesis [11], which is published in [12].

Notation 1.13.1

(A) As previously, we fix a positive integer n, a field K of characteristic 0 and consider the standard Weyl algebra

the polynomial ring

in the indeterminates Y ₁, Y ₂, …, Y _n, Z ₁, Z ₂, …, Z _n with coefficients in the field K and the isomorphism of K-vector spaces

(B) We also write

for the set of all weights. If

we also use the suffix \( \underline {\omega }\) instead of the suffix \( \underline {v} \underline {w}\) in all the previously introduced notations. So we write for example

Observe, that

where the arithmetic operations are performed in \(\mathbb {N}_0^{2n}\).

Now, we introduce the concept of admissible orderings which are compatible with a given weight.

Definition and Exercise 1.13.2

(A) (Weight Compatible Orderings) We fix a weight and an admissible ordering of the set \(\mathbb {E}\) of elementary differential operators in \(\mathbb {W}\) (see Definition, Reminder and Exercise 1.12.2 (C)):

We say that ≤ is compatible with the weight \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) (or \( \underline {\omega }\)-compatible), if for all \(d,e \in \mathbb {E}\) we have:

So, ≤ is compatible with \( \underline {\omega } = ( \underline {v}, \underline {w})\) if and only if for all \( \underline {\nu }, \underline {\mu }, \underline {\nu }', \underline {\mu }' \in \mathbb {N}_0^n\) we have the following implication:

We set

(B) (Weighted Admissible Orderings) Keep the notations and hypotheses of part (A). We define a new binary relation

on \(\mathbb {E}\), by setting, for all \(d,e \in \mathbb {E}\):

Prove that for each weight \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) and each \(\leq \in \mathrm {AO}(\mathbb {E})\) the following statements hold.

1. (a)

   \(\leq ^{ \underline {\omega }} \in \mathrm {AO}^{ \underline {\omega }}(\mathbb {E})\).

2. (b)

   \((\leq ^{ \underline {\omega }})^{ \underline {\omega }}\quad = \quad \leq ^{ \underline {\omega }}\).

3. (c)

   \(\leq \in \mathrm {AO}^{ \underline {\omega }}(\mathbb {E}) \mbox{ if and only if } \leq \quad = \quad \leq ^{ \underline {\omega }}\).

The admissible ordering \(\leq ^{ \underline {\omega }} \in \mathrm {AO}(\mathbb {E})\) is called the \( \underline {\omega }\) -weighted ordering associated to ≤.

Another important concept, which was already mentioned in the introduction to this section, is the notion of symbol of a differential operator. We now will introduce this notion after a few preparatory steps.

Definition and Exercise 1.13.3

(A) Let \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \), let \(i \in \mathbb {N}_0\) and let

We set

and

Prove that for all \(d, e \in \mathbb {W}\), all \(i,j \in \mathbb {N}_0\) and for all weights \( \underline {\omega } = ( \underline {v}, \underline {w})\in \varOmega \) the following statements hold:

1. (a)

   If \(i > \mathrm {deg}^{ \underline {\omega }}(d)\), then \(d^{ \underline {\omega }}_i = 0\).

2. (b)

   \(d^{ \underline {\omega }}_i =(d^{ \underline {\omega }}_i )^{ \underline {\omega }}_i\).

3. (c)

   \((d+e)^{ \underline {\omega }}_i = d^{ \underline {\omega }}_i + e^{ \underline {\omega }}_i\).

4. (d)

   If d, e ≠ 0, \(i := \mathrm {deg}^{ \underline {\omega }}(d)\) and \( j:= \mathrm {deg}^{ \underline {\omega }}(e)\), then

   $$\displaystyle \begin{aligned}\mathrm{supp}^{\underline{\omega}}_{i+j}(de) = \{(\underline{\nu} + \underline{\nu}',\underline{\mu} + \underline{\mu}') \mid (\underline{\nu},\underline{\mu}) \in \mathrm{supp}^{\underline{\omega}}_i(d) \mbox{ and }(\underline{\nu}',\underline{\mu}') \in \mathrm{supp}^{\underline{\omega}}_j(e)\}.\end{aligned}$$
5. (e)

   If d, e ≠ 0, \(i := \mathrm {deg}^{ \underline {\omega }}(d)\) and \( j:= \mathrm {deg}^{ \underline {\omega }}(e)\), then

   $$\displaystyle \begin{aligned}(de)^{\underline{\omega}}_{i+j} = \sum_{(\underline{\nu},\underline{\mu}) \in \mathrm{supp}^{\underline{\omega}}_i(d), (\underline{\nu}',\underline{\mu}') \in \mathrm{supp}^{\underline{\omega}}_j(e)} c^{(d)}_{\underline{\nu}\underline{\mu}}c^{(e)}_{\underline{\nu}'\underline{\mu}'}\underline{X}^{\underline{\nu} + \underline{\nu}'}\underline{\partial}^{\underline{\mu} + \underline{\mu}'}.\end{aligned}$$

(B) Keep the notations and hypotheses of part (A). We set

Prove on use of statements (a)–(e) of part (A) that for all \(d, e \in \mathbb {W}\), all \(i,j \in \mathbb {N}_0\) and for all weights \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) the following statements hold:

1. (a)

   \(\sigma ^{ \underline {\omega }}_i(d) := \sigma ^{ \underline {\omega }}_i(d^{ \underline {\omega }}_i)\).

2. (b)

   If \(i > \mathrm {deg}^{ \underline {\omega }}(d)\), then \(\sigma ^{ \underline {\omega }}_i(d) = 0\).

3. (c)

   \(\sigma ^{ \underline {\omega }}_i(d) = \sigma ^{ \underline {\omega }}_i(d^{ \underline {\omega }}_i)\).

4. (d)

   \(\sigma ^{ \underline {\omega }}_i(d + e) = \sigma ^{ \underline {\omega }}_i(d) + \sigma ^{ \underline {\omega }}_i(e)\).

(C) (The Symbol of a Differential operator with Respect to a Weight) Keep the notations of part (A), (B). We define the \( \underline {\omega } = ( \underline {v}, \underline {w})\)-symbol of the differential operator \(d \in \mathbb {W}\) by

Prove that for all \(d,e \in \mathbb {W} \setminus \{0\}\) the following statements hold.

1. (a)

   \(\sigma ^{ \underline {\omega }}(d) = \varPhi (d^{ \underline {\omega }}_{\mathrm {deg}^{ \underline {\omega }}(d)}) = \sigma ^{ \underline {\omega }}\big (d^{ \underline {\omega }}_{\mathrm {deg}^{ \underline {\omega }}(d)}(d)\big )\).

2. (b)

   \(\sigma ^{ \underline {\omega }}(d+e) = \begin {cases} \sigma ^{ \underline {\omega }}(d)+\sigma ^{ \underline {\omega }}(e) &\mbox{ if } \mathrm {deg}^{ \underline {\omega }}(d) = \mathrm {deg}^{ \underline {\omega }}(e) = \mathrm {deg}^{ \underline {\omega }}(d+e) \\ \sigma ^{ \underline {\omega }}(d) &\mbox{ if } \mathrm {deg}^{ \underline {\omega }}(d) > \mathrm {deg}^{ \underline {\omega }}(e).\end {cases}\)

First, we now prove that symbols behave well with respect to products of differential operators.

Proposition 1.13.4 (Multiplicativity of Symbols)

Let \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) and let \(d,e \in \mathbb {W}\) . Then

Proof

If d = 0 or e = 0, our claim is obvious. So, let d, e ≠ 0. We write \(i := \mathrm {deg}^{ \underline {\omega }}(d)\) and \(j:= \mathrm {deg}^{ \underline {\omega }}(e)\). Observe that \(\mathrm {deg}^{ \underline {\omega }}(de) = i+j\). So, by Definition and Exercise 1.13.3 (A)(e) we have

In Definition and Remark 1.11.5 we have seen, that each left ideal L of the standard Weyl algebra \(\mathbb {W}\) induces a graded ideal in the associated graded ring with respect to a given weight. These induced ideals will play a crucial role in our future considerations. We just revisit now these ideals.

Reminder, Definition and Exercise 1.13.5

(A) (Induced Graded Ideals) Let \(L \subset \mathbb {W}\) be a left ideal, let \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) be a weight and let us consider the \( \underline {\omega }\)-graded ideal (see Definition and Remark 1.11.5)

where

is the filtration induced on L by the weighted filtration \(\mathbb {W}^{ \underline {\omega }}_\bullet \). We now consider the \( \underline {\omega }\)-graded ideal of \(\mathbb {P}^{ \underline {\omega }} = \mathbb {P}\) given by

where

is the canonical isomorphism of graded rings of Theorem 1.9.4. We call \(\overline {\mathbb {G}}^{ \underline {\omega }}(L)\) the (\( \underline {\omega }\)-graded) ideal induced by L in \(\mathbb {P}\).

(B) Let the notations and hypotheses be as part (A). Fix \(i \in \mathbb {N}_0\) and consider the i-th \( \underline {\omega }\)-graded part

of the ideal \(\overline {\mathbb {G}}^{ \underline {\omega }}(L) \subseteq \mathbb {P}\). Prove the following statements:

1. (a)

   Let d ∈ L with \(\mathrm {deg}^{ \underline {\omega }}(d) = i\) and let \(\overline {d} := d+\mathbb {W}^{ \underline {\omega }}_{i-1} \in \mathbb {G}^{ \underline {\omega }}(L)_i\). Then it holds

   $$\displaystyle \begin{aligned}(\eta^{\underline{\omega}})^{-1}(\overline{d}) = \varPhi(d^{\underline{\omega}}_i) = \sigma^{\underline{\omega}}(d) \in \overline{\mathbb{G}}^{\underline{\omega}}(L)_i.\end{aligned} $$
2. (b)

   Each element \(h \in \mathbb {G}^{ \underline {\omega }}(L)_i \setminus \{0\}\) can be written as

   $$\displaystyle \begin{aligned}h = \sigma^{\underline{\omega}}(d), \mbox{ with } d \in L \mbox{ and } \mathrm{deg}^{\underline{\omega}}(d) = i.\end{aligned} $$

(C) (The Induced Exact Sequence Associated to a Left Ideal with Respect to a Weight) Keep the above notations and hypotheses. Prove the following statements:

1. (a)

   There is a short exact sequence of graded \(\mathbb {P}^{ \underline {\omega }}\)-modules

   $$\displaystyle \begin{aligned}0 \longrightarrow \overline{\mathbb{G}}^{\underline{\omega}}(L) \longrightarrow \mathbb{G}^{\underline{\omega}} \longrightarrow \mathrm{Gr}_{\mathbb{W}^{\underline{\omega}}_\bullet K\overline{1}}(\mathbb{W}/L) \longrightarrow 0,\end{aligned}$$

   where \(\overline {1} := 1 + L \in \mathbb {W}/L\) and \(\mathbb {W}^{ \underline {\omega }}_\bullet K\overline {1}\) is the \( \underline {\omega }\) -filtration induced on the cyclic D-module \(\mathbb {W}/L\) by its subspace \(K\overline {1}\).

2. (b)

   \(\mathrm {Ann}_{\mathbb {P}}\big (\mathrm {Gr}_{\mathbb {W}^{ \underline {\omega }}_\bullet K\overline {1}}(\mathbb {W}/L)\big ) = \overline {\mathbb {G}}^{ \underline {\omega }}(L)\).

3. (c)

   \(\mathbb {V}^{ \underline {\omega }}(\mathbb {W}/L) = \mathrm {Var}\big (\overline {\mathbb {G}}^{ \underline {\omega }}(L)\big ).\)

We call this sequence the short exact sequence associated to the left ideal L with respect to the weight \( \underline {\omega }\).

Now, we are ready to formulate and to prove a result which we already announced in the introduction to this section. It relates the symbols of the members of a Gröbner bases of a left ideal with the induced ideal with respect to a given weight.

Proposition 1.13.6 (Generation of the Induced Ideal by the Symbols of a Gröbner Basis)

Let \( \underline {\omega } \in \varOmega \) , let \(L \subseteq \mathbb {W}\) be a left ideal, let \(\leq \in \mathrm {AO}(\mathbb {E})\) and let G be a \(\leq ^{ \underline {\omega }}\) -Gröbner basis of L. Then it holds

1. (a)

   \(\overline {\mathbb {G}}^{ \underline {\omega }}(L) = \sum _{g \in G} \mathbb {P}\sigma ^{ \underline {\omega }}(g)\).

2. (b)

   For each \(h \in \overline {\mathbb {G}}^{ \underline {\omega }}(L) \setminus \{0\}\) there is some g ∈ G ∖{0}and some monomial \(m =\underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \in \mathbb {P}\) such that

   $$\displaystyle \begin{aligned}\mathrm{LM}_\leq\big(\varPhi^{-1}(h)\big) = m\mathrm{LM}_\leq\big(\varPhi^{-1}(\sigma^{\underline{\omega}}(g))\big).\end{aligned} $$

Proof

(a): As the ideal \(\overline {\mathbb {G}}^{ \underline {\omega }}(L) \subseteq \mathbb {P}^{ \underline {\omega }}\) is graded, it suffices to show, that for each \(i \in \mathbb {N}_0\) and each \(h \in \overline {\mathbb {G}}^{ \underline {\omega }}(L)_i\setminus \{0\}\) we have \(h \in \sum _{g \in G} \mathbb {P}\sigma ^{ \underline {\omega }}(g)\). So, fix \(i \in \mathbb {N}_0\) and assume that \(h \notin \sum _{g \in G} \mathbb {P}\sigma ^{ \underline {\omega }}(g)\) for some \(h \in \overline {\mathbb {G}}^{ \underline {\omega }}(L)_i\setminus \{0\}\). Then, by Reminder, Definition and Exercise 1.13.5 (B)(b), the set

is not empty. Choose \(d \in \mathfrak {S}\) such that

As G is a \(\leq ^{ \underline {\omega }}\)-Gröbner basis of L we find some g ∈ G and some \(u \in \mathbb {E}\) such that \(\mathrm {LM}_{\leq ^{ \underline {\omega }}}(d) = \mathrm {LM}_{\leq ^{ \underline {\omega }}}(ug)\) (see Definition, Reminder and Exercise 1.12.6 (C)(b)). With

it follows that \(\mathrm {LE}_{\leq ^{ \underline {\omega }}}(d) = \mathrm {LE}_{\leq ^{ \underline {\omega }}}(vg)\), hence

\(\mbox{ and } \mathrm {deg}^{ \underline {\omega }}(vg) = i.\) So, by Definition, Reminder and Exercise 1.12.2 (D)(d) we may conclude that either

1. (1)

   \(\mathrm {deg}^{ \underline {\omega }}(d-vg) < i\), or else

2. (2)

   \(\mathrm {deg}^{ \underline {\omega }}(d-vg) = i\) and \(\mathrm {LE}_{\leq ^{ \underline {\omega }}}(d-vg) < \mathrm {LE}_{\leq ^{ \underline {\omega }}}(d)\).

In the case (1) we have (see Definition and Exercise 1.13.3 (C)(b) and Proposition 1.13.4)

and hence get a contradiction.

So, assume that we are in the case (2). As d − vg ∈ L it follows by our choice of d, that \(\sigma ^{ \underline {\omega }}(d-vg) \in \sum _{g \in G} \mathbb {P}\sigma ^{ \underline {\omega }}(g)\). Observe that we have

So, by Definition and Exercise 1.13.3 (C)(b) and by Proposition 1.13.4 we have

and this is again a contradiction.

(b): We find some \(i \in \mathbb {N}_0\) such that \(\mathrm {LM}_\leq \big (\varPhi ^{-1}(h)\big ) = \mathrm {LM}_\leq \big (\varPhi ^{-1}(h^{ \underline {\omega }}_i(h))\big )\). As the ideal \(\overline {\mathbb {G}}^{ \underline {\omega }}(L) \subseteq \mathbb {P}^{ \underline {\omega }}\) is graded, we have \(h^{ \underline {\omega }}_i(h) \in \overline {\mathbb {G}}^{ \underline {\omega }}(L)\). So we may assume, that \(h \in \overline {\mathbb {G}}^{ \underline {\omega }}(L)_i\setminus \{0\}\). Now, by Reminder, Definition and Exercise 1.13.5 (B), we find some d ∈ L with \(\mathrm {deg}^{ \underline {\omega }}(d) = i\) and \(\varPhi ^{-1}(h) = d^{ \underline {\omega }}_i\), whence

As G is a \(\leq ^{ \underline {\omega }}\)-Gröbner basis of L, we find some g ∈ G ∖{0} with \(\mathrm {deg}^{ \underline {\omega }}(g) = j\) and some monomial \(m =\underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \in \mathbb {P}\) such that (see Definition, Reminder and Exercise 1.12.6 (C)(c) and also Definition and Exercise 1.13.3 (C)(a))

and so we get our claim.

Now, we are ready to prove our first basic finiteness result. It says that the set of all induced ideals of a given left ideal in the Weyl algebra is finite.

Corollary 1.13.7 (Finiteness of the Set of Induced Ideals)

Let \(L \subseteq \mathbb {W}\) be a left ideal. Then, the following statements hold:

1. (a)

   \(\#\{\overline {\mathbb {G}}^{ \underline {\omega }}(L) \mid\underline {\omega } \in \varOmega \} < \infty \).

2. (b)

   \(\#\{\mathbb {V}^{ \underline {\omega }}(\mathbb {W}/L) \mid\underline {\omega } \in \varOmega \} < \infty \).

Proof

(a): Let G be an universal Gröbner basis of L. Then, by Proposition 1.13.6, for each \( \underline {\omega } \in \varOmega \) we have \(\overline {\mathbb {G}}^{ \underline {\omega }}(L) = \sum _{g\in G} \mathbb {P}\sigma ^{ \underline {\omega }}(g)\). For each g ∈ G we write

Then, for each \( \underline {\omega } \in \varOmega \) we have

Therefore

It follows that

(b): This follows immediately from statement (a) on use of Reminder, Definition and Exercise 1.13.5 (C)(c).

The second statement of the previous result says that a given cyclic D-module has only finitely many characteristic varieties, if \( \underline {\omega }\) runs through all weights. Our first main theorem says, that this finiteness statement holds indeed for arbitrary D-modules. To prove this, we first have to investigate the behavior of characteristic varieties in short exact sequences of D-modules. This needs some preparations.

Exercise and Definition 1.13.8

(A) Let \( \underline {\omega } \in \varOmega \) and let

be an exact sequence of D-modules. Let V ⊆ U be a finitely generated K-vector subspace such that \(U = \mathbb {W}V\). We endow Q with the filtration

Prove the following statements:

1. (a)

   For each \(i \in \mathbb {N}_0\) there is a K-linear map

   $$\displaystyle \begin{aligned}\overline{\iota}_i:Q_i/Q_{i-1} \longrightarrow \mathbb{W}^{\underline{\omega}}_iV/\mathbb{W}^{\underline{\omega}}_{i-1}V, \quad q + Q_{i-1} \mapsto \iota(q) + \mathbb{W}^{\underline{\omega}}_{i-1}V.\end{aligned}$$
2. (b)

   For each \(i \in \mathbb {N}_0\) there is a K-linear map

   $$\displaystyle \begin{aligned}\overline{\pi}_i:\mathbb{W}^{\underline{\omega}}_iV/\mathbb{W}^{\underline{\omega}}_{i-1}V \longrightarrow \mathbb{W}^{\underline{\omega}}_i\pi(V)/\mathbb{W}^{\underline{\omega}}_{i-1}\pi(V), \quad q + \mathbb{W}^{\underline{\omega}}_{i-1}V \mapsto \pi(q) + \mathbb{W}^{\underline{\omega}}_{i-1}\pi(V).\end{aligned}$$
3. (c)

   For each \(i \in \mathbb {N}_0\) it holds

   $$\displaystyle \begin{aligned}\pi^{-1}\big(\mathbb{W}^{\underline{\omega}}_{i-1}\pi(V)\big) = \iota(Q) + \mathbb{W}^{\underline{\omega}}_{i-1}V.\end{aligned}$$
4. (d)

   For each \(i \in \mathbb {N}_0\) there is a short exact sequence of K-vector spaces

   $$\displaystyle \begin{aligned}0\longrightarrow Q_i/Q_{i-1} \stackrel{\overline{\iota}_i}{\longrightarrow} \mathbb{W}^{\underline{\omega}}_iV/\mathbb{W}^{\underline{\omega}}_{i-1}V \stackrel{\overline{\pi}_i}{\longrightarrow}\mathbb{W}^{\underline{\omega}}_i\pi(V) /\mathbb{W}^{\underline{\omega}}_{i-1}\pi(V)\longrightarrow 0.\end{aligned}$$

(B) (The Graded Exact Sequence associated to a Short Exact Sequence of D-Modules) Keep the hypotheses and notations of part (A). Prove the following statements:

1. (a)

   For each \(i \in \mathbb {N}_0\) there is a short exact sequence of K-vector spaces

   $$\displaystyle \begin{aligned}0\longrightarrow \mathrm{Gr}_{Q_\bullet}(Q)_i \stackrel{\overline{\iota}_i}{\longrightarrow} \mathrm{Gr}_{\mathbb{W}^{\underline{\omega}}_\bullet V}(U)_i \stackrel{\overline{\pi}_i}{\longrightarrow}\mathrm{Gr}_{\mathbb{W}^{\underline{\omega}}_\bullet\pi(V)}(P)_i\longrightarrow 0.\end{aligned}$$
2. (b)

   There is an exact sequence of graded \(\mathbb {P}^{ \underline {\omega }}\)-modules

   $$\displaystyle \begin{aligned}0\longrightarrow \mathrm{Gr}_{Q_\bullet}(Q)\stackrel{\overline{\iota}}{\longrightarrow} \mathrm{Gr}_{\mathbb{W}^{\underline{\omega}}_\bullet V}(U) \stackrel{\overline{\pi}}{\longrightarrow} \mathrm{Gr}_{\mathbb{W}^{\underline{\omega}}_\bullet\pi(V)}(P)\longrightarrow 0,\end{aligned}$$

   with \(\overline {\iota } := \bigoplus _{i \in \mathbb {N}_0}\overline {\iota }_i\) and \(\overline {\pi }:= \bigoplus _{i\in \mathbb {N}_0}\overline {\pi }_i\).

The exact sequence of statement (b) is called the exact sequence induced by the exact sequence \(0\rightarrow Q\stackrel {\iota }{\rightarrow } U\stackrel {\pi }{\rightarrow } P \rightarrow 0\) and the generating vector space V  of U.

(C) Keep the previous notations and hypotheses. Prove the following statements:

1. (a)

   For each finitely generated K-vector subspace T ⊆ Q with \(Q = \mathbb {W}T\) and V ⊆ ι(T), the two filtrations Q _• and \(\mathbb {W}^{ \underline {\omega }}_\bullet T\) of Q are equivalent.

2. (b)

   \(\mathrm {Var}\big (\mathrm {Ann}_{\mathbb {P}}(\mathrm {Gr}_{Q_\bullet }(Q))\big ) = \mathbb {V}^{ \underline {\omega }}(Q)\).

Now, we can prove the crucial result, needed to extend the previous finiteness statement for characteristic varieties from cyclic to arbitrary D-modules.

Proposition 1.13.9 (Additivity of Characteristic Varieties)

Let \( \underline {\omega } \in \varOmega \) and let

be an exact sequence of D-modules. Then it holds

Proof

We fix a finitely generated K-vector subspace V ⊆ U with \(\mathbb {W}V = U\) and consider the corresponding induced short exact sequence (see Exercise and Definition 1.13.8 (B))

On use of Exercise and Definition 1.13.8 (C)(b) we obtain

Now, we are ready to prove the announced first main theorem of this section.

Theorem 1.13.10 (Finiteness of the Set of Characteristic Varieties)

Let U be a D-module. Then

Proof

We proceed by induction on the number r of generators of U. If r = 1 we have \(U \cong \mathbb {W}/L\) for some left ideal \(L \subseteq \mathbb {W}\). In this case, we may conclude by Corollary 1.13.7 (b). So, let r > 1. Then, we find a short exact of D-modules

such that Q and P are generated by less than r elements. By induction, we have

By Proposition 1.13.9 we also have

hence

As already announced in the introduction to this section, our ultimate goal is to establish a certain stability result for characteristic varieties of a given D-module. To pave the way for this, we perform a number of preparatory considerations, which are the subject of the exercises to come.

Definition and Exercise 1.13.11

(A) (Leading Forms) We consider the polynomial ring \(\mathbb {P}\). Let

We set

and consider the i − th homogeneous component of f with respect to ω, thus the polynomial

The leading form of f with respect to the weight \( \underline {\omega }\) is defined by

Prove that for all \(f, g \in \mathbb {P}\), all \(i,j \in \mathbb {N}_0\) and for all weights \( \underline {\omega } = ( \underline {v}, \underline {w}) \in \varOmega \) the following statements hold:

1. (a)

   If \(i > \mathrm {deg}^{ \underline {\omega }}(f)\), then \(f^{ \underline {\omega }}_i = 0\).

2. (b)

   \(f^{ \underline {\omega }}_i = f^{ \underline {\omega }}_i(f^{ \underline {\omega }}_i)\).

3. (c)

   \((f+g)^{ \underline {\omega }}_i = f^{ \underline {\omega }}_i + g^{ \underline {\omega }}_i\).

4. (d)

   \((fg)^{ \underline {\omega }}_i = \sum _{j+k=i} f^{ \underline {\omega }}_jg^{ \underline {\omega }}_k.\)

5. (e)

   \(\mathrm {LF}^{ \underline {\omega }}(fg) = \mathrm {LF}^{ \underline {\omega }}(f)\mathrm {LF}^{ \underline {\omega }}(g)\).

6. (f)

   LF(f) = f if and only if f is homogeneous with respect to the \( \underline {\omega }\)-grading of \(\mathbb {P}\).

7. (g)

   If \(d \in \mathbb {W}\), then \(\sigma ^{ \underline {\omega }}(d) = \mathrm {LF}^{ \underline {\omega }}\big (\varPhi (d)\big )\).

(B) (Leading Form Ideals) Keep the notations and hypotheses of part (A). If \(S \subset \mathbb {P}\) is any subset, we define the leading form ideal of S with respect to \( \underline {\omega }\) by

Let \(S \subseteq T \subseteq \mathbb {P}\) and \(\leq \in \mathrm {AO}_(\mathbb {E})\). Prove the following statements:

1. (a)

   \(\mathrm {LFI}^{ \underline {\omega }}(S) \subseteq \mathrm {LFI}^{ \underline {\omega }}(T)\).

2. (b)

   If for each t ∈ T ∖{0} there is some monomial \(m =\underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \in \mathbb {M} \subset \mathbb {P}\) and some s ∈ S such that \(\mathrm {LM}_{\leq ^{ \underline {\omega }}}\big (\varPhi ^{-1}(t)\big ) = m\mathrm {LM}_{\leq ^{ \underline {\omega }}}\big (\varPhi ^{-1}(s)\big )\), then \(\mathrm {LFI}^{ \underline {\omega }}(S) = \mathrm {LFI}^{ \underline {\omega }}(T)\).

3. (c)

   For each ideal \(I \subseteq \mathbb {P}\) it holds

   $$\displaystyle \begin{aligned}\sqrt{\mathrm{LFI}^{\underline{\omega}}(I)} = \sqrt{\mathrm{LFI}^{\underline{\omega}}(\sqrt{I})}.\end{aligned} $$
4. (d)

   If \(I, J \subseteq \mathbb {P}\) are ideals, then

   1. (1)

      \(\mathrm {LFI}^{ \underline {\omega }}(I\cap J) \subseteq \mathrm {LFI}^{ \underline {\omega }}(I) \cap \mathrm {LFI}^{ \underline {\omega }}(I)\) and \(\mathrm {LFI}^{ \underline {\omega }}(I)\mathrm {LFI}^{ \underline {\omega }}(J) \subseteq \mathrm {LFI}^{ \underline {\omega }}(IJ)\);

   2. (2)

      \(\sqrt {\mathrm {LFI}^{ \underline {\omega }}(I \cap J)} = \sqrt {\mathrm {LFI}^{ \underline {\omega }}(I) \cap \mathrm {LFI}^{ \underline {\omega }}(J)} = \sqrt {\mathrm {LFI}^{ \underline {\omega }}(I)} \cap \sqrt {\mathrm {LFI}^{ \underline {\omega }}(J)}\).

The announced Stability Theorem for Characteristic Varieties we are heading for, concerns the behavior of characteristic varieties under certain changes of the involved weights. To prepare this new type of considerations, we suggest the following exercise.

Exercise 1.13.12

(A) Prove that for all \(d \in \mathbb {W}\), all \(i,j \in \mathbb {N}_0\), all \(s \in \mathbb {N}\) and for all weights \( \underline {\alpha } = ( \underline {a}, \underline {b}), \underline {\omega } = ( \underline {v}, \underline {w})\in \varOmega \) the following statements hold (For the unexplained notations see Definition and Exercise 1.13.3):

1. (a)

   \(\mathrm {supp}\big ([d^{ \underline {\omega }}_i]^{ \underline {\alpha }}_j\big ) = \mathrm {supp}^{ \underline {\omega }}_i(d) \bigcap \mathrm {supp}^{ \underline {\alpha }}_j(d)\).

2. (b)

   \(\mathrm {supp}\big ([d^{ \underline {\omega }}_i]^{ \underline {\alpha }}_j\big ) \subseteq \mathrm {supp}^{ \underline {\alpha }+s \underline {\omega }}_{j+si}(d)\).

3. (c)

   If \(i \geq \mathrm {deg}^{ \underline {\omega }}(d)\), \(j \geq \mathrm {deg}^{ \underline {\alpha }}(d^{ \underline {\omega }}_i)\) and \(s > \mathrm {deg}^{ \underline {\alpha }}(d) - j\), then the inclusion of statement (b) becomes an equality.

4. (d)

   If \(i \geq \mathrm {deg}^{ \underline {\omega }}(d)\), \(j \geq \mathrm {deg}^{ \underline {\alpha }}\big (d^{ \underline {\omega }}_i)\) and \(s > \mathrm {deg}^{ \underline {\alpha }}(d) - j\), then

   $$\displaystyle \begin{aligned}{}[d^{\underline{\omega}}_i]^{\underline{\alpha}}_j = d^{\underline{\alpha}+s\underline{\omega}}_{j+si}.\end{aligned}$$

(B) Prove on use of statements (a)–(d) of part (A) that for all \(d \in \mathbb {W}\), all \(i,j \in \mathbb {N}_0\), all \(s \in \mathbb {N}\) and for all weights \( \underline {\omega } = ( \underline {v}, \underline {w}),\underline {\alpha } = ( \underline {a}, \underline {b}) \in \varOmega \) the following statements hold:

1. (a)

   \(\sigma ^{ \underline {\alpha }}_j(d^{ \underline {\omega }}_i) = \sum _{( \underline {\nu }, \underline {\mu }) \in \mathrm {supp}^{ \underline {\omega }}_i(d) \cap \mathrm {supp}^{ \underline {\alpha }}_j(d)} c^{(d)}_{ \underline {\nu } \underline {\mu }} \underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} = \sigma ^{ \underline {\omega }}_i(d^{ \underline {\alpha }}_j)\).

2. (b)

   If \(i \geq \mathrm {deg}^{ \underline {\omega }}(d)\), \(j \geq \mathrm {deg}^{ \underline {\alpha }}(d^{ \underline {\omega }}_i)\) and \(s > \mathrm {deg}^{ \underline {\alpha }}(d) - j\), then

   $$\displaystyle \begin{aligned}{}[\sigma^{\underline{\omega}}_i(d)]^{\underline{\alpha}}_j = \sigma^{\underline{\alpha}+s\underline{\omega}}_{j+si}(d).\end{aligned}$$

The next two auxiliary results are of fairly technical nature. But they will play a crucial role in the proof of our Stability Theorem.

Lemma 1.13.13

Let \( \underline {\alpha },\underline {\omega } \in \varOmega \) , let \(d \in \mathbb {W} \setminus \{0\}\) and let \(s \in \mathbb {N}\) such that

Then, the following statements hold:

1. (a)

   \(\mathrm {deg}^{ \underline {\alpha } + s\underline {\omega }}(d) = \mathrm {deg}^{ \underline {\alpha }}\big (\sigma ^{ \underline {\omega }}(d)\big ) + s\mathrm {deg}^{ \underline {\omega }}(d)\).

2. (b)

   \(\mathrm {LF}^{ \underline {\alpha }}\big (\sigma ^{ \underline {\omega }}(d)\big ) = \sigma ^{ \underline {\alpha }+s \underline {\omega }}(d)\).

Proof

We write

Observe, that \(\sigma ^{ \underline {\omega }}(d) = \sigma ^{ \underline {\omega }}_i(d) = \varPhi (d^{ \underline {\omega }}_i)\), so that

Now, by Exercise 1.13.12 (B)(b) we obtain

It remains to show that

As \(\mathrm {LF}^{ \underline {\alpha }}\big (\sigma ^{ \underline {\omega }}(d)\big )\neq 0\) we have \(\sigma ^{ \underline {\alpha }+s \underline {\omega }}_{j+si}(d) \neq 0\) and hence \(j+si \leq \mathrm {deg}^{ \underline {\alpha } + s\underline {\omega }}(d)\) (see Definition and Exercise 1.13.3 (B)(b)).

Assume that \(j+si > \mathrm {deg}^{ \underline {\alpha } + s\underline {\omega }}(d)\). Then, we may write \(\mathrm {deg}^{ \underline {\alpha } + s\underline {\omega }}(d) = k+si\), with k > j. It follows, that \(s > \mathrm {deg}^{ \underline {\alpha }}(d) - k\). On application of Exercise 1.13.12 (B)(b) we get that

As \(k > j = \mathrm {deg}^{ \underline {\alpha }}\big (\sigma ^{ \underline {\omega }}(d)\big )\) we have \([\sigma ^{ \underline {\omega }}_i(d)]^{ \underline {\alpha }}_k = 0\) (see Definition and Exercise 1.13.11 (A)(a)). This contradiction completes our proof.

Lemma 1.13.14

Let \(L \subseteq \mathbb {W}\) be a left ideal, let \( \underline {\alpha },\underline {\omega } \in \varOmega \) , let \(\leq \in \mathrm {AO}(\mathbb {E})\) and let G be a \((\leq ^{ \underline {\alpha }})^{ \underline {\omega }}\) -Gröbner basis of L. Then

Proof

By Reminder, Definition and Exercise 1.13.5 (B)(a) we have

If we apply Proposition 1.13.6 (b) with \(\leq ^{ \underline {\alpha }}\) instead of ≤, we see that for all t ∈ T there is some monomial \(m =\underline {Y}^{ \underline {\nu }} \underline {Z}^{ \underline {\mu }} \in \mathbb {M} \subset \mathbb {P}\) and some s ∈ S such that \(\mathrm {LM}_{\leq ^{ \underline {\alpha }}}\big (\varPhi ^{-1}(t)\big ) = m\mathrm {LM}_{\leq ^{ \underline {\alpha }}}\big (\varPhi ^{-1}(s)\big )\). By Definition and Exercise 1.13.11 (B)(b) it follows that

Now, we are ready to formulate and to prove the announced stability result.

Theorem 1.13.15 (Stability of Induced Graded Ideals, Boldini [11, 12])

Let \(L \subseteq \mathbb {W}\) be a left ideal and let \( \underline {\alpha } \in \varOmega \) . Then, there exists an integer \(\overline {s} = \overline {s}( \underline {\alpha },L) \in \mathbb {N}_0\) such that for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \) we have

Proof

Let G be a universal Gröbner basis of L. Then, by Lemma 1.13.14, for each \( \underline {\omega } \in \varOmega \) we have

Now, we set

By Lemma 1.13.13 it follows that \(\mathrm {LF}^{ \underline {\alpha }}\big (\sigma ^{ \underline {\omega }}(g)\big ) = \sigma ^{ \underline {\alpha }+s \underline {\omega }}(g)\) for all \(s \in \mathbb {N}\) with \(s > \overline {s},\) all \( \underline {\omega } \in \varOmega \) and all g ∈ G ∖{0}. So, for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \) we have

If we apply Proposition 1.13.6 (a) with \( \underline {\alpha } + s \underline {\omega }\) instead of \( \underline {\omega }\) we also get

for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \). This completes our proof.

Notation 1.13.16

If \(\mathfrak {Z} \subseteq \mathrm {Spec}(\mathbb {P})\) is a closed set we denote the vanishing ideal of \(\mathfrak {Z}\) by \(I_{\mathfrak {Z}}\), thus:

Theorem 1.13.17 (Stability of Characteristic Varieties, Boldini [11, 12])

Let U be a D-module, and let \( \underline {\alpha } \in \varOmega \) . Then, there exists an integer \(\overline {s} = \overline {s}( \underline {\alpha },U) \in \mathbb {N}_0\) such that for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \) we have

Proof

We proceed by induction on the number r of generators of U. First, let r = 1. Then we have \(U \cong \mathbb {W}/L\) for some left ideal \(L \subseteq \mathbb {W}\). By Theorem 1.13.15 we find some \(\overline {s} \in \mathbb {N}_0\) such that for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \) we have

By Reminder, Definition and Exercise 1.13.5 (C)(c) we have

By Definition and Exercise 1.13.11 (B)(c) we thus get

so that indeed—for all \(s \in \mathbb {N}\) with \(s > \overline {s}\) and all \( \underline {\omega } \in \varOmega \)—we have

Now, let r > 1. Then, we find a short exact of D-modules

such that Q and P are generated by less than r elements. By induction, we thus find a number \(\overline {s} \in \mathbb {N}_0\), such that for all \( \underline {\omega } \in \varOmega \) and all \(s \in \mathbb {N}\) with \(s > \overline {s}\) it holds

By Proposition 1.13.9 we have

and hence, moreover

By Definition and Exercise 1.13.11 (B)(d)(2) it follows from the last equality that

Therefore

It follows, that

for all \( \underline {\omega } \in \varOmega \) and all \(s \in \mathbb {N}\) with \(s > \overline {s}\). This completes the step of induction and hence proves our claim.

To formulate our Stability Theorem in a more geometric manner, we introduce the following notion.

Definition 1.13.18 (The Critical Cone)

Let \(\mathfrak {Z} \subseteq \mathrm {Spec}(\mathbb {P})\) be a closed set. Then, the critical cone of \(\mathfrak {Z}\) is defined as

where \( \underline {1} = ( \underline {1}, \underline {1}) \in \varOmega \) denotes the standard weight.

On use of the introduced terminology, we now can define our Stability Theorem as follows.

Corollary 1.13.19 (Affine Deformation of Characteristic Varieties to Critical Cones, Boldini [11, 12])

Let U be a D-module. Then, there is an integer \(\overline {s} = \overline {s}(U) \in \mathbb {N}_0\) such that for all \( \underline {\omega } \in \varOmega \) and all \(s \in \mathbb {N}\) with \(s > \overline {s}\) it holds

Proof

This is immediate by Theorem 1.13.17.