Critical cones of characteristic varieties

We show that certain characteristic varieties of a finitely generated module over a given Weyl algebra arising from weighted degree filtrations are equal to the critical cone of some other characteristic varieties. This behaviour of the characteristic varieties permits us to introduce a new invariant of the module. As a second consequence we are able to provide an easy and non-homological proof that the characteristic varieties of a module arising from weights in the natural polynomial region of the Weyl algebra all have the same Krull and Gelfand-Kirillov dimension, equal to the Gelfand-Kirillov dimension of the module. Third we give an upper bound for the number of distinct characteristic varieties of a cyclic module in terms of degrees of elements in universal Groebner bases and the above results allow us to conjecture a further upper bound.


Introduction
Let n ∈ N, let W be the n th Weyl algebra over a field K of characteristic 0, and let Ω = {ω ∈ N 2n 0 | ω i + ω i+n > 0 for 1 ≤ i ≤ n}. For each ω ∈ Ω consider the ω-degree filtration F ω W = (F ω i W ) i∈Z of W and any good F ω W -filtration F ω M = (F ω i M ) i∈Z of a left W -module M . We construct G ω W = i∈Z F ω i W/F ω i−1 W and G ω M = i∈Z F ω i M/F ω i−1 M . Then G ω W is a ring canonically isomorphic to the commutative polynomial ring K[X, Y ] in the indeterminates X 1 , . . . , X n and Y 1 , . . . , Y n , and G ω M is a finitely generated K[X, Y ]-module. For a fixed ω ∈ Ω, the radical ideal √ (0 : G ω M ) of K[X, Y ] is independent of the choice of a good F ω W -filtration Similarly, we consider the ν-degree filtrations F ν K[X, Y ] of K[X, Y ], ν ∈ N 2n 0 , and good F ν K[X, Y ]-filtrations F ν N of K[X, Y ]-modules N and construct the rings G ν K[X, Y ], canonically isomorphic to K[X, Y ], and the finitely generated K[X, Y ]modules G ν N . Again, for a fixed ν ∈ N 2n 0 , the radical ideal does not depend on the choice of a good F ν K[X, Y ]-filtration F ν N of N . The main result of this paper is that for each ν ∈ N 2n 0 there exists s 0 ∈ N 0 such that for all s ∈ N with s > s 0 and all ω ∈ Ω in K[X, Y ] it holds Observe that s 0 does not depend on ω. We can choose the lowest such s 0 in N 0 , denoted κ ν (M ). If L is a left ideal of W , we give an upper bound for κ ν (W/L) in terms of total degrees of elements of universal Gröbner bases of L, more precisely, where γ ν (L) = inf U sup u∈U deg ν (u), the infimum being taken over all universal Gröbner bases U of L.
A case with evident geometrical meaning is when ν = (1) = (1, . . . , 1) ∈ N 2n 0 . The equality (A) says that the "affine deformations" V (1)+sω (M ) of V ω (M ) stabilize for large s to the critical cone C ω (M ) = Var(0 : G (1) G ω M ) of V ω (M ). Thus the minimal limit beyond which this occurs, namely, κ(M ) = κ (1) (M ), is -surprisinglyan invariant of M . Upper bounds for the greatest total degree of Gröbner bases and of reduced Gröbner bases of a left ideal L of W are given in [1] in terms of greatest total degrees of systems of generators of L, and hence, combining both results, we obtain an upper bound for κ(W/L) also in such terms.
The critical cone C of an affine variety V ⊆ A r over an algebraically closed field F is the cone with vertex at the origin O ∈ A r tangent to V at infinity. In other words, C consists of all lines through O along whose directions V goes to infinity. To construct C, we choose an injection ι : A r P r of A r into the projective space P r over F and put where ι(V ) is the projective closure of ι(V ) in P r and ℓ P is the projective line through the points ι(O) and P . One has that C does not depend on the choice of ι. Algebraically, if I is any ideal of F [Z 1 , . . . , Z r ] that defines V , then C is defined by the ideal J generated by the homogeneous components of greatest total degree of the polynomials in I, that is, J is the leading form ideal of I by total degree. Again, C does not depend on the choice of I.
As a further consequence of the equality (A), we are able to give an easy proof that Kdim K[X,Y ] G ω M = GKdim W M for all ω ∈ Ω. Thus, without having to appeal to sophisticated homological methods as in classical proofs, we have shown in particular that the characteristic varieties V ω (M ), ω ∈ Ω, all have the same Krull dimension. The key point is that (A) allows in some sense to pass from non-finite to finite filtrations, and Gelfand-Kirillov dimension behaves well with finite discrete filtrations: GKdim G ω W G ω M = GKdim W M whenever F ω M is finite and discrete. The second point is that Gelfand-Kirillov dimension and Krull dimension agree in the category of finitely generated modules over any fixed finitely generated algebra over a field.
Fixed a left ideal L of W , we give an upper bound for the number χ(L) of distinct ideals G ω L, ω ∈ Ω, and hence of distinct ω-characteristic varieties of W/L, namely, , the infimum being taken over all universal Gröbner bases of L. The equality (A) let us conjecture a second upper bound in the case when W is the 1 st Weyl algebra, namely, where γ(L) = γ (1) (L). As mentioned afore, by [1] it follows an upper bound for γ(L) in terms of total degrees of generators of L.
In Section 1 we recall some known facts about filtered rings and modules as well as their associated graded rings and modules.
In Section 2 we introduce Weyl algebras and state some of their basic properties, which are a generalization of results that can be found for instance in [9]. The proofs remain very similar, and we omit them here.
Section 3 is about Gröbner bases in Weyl algebras. Here, too, we recall known facts, important in the next section, in particular the existence of universal Gröbner bases for left ideals, and a very tight relation between the Gröbner bases of ω-filtered left ideals and the Gröbner bases of their associated graded ideals.
In Section 4 we define ω-characteristic varieties of a left W -module M as some particular affine spectra, and not as algebraic zero sets, as it is usual, for there is no reason here to work only over algebraically closed fields. Then we prove our main result (A) about the defining annihilators of such varieties.
In section 5 we apply (A) to give an easy proof of the known result that the ωcharacteristic varieties of M all have the same Gelfand-Kirillov and Krull dimension as ω varies in Ω, namely, equal to the Gelfand-Kirillov dimension of M .
Finally in Section 6 we perform a computer experiment in order to try to classify the ω-characteristic varieties of M in the case when M = W/L for a left ideal L of W . This experiment let us conjecture an upper bound for their number, namely (D).

Filtrations and Gradings
In this section we give a small review on filtered rings and modules and their associated graded objects. Most of this material can be found or inferred from the books of Constantin Nȃstȃsescu, Freddy van Oystaeyen, and Huishi Li, among which we particularly appreciate [14]. Besides giving the very short proof of 1.26, we provide a proof of 1.28 and 1.29, too, which we did not find in the literature.
If the ring R is provided with a filtration R, we say that the ordered pair (R, R) is a filtered ring.
Let (R, R) and (S, S) be filtered rings. A homomorphism of (R, R) in (S, S) is a ring homomorphism φ of R in S such that φ(F i R) ⊆ F i S.
If the left R-module M is provided with an R-filtration M, we say that the ordered pair (M, M) is an R-filtered left R-module or simply a left (R, R)-module. Observe that a filtered ring is also a filtered left module over itself.
Let (M, M) and (N, N ) be left (R, R)-modules. An (R, R)-homomorphism of (M, M)  Definition 1.4. Let (R, R) be a filtered ring. The associated graded ring GR of R with respect to R is the commutative group i∈Z F i R/F i−1 R provided with a multiplication given by (r i + F i−1 R) i∈Z (s j + F j−1 R) j∈Z = ( i+j=k r i s j + F k−1 R) k∈Z , which indeed turns GR into a ring.
Let (M, M) be a left (R, R)-module. The associated graded left GR-module GM of M with respect to M is the commutative group i∈Z F i M/F i−1 M with a GRaction defined by (r i +F i−1 R) i∈Z (m j +F j−1 M) j∈Z = ( i+j=k r i m j +F k−1 M) k∈Z , which indeed turns GM into a left GR-module.
GR is precisely the associated graded left GR-module of R with respect to R. We denote the i th homogeneous component F i M/F i−1 M of GM by G i M. Then G 0 R is a subring of GR and each G i M is a left G 0 R-submodule of GM. Remark 1.5. Let (R, R) be a filtered ring, (X, X ) and (Y, Y) be left (R, R)modules, and φ be a homomorphism of (X, X ) in (Y, Y). We have canonical F 0 R- ∈ I, a contradiction. Therefore, under the assumption that M is discrete, we have the implication N M ⇒ GN GM, the property of strict monotony of G for discrete filtrations. Remark 1.7. Let (R, R) be a filtered ring. Assume that R is commutative, that is, r ∈ F i R ∧ s ∈ F j R ⇒ rs − sr ∈ F i+j−1 R. Then the ring GR is commutative. In this situation let (I, I) be a left ideal of (R, R) and consider the quotient module (R/I, R/I) of (R, R). Then GI = (0 : GR/I) as ideals of GR by 1.5. We convene that  For instance consider the commutative polynomial ring R = C[X] provided with the filtration R given by as rings and GM ∼ = C[X] as C[X]-modules. In view of these isomorphisms we can Remark 1.12. The converse of 1.11 is partially true. If M is discrete and U ⊆ M is such that σ M (U ) generates GM over GR, then U generates M over R.
Remark 1.13. Let (R, R) be a filtered ring. We can provide the graded ring GR with its filtration GR induced by the grading given by F i GR = j≤i G j R. Then we construct the graded ring GGR associated to the filtered ring (GR, GR). Since for each i one has a left module isomorphism F i R ∼ = F i GR over the isomorphic rings F 0 R ∼ = F 0 GR, there exists a graded ring isomorphism GR ∼ = GGR.
In a similar manner, if (M, M) is a left (R, R)-module, we find an isomorphism GM ∼ = GGM of graded left modules over the isomorphic graded rings GR ∼ = GGR, where GM is the filtration of GM given by F i GM = j≤i G j M.
Remark 1.15. In the notation of 1.14, any good R-filtration M of M is discrete as R is discrete by definition. Then (m k ) k∈N is said to be an M-Cauchy sequence if for each j ∈ Z there exists n j ∈ N such that for all k, l ≥ n j it holds m k − m l ∈ F j M. And If every M-Cauchy sequence of elements of M is M-convergent, then M is said to be complete.
This defines indeed an equivalence relation among the R-filtrations of M .   Remark 1.29. We finish this section with a remark that will be useful later on. Let R be a commutative ring and R be a filtration of R, so that R trivially is commutative. Let I be an ideal of R and provide I with its induced R-filtration, denoted I, and provide √ I with its induced R-filtration, denoted √ I. Then √ G √ I = √ GI.
Indeed let x ∈ G √ I be a homogeneous element of degree i ∈ Z. So x = x + F i−1 R for some x ∈ F i √ I = F i R ∩ √ I. We find k ∈ N such that x k ∈ I, and so x k ∈ F ki R ∩ I = F ki I, thus x k = x k + F ki−1 R ∈ GI, hence x ∈ √ GI. We have shown that G √ I ⊆ √ GI. On the other hand, by 1.6, we have GI ⊆ G √ I. Passing to the radicals, the claim follows.

Weyl Algebras
In this section let n ∈ N and K be a field of characteristic 0. We write K[X, Y ] for the commutative polynomial ring K[X 1 , . . . , X n , Y 1 , . . . , Y n ] and denote its subring For all (r, s) ∈ N 0 × N 0 we write (r | s) for the vector ω ∈ N 2n 0 with ω i = r and ω n+i = s for 1 ≤ i ≤ n. For all α, β ∈ N n 0 we write (α | β) for the vector ω ∈ N 2n 0 with ω i = α i and ω n+i = β i for 1 ≤ i ≤ n. For all t ∈ N and all α, β ∈ N t 0 we denote the sum We introduce Weyl algebras over K and state some facts about them. In doing this, we generalize certain well known results that are proved for instance in [9]; the here missing proofs of 2.4 and 2.9 are elementary but tedious computations and can be mimicked word by word from [9].
Then F ω W is a filtration of W by 2.4. We denote by G ω W the associated graded ring of W with respect to F ω W , and by G ω i W the i th homogeneous component of G ω W .
Given any ω-filtration F ω W -filtration F ω M = (F ω i M ) i∈Z of a left W -module M , we denote by G ω M the associated graded left G ω W -module associated to M with respect to F ω M , and by G ω i M the i th homogeneous component of G ω M . We write σ ω for the symbol map W → G ω W , and σ ω i for the i th symbol map Remark 2.7. Let ω ∈ Ω and v, w ∈ W . As deg ω (uv) = deg ω (u) + deg ω (v) by 2.4, it holds σ ω (uv) = σ ω (u)σ ω (v).
Remark 2.8. For all ω ∈ Ω the filtration F ω W of W is commutative by 2.4, so that the ring G ω W is commutative. Remarks 2.7 and 2.8, the canonical injection K G ω W , and the universal property of commutative polynomial rings imply the following theorem. Theorem 2.9. For each ω ∈ Ω one has an isomorphism of commutative K-algebras which is graded if we put deg(X i ) = ω i and deg(Y i ) = ω n+i for all 1 ≤ i ≤ n. Remark 2.10. By 2.9, 1.12, and 2.4, the Weyl algebras are left noetherian domains. Remark 2.11. All what we have defined and said in this section about Weyl algebras can be done and proved in the same way for the commutative polynomial ring K[X, Y ], too. In this situation we may even drop the hypothesis that the field be of characteristic 0 and may consider whole N 2n 0 instead of Ω. We shall use a similar notation as introduced above for Weyl algebras, with one exception: given any ν ∈ N 2n 0 , we shall write τ ν i for the i th symbol map , in order to distinguish them from the symbol maps of the n th Weyl algebra.

Gröbner Bases in Weyl Algebras
In this section we remind the notion of universal Gröbner bases in Weyl algebras and state their existence. The proof of this fact can be found in [5] and [6]; see also [18]. In [17] the same statement is proved for commutative polynomial rings; a similar proof exists for Weyl algebras.
We keep the notation of the previous section, and denote by M the canonical consisting of the monomials X λ Y µ , and by N the canonical K-basis {ξ λ ∂ µ | (λ, µ) ∈ N n 0 × N n 0 } of W consisting of the normal monomials ξ λ ∂ µ .
For each ω ∈ Ω we shall tacitly identify the ring G ω W with K[X, Y ] by means of the K-algebra isomorphism ψ ω of 2.9 and hence for each left ideal L consider G ω L as an ideal of K[X, Y ]. Similarly for each ν ∈ N 2n 0 we shall identify G ν K[X, Y ] with K[X, Y ] and thus for each ideal I of K[X, Y ] consider G ν I as an ideal of K[X, Y ].
we write lm (w) for the greatest normal monomial in the canonical form of w with respect to . We denote Φ(lm (w)) by LM (w). Given L ⊆ W , we often denote by LM (L) the ideal we define LM (p) and LM (I) similarly.

Characteristic Varieties over Weyl Algebras
We encounter the notion of characteristic variety and critical cone and prove our main result, from which a relation between characteristic varieties and critical cones follows. We keep the notation of the previous section.
Main Theorem 4.9. Let M be a finitely generated left W -module. For all ν ∈ N 2n 0 there exists s ν ∈ N 0 with the property that for all ω ∈ Ω and all s ∈ N with s > s ν it holds Proof. We fix any ν ∈ N 2n 0 . We find r ∈ N such that M is generated over R by r of its elements.
First, by induction over r, we prove the existence of s ν ∈ N 0 such that for all ω ∈ Ω and all s ∈ N with s > s ν it holds If r = 1, then M ∼ = W/L for a left ideal L of W . By 1.5, 1.7, 4.8 we find s ν ∈ N 0 such that for all ω ∈ Ω and all s ∈ N with s > s ν it holds √ (0 : If r > 1, we find a cyclic submodule N of M such that P = M/N is generated by r − 1 elements. As before, by 4.8 we find s ′ ν ∈ N 0 such that for all ω ∈ Ω and all s ∈ N with s > s ′ ν it holds √ (0 : G ν G ω N ) = √ (0 : G ν+sω N ). By induction we find s ′′ ν ∈ N 0 such that √ (0 : G ν G ω P ) = √ (0 : G ν+sω P ) for all ω ∈ Ω and all s ∈ N with s > s ′′ ν . By 1.5 we get for all ω ∈ Ω and all s ∈ N with s > s ν , where s ν = max {s ′ ν , s ′′ ν }, so that s ν is independent of ω. This completes the induction step. Now, by 1.28, 1.29, 1.13, it follows  Proof. Clear by 4.9.
Corollary 4.11. It holds Proof. The first statement is clear by 4.10, the second follows from 1.13.

Application 1: Dimension of Characteristic Varieties
In this section, as an application of Theorem 4.9, we aim to furnish a new proof of a classical result: fixed a finitely generated left W -module M , the characteristic varieties V ω (M ), ω ∈ Ω, all have the same Krull dimension. This is usually proved, as exposed by Ehlers in [7, Chapter V], by not trivial homological methods. It turns out indeed that Our proof descends (1) from the equality of annihilators obtained in 4.9, which in particular allows to pass in a certain sense from non-finite to finite filtrations, (2) from the preservation of the Gelfand-Kirillov dimension when passing from finitely filtered objects to their associated graded objects, see 5.5, and (3) from the equality of Krull and Gelfand-Kirillov dimension in the category of noetherian modules over a noetherian commutative K-algebra, see 5.2.
We begin with some necessary results about the Gelfand-Kirillov dimension that can be found in [12] or [15].
Reminder 5.1. Let F be a field and B be a finitely generated F -algebra. We find a generating space of B, that is, an F -module V of finite length such that F ⊆ V and B is generated as an F -algebra by V . By V i , i ∈ N 0 , we denote the Fmodule consisting of all polynomials in the (in general not commuting) elements of V with coefficients in F of total degree less than or equal to i, so that in particular , and it is independent of V . If A is any F -algebra, we define GKdim A = sup B GKdim B, where the supremum is taken over all finitely generated F -subalgebras B of A. For finitely generated F -algebras the two definitions are easily shown to be equivalent.
Let N be a finitely generated left B-module. We find a generating space of N , that is, an F -module W of finite length such that N is generated as a Bmodule by W . The Gelfand-Kirillov dimension of N is defined as GKdim B N = lim i→∞ log i (len F V i W ) ∈ [0, ∞], and it is independent of V and of W . If M is any A-module, we define GKdim A M = sup B sup N GKdim B N , where the suprema are taken over all finitely generated F -subalgebras B of A and all finitely generated Bsubmodules of M . For finitely generated modules over finitely generated F -algebras the two definitions are easily shown to be equivalent. Indeed, in our hypotheses both dimensions are exact, see [12,Theorem 6.14] for the Gelfand-Kirillov dimension, and hence we may assume that M = A/I for some ideal I. As both dimensions are preserved when changing the base ring from A Remark 6.1. Let n ∈ N. For each finitely generated left module M over the n th Weyl algebra over K and for each ν ∈ N 2n 0 there exists a minimal number κ ν (M ) ∈ N 0 such that for all ω ∈ Ω the characteristic varieties V ν+sω (M ) stabilize to Var(0 : G ν G ω M ) as soon as s > κ ν (M ).
In particular, V (1 | 1)+sω (M ) becomes precisely the critical cone C ω (M ) for all ω ∈ Ω as soon as s > κ(M ) = κ (1 | 1) (M ). Remark 6.2. Let n ∈ N. For each left ideal L of the n th Weyl algebra over K and for each ν ∈ N 2n 0 we put γ ν (L) = inf U sup u∈U {0} deg ν (u), where the infimum is taken over all universal Gröbner bases U of L. By the proof of 4.8, (a) κ ν (W/L) ≤ γ ν (L) ∈ N 0 . Clearly, (b) γ ν ′ (L) ≤ γ ν ′′ (L) whenever |ν ′ | ≤ |ν ′′ |. Finally, (c) γ kν (L) = kγ ν (L) for all k ∈ N 0 . For some numbers s 0 ∈ N 0 we repeatedly do an experiment parametrized by s 0 as follows. A computer calculates for us the intersection points among the half-lines ℓ ν,ω ⊆ Ω of the form ℓ ν,ω (s) = ν +sω, ν ∈ N 2 0 , ω ∈ Ω, for s > s 0 , and paints incident half-lines by a common colour. The points of Ω having got the same colour turn out to build cones in Ω. For instance, for s 0 = 3 the computer program painted 17 differently coloured cones, among which 9 are degenerate, that is, half-lines. For typographical reasons, in Figure 1 we depict the so obtained cones by connected regions in R 2 , alternately in black and gray. For s 0 = 3 the 9 degenerate cones are filled in black, whereas the 8 non-degenerate cones are filled in gray, and similarly in the other pictures of Figure 1.
By 4.8, as soon as s 0 ≥ γ(L), each of these cones is a subset of precisely one equivalence class of Ω/ ∼L . Thus the results of our experiment let us conjecture an upper bound for χ(L) in terms of γ(L), namely, χ(L) ≤ 2 1+γ(L) + 1.
Question 6.5. We may ask whether similar upper bounds for χ(L) as in 6.3 exist when considering a left ideal L of the n th Weyl algebra for n > 1, namely: (1) a bound in terms of n and γ(L), and (2) a bound in terms of Fibonacci numbers.