Kashiwara theorem for hyperbolic algebras.

Valery Lunts, Alexander Rosenberg

Abstract. The class of hyperbolic algebras contains Weyl algebras and their quantized versions, the quantized and classical enveloping algebras of sl(2) and many others. A purpose of this work is to obtain an analog of the Kashiwara theorem for hyperbolic algebras and to study Gelfand-Kirillov dimension of modules over them. As an application, we establish Bernstein’s inequality for the GK dimension of modules over certain hyperbolic algebras.

Contents:

Introduction

1. Hyperbolic algebras.

2. Kashiwara theorem.

3. Examples. The GK dimension of modules of the category 𝔒.

4. The GK dimension of modules over a hyperbolic ring.

5. Bernstein’s inequality. Examples.

Complement: Kashiwara theorem for hyperbolic categories.

Introduction

Let X, Y be smooth algebraic varieties and ι : Y → X be a closed embedding. An important theorem by Kashiwara says that the embedding ι induces an equivalence of the category M(Y) of D-modules on Y and the full subcategory M_{Y} (X) of M(X) formed by D-modules on X with support in Y. Being of local nature, this theorem is equivalent to a certain statement on modules over Weyl algebras which we shall also call Kashiwara theorem. One of the purposes of this work is to extend the Kashiwara theorem from Weyl algebras to a much larger class of so called hyperbolic algebras.

Let R be an associative algebra, θ : R → R an algebra automorphism, ξ a central element of R. To this data there corresponds an algebra R{θ, ξ} generated by R and the elements x, y subject to the following relations:

xy = ξ, yx = θ^{−1}(ξ); xr = θ(r)x, ry = yθ(r) for every r ∈ R

Due to the first two relations, R{θ, ξ} is called a hyperbolic algebra of rank one over R (it is called otherwise a generalized Weyl algebra [Bav]). A hyperbolic algebra of arbitrary finite rank over R is obtained by the iteration of this construction.

Hyperbolic algebras are particularly convenient for studying representations (see [R], Chapters 2 and 4). On the other hand, by pure luck, quite a few algebras of interest (such as all Weyl algebras and their quantized versions, the quantized and classical enveloping algebras of sl(2) and many others) are hyperbolic.

The first section contains examples of hyperbolic algebras.

In Section 2 we introduce an analog of the category O for hyperbolic algebras and prove the corresponding version of the Kashiwara theorem.

In Section 3 we consider some examples (those of Section 1) and apply Kashiwara theorem to estimate the Gelfand-Kirillov (GK) dimension of modules of the category O.

In Section 4 we estimate the GK dimension for arbitrary modules over a hyperbolic algebra. The main tool is the description of the spectrum of hyperbolic algebras obtained in [R1], Ch.4 and Ch.5. For readers’ convenience, we include a short discussion of the spectrum.

In Section 5, we deduce from results of Section 4 the Bernstein’s inequality when it is available.

Finally in the complementary section we discuss the categorical Kashiwara theorem. Authors are grateful to Max-Planck Institute fur Mathematik for hospitality and ex-

cellent working conditions.

1. Hyperbolic algebras.

1.1. Definition. Fix a commutative ring k. Let R by an associative k-algebra, θ automorphism of R; and let ξ be a central element of R. The hyperbolic algebra over R determined by the automorphism θ and the element ξ is the algebra R{θ, ξ} generated by R and the elements x, y subject to the following relations:

xr = θ(r)x, ry = yθ(r) for every r ∈ R  (1)
xy = ξ, yx = θ^{−1}(ξ)  (2)

Let ℜ be an associative k-algebra with unit, and let R be a subalgebra of ℜ. We say that ℜ is a hyperbolic algebra over R, or an iterated hyperbolic algebra over R, if there is an increasing filtration {R_{m}|m ≥ 0} of ℜ such that each R_{m} is a subalgebra of ℜ, R_{0} = R, ∑_{m≥0} R_{m} = ℜ, and, for any m ≥ 1, R_{m+1} is a hyperbolic algebra over R_{m}, i.e. R_{m+1} = R_{m}{θ_{m}, ξ_{m}} = R_{m}{θ_{m}, ξ_{m}; x_{m+1}, y_{m+1}} for some automorphism θ_{m} and a central element ξ_{m} of the algebra R_{m} and elements x_{m+1}, y_{m+1} of R_{m+1}. If ℜ = R_{ν}, then the number ν will be called the rank of the algebra ℜ over R. For simplicity in this work we will always assume that all ξ_{i} are elements of R.

1.1.1. Lemma. The hyperbolic algebra R{θ, ξ} of rank one over R is a free R-module with the basis formed by the monomials x^{m}, y^{n}.

Proof. This is an easy exercise. See also Lemma II.3.1.6 in [R1].

1.1.2. Corollary. Let ℜ be a hyperbolic algebra with the hyperbolic structrure R_{m+1} = R_{m}{θ_{m}, ξ_{m}; x_{m+1}, y_{m+1}}, m ≥ 1. An element r ∈ R is central iff it belongs to the center of R and θ_{m}(r) = r for all m.

Proof. The conditions are sufficient, because if θ_{m}(r) = r for all m, the element r commutes with x_{m+1} and y_{m+1} for all m. It follows from Lemma 1.1.1 that they are necessary.

1.1.3 Remark. The hyperbolic algebra R{θ, ξ} admits a Fourier transform. Namely, we replace the hyperbolic data {x, y, ξ, θ} by the data {y, x, θ^{−1}(ξ), θ^{−1}}.

1.1.4. Remark. It follows from the definition of a hyperbolic algebra and the Corollary 1.1.2 that θ_{m}(ξ_{j}) = ξ_{j} for m < j.

1.2. Some examples of hyperbolic algebras.

1.2.1. A general example. Fix a commutative ring k. Let R be an associative k-algebra with unit, {ϑ_{i} | i ∈ J} a family of pairwise commuting k-algebra automorphisms of R. Let {ξ_{i} | i ∈ J} be a family of central elements of R such that ϑ_{i}(ξ_{j}) = ξ_{j} for any i, j ∈ J, i ≠ j. Denote the data {ϑ_{i}|i ∈ J} by Θ, and {ξ_{i} | i ∈ J} by ξ. Let q = (q_{ij})_{i,j∈J} be a matrix with entrees in k^{∗} such that 1 = q_{ij}q_{ji} for all i, j ∈ J, i ≠ j. Let R{Θ, ξ} be the k-algebra generated by R and by x_{i}, y_{i}, i ∈ J, satisfying the relations:

x_{i}r = ϑ_{i}(r)x_{i}, ry_{i} = y_{i}ϑ_{i}(r) for any r ∈ R  (1)
x_{i}y_{i} = ξ_{i}, y_{i}x_{i} = ϑ_{i}^{−1}(ξ_{i}) for any i ∈ J  (2)
x_{i}y_{j} = q_{ji}y_{j}x_{i}, x_{i}x_{j} = q_{ij}x_{j}x_{i}, y_{i}y_{j} = q_{ij}y_{j}y_{i}  (3)

for all i, j ∈ J such that i ≠ j.

For any m ∈ J, denote by R_{m} the algebra generated by R and {x_{i}, y_{j}|i, j ≤ m}. We define an automorphism θ_{m} of the algebra R_{m} as an extension of ϑ_{m} by setting θ_{m}(x_{i}) = q_{mi}x_{i}, θ_{m}(y_{i}) = q_{im}y_{i}. It follows that R_{m+1} is a hyperbolic algebra over R_{m} defined by the automorphism θ_{m} and the element ξ_{m}. Clearly the union of R_{m} coincides with the algebra R{Θ, ξ}.

1.2.1.1. Remark. The special case of the algebras of Example 1.2.1, when q_{ij} = 1 for all i, j ∈ J, was introduced independently by Bavula [Bav] (who called them generalized Weyl algebras) at the end of 80s, approximately at the same time when hyperbolic algebras were introduced by the second author of this work. Generalized Weyl algebras appeared also in the work by Smith and Bell [BS].

1.2.2. The algebra of q-differential operators D_{q}. This is the simplest particular case of Example 1.2.1: J = 1, R = k. In other words, the algebra D_{q} is generated by x, y subject to the relation

xy − qyx = 1.

1.2.3. The Weyl algebras. Recall that the n-th Weyl algebra, A_{n}, over a field k is generated by the set of elements {x_{i}, y_{i} | 1 ≥ i ≥ n} subject to the relations:

x_{i}y_{i} − y_{i}x_{i} = 1 for all i ∈ J;
x_{i}y_{j} = y_{j}x_{i}, x_{i}x_{j} = x_{j}x_{i}, y_{i}y_{j} = y_{j}y_{i} for all i, j ∈ J, i ≠ j

Let R = k[ξ_{1}, ..., ξ_{n}] be the algebra of polynomials in n variables. And let θ_{i} denote the automorphism of R determined by θ_{i}(ξ_{j}) = ξ_{j} + δ_{ij}. The map sending x_{i} into x_{i} and y_{i} into y_{i} defines an isomorphism from the Weyl algebra A_{n} onto the hyperbolic algebra R{Θ, ξ}, Θ := (θ_{i}), ξ := (ξ_{i}).

1.2.4. The quantized Heisenberg and Hayashi’s Weyl algebras. Recall that the quantized Heisenberg algebra, H_{q}(J), over a field k is generated by the set of elements {x_{i}, y_{i}, z_{i} | i ∈ J} subject to the relations:

x_{i}z_{i} = qz_{i}x_{i}, z_{i}y_{i} = qy_{i}z_{i},
x_{i}y_{i} − q^{−1}y_{i}x_{i} = z_{i}
x_{i}y_{j} = y_{j}x_{i}, x_{i}x_{j} = x_{j}x_{i}, y_{i}y_{j} = y_{j}y_{i}, z_{j}z_{i} = z_{i}z_{j}

for all i, j ∈ J, i ≠ j.

T.Hayashi [Ha] defined a quantized Weyl algebra W_{q}(J) as a quotient algebra of the algebra H_{q}(J) obtained by adding the relations

(x_{i}y_{i} − qy_{i}x_{i})z_{i} = 1 = z_{i}(x_{i}y_{i} − qy_{i}x_{i})

for all i ∈ J. Clearly W_{1}(J) is the conventional Weyl algebra.

In H_{q}(J) we have the relation (x_{i}y_{i})z_{j} = z_{j}(x_{i}y_{i}) for all i, j ∈ J, and, moreover, the morphism of the algebra k[(z_{i}), (t_{i})] of polynomials in variables z_{i}, t_{i}, i ∈ J, to the quantized Heisenberg algebra H_{q}(J) which sends z_{i} into z_{i} and t_{i} into the product x_{i}y_{i} is injective. This implies that the algebra H_{q}(J) is generated by the commutative (polynomial) algebra R := k[(z_{i}), (t_{i})] and the elements x_{i}, y_{i}, i ∈ J, subject to the relations:

x_{i}r = θ_{i}(r)x_{i}, ry_{i} = y_{i}θ_{i}(r)
x_{i}y_{i} = t_{i}, y_{i}x_{i} = q(t_{i} − z_{i})
x_{i}y_{j} = y_{j}x_{i}

for all for all r ∈ R and i, j ∈ J, where j ≠ i.

Here θ_{i} is an automorphism of the algebra R defined by

θ_{i}(z_{i}) = qz_{i}, θ_{i}(t_{i}) = q^{−1}t_{i} + qz_{i},

and

θ_{i}(z_{j}) = z_{j}, θ_{i}(t_{j}) = t_{j} if j ≠ i

Note that θ_{i}^{−1}(t_{i}) = q(t_{i} − z_{i}); i.e. y_{i}x_{i} = θ_{i}^{−1}(t_{i}). These relations show that the quantized Heisenberg algebra is hyperbolic.

Now let R be the quotient algebra of the polynomial algebra k[(z_{i}), (t_{i})] by the relations

z_{i}(t_{i}(1 − q^{2}) + q^{2}z_{i}) = 1, i ∈ J.

The automorphisms θ_{i}, i ∈ J are defined by the same formulas as for the Heisenberg quantized algebra H_{q}(J); i.e.

θ_{i}(z_{i}) = qz_{i}, θ_{i}(t_{i}) = q^{−1}t_{i} + qz_{i} for all i ∈ J
θ_{i}(z_{j}) = z_{j}, θ_{i}(t_{j}) = t_{j} if j ≠ i

The morphism of the hyperbolic algebra R{Θ, t} determined by the data (Θ, t) = {θ_{i}, t_{i} | i ∈ J} to the Hayashi’s Weyl algebra which sends z_{i} into z_{i} and t_{i} into x_{i}y_{i}, i ∈ J, is an isomorphism.

1.2.5. The enveloping algebra of sl(2). Recall that the enveloping algebra of sl(2), U(sl(2)), is generated by x, y, z subject to the relations

xz = zx + αx, yz = zy − αy, xy − yx = z  (1)

where α ≠ 0.

Let R = k[z, ξ] and let the automorphism θ be determined by the equalities: θ(z) = z + α, θ(ξ) = ξ + z + α. Then the algebra U(sl(2)) defined by the relations (1) is naturally isomorphic to the hyperbolic algebra R{θ, ξ}.

1.2.6. Example: the quantum enveloping algebra of sl(2, k). The quantum enveloping algebra of the Lie algebra sl(2), U_{q}(sl(2)) is defined by the relations:

xz = qzx, zy = qyz; xy − yx = \frac{z − z^{−1}}{q − q^{−1}}  (1)

where q is an element of k − {0, 1}.

Let R = k[z, z^{−1}, ξ] and let the automorphism θ be determined by the equalities: θ(z) = qz, θ(ξ) = ξ + u(qz), where u(z) = \frac{z−z^{−1}}{q−q^{−1}} . Then the algebra U_{q}(sl(2)) defined by the relations (1) is naturally isomorphic to the hyperbolic algebra R{θ, ξ}.

1.2.7. Example: the coordinate algebra of SL_{q}(2, k). The coordinate algebra A(SL_{q}(2, k)) of the algebraic quantum group SL_{q}(2, k) (cf. [M]) is the k-algebra generated by the indeterminates x, y, u, v satisfying the equations:

qux = xu, qvx = xv, qyu = uy, qyv = vy, uv = vu,  (1)
xy − quv = 1 = yx − q^{−1}uv  (2)

Let R be the algebra k[u, v] of polynomials in u, v. Set θf(u, v) := f(qu, qv) for any polynomial f(u, v) and denote by ξ the element 1+quv. Then the relations (1), (2) become equivalent to the relations determining the algebra R{θ, ξ}.

1.3. The algebra of differential operators on a quantum space. Fix a commutative ring k. Let q = (q_{ij})_{i,j∈J} be a matrix with entries in k^{∗} satisfying the relations 1 = q_{ij}q_{ji} for all i, j ∈ J such that i ≠ j. A skew polynomial k-algebra is the k-algebra S_{q} generated by indeterminates x_{i}, i ∈ J, subject to the relations:

x_{i}x_{j} = q_{ij}x_{j}x_{i} , i ≠ j.  (1)

The algebra S_{q} is regarded as the algebra of functions on a ’quantum space’ (more specifically, a q-space).

The algebra D_{q}(S_{q}) of q-differential operators on S_{q} is the subalgebra of End_{k}(S_{q}) generated by x_{i}, ∂_{j}, i, j ∈ J. Here by x_{i} we mean the endomorphism of multiplication by x_{i}; and ∂_{j} is the partial q-derivation of R determined by

∂_{i}(k) = {0}, ∂_{i}(x_{j}r) = δ_{ij}r + q_{ji}x_{j}∂_{i}(r)  (2)

for all r ∈ S_{q}. The relations between different x_{i}, i ∈ J, are given by (1). The relations between x_{i} and ∂_{j}, i, j ∈ J:

∂_{i}x_{j} − q_{ji}x_{j}∂_{i} = δ_{ij} for all i, j ∈ J.  (3)

The relations between ∂_{i}, i ∈ J, look as follows:

∂_{i}∂_{j} = q_{ij}∂_{j}∂_{i}, for all i ≠ j.  (4)

Thus, D_{q}(R) is generated by x_{i}, ∂_{j}, i, j ∈ J, subject to the relations (1)–(4).

1.3.1. The structure of a hyperbolic algebra on D_{q}(S_{q}). For any i ∈ J, set ξ_{i} = ∂_{i}x_{i}. It follows from the relations above that ξ_{i}ξ_{j} = ξ_{j}ξ_{i} for all i, j ∈ J. Denote by R the algebra k[(ξ_{i})] of polynomials in ξ_{i}, i ∈ J. Define automorphisms θ_{i}, i ∈ J, of the algebra R by the formulas:

θ_{i}(ξ_{j}) = ξ_{j} if i ≠ j; θ_{i}(ξ_{i}) = q_{ii}ξ_{i} + 1  (1)

Then we can regard the algebra D_{q} of q-differential operators on S_{q} as a k-algebra generated by R and elements x_{i}, ∂_{i} subject to the relations:

x_{i}x_{j} = q_{ij}x_{j}x_{i}, ∂_{i}∂_{j} = q_{ij}∂_{j}∂_{i} ∂_{i}x_{j} = q_{ji}x_{j}∂_{i}  (2)

for all i, j ∈ J, i ≠ j;

∂_{i}x_{i} = ξ_{i}, x_{i}∂_{i} = θ_{i}^{−1}(ξ_{i})  (3)
∂_{i}r = θ_{i}(r)∂_{i}, rx_{i} = x_{i}θ_{i}(r)  (4)

for all i ∈ J and r ∈ R.

The relations (2), (3), (4) show that the algebra D_{q} is a particular case of Example 1.2.1.

1.4. The Quantum Weyl algebra A_{q}. This algebra appeared naturally in the context of studying differential operators on ”quantum spaces” (cf. [LR1]). It can be obtained from the algebra D_{q} of Example 1.3 as follows.

Note that there is a natural action of the group Γ := ℤ^{J} on the algebra S_{q} of qpolynomials (cf. 1.3): the canonical generator z_{i} of Γ sends x_{j} into (q_{ii}δ_{ij} + 1 − δ_{ij})x_{j}. This way we define a group homomorphism ϕ : Γ → Aut(S_{q}). By definition, the quantum Weyl algebra A_{q} is the subalgebra of End_{k}(S_{q}) generated by the algebra D_{q} (of 1.3) and the image of the group Γ in End_{k}(S_{q}).

1.5. The algebra A′_{q}. The action of Γ on S_{q} extends naturally to an action of Γ on the algebra of differential operators D_{q}: for any i, j ∈ J, the canonical generator z_{i} of Γ sends x_{j} to (q_{ii}δ_{ij} + 1 − δ_{ij})x_{j} and ∂_{j} to (q_{ii}^{−1}δ_{ij} + 1 − δ_{ij})∂_{j}. This defines a group homomorphism ϕ : Γ → Aut(D_{q}). Denote by A′_{q} the corresponding crossed product of D_{q} and Γ: A′_{q} = D_{q}#Γ. It follows from the definition that A′_{q} is generated by the algebra D_{q} and elements (z_{i}, z_{i}^{−1}|i ∈ J) satisfying the relations:

z_{i}x_{j} = b_{ij}x_{j}z_{i}, z_{i}∂_{j} = b_{ij}^{−1}∂_{j}z_{i}; z_{i}z_{j} = z_{j}z_{i}  (1)

for all i, j ∈ J. Here b_{ij} := q_{ii}δ_{ij} + (1 − δ_{ij}).

1.5.1. Remark. The algebra A_{q} is naturally a homomorphic image of A′_{q}.

1.5.2. The hyperbolic structure of A′_{q}. We take R = k[(ξ_{i})_{i∈J}] ⊗_{k} k[(z_{i})_{i∈J}] and define the automorphrphisms θ_{i} : R → R, i ∈ J, by

θ_{i}(ξ_{j}) = b_{ij}ξ_{j} + δ_{ij}, θ_{i}(z_{j}) = b_{ij}z_{j}  (1)

where b_{ij} := q_{ii}δ_{ij} + (1 − δ_{ij}). Then the algebra A′_{q} is generated by R, x_{i}, and ∂_{i}, i ∈ J, subject to the relations (2)–(4) of 1.3.1.

1.5.3. A generalization. The construction of 1.5 is readily extended to a more general setting of Example 1.2.1. Namely, we take the trivial action of Γ on the ”coefficient algebra” R; and define the action of Γ on x_{i} and y_{i} as in 1.4; i.e., for any i, j ∈ J, the canonical generator z_{i} of Γ sends x_{j} into b_{ij}x_{j} and y_{j} into b_{ij}^{−1}y_{j}, where b_{ij} := (q_{ii}δ_{ij} + 1 − δ_{ij}). This way we define a group homomorphism ϕ : Γ → Aut(R{Θ, ξ}).

1.5.4. The hyperbolic structure of A_{q}. The hyperbolic structure of A_{q} is induced by that of A′_{q} via the canonical epimorphism A′_{q} → A_{q} (cf. 1.5.1). It is possible, however, to give an explicit description of A_{q}.

For any i ∈ J, set η_{i} := ∂_{i}x_{i} − x_{i}∂_{i} = (q_{ii} − 1)θ_{i}^{−1}(ξ_{i}) + 1. Note that θ_{i}(η_{j}) = b_{ij}η_{j}; i.e. θ_{i}(η_{i}) = q_{ii}η_{i} and θ_{i}(η_{j}) = η_{j} if i ≠ j. This implies that each η_{i} is a normal element, i.e. the left (or right) ideal in R{θ, ξ} generated by η_{i} is two-sided. Denote by S the multiplicative subset of R generated by η_{i}, q_{ii} i ∈ J. Note that, for any i ∈ J, the set S is θ_{i}-stable; hence θ_{i} induces an automorphism, θ′_{i}, of the localization R′ = S^{−1}R of the algebra R at the multiplicative set S. Denote by ξ′_{i} the image of ξ_{i} in R′, i ∈ J, and set Θ′ = (θ′_{i}), ξ′ := (ξ′_{i}). This data determines a hyperbolic algebra ℜ′ := R′{Θ′, ξ′} generated by R′, (x_{i}), (∂_{i}).

1.5.4.1. Proposition. The algebra A_{q} is naturally isomorphic to the hyperbolic algebra ℜ′ defined above.

Proof. The assertion follows from the simple observation that the image of the generator z_{i} of the group Γ in Aut(S_{q}) (cf. 1.4) coincides with η_{i}.

1.5.4.2. Remark. The multiplicative set S is an Ore set in the hyperbolic algebra ℜ = R{Θ, ξ} and the localized algebra S^{−1}ℜ is naturally isomorphic to the algebra ℜ′.

1.6. Quantum Weyl algebra of Maltsiniotis. Fix again a matrix q = (q_{ij}) such that q_{ij}q_{ji} = 1 and q_{ii} = 1 for all 1 ≤ i, j ≤ n. Fix an element λ := (λ_{1}, ..., λ_{n}) of (k^{∗})^{n}. The quantum Weyl algebra of Maltsiniotis, A_{n}^{q,λ}, is generated by x_{i}, y_{j}, 1 ≤ i, j ≤ n subject to the relations:

(a) For any 1 ≤ i < j ≤ n,

x_{i}x_{j} = λ_{i}q_{ij}x_{j}x_{i}, y_{i}y_{j} = q_{ij}y_{j}y_{i}

x_{i}y_{j} = q_{ji}y_{j}x_{i}, y_{i}x_{j} = λ_{i}^{−1}q_{ji}x_{j}y_{i}

(b) For any 1 ≤ i ≤ n,

x_{i}y_{i} − λ_{i}y_{i}x_{i} = 1 + ∑_{1≤j<i}(λ_{j} − 1)y_{j}x_{j}

For any 1 ≤ i ≤ n, set y_{i}x_{i} = ξ_{i}. It follows from the relations (a) that

ξ_{i}x_{j} = x_{j}ξ_{i}, ξ_{i}y_{j} = y_{j}ξ_{i} if i < j
ξ_{i}x_{j} = λ_{j}^{−1}x_{j}ξ_{i}, ξ_{i}y_{j} = λ_{j}y_{j}ξ_{i} if i > j.

These relations imply that ξ_{i}ξ_{j} = ξ_{j}ξ_{i} for all i, j. Set R := k[ξ_{i}, ..., ξ_{n}]. For any 0 < m ≤ n, denote by R_{m} the algebra generated by R and the elements x_{i}, y_{j}, m < i, j ≤ n. Thus R_{n} = A_{n}^{q,λ}, R_{0} = R, and R_{i} ⊂ R_{j} if i < j. Denote by θ_{m} the k-algebra automorphism of R_{m} defined by

θ_{m}(x_{i}) = λ_{m}q_{mi}x_{i}, θ_{m}(y_{i}) = λ_{m}^{−1}q_{im}y_{i} for m < i ≤ n  (1)
θ_{m}(ξ_{i}) = ξ_{i} if i < m, θ_{m}(ξ_{i}) = λ_{m}^{−1}ξ_{i} if i > m,  (2)
θ_{m}^{−1}(ξ_{m}) = λ_{m}ξ_{m} + 1 + ∑_{1≤j<m}(λ_{j} − 1)ξ_{j}  (3)

It follows that R_{m+1} is a hyperbolic algebra over its subalgebra R_{m}. Explicitly, R_{m+1} = R_{m}{θ_{m}, ξ_{m}; y_{m}, x_{m}}.

1.6.1. The algebra B_{n}^{q,λ}. Note that the elements η_{i} := x_{i}y_{i} − y_{i}x_{i} = ξ_{i} − θ_{i}^{−1}(ξ_{i}) have the property:

θ_{j}(η_{i}) = η_{i} if i < j and θ_{j}(η_{i}) = λ_{j}η_{i} if i ≥ j  (1)

It follows from (1) that the elements η_{i}, i ∈ J are normal; hence the generated by these elements multiplicative subset S of A_{n}^{q,λ} is a left and right Ore set. We denote by B_{n}^{q,λ} the localization of the algebra A_{n}^{q,λ} with respect to S.

2. Kashiwara theorem for hyperbolic algebras.

2.1. The category O of a hyperbolic algebra. Fix an automorphism θ and a central element ξ of an associative algebra R. Let ℜ denote the hyperbolic algebra R{θ, ξ} = R{θ, ξ; x, y}. Denote by O^{−} be the full subcategory of ℜ − mod consisting of modules M such that any element of M is annihilated by some power of y. We denote by O the full subcategory of O^{−} consisting of modules of finite type. This choice of notations is, of course, not accidental (cf. 1.2.5 and 1.2.6).

2.1.1. Lemma. The category O^{−} is a Serre subcategory of ℜ − mod. The quotient category ℜ − mod/O^{−} is equivalent to the category R[y, y^{−1}; θ^{−1}] − mod.

Here R[y, y^{−1}; θ^{−1}] is the algebra of skew Laurent polynomials in y.

Proof. a) Note that the multiplicative set S = {y^{n} | n ≥ 0} ⊂ ℜ satisfies left and right Ore conditions. We shall check the left Ore condition (for any s ∈ S and any r ∈ ℜ, there exist s′ ∈ S and r′ ∈ ℜ such that r′s = s′r) leaving the checking of the right one to the reader. One might also observe that the latter follows automatically by duality (the Fourier transform defined in 1.1.3 above).

Clearly y is a normal element of the subalgebra R[y, θ^{−1}] of ℜ. So that it suffices to check the Ore property for s = y^{n} and r = x^{m}. Take s′ = y^{ν+n+m}. Then s′r = y^{ν+n}(y^{m}x^{m}) = (θ^{−ν−n}(y^{m}x^{m})y^{ν})y^{n}. Here we use the fact that y^{m}x^{m} is an element of R : y^{m}x^{m} = ∏_{1≤i≤m} θ^{−i}(ξ).

b) Since S is an Ore set, and O^{−} is the corresponding to S Serre subcategory of ℜ − mod, the localization at O^{−} is equivalent to the tensoring over ℜ with the algebra of fractions S^{−1}ℜ , and ℜ − mod/O^{−} ≃ S^{−1}ℜ − mod. It remains to show that the algebra S^{−1}ℜ is isomorphic to the algebra R[y, y^{−1}; θ^{−1}] of skew Laurent polynomials.

In fact, S^{−1}ℜ is obtained from ℜ by inverting y. Clearly S^{−1}ℜ contains the algebra R[y, y^{−1}; θ^{−1}]. It remains to show that the image of ℜ in S^{−1}ℜ is contained in the algebra R[y, y^{−1}; θ^{−1}].

Any element f of ℜ is uniquely expressed as ∑_{i≥1} a_{i}x^{i} +∑_{m≥0} b_{m}y^{m} (Lemma 1.1.1). The image of this element in S^{−1}ℜ equals to f′ = ∑_{i≥1} a_{i}(x^{i}y^{i})y^{−i} + ∑_{m≥0} b_{m}y^{m}. Since x^{i}y^{i} is an element of R, x^{i}y^{i} = ∏_{0≤ν<i} θ^{ν}(ξ), for any i ≥ 1, f′ is a Laurent polynomial in y.

Set A := R/Rξ. We will identify the category A − mod with the corresponding full subcategory of R − mod. There is a functor Φ : O^{−} → A − mod which assigns to any M ∈ ObO^{−} the set M_{0} := {z ∈ M | y · z = 0} with the induced action of A.

On the other hand, there is a functor Ψ : A − mod → O^{−} which assigns to any A-module V the corresponding Verma module (with the subalgebra B = R[y; θ^{−1}] of ℜ of skew polynomials playing the role of a Borel subalgebra); i.e. Ψ is the composition of the embedding A − mod → B − mod and the functor the ℜ⊗_{B}. It is convenient to use the following canonical description of the functor Ψ (which is defined uniquely up to isomorphism). Namely, Ψ assigns to any A-module V the direct sum ⊕_{n≥0}V_{n} of copies of V with the following action of ℜ: The element x acts as the identical map from V_{n} to V_{n+1} for all n ≥ 0; y acts from V_{n+1} to V_{n} as multiplication by θ^{−n−1}(ξ) for n ≥ 0, and sends V_{0} to zero. For any r ∈ R, r acts on V_{n} as θ^{−n}(r).

Thus we get a pair of adjoint functors (Ψ, Φ). We are interested to find the conditions which imply that Ψ (and hence also Φ) is an equivalence of categories.

2.2. Theorem. The following conditions are equivalent:

(a) Rξ + Rθ^{n}(ξ) = R for any n ≠ 0;

(b) the functors Φ and Ψ are mutually inverse equivalences of categories.

Proof. (b)⇒(a) . Suppose that Rξ + Rθ^{n}(ξ) is a proper ideal for some n ≠ 0. We can (and will) assume that n < 0. And let μ be a maximal left ideal in R containing Rξ + Rθ^{n}(ξ). Then V := R/μ is a simple A-module; while Ψ(V) is not a simple object of the category O^{−}. The latter follows from the observation that ⊕_{m≥−n}V_{m} is a nontrivial submodule of Ψ(V) := ⊕_{m≥0}V_{m}. But if Ψ were an equivalence of categories, it should send simple objects into simple objects.

(a)⇒(b). Fix an ℜ-module M ∈ ObO^{−}. And set M_{n} := {z ∈ M | θ^{n}(ξ)z = 0}.

(i) Note first that, if n ≠ m, then M_{n} ≠ M_{m}.

In fact, let z ∈ M_{n} ∩ M_{m} which means that θ^{n}(ξ) and θ^{m}(ξ) are elements of the ideal Ann_{R}(z). Therefore θ^{n−m}(ξ) and ξ belong to the ideal μ := θ^{−m}(Ann_{R}(z)). Clearly z ≠ 0 ⇒ Ann_{R}(z) is a proper ideal ⇒ μ is a proper ideal. But, the latter contradicts the condition (a): Rθ^{n−m}(ξ) + Rξ = R.

(ii) Clearly, for any n, M_{n} is an R-submodule of M. One can see that the action of y (resp. x) maps M_{n} into M_{n−1} (resp. M_{n+1}): if z ∈ M_{n}, i.e. θ^{n}(ξ)z = 0, then θ^{n+1}(ξ)(xz) = xθ^{n}(ξ)z = 0, θ^{n−1}(ξ)(yz) = yθ^{n}(ξ)z = 0.

(iii) If z ∈ M is such that y^{m}z = 0, m ≥ 1, then z ∈ ⊕_{0≤n<m}M_{n}.

In fact, if yz = 0, then ξz = 0, i.e. z ∈ M_{0}. Suppose that the assertion is true for m; and let y^{m+1}z = 0. Then, by the induction hypothesis, yz ∈_{0≤n<m} M_{n}. This and (ii) imply that ξz := (xy)z ∈ ⊕_{0≤n≤m}M_{n}. We claim that this inclusion implies that z ∈ ⊕_{0≤n≤m}M_{n}.

Again, this is true (by definition) if m = 0.

Note now that θ^{m}(ξ)(ξz) = ξθ^{m}(ξ)z ∈ ⊕_{0≤n<m}M_{n} which implies, by the induction hypothesis, that θ^{m}(ξ)z ∈ ⊕_{0≤n<m}M_{n}. Therefore (Rξ +Rθ^{m}(ξ))z ⊆ ⊕_{0≤n≤m}M_{n}, and the equality Rξ + Rθ^{m}(ξ) = R implies the desired inclusion: z ∈ ⊕_{0≤n≤m}M_{n}.

(iv) It follows from (i), (ii), and (iii) that any M ∈ ObO^{−} is a ℤ_{+}-graded module: M =_{0≤n} M_{n}.

(v) Note that the actions x : M_{n} → M_{n+1}, y : M_{n+1} → M_{n} are isomorphisms for any n ≥ 0. In fact, xy = ξ : M_{n+1} → M_{n+1}. By assumption, there exist r, s ∈ R such that the element sξ + rθ^{n+1}(ξ) = 1. But sξ + rθ^{n+1}(ξ) acts on M_{n+1} as sξ. Therefore the action of ξ on M_{n+1} is invertible. Similarly, one can show that yx : M_{n} → M_{n} is invertible.

(vi) It follows from (v) that for an A-module V ΦΨ(V) = V and that the adjunction morphism ΨΦ → Id is an isomorphism, i.e. the functors Φ and Ψ are mutually inverse equivalences of categories.

2.3. Kashiwara theorem for general hyperbolic algebras. Let ℜ be a hyperbolic algebra over R with a hyperbolic structure (R_{m})_{ν≥m≥0}, R_{m} = R_{m−1}{θ_{m}, ξ_{m}; x_{m}, y_{m}}. We assume that the following condition holds:

(#) θ_{j}(y_{i}) = y_{i} for all i < j.

This condition implies that y_{i}x_{j} = x_{j}y_{i} for i < j and that y_{i}y_{j} = y_{j}y_{i} for all i, j.

Denote by O^{−} the full subcategory of ℜ − mod consisting of ℜ-modules M such that, for any i, any element of M is annihilated by some power of (y_{i}). The subcategory O^{−} is a Serre subcategory of ℜ − mod.

Let A be the quotient of the algebra R by the ideal generated by all ξ_{i}. We will identify the category A − mod with the corresponding full subcategory of R − mod. Let Φ : O^{−} → A − mod denote the functor which assigns to any M ∈ ObO^{−} the subspace M_{0} := {z ∈ M | y_{i} · z = 0 for all i} with the induced action of A.

The functor Φ has a left adjoint, Ψ, which assigns to any A-module V the corresponding Verma module (with the subalgebra B = R[(y_{i}); (θ_{i}^{−1})] of ℜ of skew polynomials playing the role of a Borel subalgebra); i.e. Ψ is the composition of the embedding A − mod → B − mod and the functor the ℜ⊗_{B}.

2.3.1. Proposition. Suppose the condition (#) holds and R_{i−1}ξ_{i} + R_{i−1}θ_{i}^{N}(ξ_{i}) = R_{i−1} for all i ≥ 0 and for all N ≠ 0. Then the functor Φ : O^{−} → A − mod is an equivalence of categories with the quasi-inverse functor Ψ.

Proof. The assertion is true by trivial reasons if one of ξ_{i} is invertible in ℜ. In fact, this implies that A = 0 and that y_{i} cannot annihilate a nonzero element; hence both the categories A − mod and O^{−} consist of zero modules only. For the rest of the argument we assume that none of ξ_{i}, 1 ≤ i ≤ ν, is invertible. In particular, all the ideals R_{i−1}ξ_{i} (in R_{i−1}) are proper.

Note that, for any 0 ≤ i < ν, the ideal R_{i}ξ_{ν} is stable under the action of the automorphism θ_{i} (since θ_{i}(ξ_{ν}) = ξ_{ν}; cf. 1.1.4). Therefore the structure of a hyperbolic algebra on ℜ induces on ℜ′ := ℜ/(ξ_{ν}) a structure of a hyperbolic algebra over R′ := R/Rξ_{ν}: R′_{m} = R′_{m−1}{θ′_{m}, ξ′_{m}; x_{m}, y_{m}}, 0 ≤ m < ν, where R′_{m} := R_{m}/R_{m}ξ_{ν}, ξ′_{m} is the image of ξ_{m} in R′_{m}, θ′_{m} is the induced by θ_{m} automorphism of R′_{m}. Clearly the equalities R_{i−1}ξ_{i} + R_{i−1}θ_{i}^{N}(ξ_{i}) = R_{i−1} imply the equalities R′_{i−1}ξ_{i} + R′_{i−1}θ_{i}^{′N}(ξ′_{i}) = R′_{i−1}.

Denote by Φ_{ν} : O^{−} → ℜ′ − mod the functor

Φ_{ν}(M) = {m ∈ M|y_{ν}m = 0}.

If M ∈ ObO^{−}, then Φ_{ν}(M) ∈ ObO^{′−}, where O^{′−} is the corresponding category of y_{1}, y_{2}, ..., y_{ν−1} - locally finite ℜ′-modules (since y_{ν} commutes with the y_{i}’s). We have x_{ν}y_{j} = y_{j}x_{ν} for all j < ν. Therefore the corresponding induction functor Ψ_{ν} : ℜ′ −mod → ℜ − mod restricts to a functor

Ψ_{ν} : O^{′−} → O^{−}.

By Theorem 2.2 the functors (Ψ_{ν}, Φ_{ν}) are mutually inverse equivalences of categories O^{′−} and O^{−}. Now apply induction on ν.

3. Examples. The GK dimension of modules of the category O.

3.1. Examples.

3.1.1. The algebras D_{q} and A_{q}. Consider the algebra D_{q} (cf. 1.3). It follows from 1.3.1(1) that θ_{i}^{n}(ξ_{i}) = q_{ii}^{n}ξ_{i} + ∑_{0≤j<n} q_{ii}^{j}. Hence the ideal Rξ_{i} + Rθ_{i}^{n}(ξ_{i}) contains the element ∑_{0≤j<n} q_{ii}^{j}. Suppose that one of the following conditions holds:

(a) q_{ii} = 1 and R is a ℚ-algebra.

(b) 1 − q_{ii}^{n} is invertible for any positive integer n.

Then ∑_{0≤j<n} q_{ii}^{j} is invertible for any n ≥ 1; hence the condition of Proposition 2.3.1 holds. The same for the algebra A_{q} (cf. 1.4 and 1.5.4).

3.1.2. The Weyl algebra by Maltsiniotis. It follows from 1.6(3) that

θ_{m}^{−N}(ξ_{m}) = λ_{m}^{N}ξ_{m} + ( ∑_{0≤i<N} λ_{m}^{i})(1 + ∑_{1≤j<m}(λ_{j} − 1)ξ_{j})  (1)

It follows that if N ≥ 1, the ideal in ℜ generated by the elements ξ_{m} and θ_{m}^{N}(ξ_{m}) coincides with the ideal generated by ξ_{m} and v_{m,N} := (∑_{0≤i<N} λ_{m}^{i})(1+∑_{1≤j<m}(λ_{j}−1)ξ_{j}). Suppose that one of the following conditions holds:

(a) λ_{1} = 1 and R is a ℚ-algebra

(b) 1 − λ_{1}^{N} is invertible for any positive integer N.

Then the element v_{1,N} = ∑_{0≤i<N} λ_{m}^{i} is invertible for any N ≥ 1. But if m ≥ 2, the elements ξ_{m} and v_{m,N} generate a proper ideal in R and, therefore, in ℜ; i.e. the conditions of Proposition 2.3.1 do not hold.

3.1.2.1. Note. Suppose that, for any 1 ≤ i < n, one of the following conditions holds: (a’) λ_{i} = 1 and R is a ℚ-algebra (b’) 1 − λ_{i}^{N} is invertible for any positive integer N. Then the algebra B_{n}^{q,λ} of 1.6.1 — the localization of A_{n}^{q,λ} at the multiplicative set

generated by the elements η_{i} := x_{i}y_{i}−y_{i}x_{i} = ξ_{i}−θ_{i}^{−1}(ξ_{i}), 1 ≤ i ≤ n, satisfies the conditions of Proposition 2.3.1, since it is isomorphic to the algebra A′_{q} of 1.5 (cf. 1.6.1 and 3.1.1). To see this directly, one might use the fact that v_{m,N} := (∑_{0≤i<N} λ_{m}^{i})η_{m−1}. Each of the conditions (a’), (b’) garantees that the ideal generated by ξ_{m} and v_{m,N} is generated by ξ_{m} and η_{m−1}. And the element η_{m−1} becomes invertible in the algebra B_{n}^{q,λ}.

3.1.3. The quantized Heisenberg and the Hayashi’s Weyl algebras. In the case of the quantized Heisenberg algebra (cf. 1.2.4), we have:

θ_{i}^{−N}(ξ_{i}) = q^{N}ξ_{i} − q^{N}( ∑_{1≤i≤N} q^{−2i})z_{i}  (1)

for all nonnegative integers N. This implies that the ideal Rξ_{i} + Rθ_{i}^{N}(ξ_{i}) is generated by ξ_{i} and (∑_{1≤i≤N} q^{−2i})z_{i}. It is a proper ideal of R for all N; that is the Kashiwara theorem fails for the Heisenberg algebras.

Note that, since z_{i} is a normal element (cf. 1.2.4), S = (z_{i}^{m}|m ≥ 1, i ∈ J) is a left and right Ore set, and the localization of H_{q}(J) at S is a hyperbolic algebra R′{Θ′, ξ′}, where R′ = S^{−1}R = k[(z_{i}, z_{i}^{−1}), (ξ_{i})] and Θ′ and ξ′ are induced by Θ and ξ. Suppose one of the following conditions holds for all i ∈ J:

(a) q_{i} = 1 and R is a ℚ-algebra

(b) 1 − q_{i}^{2N−1} is invertible for any positive integer N.

Then it follows from (1) that R′ξ′_{i} + R′θ_{i}^{N}(ξ′_{i}) = R′ for all i ∈ J; i.e. the conditions of the Kashiwara theorem hold. By the same reason they hold for the Hayashi’s Weyl algebra W_{q}(J) (cf. 1.2.4).

3.1.4. The quantum coordinate algebra of SL(2). Let R{θ, ξ} be the quantum coordinate algebra of SL(2, k); i.e.

R = k[u, v], θf(u, v) = f(qu, qv) for any f ∈ R, ξ = 1 + quv

(cf. Example 1.2.7). Note that

θ^{n}(ξ) = 1 + q^{2n+1}uv for any n ≥ 1

Since ξ = 1 + quv, this implies that the conditions of Theorem 2.2 hold, if q is not a root of one.

3.1.5. The quantum and classical enveloping algebras of sl(2). Let now R{θ, ξ} = U(sl(2)); i.e. R = k[z, ξ], and θf(z, ξ) = f(z + α, ξ + z) (see 1.2.5). Then, for any n ≥ 1,

θ^{−n}(ξ) = ξ − ∑_{1≤i≤n} θ^{i−n}(u) = ξ − nz − \frac{(n − 1)n}{2}α  (1)

This implies that the ideal Rξ+Rθ^{−n}(ξ) is generated by ξ and z− \frac{n−1}{2}α. In particular, this ideal is proper for any n.

In the quantum case, U_{q}(sl(2)) = R{θ, ξ}, where R = k[z, z^{−1}ξ], and θf(z, ξ) = f(qz, ξ + qz + \frac{z−z^{−1}}{q−q^{−1}} ) (see 1.2.5). Then, for any n ≥ 1,

θ^{−n}(ξ) = ξ + z^{−1}q^{2}(1 − q^{−n})((q^{2} − 1)(1 − q))^{−1}(z^{2} − q^{−n+1})  (1)

This implies that the ideal Rξ + Rθ^{−n}(ξ) is proper for any n.

3.2. The GK dimension of modules of the category O^{−}. For any k-algebra A, we denote by GK_{l}(A) the minimal value of GK-dimension of nonzero modules:

GK_{l}(A) = inf{GK(M)|M is a nonzero A-module}

It follows that GK_{l}(A) = 0 iff there are nonzero A-modules of finite type over k. Otherwise GK_{l}(A) ≥ 1 (cf. [McR], Proposition 8.1.17).

Let ℜ = R{θ, ξ; x, y} be a hyperbolic algebra of the rank one over R. Set S = {y^{n} | n ≥ 1}. For any ℜ-module M, t_{S}M denotes the S-torsion of M.

3.2.1. Proposition. With the notations above, suppose that Rξ + Rθ^{N}(ξ) = R for all N ≠ 0. Then, for any ℜ-module M having nonzero S-torsion, GK(M) ≥ 1+GK_{l}(R/Rξ).

Proof. Let t_{S}M ≠ 0; or, equivalently, the R/Rξ-module M_{0} := {v ∈ M|y · v = 0} is not equal to zero. By (the argument of) Theorem 2.2, t_{S}M ≃ ℜ ⊗_{𝔅} M_{0}, where 𝔅 is the subalgebra R[y, y^{−1}; θ^{−1}] of ℜ. Since ℜ ⊗_{𝔅} M_{0} ≃ (⊕_{n≥0}x^{n}M_{0}, μ) (here μ denotes the action of ℜ),

GK_{ℜ}(t_{S}M) = GK_{R/Rξ}(M_{0}) + 1 ≥ GK_{l}(R/Rξ) + 1

Let ℜ be a hyperbolic algebra of rank ν over R with the hyperbolic structure, R_{m+1} = R_{m}{θ_{m}, ξ_{m}; x_{m}, y_{m}}. Denote by R_{−1} the quotient of the algebra R by the ideal generated by the elements ξ_{m}, 1 ≤ m ≤ ν. Let O^{−} be the Serre subcategory of ℜ − mod defined in 2.3: an ℜ-module M belongs to O^{−} iff, for any 1 ≤ i ≤ ν, any element of M is annihilated by some power of y_{i}.

3.2.2. Proposition. In the above notation suppose that the assumptions of the Proposition 2.3.1 are satisfied. Then

inf{GK(M)|M ∈ ObO^{−}} = ν + GK_{l}(R_{−1})

In particular, for any ℜ-module M with the nonzero submodule in O^{−}, GK(M) ≥ ν + GK_{l}(R_{−1}).

Proof. The assertion follows from Proposition 2.3.1 (see the argument of Proposition 3.2.1).

4. The GK-dimension of modules over hyperbolic algebras.

Theorem 2.2 and Proposition 2.3.1 allow to estimate, under certain conditions, the GK dimension of nonzero modules of the category O^{−}. To get estimates for the GK-dimension of arbitrary modules over a hyperbolic algebra, we need some more subtle tools, those of the noncommutative local algebra developed in [R1], [R2]; in particular we need results on the spectrum of category of modules over a hyperbolic algebra ([R1], Ch.4). For the reader’s convenience, we sketch below the necessary preliminaries on the spectrum. Then we apply the spectrum to obtain estimates on GK-dimension of modules which allow to establish easily the Bernstein’s inequality for all known examples of hyperbolic algebras having this property.

4.0. Preliminaries on the spectrum. Fix an abelian category A. For any two objects X, Y of A, we write X ≻ Y if Y is a subquotient of a finite direct sum of copies of X (cf. Note 2.5.1). For any X ∈ ObA, denote by ⟨X⟩ the full subcategory of A such that Ob⟨X⟩ = ObA − {Y ∈ ObA | Y ≻ X}. It is easy to check that X ≻ Y iff ⟨Y ⟩ ⊆ ⟨X⟩. This observation provides a convenient realization of the quotient of (ObA, ≻) with respect to the equivalence relation induced by ≻: X ≈ Y if X ≻ Y ≻ X. Namely, (ObA, ≻)/ ≈ is isomorphic to ({⟨X⟩ | X ∈ ObA}, ⊇).

Set SpecA = {P ∈ ObA | P ≠ 0, and for any nonzero subobject X of P, X ≻ P }. The spectrum, SpecA, of the category A is the preordered set of equivalence (with respect to ≻ ) classes of objects of SpecA. The canonical realization of (ObA, ≻)/ ≈ induces a canonical realization of SpecA : (SpecA = {⟨P ⟩ | P ∈ SpecA}, ⊇).

4.0.1. Proposition. For any P ∈ SpecA, the subcategory ⟨P ⟩ is a Serre subcategory of A. If A is a category with the property (sup), then the converse is true: if X is an object of A such that ⟨X⟩ is a Serre subcategory of A, then X is equivalent (in the sense of ≻) to a P ∈ SpecA; i.e. ⟨X⟩ = ⟨P ⟩.

Proof. See Proposition 2.3.3 and 2.4.7 in [R1].

A nonzero object X of a category A is called quasifinal if, for any nonzero object Y of A, Y ≻ X. The category A having a quasifinal objects is called local.

One can check that all simple objects of a local category (if any) are isomorphic to each other. In particular, the category of left modules over a commutative ring R is local iff the ring R is local.

4.0.2. Proposition. And the quotient category A/⟨P ⟩ is local.

Proof. See Proposition 3.3.1 and Corollary 3.3.2 in [R1].

4.0.3. Proposition. (a) For any topologizing (i.e. full and closed with respect to taking direct sums and subquotients) subcategory 𝕋 of A, the inclusion functor 𝕋 ⟶ A induces an embedding Spec𝕋 ⟶ SpecA.

(b) For any exact localization Q : A ⟶ A/𝕊 and for any P ∈ SpecA, either P ∈ Ob𝕊, or Q(P) ∈ SpecA/𝕊; hence Q induces an injective map from SpecA−Spec𝕊 to SpecA/𝕊.

4.0.4. Localizations at subsets of the spectrum. For any subset U of SpecA, denote by ⟨⟨U⟩⟩ the intersection ⋂_{⟨P ⟩∈U}⟨P ⟩. Being the intersection of a set of Serre subcategories, ⟨⟨U⟩⟩ is a Serre subcategory. A localization at the subset U is a localization at the Serre subcategory ⟨⟨U⟩⟩.

4.0.5. The support of an object. For any M ∈ ObA, the support of M, Supp(M), consists of all ⟨P ⟩ ∈ SpecA such that M ∉ Ob⟨P ⟩.

4.0.6. Associated points. Let M ∈ ObA. An element ⟨P ⟩ ∈ SpecA is called an associated point of M if there exists a monomorphism P → M. The set of all associated points of M is denoted by Ass(M). It follows that Ass(M) ⊂ Supp(M).

4.0.7. The spectrum and the GK-dimension. Suppose that A = R − mod for a finitely generated k-algebra R. The GK-dimension (as well as any other known dimension function) has the property:

(#) For any M, M′ ∈ ObA such that M ≻ M′, GK(M) ≥ GK(M′).

The immediate consequences of this properties:

(a) GK(M) depends only on the equivalence class of M, i.e. GK(M) = GK(⟨M⟩)

(b) GK(M) ≥ sup(GK(⟨P ⟩)|⟨P ⟩ ∈ Supp(M)) ≥ sup(GK(⟨P ⟩)|⟨P ⟩ ∈ Ass(M)).

Moreover, there is the following fact:

(c) Suppose that the module M has a property: for any nonzero submodule M′ of M, Ass(M′) ≠ 0. Then sup(GK(⟨P ⟩)|⟨P ⟩ ∈ Supp(M)) = sup(GK(⟨P ⟩)|⟨P ⟩ ∈ Ass(M)).

4.1. The case of the rank one.

4.1.1. Lemma. Let ℜ = R{θ, ξ; x, y} be a hyperbolic algebra over R. And let M be an S-torsion free ℜ-module, S = {y^{n} | n ≥ 1}.

If there is ⟨P ⟩ ∈ Ass(_{R}M) such that the orbit {θ^{n}(⟨P ⟩)|n ∈ ℤ} is infinite, then GK(M) ≥ GK_{R}(P) + 1.

Proof. If the orbit {θ^{n}(⟨P ⟩)|n ∈ ℤ} is infinite, then by Theorem IV.6.4 in [R1], the canonical morphism of the R[y, y^{−1}; θ^{−1}]-module ⊕_{n∈ℤ}θ^{n}(P) to M is a monomorphism. Therefore GK(M) ≥ GK_{B}(⊕_{n∈ℤ}θ^{n}(P)) = GK_{R}(P) + 1.

4.1.2. Proposition. Let ℜ = R{θ, ξ; x, y} be a hyperbolic algebra over R such that Rξ + Rθ^{N}(ξ) = R for all N ≠ 0. Suppose that, for any nonzero R-module V, Ass(V) ≠ ∅. Then

GK_{l}(ℜ) = min{1 + GK_{l}(R_{−1}), 1 + GK_{∞}, GK_{f}}  (1)

where

GK_{∞} := inf{GK(P)|P ∈ Spec(R − mod) and the orbit (θ^{n}(P)|n ∈ ℤ) is infinite }

GK_{f} := inf{GK(P)|P ∈ Spec(R − mod) and the orbit (θ^{n}(P)|n ∈ ℤ) is finite }

Proof. (a) Suppose P ∈ Spec(R − mod) and the orbit (θ^{n}(P)|n ∈ ℤ) is infinite. Then GK(ℜ ⊗_{R} P) = GK(P) + 1.

(b) If the orbit (θ^{n}(P)|n ∈ ℤ) is finite. Then GK(ℜ ⊗_{R} P) = GK(P).

It follows from (a) and (b) that GK_{l}(ℜ) ≤ min{1 + GK_{l}(R_{−1}), 1 + GK_{∞}, GK_{f}}. Let (M, m) be an ℜ-module. By assumption, Ass(M) ≠ ∅; i.e. there exists a

monomorphism P → M, P ∈ Spec(R − mod). Clearly GK(M) ≥ GK(P). If the orbit {θ^{n}(⟨P ⟩)|n ∈ ℤ} is infinite, then by Theorem IV.6.4 in [R1], the canonical morphism of the R[y; θ^{−1}]-module ⊕_{n∈ℤ}θ^{n}(P) to M is a monomorphism. Therefore, in this case, GK(M) ≥ GK_{B}(⊕_{n∈ℤ}θ^{n}(P)) = GK_{R}(P) + 1.

4.1.3. Corollary. Let ℜ = R{θ, ξ; x, y} be a hyperbolic algebra over R such that Rξ + Rθ^{N}(ξ) = R for all N ≠ 0. Suppose that the following conditions hold:

(a) For any nonzero R-module V, Ass(V) ≠ ∅.

(b) For any ⟨P ⟩ ∈ Spec(R − mod) such that the orbit {θ^{n}(⟨P ⟩)|n ∈ ℤ} is finite, GK(P) ≥ 1.

Then, for any nonzero ℜ-module M, GK(M) ≥ 1.

4.2. The general case. Let ℜ be a hyperbolic algebra over R with the hyperbolic structure (R_{m})_{0≤m≤ν}, R_{0} = R, R_{m+1} = R_{m}{θ_{m}, ξ_{m}; x_{m+1}, y_{m+1}}. We assume that R contains all ξ_{m} (cf. 1.1.3). We assume in addition that the algebra R is θ_{m}-stable for all m. Denote by ϑ_{m} the automorphism of the algebra R induced by θ_{m}. Let Γ denote the group ℤ^{ν} and Θ the group homomorphism Γ → Aut(R) which sends the m-th canonical generator of Γ into ϑ_{m}. For any P ∈ Spec(R − mod), denote by Γ_{P} the stabilizer of P; i.e. Γ_{P} := {γ ∈ Γ|Θ_{γ}(P) = P }. Finally, we define the ℚ-rank of an abelian group H by rk_{ℚ}(H) := dim_{ℚ}ℚ ⊗ H.

4.2.1. Proposition. Let ℜ be a hyperbolic algebra over R satisfying the assumptions above. Suppose that R_{i}ξ_{i} +R_{i}θ_{i}^{N}(ξ_{i}) = R_{i} for all i ≥ 0 and for all N ≠ 0. Let 𝔐 = (M, m) be an ℜ-module, and let P ∈ Ass(M). Then GK(𝔐) ≥ rk_{ℚ}(Γ/Γ_{P}) + GK(P).

Proof. Since P ∈ Spec(R − mod), either P is annihilated by all ξ_{i}, or there is i, 1 ≤ i ≤ N, such that ξ_{i} acts injectively on P.

(a) In the first case, the stabilizer Γ_{P} of P is trivial, hence rk_{ℚ}(Γ/Γ_{P}) = rk_{ℚ}(Γ) = ν. And it follows from Proposition 2.3.1 that GK(M) ≥ ν + GK(P) = rk_{ℚ}(Γ/Γ_{P}) + GK(P).

(b) Consider the second case. Take the maximal i such that ξ_{i} acts injectively on P. (b1) Suppose that i < ν. Let S_{i} denote the multiplicative set of monomials in y_{j},

where i < j ≤ ν. This is a (right and left) Ore set. Denote by 𝔐′ = (M′, m′) the S_{i}- torsion of M. Clearly P is a subobject of M′. Let 𝔐′_{i} denote the image of 𝔐′ under the canonical functor ℜ − mod → R_{i} − mod. By induction hypothesis, there exists a subobject P′ → 𝔐′_{i} such that P′ ∈ Spec(R_{i} − mod) and P is a subobject of P′_{0}. The fact that ξ_{j} annihilate P for j > i implies that they annihilate the R_{i}-submodule P′. Hence GK(𝔐) ≥ ν − i + GK(P′). By induction hypothesis, GK(P′) ≥ rk_{ℚ}(Γ′/Γ_{P}) + GK(P). Here Γ′ = ℤ^{i}. It follows that ν − i + rk_{ℚ}(Γ′/Γ_{P}) = rk_{ℚ}(Γ/Γ_{P}).

(b2) Suppose that i = ν; i.e. the action of ξ_{ν} is injective. We have two cases: the orbit Ω_{ν,P} = (ϑ_{ν}^{n}(P)|n ∈ ℤ) might be either finite, or infinite.

(b2.1) Suppose that the orbit Ω_{ν,P} is finite. By induction hypothesis, we have GK(𝔐) ≥ GK(𝔐_{ν−1}) ≥ rk_{ℚ}(Γ′′/Γ′′_{P}) + GK(P), where Γ′′ = ℤ^{ν−1}. Since the orbit Ω_{ν,P} is finite, rk_{ℚ}(Γ′′/Γ′′_{P}) = rk_{ℚ}(Γ/Γ_{P}).

(b2.2) Suppose that the orbit Ω_{ν,P} is infinite. We can assume that it contributes to the rank rk_{ℚ}(Γ/Γ_{P}). By induction hypothesis, there exists a subobject P′ → 𝔐′_{ν−1} such that P′ ∈ Spec(R_{ν−1} − mod) and P is a subobject of P′_{0}. The fact that Ω_{ν,P} contributes to the rank means that the orbit (θ_{ν}^{n}(P)|n ∈ ℤ) is infinite. This implies that the canonical morphism ⊕_{n∈ℤ}θ^{n}(P′) → 𝔐 induced by the embedding P′ → 𝔐′_{ν−1} is a monomorphism. Therefore GK(𝔐) ≥ 1 + GK(P′). By induction hypothesis, GK(P′) ≥ rk_{ℚ}(Γ/Γ_{P}) − 1 + GK(P); hence the required inequality: GK(𝔐) ≥ rk_{ℚ}(Γ/Γ_{P}) + GK(P).

4.2.2. Proposition. Let ℜ be a hyperbolic algebra over R satisfying the assumptions above. Suppose that the following conditions hold:

(a) R_{i}ξ_{i} + R_{i}θ_{i}^{N}(ξ_{i}) = R_{i} for all i ≥ 0 and for all N ≠ 0.

(b) For any nonzero R-module V, Ass(V) ≠ ∅.

Then

GK_{l}(ℜ) = min{rk_{ℚ}(Γ/Γ_{P}) + GK(P)|P ∈ Spec(R − mod)}  (1)

or, what is the same

GK_{l}(ℜ) = min{ν + GK_{l}(R_{−1}), rk_{ℚ}(Γ/Γ_{P}) + GK(P)|P ∈ U(ξ)}  (1′)

where U(ξ) is the complement to the Zariski closed set defined by ξ = (ξ_{i}|1 ≤ i ≤ ν); i.e. U(ξ) = Spec(R − mod) − Spec(R_{−1} − mod).

Proof. The inequality

GK_{l}(ℜ) ≥ min{rk_{ℚ}(Γ/Γ_{P}) + GK(P)|P ∈ Spec(R − mod)}

follows from Proposition 4.2.1. It remains to show the inverse inequality. The latter follows from the description of the spectrum of a hyperbolic algebra (cf. [R1], Ch. IV and Ch. V).

4.2.2.1. Corollary. Under the assumptions (a), (b) of Proposition 4.2.2, the following conditions are equivalent:

(i) GK_{l}(ℜ) = 0

(ii) There exists P ∈ Spec(R − mod) of finite type over k and such that the orbit of P under the action of Γ is finite.

Proof. It follows from 4.2.2 that GK_{l}(ℜ) = 0 iff there exists P ∈ Spec(R − mod) such that rk_{ℚ}(Γ/Γ_{P}) = 0 and GK(P) = 0. The latter equality implies that P is of finite type over k. The equality rk_{ℚ}(Γ/Γ_{P}) = 0 means that the orbit of P under the action of Γ is finite.

5. The Bernstein’s inequality. Examples.

5.1. The Bernstein’s inequality. Let ℜ be a hyperbolic k-algebra over R with the hyperbolic structure (R_{m})_{0≤m≤ν}, R_{0} = R, R_{m+1} = R_{m}{θ_{m}, ξ_{m}; x_{m+1}, y_{m+1}}. We shall say that the Bernstein’s inequality holds for an ℜ-module M if GK(M) ≥ ν. We shall say that the Bernstein’s inequality holds for ℜ if it holds for any nonzero ℜ-module.

Thus Theorem 2.2 and Proposition 2.3.1 assert that, under certain conditions, the Bernstein’s inequality holds for all nonzero modules of the category O^{−}.

5.1.1 Proposition. Let ℜ be a hyperbolic algebra over R satisfying the assumptions above. Suppose that the following conditions hold:

(a) R_{i}ξ_{i} + R_{i}θ_{i}^{N}(ξ_{i}) = R_{i} for all i ≥ 0 and for all N ≠ 0.

(b) For any nonzero R-module V, Ass(V) ≠ ∅.

(c) For any P ∈ Spec(R − mod), GK(P) ≥ rk_{ℚ}(Γ_{P}).

Then the Bernstein’s inequality holds for any nonzero ℜ-module M.

Proof. The assertion follows from Proposition 4.2.2.

5.1.2. Note. Any of the conditions (a) and (b) of Proposition 5.1.1 cannot be dropped. For instance, the condition (a) (i.e. that of Theorem 2.2) holds for the algebra D_{q} of q-differential operators (cf. 1.3 and 3.1.1) if q is not a root of one, but this algebra has a family of one dimensional modules. In fact, D_{q} = R{θ, ξ}, where R = k[ξ], θ(ξ) = qξ + 1. One can check that η := (1 − q)ξ − 1 is an eigen vector of θ: θ(η) = qη. Therefore D_{q}η is a two-sided ideal, and D_{q}/D_{q}η ≃ k[x, y]/((q − 1)xy − 1) ≃ k[x, x^{−1}].

Similarly, the quantum coordinate algebra of SL(2), A(SL_{q}(2)) (cf. 3.1.4), satisfies the condition (a) if q is not a root of one, but it also has a family of one dimensional representations.

On the other hand, the enveloping algebra of sl(2) over the field of zero characteristic and the quantized enveloping algebra U_{q}(sl(2)) in the case when q is not a root of one (cf. 3.1.5) satisfy both the condition (b) and have finite dimensional representations.

5.2. The ’classical’ Bernstein’s inequality. Consider the n-th Weyl algebra A_{n} over a field k of zero characteristic as a hyperbolic algebra over the algebra R = k[ξ_{1}, ..., ξ_{n}], θ_{i}(ξ_{j}) = ξ_{j} + δ_{i}j (cf. example 1.2.3). Then Γ = ℤ^{n}, and, for any P ∈ Spec(R − mod), GK(P) ≥ rk_{ℚ}(Γ_{P}); i.e. the conditions of Proposition 5.1.1 hold. Therefore we have obtained, among other things, another proof of the Bernstein’s inequality: GK(M) ≥ n for any nonzero module M.

5.3. The Bernstein’s inequality for Hayashi’s Weyl algebras. Consider the n-th Hayashi’s Weyl algebra with the hyperbolic structure of 1.2.4; i.e. R is the quotient of the algebra k[(z_{i}), (ξ_{i})] of polynomials by the relations z_{i}(ξ_{i}(1 − q^{2}) + q^{2}z_{i}) = 1, i ∈ J., and the automorphisms ϑ_{i}, i ∈ J are defined by the same formulas as for the Heisenberg quantized algebra H_{q}(J); i.e.

ϑ_{i}(z_{i}) = qz_{i}, ϑ_{i}(ξ_{i}) = q^{−1}ξ_{i} + qz_{i} for all i ∈ J  (1)
ϑ_{i}(z_{j}) = z_{j}, ϑ_{i}(ξ_{j}) = ξ_{j} if j ≠ i

It follows from (1) that, for any P ∈ Spec(R − mod), GK(P) ≥ rk_{ℚ}(Γ_{P}); i.e. the conditions of Proposition 5.1.1 hold. Therefore in the case of Hayashi’s algebras we also have the Bernstein’s inequality: GK(M) ≥ n for any nonzero module M.

5.4. The algebra of differential operators on a quantum space. Consider the algebra D_{q} of differential operators on a quantum space of dimension n (cf. 1.3) regarded as a hyperbolic algebra over the polynomial algebra R = k[(ξ_{i})] (see 1.3.1). The restriction of the automorphisms θ_{i} to the algebra R is determined by

θ_{i}(ξ_{j}) = ξ_{j} if i ≠ j; θ_{i}(ξ_{i}) = q_{ii}ξ_{i} + 1  (1)

If q_{ii} ≠ 1 for all i, then R has one dimensional (hence simple) Γ-stable modules. This modules induce one-dimensional (in particular of the GK dimension zero) modules over the algebra D_{q}.

5.5. The quantum Weyl algebra. In the case of the quantum Weyl algebra A_{q} (cf. 1.5, in particular 1.5.4), one can check that, for any P ∈ Spec(R − mod), GK(P) ≥ rk_{ℚ}(Γ_{P}). Hence the Bernstein’s inequality holds: GK(M) ≥ n for any nonzero module M.

5.6. The quantum Weyl algebra by Maltsiniotis. Consider the quantum Weyl algebra by Maltsiniotis A_{n}^{q,λ} (cf. 1.6). In this case, like in the case of the algebra D_{q} of 1.3, there are, in general, finite dimensional nonzero modules. An appropriate canonical localization (the one explained in 3.1.2.1) of A_{n}^{q,λ} is isomorphic to the algebra A_{q}. Therefore the Bernstein’s inequality holds for all modules over A_{n}^{q,λ} which are not in the kernel of this localization.

5.7. Note. The Bernstein’s inequality for the Hayashi’s Weyl algebras and the localized Weyl algebra by Maltsiniotis (the algebra A_{q}) was obtained by L. Rigal [Ri1], [Ri2] by different methods.

Complements: Kashiwara theorem for hyperbolic categories.

Although Theorem 2.2 suffices for our immediate needs, it is more convenient to have its relative analog which allows to work, for example, with categories of sheaves of hyperbolic algebras and modules over them. With more reason that both assumptions and the argument look more naturally in the case of hyperbolic categories than in the case of hyperbolic rings.

C.1. Hyperbolic categories. Let θ be an auto-equivalence of an additive category A; and let ξ be an endomorphism of the identical functor of A.

Denote by A{θ, ξ} the category objects of which are triples (γ, M, η), where M ∈ ObA and γ : M ⟶ θ(M), η : θ(M) ⟶ M are arrows such that

η ∘ γ = ξ(M) and γ ∘ η = ξθ(M).

Morphisms from (γ, M, η) to (γ′, M′, η′) are those morphisms f from M to M′ for which the diagram

M ⟶^{γ} θ(M)  ⟶^{η} M
f ↓  ↓θ(f)  ↓ f
M′ ⟶^{γ′} θ(M′)  ⟶^{η′} M′

is commutative.

The category A{θ, ξ} will be called hyperbolic.

C.1.1. Example. Let R be an associative ring, ϑ an automorphism of R, ξ′ ∈ R a central element, and R{ϑ, ξ′} the related to this data hyperbolic ring.

The category R{ϑ, ξ′} − mod is hyperbolic.

Namely, R{ϑ, ξ′}-mod is equivalent to the category A{θ, ξ}, where A = R-mod, θ is an auto-equivalence of the category A induced by the automorphism ϑ (cf. 2.1), ξ is the endomorphism of the identical functor, Id_{A}, which assigns to every R-module M the action of the element ξ′ on M; i.e. ξ(w) := ξ′ · w for each w ∈ M.

C.2. Kashiwara theorem for hyperbolic categories. Fix a hyperbolic category A{θ, ξ} over an abelian category A. Fix an object (s, M, t) in A{θ, ξ}; and denote by M^{(n)} the kernel of the morphism s^{(n)} := θ^{n}s ∘ … ∘ θs ∘ s : M → θ^{n+1}(M). Set M^{(∞)} := sup{M^{(n)} | n ≥ 0}.

C.2.1. Lemma. The subobject M^{(∞)} of M has (necessarily unique) structure of a subobject of (s, M, t).

Proof. 1) Note that t sends M^{(n)} into M^{(n+1)}.

In fact,

s^{(n+1)} ∘ t = θ(s^{(n)}) ∘ s ∘ t = θ(s^{(n)}) ∘ ξθ(M) = ξθ^{n+1}(M) ∘ θ(s^{(n)}).

Therefore, if ι^{(n)} is the monomorphism M^{(n)} → M, then

s^{(n+1)} ∘ t ∘ θι^{(n)} = ξθ^{n+1}(M) ∘ θ(s^{(n)}) ∘ θι^{(n)} = ξθ^{n+1}(M) ∘ θ(s^{(n)} ∘ ι^{(n)}) = 0

which is required to show.

2) Even more transparent is the fact that the morphism s sends the subobject M^{(n)} into θ(M^{(n−1)}), n ≥ 1.

Indeed, since θ is left exact, θι^{(n−1)}) : θ(M^{(n−1)}) → θ(M) is the kernel of θ(s^{(n−1)}). But we have: θ(s^{(n−1)}) ∘ s ∘ ι^{(n)} = s^{(n)} ∘ ι^{(n)} = 0.

2) Since the category A has the property (sup), it follows that t sends the subobject θ(M^{(∞)}) to M^{(∞)}, and s sends M^{(∞)} to θ(M^{(∞)}).

Consider the full subcategory 𝕊 of A{θ, ξ} generated by objects (s, M, t) such that the natural monomorphism M^{(∞)} → M is an isomorphism. One can check that 𝕊 is a thick subcategory of A{θ, ξ}.

C.2.2. Corollary. The subcategory 𝕊 is a coreflective (hence Serre) subcategory of A{θ, ξ}.

Proof. One can see that the map assigning to any object (s, M, t) of A{θ, ξ} the object (s^{(∞)}, M^{(∞)}, t^{(∞)}) defined in Lemma C.2.1 extends naturally to a functor from A{θ, ξ} to 𝕊 which is right adjoint to the embedding 𝕊 → A{θ, ξ}.

Consider the Verma functor M_{+} : A_{ξ} → A{θ, ξ} defined in [R4], 5.8. Here A_{ξ} is the full (actually, topologizing) subcategory of A generated by V ∈ ObA such that ξ(V) = 0. The functor M assigns to any object V the triple (g_{+}, θ_{+}(V), h_{+}), where θ_{+} :=_{n≥0} θ^{n}, h_{+} : θ ∘ θ_{+} → θ_{+} and g_{+} : θ_{+} → θ ∘ θ_{+} are defined by:

h_{+i} = id : θ ∘ θ^{i−1} → θ^{i}, g_{+i} = ξθ^{i} : θ^{i} → θ ∘ θ^{i−1}

for i ≥ 1.

One can check that the functor M_{+} takes values in the Serre subcategory 𝕊; so that it induces a functor M from A_{ξ} to 𝕊.

On the other hand, there is a functor Φ : 𝕊 → A_{ξ} which assigns to any object (s, M, t) the kernel of s.

We leave to a reader to check that the functor Φ is right adjoint to the functor M. It follows from the definition of M that M is exact and the adjunction arrow Id_{A_{ξ}} → Φ ∘ M is an isomorphism. The latter means exactly that the functor M is fully faithful.

Dually, define the full subcategory 𝕊 of A{θ, ξ} generated by all those objects (s, M, t) for which the canonical arrow from M_{(∞)} := sup_{n≥0} (Ker(t ∘ θt ∘ … ∘ θ^{n}t)) to M is an isomorphism. We have a functor M from A_{ξθ} to A{θ, ξ} which takes values in 𝕊 , hence induces a functor M′ : A_{ξθ} → 𝕊 . Formally, all this can be obtained by switching to the adjoint hyperbolic category A{θˆ , ξˆ } and using the canonical equivalence of categories A{θ, ξ} → A{θˆ , ξˆ } (cf. [R4], 5.8). Recall that θˆ is an adjoint to θ functor, and ξˆ := σ ∘ θˆ ξθ ∘ σ^{−1}, where σ : θˆ ∘ θ → Id_{A} is an adjunction isomorphism.

C.2.3. Theorem. The following conditions are equivalent:

(a) If V ∈ ObA is such that ξ(V) = 0 and ξθ^{n}(V) = 0 for some n > 0, then V = 0;

(b) the functor M : A_{ξ} → 𝕊 is an equivalence of the categories.

(c) the functor M′ : A_{ξθ} → 𝕊 is an equivalence of the categories.

Proof. (a) ⇔ (b). Suppose there exists a nonzero object V of the category A such that ξ(V) = 0 and ξθ^{n}(V) = 0 for some n ≠ 0. By assumption on A, Supp(V)≠ ; i.e. there exists P ∈ SpecA such that V ≻ P. The latter implies that the object P satisfies the same conditions: ξ(P) = 0, ξθ^{n}(P) = 0; in particular, P ∈ SpecA ∩ ObA_{ξ} = SpecA_{ξ}. It follows from Theorem 6.2 in [R4] that M_{+}(P) ∉ SpecA{θ, ξ}. Since 𝕊 is a topologizing subcategory of A{θ, ξ} and M(P) belongs to 𝕊, M(P) ∉ Spec𝕊(=SpecA{θ, ξ}∩Ob𝕊). But, if the functor M were an equivalence of categories, it should send objects of the spectrum to objects of the spectrum.

(b) ⇔ (a). (i) Note first that the group Aut(A) of isomorphy classes of autoequivalences of A acts (by conjugations) on the center C(A) := End(Id_{A}) of A. This action is defined as follows.

Given an element λ ∈ C(A) and an auto-equivalence Θ of A set adΘ(λ) equal to the composition ΘλΘˆ with the adjunction isomorphisms Id → Θ ∘ Θˆ and Θ ∘ Θˆ → Id. One can see that adΘ coincides with adΘ′ if Θ is isomorphic to Θ′.

We need this here for the case λ = ξ and Θ = θ^{n}, n ∈ ℤ_{+}. To simplify the notations, we shall write ξ_{n} instead of adθ^{n}(ξ).

Fix (s, M, t) ∈ Ob𝕊. For any n ∈ ℤ_{+}, set M_{n} := Kerξ_{n}(M).

(ii) Note that M_{n} ∩ M_{m} = 0 if n ≠m.

In fact, the object W := M_{n} ∩ M_{m} has the property ξ_{n}(W) = 0 = ξ_{m}(W); or, equivalently, ξθ ^{ˆ n}(W) = 0 and ξθ ^{ˆ m}(W) = 0. Suppose that n > m. The equalities above are equivalent to ξ(θ ^{ˆ n}(W)) = 0, ξθ^{m−n}(θ ^{ˆ n}(W)) = 0. By the condition (a), they imply that the object θ ^{ˆ n}(W) equals to zero, or, which is the same, W = 0.

(iii) For any n ≥ 1, ξ(M_{n}) is an isomorphism.

Indeed, the condition (a) is equivalent to the following one: Kerξ_{n} ∩ Kerξ = 0 for all n > 0. Since ξ_{n}(M_{n}) = 0, we have: 0 = Kerξ_{n}(M_{n}) ∩ Kerξ(M_{n}) = Kerξ(M_{n}); i.e. ξ(M_{n}) is a monomorphism.

On the other hand, the object W := Cokξ(M_{n}), being a quotient object of M_{n}, has the property ξ_{n}(W) = 0 = ξ(W). Therefore W = 0; i.e. ξ(M_{n}) is also an epimorphism.

(iii′) Similarly, one can see that the condition (a) is equivalent to the requirement Kerξθ ∩ Kerξ_{n} = 0 for all n ≥ 0. This implies, by the same argument as in (iii), that ξθ(M_{n}) is an isomorphism for any n ≥ 0.

(iv) The arrow t : θ(M) → M induces, for any n ≥ 0, an isomorphism from θ(M_{n}) to M_{n+1}; and s : M → θ(M) induces, for any n ≥ 1, an isomorphism from M_{n} to θ(M_{n−1}).

Denote by ι_{n} the monomorphism M_{n} → M. We have:

ξ_{n+1}(M) ∘ t ∘ θι_{n} = t ∘ ξ_{n+1}θ(M) ∘ θι_{n} = 0

because ξ_{n+1}θ = θξ_{n}; so that ξ_{n+1}θ(M) ∘ θι_{n} = θ(ξ_{n}(M) ∘ ι_{n}) = 0. This shows that t maps θ(M_{n}) to M_{n+1}. Similarly, the equalities

0 = s ∘ ξ_{n}(M) ∘ ι_{n} = ξ_{n}θ ∘ s ∘ ι_{n} = θξ_{n−1} ∘ s ∘ ι_{n}

imply that s maps M_{n} to θ(M_{n−1}), n > 0.

(v) For any n ≥ 0, the induced by s and t morphisms

s_{n} : M_{n+1} → θ(M_{n}) and t_{n} : θ(M_{n}) → M_{n+1}

are isomorphisms.

In fact, s_{n} ∘t_{n} = ξθ(M_{n}), and t_{n} ∘s_{n} = ξ(M_{n+1}). But, by (iii) and (iii′), both ξθ(M_{n}) and ξ(M_{n+1}) are isomorphisms for any n ≥ 0.

(vi) For any n ≥ 0, M^{(n)} := Ker(s^{(n)}), where s^{(n)} := θ^{n}s∘…∘θs∘s : M → θ^{n+1}(M), is contained in the sum of the subobjects M_{m}, 0 ≤ m ≤n.

We have: M^{(0)}:=Ker(s)⊆ Ker(t ∘ s) = Ker(ξ(M)) := M_{0}; i.e. the assertion is true for n = 0. For any n ≥ 1,

Ker(s^{(n)}) = Ker(θ^{n}s ∘ s^{n−1}) ⊆ Ker(θ^{n}ξ(M) ∘ s^{n−1})
= Ker(ξ_{n}θ^{n}(M) ∘ s^{n−1}) = Ker(s^{n−1} ∘ ξ_{n}(M)).

Continuing this way, we obtain that Ker(s^{(n)}) ⊆ Ker(ξ∘ξ_{1}∘…∘ξ_{n})(M). Thus, it suffices to show that the Ker((ξ ∘ξ_{1} ∘…∘ξ_{n}))(M) = ∑_{0≤m≤n} M_{m} := ∑_{0≤m≤n} Ker(ξ_{m}(M)).

This follows (by induction argument) from the fact that, for any m > 0, ξ(M_{m}) is an isomorphism; hence ξ(∑_{1≤m≤n} M_{m}) = ξ(Ker(ξ_{1} ∘ … ∘ ξ_{n})(M))) is an isomorphism.

Note that, since the inclusion ∑_{0≤m≤n} M_{m} ⊆ Ker(s^{(n)}) holds, we have, actually, showed that Ker(s^{(n)}) = ∑_{0≤m≤n} M_{m} = Ker((ξ ∘ ξ_{1} ∘ … ∘ ξ_{n})(M)).

(vii) Combining all above, we see that the natural morphism from M(M_{0}) to (s, M, t) is an isomorphism. Thanks to the arbitrarity of (s, M, t) ∈ Ob𝕊, this means that the adjunction arrow η : M ∘ Φ → Id_{𝕊} is an isomorphism; i.e. the functor M is an equivalence of categories.

(a)⇒ (c). Note that the condition (a) is equivalent to the same condition in the adjoint hyperbolic category A{θˆ , ξˆ }. Since (c) in A{θ, ξ} is the same as (b) in A{θˆ , ξˆ }, the implications (a)⇒ (c) follow from those (a)⇒ (b).

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