POINCARÉ SERIES OF MULTI-FILTERED ALGEBRAS AND PARTITIVITY

J. GÓMEZ TORRECILLAS AND T. H. LENAGAN

Abstract. We prove that if an algebra R over a field can be endowed with a pointed and locally finite ℕ^{n}-filtration such that the associated ℕ^{n}-graded algebra T is semicommutative, then R is left and right finitely partitive. In order to do this, we consider a multi-variable Poincaré Series for every finitely generated graded T -module and show that this Poincaré Series is a rational function. Our methods apply to some iterated Ore extensions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of 𝔰𝔩(ν + 1).

Introduction

The notion of a multi-filtration has been used recently in [3] to obtain the exactness of the Gelfand-Kirillov dimension of certain associative algebras over a field that have no “obvious” filtration with a simpler associated ℕ-graded algebra. A typical situation that arises is when the ℕ^{n}-graded algebra, G_{F}(R), associated to a multifiltration F on an algebra R, is semi-commutative. This is the case for the quantum coordinate ring, O\underline{q}(M_{n}(k)), of quantum n×n matrices over k, or the quantized universal enveloping algebras U_{q}(𝔰𝔩(n+1)). In this paper, we show that for these algebras the Gelfand-Kirillov dimension is left and right partitive, in addition to being exact.

On the way, we develop some tools which may be of independent interest. In Section 1, we consider a natural notion of a multi-variable Poincaré Series for finitely generated modules over a locally finite pointed ℕ^{n}-graded algebra. In the case that the algebra is semi-commutative, then we show that the Poincaré series is a rational function of n variables, Theorem 1.5. This is then applied to G_{F}(R) in Section 2, in conjunction with the main result of [3] to obtain the result that R is a left and right partitive algebra whenever the ℕ^{n}-filtration is finite dimensional and the associated multigraded algebra is semicommutative, Theorem 2.8. In order to achieve these results, we introduce a notion of multiplicity for ℕ^{n}-filtered modules, see 1.7 and 2.6.

1. Poincaré series

In this note, all algebras are associative with identity. By ℕ^{n} we will denote the free abelian monoid with n generators. Let R = ⊕_{α∈ℕ^{n}}R_{α} be an ℕ^{n}-graded algebra over a field

1991 Mathematics Subject Classification. 16P90,16W60,16W50.

2 J. GÓMEZ TORRECILLAS AND T. H. LENAGAN

k. Assume that R is pointed and locally finite, in the sense that R_{0} = k and that R_{α} is a finite-dimensional k–vector space for every α ∈ ℕ^{n}. (Note that many authors used connected instead of pointed). If r ∈ R_{α} then we say that deg(r) = α. Every finitely generated ℕ^{n}–graded left R–module M = ⊕_{α∈ℕ^{n}}M_{α} is locally finite; that is, dim_{k} M_{α} < ∞ for every multi-index α. Consider the function d_{M} : ℕ^{n} → ℕ given by d_{M}(α) = dim_{k} M_{α}.

Definition 1.1. The multi-variable Poincaré series of a finitely generated ℕ^{n}-graded left R–module M is given by

P (M; \underline{t}) = ∑_{α∈ℕ^{n}} d_{M}(α)\underline{t}^{α}

where \underline{t}^{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}, for t_{1}, …, t_{n} commuting indeterminates over ℤ and α = (α_{1}, …, α_{n}) ∈ ℕ^{n}.

The aim of this section is to prove that the Poincaré series is a rational function in the variables t_{1}, …, t_{n} for certain non-commutative algebras that resemble affine commutative graded algebras.

Definition 1.2. We will say that two elements x, y of the algebra R are semi-commuting if there is 0 ≠ q ∈ k such that yx = qxy. The graded k-algebra R is called semi-com-mutative if R is generated as k–algebra by a finite set of homogeneous semi-commuting elements.

Note that a consequence of the definition is that such an algebra R is noetherian, since it can be presented as a factor algebra of an iterated Ore extension of the base field k. The aim of this section is to prove the following non-commutative version of the Hilbert-Serre Theorem.

Theorem. Let M be a finitely generated ℕ^{n}–graded left module over a semi-commutative ℕ^{n}–graded k-algebra R. Let x_{1}, …, x_{s} be a set of semi-commuting homogeneous generators of R, with deg(x_{j}) = μ^{j} ∈ ℕ^{n}. There exists a polynomial f(\underline{t}) ∈ ℤ[t_{1}, …, t_{n}] such that

P (M; \underline{t}) = \frac{f(\underline{t})}{∏_{j=1}^{s}(1 − \underline{t}^{μ^{j}})}

To give a proof of this theorem, we need some notation. For each non-empty subset J of {1, …, n}, let G_{J} denote the submonoid of ℕ^{n} whose elements are the multi-indices α such that α_{j} = 0 for every j ∈ J. Observe that G_{J} is isomorphic to the monoid ℕ^{n−d}, where d is the cardinal of J. Define

R^{[J]} = ⨁_{α∈G_{J}} R_{α}.

It is immediate that R is a G_{J}–graded subalgebra of R.

Lemma 1.3. The graded algebra R^{[J]} is generated by the set {x_{j} : deg(x_{j}) ∈ G_{J}}. In particular, R^{[J]} is a semi-commutative graded algebra.

POINCARÉ SERIES OF MULTI-FILTERED ALGEBRAS AND PARTITIVITY 3

Proof. Pick r ∈ R^{[J]} an homogeneous element with deg(r) = α ∈ G_{J}. Since R is generated by the semi-commuting elements x_{1}, …, x_{s}, we can write r = ∑_{β∈ℕ^{s}} a_{β}x_{1}^{β_{1}} ⋯ x_{s}^{β_{s}} for some a_{β} ∈ k. But the x_{i}’s are homogeneous, so that for every β ∈ ℕ^{s} such that a_{β} ≠ 0, we have that α = deg(x_{1}^{β_{1}} ⋯ x_{s}^{β_{s}}) = ∑_{i=1}^{s} β_{i} deg(x_{i}). Now, β_{i} ≠ 0 does imply deg(x_{i}) ∈ G_{J}, as desired.

1.4. For every α = (α_{1}, …, α_{n}) ∈ ℕ^{n}, define α|J = (a_{1}, …, a_{n}) ∈ G_{J} by requiring that a_{i} = 0 if i ∈ J and a_{i} = α_{i} if i ∉ J. Every ℕ^{n}–graded left R–module M = ⊕_{α∈ℕ^{n}}M_{α} is a left R^{[J]}–module, by restriction of scalars. For every γ ∈ G_{J}, consider the vector subspace of M given by M_{(γ)} = ⊕_{α|J=γ} M_{α}. The decomposition M = ⊕_{γ∈G_{J}} M_{(γ)} is a G_{J}–grading for the R^{[J]}–module M. For each μ ∈ ℕ^{n} set S_{J,μ} = {α ∈ ℕ^{n} | α_{j} < μ_{j} ∀j ∈ J}. The vector space

M^{[J],μ} = ⨁_{α∈S_{J,μ}} M_{α}

is a submodule of the left R^{[J]}–module M (we consider M^{[J],μ} = 0 when S_{J,μ} is empty). It is not difficult to check that M^{[J],μ} = ⊕_{γ∈G_{J}} (M^{[J],μ} ∩ M_{(γ)}), which implies that M^{[J],μ} is a G_{J}–graded R^{[J]}–submodule of M.

Lemma. If M is a finitely generated ℕ^{n}–graded left R–module, then M^{[J],μ} is a finitely generated G_{J}–graded left R^{[J]}–module.

Proof. It is easy to prove that (L + N)^{[J],μ} = L^{[J],μ} + N^{[J],μ} for L, N any graded R– submodules of M; and so we can assume that M is cyclic, generated by an homogeneous element m, say. If M^{[J],μ} ≠ 0 then there is a nonzero homogeneous r in R such that 0 ≠ rm ∈ M^{[J],μ}. Therefore deg(r)+deg(m) = deg(rm), and, thus, deg(m)_{j} < μ_{j} for every j ∈ J. This implies that m ∈ M^{[J],μ}. Since R is generated as an algebra by homogeneous semi-commuting elements x_{1}, …, x_{s} it follows that the left R–module Rm = M^{[J],μ} is spanned as vector space over k by the subset Y consisting of the elements of the form x_{1}^{b_{1}} ⋯ x_{s}^{b_{s}}m such that ∑_{k=1}^{s} b_{k} deg(x_{k})_{j} + deg(m)_{j} < μ_{j} for every j ∈ J. Let X be the subset of Y consisting of those x_{1}^{b_{1}} ⋯ x_{s}^{b_{s}}m such that b_{i} = 0 for every x_{i} ∈ R^{[J]}. As the elements x_{i} are semi-commuting, it follows that X generates M as a left R^{[J]}–module. It is clear that X is a finite set.

We are now ready to give a proof of the theorem.

Theorem 1.5. Let M be a finitely generated ℕ^{n}–graded left module over a semi-commutative ℕ^{n}–graded k-algebra R. Let x_{1}, …, x_{s} be a set of semi-commuting homogeneous generators of R, with deg(x_{j}) = μ^{j} ∈ ℕ^{n} for j = 1, …, s. There exists a polynomial f(\underline{t}) ∈ ℤ[t_{1}, …, t_{n}] such that

P (M; \underline{t}) = \frac{f(\underline{t})}{∏_{j=1}^{s}(1 − \underline{t}^{μ^{j}})}

4 J. GÓMEZ TORRECILLAS AND T. H. LENAGAN

Proof. We proceed by induction on s. For any n and for s = 0, the algebra R is just k and the module M is finite-dimensional, so that the Poincaré series is a polynomial. Now, let s > 0. Assume that the result is true for any n and s′ < s. Write μ = deg(x_{s}) and consider the subalgebra B of R generated by x_{1}, … x_{s−1}. Since the semi-commuting generators of B are homogeneous, it is clear that B is ℕ^{n}-graded in a straightforward way, and, in fact, the inductive hypothesis applies to B, since B has fewer than s semi-commuting generators. Now, for an ℕ^{n}–graded left R–module M consider the k–linear graded map M → M(μ) given by multiplication by x_{s}. Here, M(μ) denotes the μ–shifting of M. The fact that x_{s} semi-commutes with the rest of the generators of R gives that the kernel K and the image L of this mapping are R–submodules of M and M(μ), respectively. Hence,

P (M(μ); \underline{t}) − P (M; \underline{t}) = P (C; \underline{t}) − P (K; \underline{t})

where C is the cokernel of our mapping M → M(μ). Now, K and C are finitely generated graded left R–modules, since R is a noetherian algebra, and, they are finitely generated graded left B–modules, since x_{s} annihilates both of them. By induction on s it follows that

P (C; \underline{t}) − P (K; \underline{t}) = \frac{g(\underline{t})}{∏_{j=1}^{s−1}(1 − \underline{t}^{μ^{j}})}

for some polynomial g(\underline{t}) ∈ ℤ[t_{1}, …, t_{n}]. Call this rational function q(\underline{t}); so that

P (M(μ); \underline{t}) = P (M; \underline{t}) + q(\underline{t}).

On the other hand,

(1) P (M(μ); \underline{t}) = ∑_{β∈ℕ^{n}} d_{M(μ)}(β)\underline{t}^{β}

= ∑_{β∈ℕ^{n}} d_{M}(β + μ)\underline{t}^{β} = ∑_{β+μ} d_{M}(β + μ)\underline{t}^{β+μ}\underline{t}^{−μ}

= \underline{t}^{−μ} ( ∑_{α∈ℕ^{n}} d_{M}(α)\underline{t}^{α} − ∑_{α∉μ+ℕ^{n}} d_{M}(α)\underline{t}^{α})

= \underline{t}^{−μ} (P (M; \underline{t}) − P (M; μ, \underline{t})) ,

where

P (M; μ, \underline{t}) = ∑_{α∉μ+ℕ^{n}} d_{M}(α)\underline{t}^{α}.

By comparing equation (1) with the earlier formula for P (M(μ); \underline{t}), we see that

(1 − \underline{t}^{μ})P (M; \underline{t}) = \underline{t}^{μ}q(\underline{t}) + P (M; μ, \underline{t}). (2)

POINCARÉ SERIES OF MULTI-FILTERED ALGEBRAS AND PARTITIVITY 5

It remains to show that P (M; μ, \underline{t}) can be written as a rational function of the required form; that is, P (M; μ, \underline{t}) can be written as a rational function with denominator ∏_{j=1}^{s−1}(1 − \underline{t}^{μ^{j}}). From the formula

∑_{J⊆{1,...,n}}(−1)^{|J|} = 0,

where |J| denotes the cardinal of J, we get

P (M; μ, \underline{t}) = ∑_{∅≠J⊆\{1,...,n\}}(−1)^{|J|+1}P (M^{[J],μ}; \underline{t}) (3)

Consider a fixed nonempty subset J, and suppose that x_{s} ∈ R^{[J]}. Then μ_{i} = 0, for each i ∈ J, where μ = (μ_{1}, …, μ_{n}) = deg(x_{s}). Recall that

M^{[J],μ} = ⨁_{α∈S_{J,μ}} M_{α},

where

S_{J,μ} = {α ∈ ℕ^{n} | α_{j} < μ_{j} ∀j ∈ J}.

Now, α ∈ S_{J,μ} means that α_{i} < μ_{i}, for each i ∈ J. Thus, α_{i} < 0, for each i ∈ J; and so M^{[J],μ} = 0. Hence, P (M^{[J],μ}; \underline{t}) = 0, whenever x_{s} ∈ R^{[J]}, and the only nonzero contributions to the right hand side of (3) occur when x_{s} ∉ R^{[J]}. In these cases, the inductive hypothesis applies, and P (M^{[J],μ}; \underline{t}) can be written as a rational function with denominator ∏_{j=1}^{s−1}(1 − \underline{t}^{μ^{j}}), since R^{[J]} is generated by a subset of \{x_{1}, …, x_{s−1}\}, by Lemma 1.3 and M^{[J],μ} is a finitely generated R^{[J]}-module, by Lemma 1.4. This fact, together with (2), proves the theorem.

1.6. For each multi-index α ∈ ℕ^{n}, we set |α| := α_{1} + ⋯ + α_{n}. Define, for m ∈ ℕ, R_{m} = ⊕_{|α|=m}R_{α}. The vector subspace decomposition R = ⊕_{m∈ℕ}R_{m} is an ℕ–grading of R. Moreover, if M is an ℕ^{n}–graded left R–module, then M is an ℕ–graded left R–module by putting M_{m} = ⊕_{|α|=m}M_{α} for every natural number m. If M is finitely generated as a left R–module, then it makes sense to define the classical Poincaré series of M with respect to the grading M = ⊕_{m∈ℕ}M_{m} by

P_{c}(M; t) = ∑_{m∈ℕ}dim_{k} M_{m}t^{m}

But it is clear that P_{c}(M; t) = P (M; t, …, t), so that Theorem 1.5 implies that this series is rational of the form

P_{c}(M; t) = \frac{f(t)}{∏_{i=1}^{s}(1 − t^{|μ^{i}|})}, (4)

where f(t) = f(t, …, t) ∈ ℤ[t].

6 J. GÓMEZ TORRECILLAS AND T. H. LENAGAN

1.7. Consider, for the graded R–module M = ⊕_{m∈ℕ}M_{m}, the mapping m ↦ g(m) = ∑_{l=0}^{m} dim_{k}(M_{l}). By [6, Lemma I.2.5], the Gelfand-Kirillov dimension of M is the degree of growth of the function g(m). Now, the series P (t) = ∑_{m∈ℕ} g(m)t^{m} = P_{c}(M; t)/(1 − t). By (4) and [6, Lemma III.1.3] there exists a natural number ν and there exist functions c_{l} : ℤ/νℤ → ℚ (l = 0, 1, …, d) with c_{d} constant and positive such that g(m) = ∑_{l=0}^{d} c_{l}(m+ νℤ)m^{l}, for m big enough. In fact, ν is the least common multiple of |μ^{1}|, …, |μ^{s}|, so that it does not depends on the module M. Moreover, d + 1 is the order of the pole of P (t) at t = 1. In particular, we obtain the following well-known result.

Proposition 1.8. The Gelfand-Kirillov dimension of any finitely generated ℕ^{n}–graded left R–module is the order of the pole at t = 1 of the Poincaré series P_{c}(M; t).

1.9. Keep the notation of 1.7. Put h(m) = g(νm), for every m ∈ ℕ. The function h is a polynomial of degree d with rational coefficients, for sufficiently large m. Moreover, the leading coefficient of h(m) is ν^{d}c_{d}. By [5, Lemma 1.5], d!ν^{d}c_{d} is a positive integer.

Definition. The positive integer e(M) = d!ν^{d}c_{d} will be called the multiplicity of the ℕ^{n}–graded left R–module M.

2. Finitely partitive algebras

In this section, R will denote an algebra over the field k. We do not assume that R is graded nor semi-commutative. We recall the definitions of multi-filtered algebras and modules, in the sense of [3].

Definition 2.1. An admissible order ‘≤’ on (ℕ^{n}, +) is a total order such that,

(a) 0 = (0, …, 0) ≤ α for every α ∈ ℕ^{n}.

(b) For all α, β, γ ∈ ℕ^{n} with α ≤ β it follows α + γ ≤ β + γ.

Remark 2.2. By Dickson’s Lemma (see, e.g., [2, Corollary 4.48]), admissible orders on ℕ^{n} are good orders; that is, any non empty subset of ℕ^{n} has a first element.

¿From now on, fix an admissible order ‘≤’ on (ℕ^{n}, +).

Definition 2.3. A multi-filtration on R is a family F = {F_{γ}(R)|γ ∈ ℕ^{n}} of vector subspaces of R satisfying the following axioms.

1. F_{γ}(R) ⊆ F_{δ}(R) if γ ≤ δ.

2. F_{γ}(R)F_{δ}(R) ⊆ F_{γ+δ}(R).

3. ⋃_{α∈ℕ^{n}} F_{α}(R) = R.

4. 1 ∈ F_{0}(R).

In what follows, we assume that the algebra R is endowed with a fixed multi-filtration.

Definition 2.4. A multi-filtration on a left R–module M is a family F = {F_{γ}(M)|γ ∈ ℕ^{n}} of vector subspaces of M satisfying the following axioms.

1. F_{γ}(M) ⊆ F_{δ}(M) if γ ≤ δ.

2. F_{γ}(R)F_{δ}(M) ⊆ F_{γ+δ}(M).

3. ⋃_{α∈ℕ^{n}} F_{α}(M) = M.

The multi-filtered algebra R has an associated ℕ^{n}–graded algebra defined as follows.

Given a multi-filtered left R–module M, and γ ∈ ℕ^{n}, write

V_{γ}(M) = ⋃_{γ′<γ} F_{γ′}(M).

It is understood that V_{0} = {0}. Consider the vector space

G_{γ}(M) = F_{γ}(M)/V_{γ}(M),

and define

G(M) = G_{F}(M) = ⨁_{γ∈ℕ^{n}} G_{γ}(M).

For r + V_{γ}(R) ∈ G_{γ}(R) and m + V_{δ}(M) ∈ G_{δ}(M), the product

(r + V_{γ}(R))(m + V_{δ}(M)) = rm + V_{γ+δ}(M).

is well-defined. Thus, if M = R, then we have a product in G(R) which makes G(R) into an ℕ^{n}–graded algebra; moreover, G(M) becomes an ℕ^{n}–graded left G(R)–module.

Definition 2.5. Let F_{α}(R) be an ℕ^{n}–filtration over a k–algebra R. We say that this multi-filtration is locally finite if the vector subspaces F_{α}(R) are finite-dimensional and pointed if F_{0}(R) = k.

2.6. Assume that R is endowed with a locally finite, pointed multi-filtration. This implies that G(R) is pointed and locally finite. Moreover, suppose that G(R) is a semi-commutative multi-graded algebra in the sense of Definition 1.2. It follows from [3, Theorems 2.5 and 2.10] that R is noetherian and that Gelfand-Kirillov dimension is exact on finitely generated modules. The aim of this section is to show that Gelfand-Kirillov dimension is finitely partitive (see definition 2.8). Let M be a left R–module endowed with an ℕ^{n}–filtration F = {F_{α}(M)}. Assume that the associated ℕ^{n}–graded left G(R)–module G_{F}(M) is finitely generated. It makes sense consider the Poincaré series P (G_{F}(M; \underline{t})). By 1.6, 1.7 and 1.9 we can associate to G_{F}(M) the multiplicity when considered as ℕ–graded module. Since, once the multi-filtration on R is fixed, this multiplicity depends on the multi-filtration F in M, we will denote it by e_{F}(M). We know that it is a positive integer.

Proposition 2.7. Let 0 → M_{1} → M_{2} → M_{3} → 0 be an exact sequence of homomorphisms of R–modules. Assume given ℕ^{n}–filtrations F_{1}, F_{2} and F_{3} on M_{1}, M_{2} and M_{3}, respectively, such that the homomorphisms in the short exact sequence are filtered and that the associated sequence of ℕ^{n}–graded G(R)–modules

0 → G_{F_{1}}(M_{1}) → G_{F_{2}}(M_{2}) → G_{F_{3}}(M_{3}) → 0 (5) is exact. If G_{F_{2}}(M_{2}) is finitely generated then

e_{F_{2}}(M_{2}) = \begin{cases}e_{F_{1}}(M_{1}) + e_{F_{3}}(M_{3}) & if GKdim(M_{1}) = GKdim(M_{3}) \\ e_{F_{1}}(M_{1}) & if GKdim(M_{1}) > GKdim(M_{3}) \\ e_{F_{3}}(M_{3}) & if GKdim(M_{1}) < GKdim(M_{3}) \end{cases}

Proof. The G(R)–modules G_{F_{i}}(M_{i}) are finitely generated since G(R) is noetherian. The exactness of the sequence (5) gives

P (G_{F_{2}}(M_{2}); \underline{t}) = P (G_{F_{1}}(M_{1}); \underline{t}) + P (G_{F_{3}}(M_{3}); \underline{t}) (6)

Now consider the exact sequence (5) as a sequence of ℕ–graded G(R)–modules (see 1.6). If g_{i}(m) denotes the growth function of G_{F_{i}}(M_{i}) (see 1.7), then, by (6), g_{2}(m) = g_{1}(m)+g_{3}(m) for m big enough. By [6, Lemma III.1.3] (see also 1.7), there is a natural number ν and functions c_{l}^{i} : ℤ/νℤ → ℚ, for l = 0, 1, …, d_{i} and i = 1, 2, 3 such that g_{i}(m) = ∑_{l=0}^{d_{i}} c_{l}^{i}(m + νℤ)m^{l}, where d_{i} = GKdim(G_{F_{i}}(M_{i})). By [3, Theorems 2.8, 2.10] GKdim(M_{i}) = d_{i} for i = 1, 2, 3 and d_{2} = max{d_{1}, d_{3}}. The result can be easily deduced now.

Definition 2.8. Let R be an affine algebra over k of finite Gelfand-Kirillov dimension. The algebra R is said to be left finitely partitive if for every finitely generated left R–module M there is an integer e > 0 such that, for every strictly descending chain

M = M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{m} (7)

of submodules with GKdim(M_{i}/M_{i+1}) = GKdim(M), it follows m ≤ e.

Theorem 2.9. Let R be an ℕ^{n}–filtered algebra such that F_{0}(R) = k. Assume that the multi-filtration is finite and that G(R) is a semi-commutative ℕ^{n}-graded algebra. Then R is left and right partitive.

Proof. By [3, Corollary 2.12], the Gelfand-Kirillov dimension of R is a non-negative integer. Let M = M_{0} be a nonzero finitely generated left R–module, and consider a descending chain as in (7). Define recursively a locally finite ℕ^{n}–filtration F_{i} = {F_{α}(M_{i}) | α ∈ ℕ^{n}} on M_{i} for every i = 1, …, m as follows. As the first step, the multi-filtration F_{0} can be given by F_{α}(M) := F_{α}(R)U, for U a finite-dimensional generating subspace of M. Now assume, for some i ≥ 0, that F_{α}(M_{i}) has been defined for every α ∈ ℕ^{n}. Define F_{α}(M_{i+1}) := M_{i+1} ∩ F_{α}(M_{i}). Analogously, we will define locally finite filtrations G_{i+1} = {F_{α}(M_{i}/M_{i+1}) | α ∈ ℕ^{n}}, for each M_{i}/M_{i+1}, 0 ≤ i < m. This filtration is given by F_{α}(M_{i}/M_{i+1}) := (F_{α}(M_{i}) + M_{i+1})/M_{i+1}, for every α ∈ ℕ^{n} and for 0 ≤ i < m. For each i ∈ {0, …, m − 1} the sequence of associated ℕ^{n}–graded G_{F}(R)–modules

0 → G_{F_{i+1}}(M_{i+1}) → G_{F_{i}}(M_{i}) → G_{G_{i+1}}(M_{i}/M_{i+1}) → 0

is exact. Moreover, since G_{F}(R) is a noetherian algebra and G_{F_{0}}(M) is a finitely generated left G_{F(R)}–module, it follows that G_{F_{i}}(M_{i}) is finitely generated for every i ∈ {0, …, m}. By Proposition 2.7, we have that the associated multiplicities satisfy

e_{F_{i}}(M_{i}) = e_{F_{i+1}}(M_{i+1}) + e_{G_{i+1}}(M_{i}/M_{i+1})

for every i < m. Therefore

e_{F_{0}}(M) = ∑_{i=0}^{m−1} e_{G_{i+1}}(M_{i}/M_{i+1}) + e_{F_{m}}(M_{m})

The fact that the multiciplities of nonzero modules are positive integers implies that if we put e = e_{F_{0}}(M) then m ≤ e.

Remark 2.10. The Krull dimension of every module over a finitely partitive algebra is bounded by its Gelfand-Kirillov dimension, see, for example, [9, Corollary 8.4.9], and this last is an integer when R satisfies the hypotheses of Theorem 2.9.

3. Examples

This section is devoted to providing some non-trivial examples of multi-filtered algebras by finite filtrations having a pointed semi-commutative associated graded algebra.

Definition 3.1. An admissible order ≤ on ℕ^{n} is called locally finite if the interval [0, α] = {γ ∈ ℕ^{n} : γ ≤ α} is finite for every α ∈ ℕ^{n}.

A source of examples is given by the following proposition.

Proposition 3.2. [3, Proposition 3.2] Let R be a k–algebra finitely generated by x_{1}, …, x_{n} and set X^{α} = x_{1}^{α_{1}} … x_{n}^{α_{n}} for all α ∈ ℕ^{n}. If there is a locally finite admissible order ‘≤’ and nonzero scalars q_{ji} ∈ K such that

x_{j}x_{i} = q_{ji}x_{i}x_{j} + ∑_{γ<ϵ_{i}+ϵ_{j}} c_{γ}X^{γ}

where ϵ_{i} = (0, …, 1^{(i)}, …, 0), then

1. R is endowed with a locally finite multi-filtration given by

F_{α}(R) = ∑_{β≤α} kX^{β}

2. G(R) is a semi-commutative algebra with locally finite associated filtration. In particular, R is a finitely partitive algebra.

This proposition applies, [3, Example 3.3], to any iterated Ore extension of the form R = k[x_{1}][x_{2}; σ_{2}, δ_{2}] … [x_{n}; σ_{n}, δ_{n}], satisfying σ_{j}(x_{i}) = q_{ij}x_{i}, for every i < j ≤ n, where q_{ij}’s are nonzero scalars in k. Some examples of these iterated Ore extensions are the algebras H(p, λ) defined in [1], which include the quantum coordinate algebras of M_{n}(k); and the multiparameter quantized Weyl algebra R = A_{n}^{Q,Γ}(k) introduced in [7] (see also [4]). The iterated differential operator algebras of [10] are also covered. It can be also shown [3, Example 3.4] that the quantized universal enveloping algebra U_{q}(𝔰𝔩(ν + 1)), with q^{8} ≠ 1 can be endowed with a finite filtration with pointed semi-commutative associated graded algebra.

4. Filtrations induced by multifiltrations

Let F be a multifiltration on the k-algebra R. One might ask whether there is an ℕfiltration E that can be induced from F in such a way that the associated graded algebras G_{E}(R) and G_{F}(R) would be isomorphic. If so, this would simplify some calculations. There is an obvious way to try to achieve this aim. Let R be an algebra endowed with an ℕ^{n}-filtration F = {F_{α}(R) | α ∈ ℕ^{n}}. Let T = G_{F}(R) = ⊕_{α∈ℕ^{n}} T_{α} be the associated ℕ^{n}-graded algebra. For each nonzero r ∈ R, we define exp(r) = min{α ∈ ℕ^{n} | r ∈ F_{α}(R)}, and put in_{F}(r) = r + V_{exp(r)}(R), the initial form of r with respect to F.

For each multi-index α ∈ ℕ^{n}, we set |α| := α_{1} + ⋯ + α_{n}. If we set

R_{m} = {r ∈ R | in_{F}(r) ∈ ⊕_{|k|≤m} T_{k}},

for each m ∈ ℕ, then one might hope that {R_{m}} defines a filtration on R. However, this is not the case in general, the problem being due to possible failure of the implication: α < β ⇒ |α| ≤ |β|, as the following example shows.

Consider the ordering, <, on ℕ^{2}, given by

(α_{1}, α_{2}) < (β_{1}, β_{2}) ⇔ \{ \begin{matrix}α_{1} + 2α_{2} < β_{1} + 2β_{2} or \\ α_{1} + 2α_{2} = β_{1} + 2β_{2} and α_{2} < β_{2} \end{matrix} .

Then (3, 1) < (0, 3), while |(3, 1)| > |(0, 3)|.

A weighted admissible order on ℕ^{n}, is defined as follows, see, for example, [3]. Let w = (w_{1}, …, w_{n}) be a vector in ℝ^{n} with w_{i} > 0 for every i = 1, …, p. Consider the usual dot product ⟨−, −⟩ on ℝ^{n}. By <_{lex} we will denote the lexicographic order in ℕ^{n} with ϵ_{1} < ϵ_{2} < ⋯ < ϵ_{n}. The binary relation in ℕ^{n} defined for α, β ∈ ℕ^{n} as α <_{w} β if ⟨α, w⟩ < ⟨β, w⟩ or ⟨α, w⟩ = ⟨β, w⟩ and α <_{lex} β is a locally finite admissible order in ℕ^{n}. We will say that w is the weight vector for the order <_{w}.

Note that, in the situation of Proposition 3.2, locally finite admissible orders lead to locally finite multifiltrations.

Assume that we have a weighted ordering ≤_{w} on ℕ^{n}, with w ∈ ℕ^{n}, and let R be an algebra endowed with an ℕ^{n}-filtration F = {F_{α}(R) | α ∈ ℕ^{n}}. Let T = G_{F}(R) = ⊕_{α∈ℕ^{n}} T_{α} be the associated ℕ^{n}-graded algebra. For each m ∈ ℕ, define T_{m} = ⊕_{|α|_{w}=m} T_{α}, where |α|_{w} = ⟨w, α⟩ = ∑_{i} w_{i}α_{i}. Then T = ⊕_{m∈ℕ} T_{m} is an ℕ-graded algebra. In general, exp(rs) ≤_{w} exp(r) + exp(s) and, in fact,

exp(rs) <_{w} exp(r) + exp(s) ⇔ rs ∈ V_{exp(r)+exp(s)},

for each r, s ∈ R. Observe that

|exp(rs)|_{w} ≤_{w} |exp(r) + exp(s)|_{w} = |exp(r)|_{w} + |exp(s)|_{w}.

Now, we use the multifiltration F to induce an ℕ-filtration on R. If we set

R_{m} = {r ∈ R | in_{F}(r) ∈ ⊕_{k≤m} T_{k}} = {r ∈ R | |exp(r)|_{w} ≤ m},

for each m ∈ ℕ, then {R_{m}}_{m∈ℕ} is a filtration on R, by virtue of the second description in the display above.

Denote by S = ⊕_{m∈ℕ} S_{m} the associated graded algebra for this filtration; that is,

S_{m} = \frac{R_{m}}{R_{m−1}} .

Thus, for each m ∈ ℕ, we have

T_{m} = ⨁_{|α|=m} \frac{F_{α}}{V_{α}} , and S_{m} = \frac{⋃_{|α|_{w}≤m} F_{α}(R)}{⋃_{|α|_{w}≤m−1} F_{α}(R)}.

Unfortunately, even in this setting, the new associated graded algebra S may fail to be semi-commutative, even if the associated multigraded algebra T is commutative. The following example shows that the graded algebras S and T are not necessarily isomorphic and illustrates the above point.

Example 4.1. Let R be the algebra over the field k generated by indeterminates X, Y, subject to the relation XY = Y X + X^{2}. Thus, R is the co-ordinate ring of the Jordan plane. Consider the ordering, ≤_{grlex}, on ℕ^{2}, given by

(α_{1}, α_{2}) ≤_{grlex} (β_{1}, β_{2}) ⇔ \{ \begin{matrix}α_{1} + α_{2} < β_{1} + β_{2} or \\ α_{1} + α_{2} = β_{1} + β_{2} and α_{2} ≤ β_{2} \end{matrix} .

Thus, the ordering begins

(0, 0) <_{grlex} (1, 0) <_{grlex} (0, 1) <_{grlex} (2, 0) <_{grlex} (1, 1) <_{grlex} (0, 2) <_{grlex} ….

In fact, the ordering ≤_{grlex} is the same as the ordering deteermined by the weighting (1, 1). The algebra R is ℕ^{2}-filtered by

F_{α}(R) = ∑_{β≤_{grlex} α} kx^{β_{1}}y^{β_{2}}

and T = G_{F}(R) is commutative; in fact, T is just the polynomial algebra in two variables over k.

However, we will show that the ring S is not commutative. Note that S_{0} = k, while

S_{1} = \frac{F_{(0,1)}(R)}{k} = \frac{k + kX + kY}{k}

and

S_{2} = \frac{F_{(0,2)}(R)}{F_{(0,1)}(R)} = \frac{k + kX + kY + kX^{2} + kXY + kY^{2}}{k + kX + kY} .

Set x = X+S_{0} ∈ S_{1} and y = Y +S_{0} ∈ S_{1}. Then yx = Y X+S_{1} = (XY +X^{2})+S_{1} = xy+x^{2}, and x^{2} ≠ 0. Thus, S is not commutative. In fact, one can check that S ≅ R.

In connection with this example, it is interesting to note that even beginning with a graded algebra (R is graded, since the defining relation is homogeneous), a multifiltration can be used to produce an associated graded algebra with nicer properties.

References

[1] M. Artin, W. Schelter, and J. Tate, Quantum deformations of GL_{n}, Comm. Pur. Appl. Math. 44 (1991), 879–895.

[2] T. Becker and V. Weispfenning, Gröbner basis. A computational approach to commutative algebra, Springer, New York, 1993.

[3] J. Gómez Torrecillas, Gelfand–Kirillov dimension of multi-filtered algebras., P. Edinburgh Math. Soc (to appear).

[4] K.R. Goodearl and T.H. Lenagan, Catenarity in quantum algebras, J. Pure Appl. Algebra 111 (1996), 123–142.

[5] G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman Pub. Inc., London, 1985.

[6] M. Lorenz, Gelfand-Kirillov dimension and Poincaré series, Cuadernos de Algebra, vol. 7, Universidad de Granada, 1988.

[7] G. Maltsiniotis, Calcul difféntiel quantique, Université de Paris, VII, 1992.

[8] J.C. McConnell Quantum groups, filtered rings and Gelfand-Kirillov dimension, Noncommutative ring theory (Athens, OH, 1989), 139–147, Lecture Notes in Math., 1448 , Springer, Berlin, 1990.

[9] J.C. McConnell and J.C. Robson, Noncommutative noetherian rings, J. Wiley and Sons, Chichester, New York, 1998.

[10] G. Sigurdsson, Ideals in universal enveloping algebras of solvable Lie algebras, Comm. Algebra 15 (1987), 813–826.

Dept. Algebra, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain

Dept. Mathematics, University of Edinburgh, J.C.M.B., Kings Buildings, Mayfield Road, Edinburgh EH9 3JC, Scotland, U.K.

  E-mail address: torrecil@ugr.es
  E-mail address: tom@mathematics.edinburgh.ac.uk