Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 351-363.

Gelfand-Kirillov Dimension in some Crossed Products

Thomas Guédénon^{∗}

U.F.R. de Mathématiques, Université des Sciences et Technologies de Lille 59655 Villeneuve d’Ascq, France

Abstract. Let k be a field, R a k-algebra and A = R[θ_{1}, θ_{2}, …, θ_{n}] a Poincaré-Birkhoff-Witt extension of R. If each θ_{i} acts locally finitely on R, then we show that GKdim(A) = GKdim(R) + n. From this we deduce some results concerning incomparability, prime length and Tauvel’s height formula in the crossed product R⋆g where g is a finite-dimensional Lie algebra acting as derivations on R. Similar results are obtained for R ⋆ G, where G is a free abelian group of finite rank. As a corollary of the results for G, Tauvel’s height formula is established in R ⊗_{k} P (λ) where P (λ) is the quantum torus.

0. Introduction

All rings (except the Lie algebras) in this paper are associative with identity. When we say that a ring is noetherian we mean left and right noetherian.

For the basic material concerning the Gelfand-Kirillov dimension (denoted by GKdim) we refer to [8]. We refer the reader to [5, 7, 12] for some definitions and undefined terminologies.

Let k be a field. We say that Tauvel’s height formula holds in a noetherian k-algebra A with finite Gelfand-Kirillov dimension provided

ht(P) + GKdim(A/P) = GKdim(A)

for any prime ideal P of A where ht(P) denotes the height of P.

Throughout, we fix a field k, an algebra R over k and a positive integer n.

∗Permanent address: 21, Avenue de Versailles 75016 Paris, France; e-mail: guedenon@caramail.com 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Let g be a k-Lie algebra of finite dimension n. We suppose that g acts by derivations on R and we denote by A = R ⋆ g the crossed product of R by the enveloping algebra U(g) of g (see [1, 12, 13]).

For any g-prime ideal P of R, g-ht(P) denotes the g-height of P; i.e.

sup{s : there is a chain of g-prime ideals P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{s} = P of R}.

We denote by spec_{g}(R) the set of g-prime ideals of R. If k has characteristic zero and if R is noetherian or g-locally finite with g solvable, any g-prime ideal of R is prime [2, Proposition 4.4]; so g-ht(P) ≤ ht(P).

If X ∈ g, we denote by X̄ the canonical image of X in R ⋆ g and we set δ(X) = δ_{X}. We recall that there exist a linear map δ from g to the k-Lie algebra of k-derivations of R and a bilinear map t : g × g → R such that [X̄, Ȳ] − \overline{[X, Y]} = t(X, Y). If t is the 0-map, R ⋆ g is usually denoted by R#U(g); in this case R is a g-module.

We say that condition (C_{1}) (resp. (C_{2})) is satisfied in spec_{g}(R) if Tauvel’s height formula holds for all g-prime ideals of R (resp. if the g-height and the height are finite and coincide on the g-prime ideals of R).

Let R be finitely generated over k and A = R[θ_{1}, θ_{2}, …, θ_{n}] a PBW extension of R. In [11], J. Matczuk showed that GKdim(A) = GKdim(R) + n. This result generalizes [12, 8.2.10] and from this, he deduced some results concerning incomparability and prime length. On the other hand a result of M. Lorenz [10] states that if R is δ-locally finite, then GKdim(R[θ]) = GKdim(R) + 1 where δ is the derivation of R determined by θ.

In the first section of this paper, we show that all the results in [11] are also true if we replace the “finitely generated” hypothesis on R by the assumption that each θ_{i} acts locally finitely on R. This result generalizes [10] and it enables us to obtain the results in [11] with their g-invariant version in the ring R ⋆ h if R is g-locally finite or finitely generated, where h is an ideal of g. If further k is of characteristic 0, g is completely solvable, condition (C_{1}) is satisfied in spec_{g}(R) and R is g-hypernormal, we have improved the preceding results. To close this section, we show that if R is noetherian of finite Gelfand-Kirillov dimension and if conditions (C_{1}) and (C_{2}) hold in spec_{g}(R), then conditions (C_{1}) and (C_{2}) also hold in spec_{g}(R ⋆ g_{i}) for each i; where the g_{i} are terms of a composition series of g (see the definition after Corollary 1.11).

In the second section, G is a free abelian group of finite rank n. We prove similar statements as in Section 1.

1. When the action comes from a Lie algebra

In this section the characteristic of k is arbitrary. A Poincaré-Birkhoff-Witt extension (or PBW extension) of R is a ring extension A ⊇ R generated by elements {θ_{1}, …, θ_{n}} such that

- [θ_{i}, r] ∈ R for each i = 1, 2, …, n.

- [θ_{i}, θ_{j}] ∈ R + Rθ_{1} + Rθ_{2} + ⋯ + Rθ_{n}.

- A is free as (left and right) R-module with basis θ_{1}^{i_{1}}θ_{2}^{i_{2}}...θ_{n}^{i_{n}}; where the i_{j} are positive integers [11].

The Poincaré-Birkhoff-Witt theorem implies that any crossed product R ⋆ g is a PBW extension of R. We will denote by δ_{i} the derivation of R determined by θ_{i}; i.e. δ_{i}(r) = [θ_{i}, r] for all r ∈ R and we will set Δ = {δ_{1}, …, δ_{n}}. We say that R is Δ-locally finite if every element of R is contained in a finite-dimensional Δ-stable subspace of R.

Let A = R[θ_{1}, θ_{2}, …, θ_{n}] be a PBW extension of R. Our aim in the next theorem is to give a relation between the Gelfand-Kirillov dimension of R and A.

Theorem 1.1. Suppose that A = R[θ_{1}, θ_{2}, …, θ_{n}] is a PBW extension of R. If R is Δ-locally finite, then GKdim(A) = GKdim(R) + n.

Proof. Denote by X the k-linear subspace of A spanned by 1, θ_{1}, θ_{2}, …, θ_{n}. For any l ≥ 1, let A_{l} ⊆ X^{l} denote the subspace spanned by all monomials of the form θ_{1}^{i_{1}}θ_{2}^{i_{2}}...θ_{n}^{i_{n}} where i_{1} + i_{2} + ⋯ + i_{n} ≤ l, 0 ≤ i_{j} for j = 1, …, n.

Let W be a finite-dimensional k-vector subspace of A. Using locally finite-dimensionality of Δ, we can choose a finite-dimensional Δ-invariant subspace V of R such that W ⊆ ∑_{i_{1}+i_{2}+···+i_{n}≤t} V θ_{1}^{i_{1}}θ_{2}^{i_{2}}...θ_{n}^{i_{n}} for some t ∈ ℕ where the i_{j}′s ∈ ℕ and [θ_{i}, θ_{j}] ∈ V + V θ_{1} + ⋯ + V θ_{n} = V X for all 0 ≤ i, j ≤ n. Thus, eventually replacing V by k.1 + V, we may assume that 1 ∈ V. Now, it is clear that each power of V is Δ-stable and V^{l} ⊆ V^{l+1} for any l ∈ ℕ. We can show as in the proof of [11, Theorem A] that between the subspaces V, X and A_{l} the following relations

X^{t}V ⊆ V X^{t} + V^{2}X^{t−1} + ⋯ + V^{t}X + V^{t+1}

and

X^{t} ⊆ V^{t−1}A_{t}

hold for any t ≥ 1. From these relations, we deduce that W^{′t} ⊆ V^{t}X^{t} for any t ≥ 1; where W′ = X + V ⊆ V X. Now the same argument as in [11] shows that GKdim(k[W′]) ≤ GKdim(R) + n. It is clear that W ⊆ k[W′]; so GKdim(k[W]) ≤ GKdim(R) + n. This means that GKdim(A) ≤ GKdim(R) + n.

Let V′ be a finite-dimensional k-vector subspace of R. Since R is Δ-locally finite, we can find a finite-dimensional Δ-stable subspace V of R such that V′ ⊆ V and [θ_{i}, θ_{j}] ∈ V + V θ_{1} + ⋯ + V θ_{n} = V X for all 0 ≤ i, j ≤ n. We may suppose that 1 ∈ V. Set W = V + X ⊆ V X. It is easy to see that V^{′t}A_{t} ⊆ V^{t}A_{t} ⊆ W^{2t} for any t ≥ 1. This implies that

GKdim(k[V′]) + n ≤ GKdim(k[V]) + n ≤ GKdim(k[W]) ≤ GKdim(A).

It follows that GKdim(R) + n ≤ GKdim(A).

Remark 1.2. All the results in [11] remain true if we replace the “finitely generated” hypothesis on R by the assumption that R is Δ-locally finite.

Let b ∈ R. Then U(g).b is a g-stable k-vector subspace of R. We say that b is g-finite if U(g).b has a finite dimension. It is clear that an element of R ⋆ h is g-finite if and only if it is g/h-finite. If b and b′ are two g-finite elements of R, then bb′ is a g-finite element of R. We say that R is g-locally finite if all its elements are g-finite.

The enveloping algebra of an ideal h of g is g-locally finite for the left adjoint action of g. If R is g-locally finite, any g-invariant factor of R is g-locally finite. If g acts trivially on R, then R is g-locally finite.

Let (X_{1}, X_{2}, …, X_{n}) be a basis of g. Then R is g-locally finite if and only if R is Δ-locally finite, where Δ = {δ_{X_{i}}; 1 ≤ i ≤ n}.

Lemma 1.3. Let h be an ideal of g of dimension l and let Y ∈ h.

(1) If the k-bilinear map t is identically zero, then Ȳ is a g-finite element of R#U(h).

(2) If R is g-locally finite, then Ȳ is a g-finite element of R ⋆ h.

Proof. Let (Y_{1}, Y_{2}, …, Y_{l}) be a basis of h and X any element of g.

(1) We have δ_{X}(Ȳ) = \overline{[X, Y]} ∈ \overline{kY_{1} + ⋯ + kY_{l}} = k\overline{Y_{1}}+⋯+k\overline{Y_{l}}. It is clear that k\overline{Y_{1}}+⋯+k\overline{Y_{l}} is a finite-dimensional g-stable subspace of R#U(h).

(2) The image Imt of t is a finite-dimensional subspace of R. Since R is g-locally finite, Imt is contained in a finite-dimensional g-stable subspace V of R. Now, δ_{X}(Ȳ) = \overline{[X, Y]}+t(X, Y) ∈ \overline{kY_{1} + ⋯ + kY_{l}} + V = k\overline{Y_{1}} + ⋯ + k\overline{Y_{l}} + V and k\overline{Y_{1}} + ⋯ + k\overline{Y_{l}} + V is a finite-dimensional g-stable subspace of R ⋆ h.

Corollary 1.4. Let h be an ideal of g. If R is g-locally finite, then so is R ⋆ h.

The following is a generalization of the well known result [12, 8.2.7] which asserts that GKdim(R ⊗_{k} U(g)) = GKdim(R) + n.

Corollary 1.5. Let h be an ideal of g of dimension l. Suppose that R is g-locally finite. Then GKdim(R ⋆ h) = GKdim(R) + l.

Proof. R is h-locally finite, since it is g-locally finite. The result follows from Theorem 1.1.

As an application of Corollary 1.5 we shall show some results concerning incomparability, prime length and Tauvel’s height formula. We denote by dim the classical Krull dimension and by g-dim its g-invariant version; i.e. the maximal length of a chain of g-prime ideals of R. If k is of characteristic zero and if R is noetherian or g-locally finite with g solvable, then g-dim(R) ≤ dim(R).

Corollary 1.6. Let R be noetherian of finite Gelfand-Kirillov dimension, h an ideal of g of dimension l, A = R ⋆g and B = R ⋆h. Suppose that R is finitely generated or g-locally finite.

(1) Let k be of characteristic zero and P be a g-prime ideal of B such that P ∩ R = 0. Then g-ht(P) ≤ ht(P) ≤ l. If R is g-simple, then g-dim(B) ≤ l.

(2) Let P be a prime ideal of A such that P ∩ R = 0. Then ht(P) ≤ n. If R is g-simple, then dim(A) ≤ n.

Proof. (1) The lower bound was previously noted. Since R = R/(P ∩ R) is a subalgebra of B/P, we have GKdim(R) ≤ GKdim(B/P). Corollary 1.5 in the g-locally finite case ([12, 8.2.10] in the finitely generated case) implies that GKdim(B) − GKdim(B/P) ≤ l. By [8, Corollary 3.16], ht(P) ≤ l.

If R is g-simple, by the preceding paragraph, ht(Q) ≤ l for any g-prime ideal Q of B. Thus g-dim(B) ≤ l.

(2) Note that the g-prime ideals of R ⋆ g are precisely its prime ideals and that in (1) we have assumed char(k) = 0 to be sure that all the g-prime ideals of B = R ⋆ h are prime.

The next result bounds g-dim(B) in terms of dim_{k}(h) and of g-dim(R). Although, the bound is surely not sharp.

Proposition 1.7. Let R be noetherian of finite Gelfand-Kirillov dimension, h an ideal of g of dimension l, A = R ⋆ g and B = R ⋆ h. Suppose that R is finitely generated or g-locally finite.

(1) Let k be of characteristic zero. Suppose that P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{l+1} is a strictly increasing chain of g-prime ideals of B, then P_{0} ∩ R ⊂ P_{l+1} ∩ R and g-dim(B) < (l + 1)(gdim(R) + 1).

(2) Suppose that P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{n+1} is a strictly increasing chain of prime ideals of A, then P_{0} ∩ R ⊂ P_{n+1} ∩ R and dim(A) < (n + 1)(g-dim(R) + 1).

Proof. (1) Suppose that P_{0} ∩ R = P_{l+1} ∩ R = I. Then I is a g-prime ideal of R and B/IB ≃ (R/I) ⋆ h. Set R̄ = R/I and B̄ = B/IB. In B̄, we have a strictly increasing chain of g-prime ideals \overline{P_{0}} ⊂ \overline{P_{1}} ⊂ ⋯ ⊂ \overline{P_{l+1}} of length l + 1 such that \overline{P_{0}} ∩ R̄ = \overline{P_{l+1}} ∩ R̄ = Ī = 0; where \overline{P_{i}}’s denote the natural images of P_{i}’s in B̄. It follows that g-ht(\overline{P_{l+1}}) ≥ l + 1. By Corollary 1.6(1), g-ht(\overline{P_{l+1}}) ≤ l and we get a contradiction.

Let P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{s} be a strictly increasing chain of g-prime ideals of B. By the preceding paragraph,

P_{0} ∩ R ⊂ P_{l+1} ∩ R ⊂ P_{2(l+1)} ∩ R ⊂ P_{3(l+1)} ∩ R ⊂ ⋯

is a strictly increasing chain of g-prime ideals of R. Since this chain can contain at most (1+g-dim(R)) g-prime ideals, we conclude that s < (l + 1)(g-dim(R) + 1).

(2) The proof is similar to that of (1). Use Corollary 1.6(2) in the place of Corollary 1.6(1).

Remarks 1.8. (1) Under the assumption that R is finitely generated, the results in Corollary 1.6(2) and Proposition 1.7(2) are consequences of [11]. In the g-locally finite case, they are new.

(2) If the characteristic of k is 0, Passman [14, Corollary 4.4] established (1.7(2)) in a more general setting without the assumptions that R has finite Gelfand-Kirillov dimension and is “finitely generated” or “g-locally finite”.

(3) Let g be abelian. If the characteristic of k is 0 (resp. p > 0), Chin [1, Theorem 2.11] (resp. Chin and Quinn [2, Theorem 1.10]) established the first assertion of (1.7(2)) without the assumptions that R is noetherian of finite Gelfand-Kirillov dimension and is finitely generated or g-locally finite.

(4) If the characteristic of k is 0 and g is solvable, Chin [1, Corollary 2.19] established the first assertion of (1.7(2)) for a chain of prime ideals

P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{l}

such that l ≥ 2^{n} without the assumptions that R is noetherian with finite Gelfand-Kirillov dimension and is finitely generated or g-locally finite.

The ring R is g-hypernormal if for any pair of distinct comparable g-invariant ideals I ⊂ J in R, the factor J/I contains a nonzero normal element of (R/I) ⋆ g. If R is g-simple, R is g-hypernormal. Let k be algebraically closed of characteristic 0. If g is solvable and if R is commutative g-locally finite then R is g-hypernormal.

We say that condition (C_{1}) (resp. (C_{2})) is satisfied in spec_{g}(R) if Tauvel’s height formula is valid for all g-prime ideals of R (resp. if the g-height and the height are finite and coincide on the g-prime ideals of R).

By [5, Corollaire 2.8], condition (C_{2}) is satisfied in spec_{g}(R) if char(k) = 0 and R is noetherian g-hypernormal.

If R is g-simple, conditions (C_{1}) and (C_{2}) are trivially satisfied in spec_{g}(R).

Proposition 1.7 can be improved under additional hypotheses.

Corollary 1.9. Let k be of characteristic 0, R noetherian of finite Gelfand-Kirillov dimension, h an ideal of g of dimension l, A = R ⋆ g and B = R ⋆ h. Suppose that R is finitely generated or g-locally finite. Assume also that condition (C_{1}) is satisfied in spec_{g}(R).

(1) If P is a g-prime ideal of B, then g-ht(P) ≤ ht(P) ≤ ht(P ∩ R) + l.

(2) Then g-dim(B) ≤ dim(R) + l. In particular, dim(A) ≤ dim(R) + n.

Proof. (1) Set Q=P ∩R. So R/Q is a subalgebra of B/P and GKdim(R/Q)≤GKdim(B/P). By condition (C_{1}), GKdim(R)−ht(Q) ≤ GKdim(B/P). Corollary 1.5 in the g-locally finite case and [12, 8.2.10] in the finitely generated case imply that GKdim(B) − GKdim(B/P) ≤ ht(Q) + l, hence the upper bound follows from [8, Corollary 3.16].

Corollary 1.10. Let k be of characteristic 0, R noetherian g-hypernormal of finite Gelfand-Kirillov dimension, h an ideal of g of dimension l, A = R ⋆ g and B = R ⋆ h. Suppose that R is finitely generated or g-locally finite. Assume also that condition (C_{1}) and (C_{2}) are satisfied in spec_{g}(R).

(1) If P is a g-prime ideal of B, then g-ht(P) ≤ ht(P) ≤ g-ht(P ∩ R) + l.

(2) Then g-dim(B) ≤ g-dim(R) + l. In particular, dim(A) ≤ g-dim(R) + n.

Proof. (1) By condition (C_{2}), we have g-ht(P ∩ R) = ht(P ∩ R). The results follow from Corollary 1.9(1).

Corollary 1.11. Let k be algebraically closed of characteristic 0, g solvable, h an ideal of g of dimension l, R commutative noetherian g-locally finite with finite Gelfand-Kirillov dimension, A = R ⋆ g and B = R ⋆ h. Suppose that condition (C_{1}) is satisfied in spec_{g}(R).

(1) If P is a g-prime ideal of B, then g-ht(P) ≤ g-ht(P ∩ R) + l.

(2) Then g-dim(B) ≤ g-dim(R) + l. In particular, dim(A) ≤ g-dim(R) + n.

Proof. By [5, Corollaire 2.18] and [6, Remark 0.1], B is g-hypernormal.

Now we shall study Tauvel’s height formula in R ⋆ g. Our aim is to obtain the results of [4] in the case where the ring R is g-locally finite instead of finitely generated.

Let g be completely solvable. We fix a composition series of g; i.e. a chain

0 = g_{0} ⊂ g_{1} ⊂ ⋯ ⊂ g_{n} = g

of ideals of g such that g_{i+1}/g_{i} has dimension one. We shall set R_{i} = R ⋆ g_{i}; 0 ≤ i ≤ n and B = R_{m}; so R_{0} = R and R ⋆ g_{n} = R ⋆ g. Choose X_{i} in g_{i} − g_{i−1} such that X_{i} + g_{i−1} is a basis of g_{i}/g_{i−1}. So (X_{1}, X_{2}, …, X_{n}) is a basis of g. We have R_{i} ≃ R_{i−1}[θ_{i}, δ_{i}]; where δ_{i}(r) = δ_{X_{i}}(r) for any r ∈ R_{i−1}.

Suppose that condition (C_{2}) is satisfied in spec_{g}(R). Let P be a g-prime ideal of B = R ∗ g_{m}, 0 ≤ m ≤ n and Q = P ∩ R. Set l = g-ht(Q). So there exists a strictly saturated increasing chain of g-prime ideals of R ending at Q

Q_{0} ⊂ Q_{1} ⊂ ⋯ ⊂ Q_{l} = Q.

Set P_{i} = Q_{i}B; 0 ≤ i ≤ l and P_{l+i} = (P ∩ R_{i})B; 0 ≤ i ≤ m. So P_{l} = Q_{l}B = QB and P_{l+m} = P. Consider the chain of g-prime ideals of B ending at P

P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{l} ⊆ P_{l+1} ⊆ ⋯ ⊆ P_{l+m} = P  (α)

Proposition 1.12. Let k be of characteristic 0, g completely solvable, R noetherian g-locally finite with finite Gelfand-Kirillov dimension, B = R ∗ g_{m}; 0 ≤ m ≤ n and P a g-prime ideal of B. Suppose that conditions (C_{1}) and (C_{2}) are satisfied in spec_{g}(R). Then the length of the chain (α) is GKdim(B) − GKdim(B/P).

Proof. The proof is similar to that of [4, Proposition 3.1] using “g-locally finite” in the place of “finitely generated”. If m = 0, the result is true by the hypotheses. Assume that the result is true in R_{i}, 0 ≤ i < m. Set B′ = R ∗ g_{m−1}, P′ = P ∩ B′; so P ∩ R_{i} = P′ ∩ R_{i} for 0 ≤ i ≤ m − 1. Set P′_{i} = Q_{i}B′; 0 ≤ i ≤ l and P′_{l+i} = (P′ ∩ R_{i})B′; 0 ≤ i ≤ m − 1; so P′ = P′_{l+m−1}. By the induction hypothesis, the length of the chain

P′_{0} ⊂ P′_{1} ⊂ ⋯ ⊂ P′_{l} = QB′ ⊆ P′_{l+1} ⊆ ⋯ ⊆ P′_{l+m−1} = P′  (β)

is GKdim(B′)−GKdim(B′/P′). By Corollary 1.5, its length is GKdim(B)−GKdim(B/P′B). Clearly, P_{i} = P′_{i}B for 0 ≤ i ≤ l; P_{l+i} = P′_{l+i}B and P_{l+i} ∩ B′ = P′_{l+i} for 0 ≤ i ≤ m − 1. We deduce that P_{i} = P_{i+1} if and only if P′_{i} = P′_{i+1}, 0 ≤ i ≤ l and P′_{l+i+1} = P′_{l+i} if and only if P_{l+i+1} = P_{l+i}. It follows that the chain of g-prime ideals of B

P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{l} ⊆ P_{l+1} ⊆ ⋯ ⊆ P_{l+m−1} = P′B

has the same length as the chain (β). Thus, its length is GKdim(B) − GKdim(B/P′B). If P = P′B, then by Corollary 1.5, the result is true.

If P′B ⊂ P, the chain (α) has for length GKdim(B)−GKdim(B/P′B)+1. Now we shall show that GKdim(B/P) = GKdim(B/P′B) − 1. As B′/P′ is a subalgebra of B/P, we have GKdim(B′/P′) ≤ GKdim(B/P); hence GKdim(B/P′B)−1 ≤ GKdim(B/P), by Corollary 1.5. On the other hand, P/P′B is a nonzero g-invariant ideal of the prime noetherian ring B/P′B. By [12, 2.3.5 (ii)], P/P′B contains a regular element. By [8, Proposition 3.15], GKdim(B/P) ≤ GKdim(B/P′B) − 1 and the proposition is proved.

We are now ready to prove Tauvel’s height formula in R ⋆ g.

Proposition 1.13. Let k be of characteristic 0, g completely solvable, R noetherian glocally finite with finite Gelfand-Kirillov dimension, B = R ∗ g_{m}; 0 ≤ m ≤ n and P a g-prime ideal of B. Suppose that conditions (C_{1}) and (C_{2}) are satisfied in spec_{g}(R). Then GKdim(B) = GKdim(B/P) + ht(P).

Proof. By (1.12), GKdim(B) − GKdim(B/P) ≤ htP. By [8, Corollary 3.16], GKdim(B) − GKdim(B/P) ≥ htP.

The following result is not already known.

Corollary 1.14. Let k be of characteristic 0, g completely solvable, R noetherian g-locally finite g-simple with finite Gelfand-Kirillov dimension. Set B = R ∗ g_{m}, 0 ≤ m ≤ n and let P be a g-prime ideal of B. Then we have GKdim(B) = GKdim(B/P) + ht(P).

Corollary 1.15. Let k be of characteristic 0, g completely solvable, R noetherian g-locally finite with finite Gelfand-Kirillov dimension, B = R∗g_{m}; 0 ≤ m ≤ n and P a g-prime ideal of B. Suppose that conditions (C_{1}) and (C_{2}) are satisfied in spec_{g}(R). Then g-ht(P) = ht(P).

Proof. Let h′ be an ideal of g of codimension 1 in h. Assume that the property is true in B′ = R ⋆ h′. Set B = R ⋆ h and P′ = P ∩ B′. So B = B′ ⋆ (h/h′), B/P′B = (B′/P′) ⋆ (h/h′), g-ht(P′) = ht(P′) and (cf [4, Lemme 4.9]),

ht(P′) = g − ht(P′) ≤ g − ht(P′B) ≤ g − ht(P)  (∗∗)

By Corollary 1.5, GKdim(B/P′B) = GKdim(B′/P′)+1 and GKdim(B) = GKdim(B′)+1, since B′ and B′/P′ are h/h′-locally finite. We deduce from Proposition 1.13 that ht(P′B) = ht(P′).

If P = P′B, (∗∗) implies that ht(P) ≤ g-ht(P), as claimed.

If P′B ⊂ P then ht(P′) = ht(P′B) < ht(P). As B′/P′ is a subalgebra of B/P, we have GKdim(B′/P′) ≤ GKdim(B/P). On the other hand, Proposition 1.13 implies that ht(P) ≤ ht(P′) + 1. It follows that ht(P) = ht(P′) + 1. By [4, Lemme 4.9], g-ht(P) ≥ght(P′B) + 1. But g-ht(P′B) = ht(P′B) = ht(P′), so g-ht(P) ≥ ht(P′) + 1 ≥ ht(P) and the proof is complete.

2. When the action comes from an abelian group

In this section R is a k-algebra. Let G be a group and kG the group algebra of G. We suppose that G acts on R by k-algebra automorphisms. Denote by R ⋆G the crossed product of R by G. For any x ∈ G, we denote by x̄ the canonical image of x in R ⋆ G. We recall that there exists a map t : G × G → U(R); (U(R) is the multiplicative group of unit elements of R) such that

x̄ȳ = \overline{xy}t(x, y)

and x̄r = x(r)x̄. It follows that

x(ȳ) = \overline{x(y)}t(xy, x^{−1})t(x, x^{−1})^{−1}x(t(x, y))

where x(y) = xyx^{−1}.

If t(x, y) = 1 ∀ x, y ∈ G then R is a G-module, R ⋆ G is usually denoted by R#G [12] and is the k-algebra generated by R and kG.

For any G-prime ideal P of R, G-ht(P) denotes the G-height of P. We denote by G-dim the G-invariant version of dim.

Let P be a G-prime ideal of R. Then P is semi-prime invariant and G permutes transitively the minimal prime ideals over P. It follows that ht(P) = ht(I) and GKdim(R/P) = GKdim(R/I), where I is any minimal prime ideal over P.

From now on G is a free abelian group of finite rank n with basis x_{1}, x_{2}, …, x_{n}.

We denote by G_{i} the subgroup of G whose basis is x_{1}, x_{2}, …, x_{i}. We will set R_{i} = R ⋆ G_{i}; 0 ≤ i ≤ n. So we have R_{0} = R, R ⋆ G_{n} = R ⋆ G and R_{i} = R_{i−1}[x̄_{i}, x̄_{i}^{−1}, ϕ_{i}]; where ϕ_{i}(r) = x_{i}(r) = x̄_{i}rx̄_{i}^{−1} for any r ∈ R_{i−1}. If Imt is contained in the center of R, the ϕ_{i} are commuting automorphisms of R.

In the remainder of this section we suppose (except 2.7) that the image of t is contained in a field W which is a finite-dimensional G-stable subalgebra of R. This condition is satisfied if the image Imt of t is contained in k; in particular if Imt is identically 1.

Lemma 2.1.

(1) If y ∈ G_{i}, then ȳ is a G-finite element of R_{i}.

(2) If R is G-locally finite, then R_{i} is G-locally finite.

Proof. (1) Since G is abelian we have

x(ȳ) = ȳt(xy, x^{−1})t(x, x^{−1})^{−1}x(t(x, y)).

(Note in passing that we have also x(ȳ) = ȳt(y, x^{−1})t(x, y).) So x(ȳ) ∈ ȳW and ȳW is a finite-dimensional G-stable k-vector subspace of R.

(2) Suppose that R_{i−1} is G-locally finite. Let u ∈ R_{i−1}. Then g(u) ∈ V where V is a finitedimensional G-stable subspace of R and g(ux̄_{i}) = g(u)g(x̄_{i}) ∈ V x̄_{i}W. Now it is easy to see that V x̄_{i}W is a finite-dimensional G-stable subspace of R_{i}. Then R_{i} is G-locally finite.

Lemma 2.2. Let R be G-locally finite. Then GKdim(R_{i}) = GKdim(R_{i−1}) + 1; 1 ≤ i ≤ n.

Proof. By (2.1), R_{i} is G-locally finite. So R_{i} is ϕ_{i+1}-locally finite. By [9, Proposition 1], GKdim(R_{i+1}) = GKdim(R_{i}) + 1.

Let λ = (λ_{ij}) be an n × n matrix of nonzero elements of k such that λ_{ii} = 1 and λ_{ji} = λ_{ij}^{−1} for 1 ≤ i, j ≤ n. The multiparameter coordinate ring of quantum affine n-space is the kalgebra O_{λ}(k^{n}) generated by elements x_{1}, …, x_{n} subject only to the relation x_{i}x_{j} = λ_{ij}x_{j}x_{i} for 1 ≤ i, j ≤ n. Denote by P (λ) the localisation of O_{λ}(k^{n}) with respect to the multiplicative set generated by the x_{i}. The P (λ) are exactly the k-algebras which are crossed product k⋆ℤ^{n} of the field k by the group ℤ^{n}.

We deduce from (2.2) that the Gelfand-Kirillov dimension of R⊗_{k}P (λ) is GKdim(R)+n, since R ⊗_{k} P (λ) ≃ R ⋆ G where G acts trivially on R and Imt is contained in k.

Let P be a G-prime ideal of R. Then G permutes transitively the minimal prime ideals over P. A consequence of this fact is that ht(P) = ht(I) for any minimal prime I over P.

Proposition 2.3. Let R be noetherian G-locally finite with finite Gelfand-Kirillov dimension. Set A = R ⋆ G and B = R ⋆ G_{m}.

(1) Let P be a G-prime ideal of B such that P ∩ R = 0. Then ht(P) ≤ m.

(2) If R is G-simple, then G-dim(B) ≤ m.

(3) Suppose that P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{m+1} is a strictly increasing chain of G-prime ideals of B, then P_{0} ∩ R ⊂ P_{m+1} ∩ R.

(4) Then G-dim(B) < (m + 1)(G-dim(R) + 1).

Proof. (1) Since R = R/(P ∩R) is a subalgebra of B/P, we have GKdim(R)≤GKdim(B/P). Lemma 2.2 implies that GKdim(B) − GKdim(B/P) ≤ m. Note that GKdim(B/P) = GKdim(B/I) for any minimal prime ideal I over P. By [8, Corollary 3.16], ht(I) ≤ GKdim(B) − GKdim(B/I). But ht(I) = ht(P); so ht(P) ≤ m.

(2) By [7, Proposition 3.6, Corollaire 3.8], B is G-hypernormal and G-ht(P) = ht(P).

(3) and (4) Adapt the proof of Proposition 1.7 (1).

Remarks 2.4. (1) If G is abelian finitely generated, Chin [1, Theorem 3.9] established (2.3(3)) for m = n without the assumptions that R is noetherian G-locally finite of finite Gelfand-Kirillov dimension.

(2) If G is nilpotent finitely generated of Hirsch number n, Chin [1, Theorem 3.10] established (2.3(3)) for m = n and for a chain of prime ideals

P_{0} ⊂ P_{1} ⊂ ⋯ ⊂ P_{l}

such that l ≥ 2^{n} without the assumptions that R is noetherian G-locally finite with finite Gelfand-Kirillov dimension.

We say that condition (C_{1}) is satisfied in spec_{G}(R) if Tauvel’s height formula is valid for all G-prime ideals of R.

By [7, Remarque 3.5(ii)], G-ht(P) ≤ ht(P) for any G-prime ideal P of R if R is commutative noetherian. Proposition 2.3(3) can be improved under additional hypothesis.

Corollary 2.5. Let R be commutative noetherian G-locally finite with finite Gelfand-Kirillov dimension. Set A = R ⋆ G and B = R ⋆ G_{m}. Suppose that condition (C_{1}) is satisfied in spec_{G}(R).

(1) Let P be a G-prime ideal of B, then G-ht(P) ≤ ht(P) ≤ ht(P ∩ R) + m.

(2) Then G-dim(B) ≤ dim(R) + m.

Proof. (1) Adapt the proof of Corollary 1.9.

Lemma 2.6. Let k be algebraically closed.

(1) Assume that W is contained in the center of R. If I is a nonzero G-invariant ideal of R, then there exists in I a nonzero element u such that x(u) = λ_{x}u for any x ∈ G; where λ_{x} ∈ k.

(2) If R is commutative, then R is G-hypernormal.

Proof. (1) See the proof of [7, Lemme 4.12].

Corollary 2.7. Let k be algebraically closed, R commutative noetherian G-locally finite with finite Gelfand-Kirillov dimension. Set A = R ⋆ G and B = R ⋆ G_{m}. Suppose that condition (C_{1}) is satisfied in spec_{G}(R).

(1) Let P be a G-prime ideal of B, then G-ht(P) ≤ G-ht(P ∩ R) + m.

(2) Then G-dim(B) ≤ G-dim(R) + m. In particular, dim(A) ≤ G-dim(R) + n.

Proof. By Corollary 2.6(2), R is G-hypernormal. By [7, Corollaire 3.8], B is G-hypernormal. By [7, Proposition 3.6], G-ht(P ∩ R) = ht(P ∩ R) and G-ht(P) = ht(P). The results follow from Corollary 2.5.

The following proposition and its two corollaries are generalizations of [7, Proposition 4.14, Corollaries 4.15, 4.16]

Proposition 2.8. Let k be algebraically closed, R commutative noetherian G-locally finite with finite Gelfand-Kirillov dimension. Set A = R ⋆G and B = R ⋆G_{m}; 0 ≤ m ≤ n. Suppose that condition (C_{1}) is satisfied in spec_{G}(R). Then Tauvel’s height formula holds in spec_{G}(B).

Proof. The result follows from [7, Corollaire 4.5].

From Proposition 2.8, we get the two following corollaries.

Corollary 2.9. Let k be algebraically closed, R commutative, finitely generated, G-locally finite and G-prime with finite Gelfand-Kirillov dimension. Set A = R ⋆ G and B = R ⋆ G_{m}; 0 ≤ m ≤ n. Then Tauvel’s height formula holds in spec_{G}(B).

Corollary 2.10. Let k be algebraically closed, R commutative, noetherian, G-locally finite and G-simple with finite Gelfand-Kirillov dimension. Set A = R ⋆ G and B = R ⋆ G_{m}; 0 ≤ m ≤ n. Then Tauvel’s height formula holds in spec_{G}(B).

As last application of [7, Proposition 4.5], we get the following result which was proved for R = k in [3].

Proposition 2.11. Let R be noetherian hypernormal with finite Gelfand-Kirillov dimension. If Tauvel’s height formula holds in R then it also holds in R ⊗_{k} P (λ).

3. When the action comes from an abelian monoid

Let S = R[x_{1}, ϕ_{1}][x_{2}, ϕ_{2}] … [x_{n}, ϕ_{n}] be an iterated skew polynomial ring over k. For 1 ≤ i ≤ n set S_{i} = R[x_{1}, ϕ_{1}][x_{2}, ϕ_{2}] … [x_{i}, ϕ_{i}] with S_{0} = R, S_{n} = S and assume that the following conditions hold:

(1) for 1 ≤ i ≤ n, ϕ_{i} is a k-algebra automorphism of R,

(2) for 1 ≤ i < j ≤ n, ϕ_{j} is a k-algebra automorphism of S_{i},

(3) for 1 ≤ i < j ≤ n, there exists λ_{i,j} ∈ k^{∗} such that ϕ_{j}(x_{i}) = λ_{i,j}x_{i}.

Set Φ_{i,j} = {ϕ_{i}, ϕ_{i+1}, …, ϕ_{j}}; 1 ≤ i < j ≤ n. We say that R is Φ_{i,j}-locally finite if each element of R is contained in a finite-dimensional ϕ_{l}-stable subspace of R; l = i, i + 1, …, j.

Lemma 3.1. If R is Φ_{1,n}-locally finite, then S_{i} is Φ_{i+1,n}-locally finite.

Proposition 3.2. If R is Φ_{1,n}-locally finite, then GKdim(S) = GKdim(R) + n.

Corollary 3.3. The Gelfand-Kirillov dimension of R ⊗_{k} O_{λ}(k^{n}) is GKdim(R) + n.

Proof. R ⊗_{k} O_{λ}(k^{n}) ≃ R[x_{1}, ϕ_{1}][x_{2}, ϕ_{2}] … [x_{n}, ϕ_{n}] where each ϕ_{i} acts trivially on R.

The result in Corollary (3.3) is known for R = k (see [3]).

References

[1] Chin, W.: Prime ideals in differential operator rings and crossed products of infinite groups. J. Algebra 106 (1987), 78–104. Zbl 0611.16023

[2] Chin, W.; Quinn, D.: Actions of divided power Hopf algebras. J. Algebra 144 (1991), 371–389. Zbl 0737.16023

[3] Goodearl, K. R.; Lenagan, T. H.: Catenarity in quantum algebras. J. Pure and Appl. Algebra 111 (1996), 123–142. Zbl 0864.16018

[4] Guédénon, T.: La formule des hauteurs de Tauvel dans les anneaux d’opérateurs différentiels. Comm. Algebra 21(6) (1993), 2077–2100. Zbl 0796.16018

[5] Guédénon, T.: Localisation, caténarité et dimensions dans les anneaux munis d’une action d’algèbre de Lie. J. Algebra 178 (1995), 21–47. Zbl 0837.17004

[6] Guédénon, T.: Localisation in rings equipped with a solvable Lie algebra action. Comm. Algebra, 28(7) (2000), 3533–3543. Zbl pre01475126

[7] Guédénon, T.: Anneaux munis d’une action de groupe superrésoluble. Algebras, Groups and Geometries 17(1) (2000), 17–48.

[8] Krause, G.; Lenagan, T. H.: Growth of Algebras and Gelfand-Kirillov dimension. Research Notes in Math. 116, Pitman, London 1985. Zbl 0564.16001

[9] Leroy, A.; Matczuk, J.; Okninski, J.: On the Gelfand-Kirillov dimension of normal localizations and twisted polynomial rings. In: Perspectives in Rings Theory (F. Van Oystaeyen and L. Le Bruyn, eds.), Kluwer Academic Publishers 1988, 205–214.

Zbl 0692.16018 [10] Lorenz, M.: On the Gelfand-Kirillov dimension of skew polynomial rings. J. Algebra 77

(1982), 186–188. Zbl 0491.16004 [11] Matczuk, J.: The Gelfand-Kirillov dimension of Poincaré Birkhoff Witt extensions. In:

Perspectives in Rings Theory (F. Van Oystaeyen and L. Le Bruyn, eds.), Kluwer Academic Publishers 1988, 221–226. Zbl 0691.16005

[12] McConnell, J. C.; Robson, J.: Noncommutative Noetherian Rings. Wiley, Chichester New York 1987. Zbl 0644.16008

[13] Montgomery, S.: Hopf algebras and their actions on rings. CBMS Lectures in Math., Vol. 82, AMS Providence, RI, 1993. Zbl 0793.16029

[14] Passman, D. S.: Prime ideals in enveloping rings. Trans. Amer. Math. Soc. 302 (1987), 535–560. Zbl 0628.16020

Received September 2, 2000