Version 1.21, February, 1998. To appear in Trans. Amer. Math. Soc.

THE DIXMIER-MOEGLIN EQUIVALENCE IN QUANTUM

COORDINATE RINGS AND QUANTIZED WEYL ALGEBRAS

K. R. Goodearl and E. S. Letzter

Abstract. We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of n × n matrices, of Weyl algebras, and of Euclidean and symplectic space. The framework for this analysis is based upon certain iterated skew polynomial algebras A over infinite fields k of arbitrary characteristic. Our main result is the verification, for A, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal P of A is primitive if and only if the center of the Goldie quotient ring of A/P is algebraic over k, if and only if P is a locally closed point – with respect to the Jacobson topology – in the prime spectrum of A.

These equivalences are established with the aid of a suitable group H acting as automorphisms of A. The prime spectrum of A is then partitioned into finitely many “H-strata” (two prime ideals lie in the same H-stratum if the intersections of their H-orbits coincide), and we show that a prime ideal P of A is primitive exactly when P is maximal within its H-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which H acts transitively on the set of rational ideals within each H-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of A.

For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.

1. Introduction

For several classes of finitely generated noncommutative algebras, a long-standing common goal has been to classify the primitive ideals. Toward this end it has often been necessary to first identify the primitive ideals within the larger set of prime ideals, and one of the most famous results along these lines was proved by Dixmier [8] and Moeglin [29]: If U is the enveloping algebra of a finite dimensional complex Lie algebra, and P is a prime ideal of U, then P is primitive if and only if it is rational (i.e., ℂ is the center of the Goldie quotient ring of U/P), if and only if P is a locally closed point in the prime spectrum of U. Following [40], we will say that any noetherian algebra whose primitive ideals can be classified in this fashion satisfies the Dixmier-Moeglin Equivalence; over an arbitrary field k, rationality occurs when the center of the appropriate Goldie quotient ring is algebraic over k (cf. [19; 20; 38]). Our first aim in this paper is to verify the equivalence for a class of iterated skew polynomial rings that includes quantum matrices and quantum Weyl algebras, both considered here in the “non-root-of-unity” case; other

1991 Mathematics Subject Classification. 16S36, 16P40, 81R50. The research of both authors was supported in part by the National Science Foundation.

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examples include quantizations of affine, symplectic, and Euclidean spaces. This result can be viewed as an analog of the Dixmier-Moeglin Equivalence for enveloping algebras of solvable Lie algebras, whose proof preceded the general case (cf. [9, 4.5.7]), because such enveloping algebras may be constructed as iterated differential operator rings.

Another motivating context for our work arises from quantizations of the function algebra on a complex connected semisimple Lie group G. Recently, detailed descriptions of the prime and primitive spectra have been obtained by Joseph [23; 24; 25] for R_{q}[G] and by Hodges-Levasseur-Toro [17] for ℂ_{q,p}[G]; both of these studies follow the earlier conjectures in [15; 16] and are concerned with the case when the parameter q is not a root of unity. A key qualitative feature of these algebras is that the prime spectrum of the algebra divides into finitely many “H-strata” (see also [6, 2.4]), where H is a maximal torus of G acting as algebra automorphisms, and where two prime ideals are in the same stratum exactly when their H-orbits have the same intersection. It is further proved by these authors that H acts transitively on each stratum of primitive ideals (in the algebraically closed case) and that a prime ideal is primitive if and only if it is maximal within its stratum. For the iterated skew polynomial rings mentioned in the first paragraph, we establish a similar description, which turns out to be an intermediate step toward proving the Dixmier-Moeglin Equivalence in our desired setting. For R_{q}[G] or ℂ_{q,p}[G], the equivalence follows directly from the results of Joseph and Hodges-Levasseur-Toro; see (2.4).

Our approach can be divided into two main steps. In the first we consider an affine algebraic group H acting rationally (cf. (2.5)) by automorphisms on a noetherian algebra A. We then assume that there exist only finitely many H-strata in Spec A, and we prove that the remaining qualitative properties discussed in the preceding paragraph follow for A. More precisely, we establish the equivalence of the following four conditions for prime ideals P of A:

(1) P locally closed in Spec A;

(2) P primitive;

(3) P rational;

(4) P maximal within its H-stratum. It is well known that (1) ⟹ (2) in Jacobson rings, and that (2) ⟹ (3) in the presence of the Nullstellensatz. Under the assumption that Spec A contains only finitely many Hstrata, it is easy to show that (4) ⟹ (1). Thus our main effort is devoted to establishing (3) ⟹ (4) under suitable additional hypotheses. Our proof of this implication rests upon a result of Moeglin and Rentschler, asserting – given certain extra restrictions – that H must act transitively on each stratum of rational ideals; included among the required added hypotheses here is the assumption of an algebraically closed base field of characteristic zero. (A positive characteristic version of this theorem has recently been obtained by Vonessen [46].) Descent techniques, partly modelled on those of Irving and Small [20], now allow us to obtain the equivalence of (1)–(4) for suitable algebras and groups over non-algebraically-closed fields. (Over algebraically closed fields of characteristic zero, the implication (3) ⟹ (1) is already present in unpublished work of Moeglin and Rentschler [31]; our paper does not rely on the results therein, and the methods we employ are different.) In the second main step, finally, we use techniques from [12] to prove that the skew polynomial algebras in our setting have only finitely many H-ideals, and so only finitely many H-strata, where H is a suitable affine algebraic group acting rationally by automorphisms.

For rational torus actions we are able to prove, furthermore, that the strata are homeomorphic to the spectra of commutative Laurent polynomial algebras occurring naturally as the centers of suitable localizations; see Section 6.

Examples of finitely generated noetherian algebras that do not satisfy the Dixmier-Moeglin Equivalence may be found in [19] and references therein.

2. Finite Stratification

The first goal of this section is to prove, under suitable hypotheses, that an algebra with finitely stratified prime spectrum satisfies the Dixmier-Moeglin Equivalence. This part of our analysis does not explicitly require a rational action of an algebraic group; the major nontrivial assumption of the first theorem is that the group giving the stratification acts transitively on the rational ideals within each stratum. In the second part of the section, we appeal to work of Moeglin-Rentschler and Vonessen to obtain this hypothesis in the algebraically closed case, under certain additional assumptions, where now the acting group is assumed to be affine algebraic. In characteristic zero, results of Irving and Small then show that the Dixmier-Moeglin Equivalence descends to the non-algebraically closed case. We provide an alternate argument, based on algebraic group actions, which allows descent also in positive characteristic. Over algebraically closed fields of characteristic zero, it already follows from an unpublished paper of Moeglin and Rentschler [31] that the Dixmier-Moeglin Equivalence holds when the prime spectrum is finitely stratified by a rational action of an affine algebraic group.

2.1 Notation and Preliminaries. Throughout this section, A will denote a noetherian algebra, over a field k, equipped with a group H acting on A by k-algebra automorphisms. The group of all k-algebra automorphisms of A will be denoted Aut_{k} A.

(i) Tensor products will be assumed over k, unless indicated otherwise. If k′ is a field extension of k, then we will always identify A with its image A ⊗ 1 in A ⊗ k′. Observe in this setting that if P is a prime ideal of A ⊗ k′, then P ∩ A is a prime ideal of A. Further information concerning extensions of scalars and prime ideals may be found, for example, in [43]. The reader is referred to [9] and [28] for additional background information.

(ii) The set of (left) primitive ideals of A will be denoted Prim A, and the set of prime ideals of A will be referred to as Spec A. Both Prim A and Spec A will be equipped with the Jacobson topology. As before, a prime ideal P of A is rational if the center of the Goldie quotient ring of A/P is algebraic over k, and P is said to be locally closed if it is a locally closed point of Spec A. (In other words, P is locally closed if and only if it is strictly contained in the intersection of all those prime ideals that properly contain it – following the convention that the intersection of an empty set of ideals is equal to A). The set of rational ideals of A will be denoted Rat A and will also be provided the Jacobson topology (i.e., Rat A is a subspace of Spec A). Crucial to the analysis below are the induced actions of H on Spec A, Prim A, and Rat A.

As previously stated, A satisfies the Dixmier-Moeglin Equivalence provided the sets of rational, locally closed, and primitive ideals all coincide. Note that in [30], a prime ideal P of A is called “rational” only if Z(Fract(A/P)) coincides with k (rather than allowing this field to be any algebraic extension of k). However, the theorem we shall use from that paper, namely [30, 2.12ii] and its extension in [46], will only be applied in the case when k is algebraically closed. Hence, the conflict in terminology does not affect our work here.

(iii) Following [28, Chapter 9], we will say that A satisfies the Nullstellensatz (over k) if A is a Jacobson ring and if all of the endomorphism rings of simple A-modules are algebraic over k. It follows from [9, 4.1.6], when A satisfies the Nullstellensatz, that every primitive ideal of A is rational. It is immediate, of course, that a locally closed prime ideal in a Jacobson ring must be primitive.

(iv) An H-stable ideal (or, more briefly, H-ideal) I is said to be H-prime when I contains no product of H-ideals that all properly contain it. Next, if J is an ideal of A, set

(J : H) = ⋂_{h∈H} h(J) .

When P is a prime ideal of A, it is easy to verify that (P : H) is an H-prime ideal of A. Conversely, every H-prime ideal I of A is the intersection of the (finite) H-orbit of prime ideals minimal over I (cf. [10, Remarks 4*,5*, p. 338]). The set of H-prime ideals of A is denoted H -Spec A.

The H-stratum in Spec A of a prime ideal P is then the set of prime ideals P′ for which (P′ : H) = (P : H). The H-strata in Rat A and Prim A are similarly defined. Note that every H-stratum in Spec A is a union of H-orbits.

2.2. (i) Assume that A is a Jacobson ring. Set Y = { (P : H) | P ∈ Prim A }. By (2.1iv), Y is a subset of H -Spec A. Now let I be an H-prime ideal of A, and set J equal to the intersection of all of the ideals in Y that properly contain I. Assume further that I is strictly smaller than J; this assumption will be valid, for example, when Y is finite. Since I is a semiprime ideal and A is a Jacobson ring, we may now choose a primitive ideal Q that contains I but does not contain J. Thus I = (Q : H), and so I ∈ Y. It follows, for example, that if Y is finite, then Y = H -Spec A; hence, Y is finite if and only if H -Spec A is finite. In particular, H -Spec A is finite if Prim A consists of only finitely many H-orbits; a partial converse to this assertion will be discussed in (2.7).

(ii) Assume now that H -Spec A is finite (but that A is not necessarily a Jacobson ring). Let P be a prime ideal that is maximal within its H-stratum, and set I = (P : H). Letting J denote the intersection of all the H-prime ideals of A that properly contain I, it follows that I is strictly contained in J. Observe that every prime ideal of A that properly contains P also contains J, and so P is locally closed.

2.3 Proposition. Assume that A satisfies the Nullstellensatz, that H acts transitively on each H-stratum in Rat A, and that H -Spec A is finite.

(i) A satisfies the Dixmier-Moeglin Equivalence.

(ii) In A, an arbitrary prime ideal is primitive if and only if it is maximal within its H-stratum.

Proof. Let P be a prime ideal of A. It follows from (2.1iii) and (2.2ii) that P is locally closed if it is maximal within its H-stratum, that P is primitive if it is locally closed, and that P is rational if it is primitive. It remains to show that if P is rational, then P is maximal within its H-stratum.

Choose a prime ideal P′ ⊇ P maximal within the H-stratum of P. It follows from (2.2ii) that P′ is locally closed, and from (2.1iii) that it is rational. Because P and P′ are rational ideals in the same H-stratum, they are contained within the same H-orbit. Hence, P = P′, and therefore P is maximal within its H-stratum, as desired. (The last equality follows, for example, from the fact that the classical Krull dimensions of the isomorphic algebras A/P and A/P′ must be the same.) □

2.4 Remarks. Let G be a connected, simply connected, semisimple, complex Lie group with maximal torus H.

(i) Define the quantum function algebra R = R_{q}[G] over \overline{k(q)}, as in [25, Chapter 9]. In particular, R is a noetherian \overline{k(q)}-algebra satisfying the Nullstellensatz [25, 9.2.2], where k is assumed to have characteristic zero. An explicit structure theory for Prim R has been established by Joseph [23; 24; 25, Chapters 9, 10], following conjectures (verified for ℂ_{q}[SL_{n}]) by Hodges-Levasseur [15; 16]. Key to these results is the action on R, by its character group R^{∧}, via “winding” automorphisms [25, 1.3.4]. Furthermore, R^{∧} may be identified with H [25, 10.3.12]. From [25, 10.3.2, 10.3.11] it follows that H acts transitively on each H-stratum in Rat R and that there are only finitely many H-orbits in Rat R. Consequently, R_{q}[G] satisfies the hypotheses of (2.3), recalling (2.1iii) and (2.2i). However, the second conclusion of the proposition, that the primitive ideals are exactly those prime ideals maximal within their H-strata, is already a fundamental part of the theory [25, 10.3.2, 10.3.7]. Moreover, from [25, 10.3.4, 10.3.7] it follows that every rational ideal of R is primitive, and by arguing as in [15, proof of 4.3.1], one can deduce from [25, 10.3.2] (cf. [4, 5.6]) that every primitive ideal of R is locally closed.

(ii) Considerations similar to (i) apply to the multiparameter quantum function algebras ℂ_{q,p}[G], studied by Hodges-Levasseur-Toro [17]. In particular, H acts transitively on each H-stratum of rational ideals [17, 4.14], and there are only finitely many H-strata of rational ideals [17, 4.16]. The Dixmier-Moeglin Equivalence follows from [17, §4], arguing as in [15, 4.4.1; 16, 4.2].

(iii) Quantum function algebras at roots of unity (cf. [7]) are affine PI algebras, and so the primitive ideals are precisely the maximal ideals. These coincide with the rational ideals (see, e.g., [45, 2.6]), and the Dixmier-Moeglin Equivalence immediately follows.

2.5. We now recall the definition of a rational action of an algebraic group. In order to include the case of the group of k-rational points of an algebraic group defined over the algebraic closure \overline{k}, assume only that H is a Zariski-closed subgroup of GL_{n}(k) for some n. The action of H on A is then said to be rational provided A is a union of finite dimensional H-invariant linear subspaces W_{ℓ} for which the induced group homomorphisms H → GL(W_{ℓ}) are morphisms of algebraic varieties. In particular, rational actions are locally finite, but the converse does not hold. For example, given any group homomorphism ϕ : k^{×} → k^{×}, there is a locally finite action of k^{×} on the polynomial ring k[X] by k-algebra automorphisms such that α.X = ϕ(α)X for α ∈ k^{×}. This action is rational only if ϕ is given by ϕ(α) = α^{n} for some n ∈ ℤ.

2.6 Theorem. (Moeglin-Rentschler, Vonessen) Assume that the field k is algebraically closed, and that the k-algebra A is noetherian. Suppose further that H is a k-affine algebraic group and that its action on A is rational. Then H acts transitively on each H-stratum of Rat A.

Proof. The original version of this result is proved in [30, 2.12ii] (cf. [40, §2]) under several additional assumptions: that k has characteristic zero, that A is finitely generated as a k-algebra, that A ⊗ k′ is noetherian for all extension fields k′ of k, and that H is connected. The final assumption is easily removed, given that the identity component of H is a closed, connected, normal subgroup of finite index (e.g., [3, I.1.2]). Vonessen [46] has recently shown that the theorem as stated can be proved by a combination of Moeglin and Rentschler’s methods together with techniques he developed in [45]. □

Combining (2.2), (2.3), and (2.6), we obtain the following corollary; when the base field has characteristic zero, part (ii) may be deduced from [31].

2.7 Corollary. Retain the hypotheses of (2.6), and assume that A satisfies the Nullstellensatz over k = \overline{k}.

(i) The set of H-orbits in Prim A is finite if and only if H -Spec A is finite. When H -Spec A is finite, the assignment P ↦ (P : H) induces a bijection from the set of Horbits in Prim A onto H -Spec A.

(ii) Suppose that H -Spec A is finite. Then A satisfies the Dixmier-Moeglin Equivalence, and the primitive ideals of A are precisely the prime ideals maximal within their H-strata. □

The remainder of this section is devoted to showing, in certain situations, that (2.7ii) descends to non-algebraically-closed fields. Descent of the Dixmier-Moeglin Equivalence in characteristic zero already follows from the work of Irving-Small [20]; see [43, §8.4] for a more general exposition. In our situation, these results will not be needed, because the Dixmier-Moeglin Equivalence will follow from the maximality of rational ideals within their H-strata.

2.8. Let A′ be an over-ring of A, and let H′ be a group acting by automorphisms on A′. If H is a subgroup of H′, and if the action of each element h ∈ H on A′ restricts to the given action of h on A, then we say that the action of H′ on A′ extends the action of H on A. For instance, the canonical action of a torus (k^{×})^{n} on a polynomial ring k[x_{1}, …, x_{n}] extends to the canonical action of (K^{×})^{n} on K[x_{1}, …, x_{n}] whenever K is a field extension of k. We shall further say that the action of H′ densely extends that of H if (P : H′) = (P : H) for all P ∈ Spec A′. In the example above, the action of (K^{×})^{n} densely extends that of (k^{×})^{n} as long as k is infinite.

2.9 Lemma. Assume that k′ is an algebraic field extension of k, and that A′ = A ⊗ k′ is noetherian. Suppose that the action of H on A extends to an action of a group H′ on A′ by k′-algebra automorphisms. If H -Spec A is finite, then H′- Spec A′ is finite.

Proof. Let Q be a prime ideal of A, and let P_{1}, …, P_{t} denote the prime ideals of A′ minimal over Q ⊗ k′. Next, recall that every prime ideal of A′ contracts to a prime ideal of A (e.g., [43, 2.12.39]). Hence, since Q contains a product of the P_{i} ∩ A, it follows that at least one of the P_{i} contracts to Q. Furthermore, every prime ideal of A′ contracting to Q contains Q ⊗ k′, and so from Incomparability (cf. [43, 3.4.13′]) it follows that every prime ideal of A′ contracting to Q is in the set {P_{1}, …, P_{t}}. We have just shown that intersection with A produces a finite-to-one surjection from Spec A′ onto Spec A. (Cf. [47].)

As noted in (2.1iv), the H-prime ideals of A are exactly the intersections of finite Horbits in Spec A, and a similar statement holds for the H′-prime ideals of A′. It therefore follows from our assumptions that there exist only finitely many finite H-orbits in Spec A. Also, to prove the lemma it suffices to show that there exist only finitely many finite H′- orbits in Spec A′. However, it follows from the preceding paragraph that contraction to A induces a finite-to-one surjection from the set of finite H′-orbits in Spec A′ onto the set of finite H-orbits in Spec A. The lemma follows. □

2.10 Lemma. Assume that A′ = A⊗\overline{k} is noetherian, and suppose that the action of H on A extends densely to an action, by \overline{k}-algebra automorphisms, of a group H′ on A′. If every \overline{k}-rational ideal in A′ is maximal within its H′-stratum in Spec A′, then every rational ideal of A is maximal within its H-stratum in Spec A.

Proof. Suppose to the contrary that Q_{0} is a rational ideal of A which is properly contained in a prime ideal Q_{1} lying within the same H-stratum. Set E_{0} equal to the Goldie quotient ring of A/Q_{0}. Observe that \overline{E}_{0} = E_{0} ⊗_{k} \overline{k} is an Ore localization of its noetherian subring (A/Q_{0}) ⊗_{k} \overline{k}, because \overline{E}_{0} is obtained therefrom by inverting regular elements of the form s ⊗ 1. Consequently, \overline{E}_{0} is noetherian.

By the rationality of Q_{0}, we may regard Z = Z(E_{0}) as a subfield of \overline{k}, and so we may set F_{0} = E_{0} ⊗_{Z} \overline{k}. Since \overline{E}_{0} maps surjectively onto F_{0}, the latter is noetherian, and it follows from a standard lemma (e.g., [43, 1.7.27]) that F_{0} is simple. We claim that F_{0} is its own Goldie quotient ring, which can be proved as follows. Given a regular element r of F_{0}, choose a field k′ ⊆ \overline{k} such that k′ is finite over Z and such that r ∈ E′_{0} = E_{0} ⊗_{Z} k′ ⊆ F_{0}. Since E′_{0} is artinian, and since r is regular in E′_{0}, it follows that r is invertible in E′_{0}. Thus every regular element of F_{0} is invertible (in F_{0}), and the claim is established.

Now let P_{0} denote the kernel of the composition of canonical maps

A′ → (A/Q_{0}) ⊗_{k} \overline{k} → \overline{E}_{0} → F_{0},

and observe that P_{0} ∩ A = Q_{0}. Because the first and third maps in the composition are surjective and the second is an Ore localization, it follows from the preceding paragraph that P_{0} is prime and that F_{0} is the Goldie quotient ring of A′/P_{0}. Next, observe (by, e.g., [43, 1.7.24]) that Z(F_{0}) = \overline{k}. Hence P_{0} is \overline{k}-rational.

By [43, 2.12.50], there exists a prime ideal P_{1} of A′ such that P_{1} ∩A = Q_{1} and P_{0} ⊂ P_{1}. From our assumptions on A′ it follows that P_{0} and P_{1} are not in the same H′-stratum.

For i ∈ {0, 1}, set J_{i} equal to the H′-prime ideal (P_{i} : H′). As in (2.1iv), we may choose a prime ideal M_{i} minimal over J_{i} such that J_{i} = (M_{i} : H′); the finitely many prime ideals of A′ minimal over J_{i} are precisely the ideals comprising the H′-orbit of M_{i}. However, any product of all of the prime ideals minimal over J_{0} is contained in M_{1}, and so – without loss of generality – we may assume that M_{0} ⊆ M_{1}. Moreover, M_{0} ⊊ M_{1}, because J_{0} ⊊ J_{1}.

Because the action of H′ densely extends that of H, we have

J_{i} ∩ A = (P_{i} : H′) ∩ A = (P_{i} : H) ∩ A = (P_{i} ∩ A : H) = (Q_{i} : H).

Similarly, setting N_{i} = M_{i} ∩ A we have J_{i} ∩ A = (M_{i} : H′) ∩ A = (N_{i} : H). Since the H′-orbit of M_{i} is finite, so is the H-orbit of N_{i}, whence N_{i} is minimal over J_{i} ∩ A. Therefore, since J_{0} ∩ A ⊆ N_{0} ⊆ N_{1}, it follows that J_{0} ∩ A = J_{1} ∩ A if and only if N_{0} = N_{1}. By Incomparability (cf. [43, 3.4.13′]), M_{0} ⊊ M_{1} implies that N_{0} ⊊ N_{1}, and so J_{0} ∩ A ⊊ J_{1} ∩ A. But J_{i} ∩ A = (Q_{i} : H), and we assumed at the beginning of the proof that (Q_{0} : H) = (Q_{1} : H). This contradiction proves the lemma. □

2.11. Set A′ = A ⊗ \overline{k}, and suppose that H′ is a \overline{k}-affine algebraic group acting rationally (by \overline{k}-algebra automorphisms) on A′. Moeglin and Rentschler proved, for any rational ideal Q of A′, that the map H′ → Rat\overline{k} A′ given by h′ ↦ h′(Q), for h′ ∈ H′, is continuous [30, 1.5]. We shall need the corresponding fact for arbitrary prime ideals:

Lemma. For any Q ∈ Spec A′, the map H′ → Spec A′, given by h′ ↦ h′(Q) for h′ ∈ H′, is continuous.

Proof. It suffices to show, for any ideal I of A′, that the set

X = {h ∈ H′ | h(Q) ⊇ I}

is closed in H′. Write A′ = ⋃_{j∈J} V_{j} for some finite dimensional H′-invariant \overline{k}-subspaces V_{j}. Then X equals the intersection of the sets

X_{j} = {h ∈ H′ | h(Q ∩ V_{j}) ⊇ I ∩ V_{j}},

and so it is enough to show that the X_{j} are all closed in H′.

Fix j ∈ J, and choose idempotents e_{j}, f_{j} ∈ End\overline{k}(V_{j}) with images Q ∩ V_{j} and I ∩ V_{j} respectively. The set

{g ∈ GL(V_{j}) | (1 − e_{j})g^{−1}f_{j} = 0}

is clearly closed in GL(V_{j}). Since the restriction map H′ → GL(V_{j}) is continuous by hypothesis, it follows that X_{j} is closed in H′, as desired. □

We now (partially) generalize (2.7ii) to arbitrary infinite fields.

2.12. Suppose that k is infinite, that A ⊗ \overline{k} is noetherian and satisfies the Nullstellensatz over \overline{k}, and that H is the group of k-rational points of a k-affine algebraic group H′. (Thus H′ is a Zariski-closed subgroup of some GL_{n}(\overline{k}), the polynomials defining H′ can be chosen with coefficients from k, and H = H′ ∩ GL_{n}(k).) We further assume that the action of H on A is rational; equivalently, this action arises from a right comodule structure μ: A → A ⊗ O(H), where O(H) denotes the coordinate ring of H over k (see, e.g., [21, 2.8]). That H acts as k-algebra automorphisms means that μ must be a k-algebra homomorphism (cf. [30, 1.1]), i.e., A is a right O(H)-comodule algebra via μ.

Theorem. Suppose that either k is perfect or H′ is reductive. If H -Spec A is finite, then A satisfies the Dixmier-Moeglin Equivalence, and the primitive ideals of A are precisely the prime ideals maximal within their H-strata.

Proof. By (2.1iii) and (2.2ii), it suffices to prove that the rational ideals of A are maximal within their H-strata in Spec A. Set A′ = A ⊗ \overline{k}. Tensoring the comodule structure map μ: A → A ⊗ O(H) with \overline{k} and composing with the identity map tensored with the natural inclusion O(H) → O(H′), we obtain a \overline{k}-algebra homomorphism

μ′: A′ → A′ ⊗_{k} O(H) → A′ ⊗_{k} O(H′) → A′ ⊗\overline{k} O(H′),

which gives A′ the structure of a right O(H′)-comodule algebra. This yields a rational action of H′ on A′ by \overline{k}-algebra automorphisms, which extends the action of H on A, in the sense of (2.8). In view of (2.9), H′- Spec A′ is finite.

Now let H′_{0} denote the identity component of H′, a (closed connected) normal subgroup in H′ of finite index (cf. [3, I.1.2b]). Also, H′_{0} is defined over k [ibid]. Note, moreover, that the finiteness of H′- Spec A′ ensures that H′_{0}- Spec A′ is finite. Setting H_{0} = H′_{0}(k) = H′_{0} ∩ H, it follows that H_{0} is a normal subgroup in H of finite index.

Because either k is perfect or H′ is reductive, it follows from [3, V.18.3] that H_{0} is dense in H′_{0}. We claim that the action of H′_{0} on A′ densely extends that of H_{0} on A, in the sense of (2.8). To prove the claim, consider P ∈ Spec A′, and write X = {P_{1} ∈ Spec A′ | P_{1} ⊇ (P : H_{0})}, a closed subset of Spec A′. By the continuity of the map h′ ↦ h′(P) (see (2.11)), the set Y = {h′ ∈ H′_{0} | h′(P) ∈ X} is closed in H′_{0}. Since Y contains the dense subgroup H_{0}, it follows that Y must equal H′_{0}, and so (P : H_{0}) = (P : H′_{0}). The claim is thus proved. It now follows from (2.7ii) and (2.10) that every rational ideal of A is maximal within its H_{0}-stratum.

Finally, let Q_{0} and Q_{1} be prime ideals of A, lying in the same H-stratum, such that Q_{0} is rational and Q_{0} ⊆ Q_{1}. Let {h_{1}, …, h_{n}} be a transversal for H_{0} in H, and observe that

⋂_{i=1}^{n} h_{i}(Q_{1} : H_{0}) = (Q_{1} : H) = (Q_{0} : H) ⊆ (Q_{0} : H_{0}).

Further, each h_{i}(Q_{1} : H_{0}) = (h_{i}(Q_{1}) : H_{0}), an H_{0}-ideal of A (this uses the normality of H_{0} in H). Since (Q_{0} : H_{0}) is an H_{0}-prime ideal (2.1iv), it follows that h_{i}(Q_{1} : H_{0}) ⊆ (Q_{0} : H_{0}) for some i. But (Q_{0} : H_{0}) ⊆ (Q_{1} : H_{0}) because Q_{0} ⊆ Q_{1}, and h_{i}(Q_{1} : H_{0}) cannot be properly contained in (Q_{1} : H_{0}) because A is noetherian. Hence, h_{i}(Q_{1} : H_{0}) = (Q_{0} : H_{0}) = (Q_{1} : H_{0}); in particular, Q_{0} and Q_{1} belong to the same H_{0}-stratum. It follows from the preceding paragraph that Q_{0} = Q_{1}, and therefore Q_{0} is maximal within its H-stratum. □

3. Skew Polynomial Rings in One Variable

In order to apply the results of the previous section to quantum matrices and quantized Weyl algebras, we must exhibit rational actions of algebraic groups H on these algebras such that the number of H-prime ideals is finite. Our approach to the latter condition relies on the (known) structure of the given algebras as iterated skew polynomial rings. The purpose of the present section is to provide the induction step, i.e., to develop conditions under which finiteness of H -Spec passes from an algebra R to a skew polynomial extension R[y; τ, δ].

3.1 Notation and Preliminaries. Throughout this section, R will denote a noetherian algebra over a field k, and S = R[y; τ, δ] a skew polynomial extension of R. We will follow the conventions of [12, Chapter 2].

(i) We assume that τ is a k-algebra automorphism of R, and that δ is a k-linear left τ-derivation of R. Recall that the multiplication in S is determined by the rule yr = τ(r)y + δ(r), and that S is noetherian [28, 1.2.9iv].

(ii) Let H denote a group acting on S by k-algebra automorphisms. Assume that R is H-stable, that y is an H-eigenvector, and that τ coincides with the action on R of some h_{0} ∈ H. Let λ: H → k^{×} denote the H-eigenvalue of y, that is, the character of H such that h(y) = λ(h)y for h ∈ H.

(iii) It is convenient to denote by q the reciprocal of the h_{0}-eigenvalue of y, so that h_{0}(y) = q^{−1}y. For r ∈ R, observe that

τδ(r) = h_{0}(yr − τ(r)y) = q^{−1}yh_{0}(r) − h_{0}^{2}(r)q^{−1}y = q^{−1}(yτ(r) − τ^{2}(r)y) = q^{−1}δτ(r).

Thus δτ = qτδ, and so the pair (τ, δ) is a q-skew derivation in the sense of [12]. As in [12, 2.4ii], it follows that we may extend τ to an automorphism of S such that τ(y) = q^{−1}y. In the present setting, this extension is given by the action of h_{0}.

(iv) For d ∈ R, the inner τ-derivation r ↦ dr − τ(r)d on R will be denoted δ_{d}.

3.2 Lemma. If q is not a root of unity, then every H-prime ideal of S contracts to a δ-stable H-prime ideal of R.

Proof. Let I be an H-prime ideal of S. Then I ∩ R is an H-stable ideal of R and in particular is τ-stable. It follows that I ∩ R is also δ-stable [12, 2.1v].

Choose a prime ideal P minimal over I; then the H-orbit of P is finite, and (P : H) = I. Now choose Q ∈ Spec R minimal over P ∩R. The prime ideals h(Q), for h ∈ H, are minimal over the ideals h(P) ∩ R. But there are only finitely many ideals of the form h(P) ∩ R, and there are only finitely many prime ideals in R minimal over each such ideal. Thus the H-orbit of Q is finite. In particular, the τ-orbit of Q is finite.

By [12, 10.3], P contracts to either Q or the ideal J = (Q : τ). Hence, I∩R = (P ∩R : H) equals either (Q : H) or (J : H). However, (Q : H) = (J : H), because τ ∈ H. Therefore I ∩ R = (Q : H) is H-prime. □

Recall that R is said to be H-simple when its only H-stable ideals are 0 and itself.

3.3 Lemma. Assume that q is not a root of unity. Suppose that R is H-simple but S is not.

(i) There is a unique element d ∈ R such that δ = δ_{d} and h(d) = λ(h)d for all h ∈ H. (ii) There are precisely two H-prime ideals in S, namely 0 and (y − d)S.

Proof. (i) Let I be a proper nonzero H-ideal in S, and let n be the minimum degree for nonzero elements of I. The set consisting of 0 and the leading coefficients of the elements in I of degree n is then a nonzero H-ideal of R and so equals R. Hence, there exists a monic polynomial s ∈ I with degree n, say s = y^{n} + cy^{n−1} + [lower terms].

The usual analysis (as, e.g., in [11, 3.5]) of sa − τ^{n}(a)s, for a ∈ R, shows that δ = δ_{d}, where d = −\binom{n}{1}_{q}^{−1}q^{n−1}c. For h ∈ H, we have

h(s) = λ(h)^{n}y^{n} + λ(h)^{n−1}h(c)y^{n−1} + [lower terms].

Then h(s) − λ(h)^{n}s is an element of I of degree < n, whence h(s) = λ(h)^{n}s, and consequently h(c) = λ(h)c. Thus h(d) = λ(h)d for all h ∈ H.

Suppose also that e ∈ R, with δ = δ_{e} and h(e) = λ(h)e, for all h ∈ H. Set f = d − e: Then h(f) = λ(h)f for all h ∈ H, and fa = τ(a)f for all a ∈ R. Hence, fR = Rf is an H-ideal of R, and so f is either zero or a unit. However, τ(f) = q^{−1}f, and so f^{2} = q^{−1}f^{2}, whence f^{2} = 0. Therefore f = 0, and so e = d.

(ii) Set z = y − d; then S = R[z; τ], and h(z) = λ(h)z for all h ∈ H. Then zS is an H-ideal of S, and there is an H-equivariant ring isomorphism S/zS ≅ R. Therefore, S/zS is an H-simple ring, and so zS is an H-prime ideal of S. As in the proof of part (i), every nonzero H-ideal of S contains a monic polynomial. Since the product of two monic polynomials is always nonzero, S is an H-prime ring; that is, 0 is an H-prime ideal of S.

Now let P be a nonzero H-prime ideal of S, and let n be the minimum degree for nonzero elements of P. As in part (i), there is a monic (in z) polynomial p ∈ P of degree n, say p = z^{n} +p_{n−1}z^{n−1} +⋯+p_{1}z +p_{0}. Since p is monic, the Division Algorithm applies, showing that P = pS = Sp. Note that P ∩ R is a proper H-ideal of R, whence P ∩ R = 0. Thus n ≥ 1.

For a ∈ R, observe that pa − τ^{n}(a)p is an element of P with degree less than n, whence pa = τ^{n}(a)p. Thus p_{n−i}a = τ^{i}(a)p_{n−i} for all a ∈ R and all i = 1, …, n. Similarly, given h ∈ H, observe that h(p) − λ(h)^{n}p is an element of P with degree less than n, whence h(p) = λ(h)^{n}p. Thus h(p_{n−i}) = λ(h)^{i}p_{n−i} for all h ∈ H and all i = 1, …, n. Now each p_{n−i}R = Rp_{n−i} is an H-ideal of R, and therefore p_{n−i} is either zero or a unit.

We next compute that

zp = z^{n+1} + τ(p_{n−1})z^{n} + ⋯ + τ(p_{1})z^{2} + τ(p_{0})z
= z^{n+1} + q^{−1}p_{n−1}z^{n} + ⋯ + q^{−n+1}p_{1}z^{2} + q^{−n}p_{0}z,

and so zp − pz = (q^{−1} − 1)p_{n−1}z^{n} + ⋯ + (q^{−n+1} − 1)p_{1}z^{2} + (q^{−n} − 1)p_{0}z. Since zp − pz is an element in P of degree n, it must follow that zp − pz = ap, where a = (q^{−1} − 1)p_{n−1}. Thus (q^{−i} − 1)p_{n−i} = ap_{n−i+1}, for all i = 1, …, n, and ap_{0} = 0.

Suppose that p_{0} is a unit. Then a = 0, and since q is not a root of unity, it follows that p_{n−i} = 0 for all i. Thus p = z^{n}. Furthermore, because zS = Sz is an H-ideal and P is H-prime, z ∈ P. Therefore, n = 1 in this case, whence p = z and P = zS.

Now assume that p_{0} = 0. Then p = fz, where f = z^{n−1} + p_{n−1}z^{n−2} + ⋯ + p_{1}. By the minimality of n, f ∉ P. Since zS is an H-ideal, it follows that z ∈ P, and we conclude – as in the previous case – that P = zS. □

3.4 Proposition. Assume that q is not a root of unity.

(i) There are at most twice as many H-prime ideals in S as in R.

(ii) If all H-prime ideals of R are prime (completely prime), then the same is true for S.

Proof. (i) By (3.2), every H-prime ideal of S contracts to a δ-stable H-prime ideal of R. Hence, it suffices to show that at most two H-prime ideals of S can contract to any given δ-stable H-prime ideal Q in R. After passing to R/Q and localizing (see, e.g., [12, 2.3]), we may assume that R is an H-simple artinian ring. If S is H-simple, then S has just one H-prime ideal, namely 0. If S is not H-simple, then by (3.3), S has just two H-prime ideals.

(ii) As in part (i), we can reduce to the case that R is H-simple artinian. If R is prime (completely prime), the same is true of S; see, for example, [28, 1.2.9]. If there exists a nonzero H-prime ideal P in S, then by (3.3), P is of the form (y − d)S, and S/P ≅ R. In this case, P is prime (completely prime) if and only if R is a prime (completely prime) ring, and we are done. □

4. Iterated Skew Polynomial Rings

Finite stratification is proved for certain iterated skew polynomial extensions in (4.2), and the Dixmier-Moeglin Equivalence is established for them in (4.4) and (4.7).

4.1 Notation and Assumptions. Let A = k[y_{1}][y_{2}; τ_{2}, δ_{2}] ⋯ [y_{n}; τ_{n}, δ_{n}] be an iterated skew polynomial ring over the field k. Let H denote a group, acting as k-algebra automorphisms of A, for which y_{1}, …, y_{n} are all H-eigenvectors. (Note that the image of H in Aut_{k} A must therefore be abelian.) For 1 ≤ i ≤ n, set A_{i} = k[y_{1}][y_{2}; τ_{2}, δ_{2}] ⋯ [y_{i}; τ_{i}, δ_{i}].

We make the following three assumptions:

(a) There are infinitely many distinct eigenvalues for the action of H on y_{1}. (This of course implies that k must be infinite.)

(b) Each τ_{i} is a k-algebra automorphism of A_{i−1} and each δ_{i} is a k-linear τ_{i}-derivation of A_{i−1}.

(c) For 2 ≤ i ≤ n, there exists h_{i} ∈ H such that the restriction of h_{i} to A_{i−1} coincides with τ_{i} and the h_{i}-eigenvalue of y_{i} is not a root of unity.

4.2 Proposition. The H-prime ideals of A are all completely prime, and there are at most 2^{n} of them.

Proof. By assumption (a), the only H-prime ideals of A_{1} are 0 and y_{1}A_{1}. Thus the desired conclusions hold for A_{1}, and they follow for A by inductively applying (3.4). □

The fact that the H-prime ideals of A are completely prime will allow us, in Section 6, to develop an explicit picture of the H-strata in Spec A.

4.3. If k′ is an arbitrary field extension of k, then A⊗k′ is a noetherian k′-algebra satisfying the Nullstellensatz; see, for example, [28, 1.2.9iv, 9.4.21] for details.

4.4 Theorem. Assume that H is the group of k-rational points of a k-affine algebraic group H′, and that the action of H on A is rational. Suppose further that either k is perfect or H′ is reductive. Then A satisfies the Dixmier-Moeglin Equivalence, and the primitive ideals of A are precisely the prime ideals maximal within their H-strata.

Proof. By (4.2) and (4.3), the desired conclusions follow from (2.12). □

4.5. In our applications (Section 5), H will be a torus of the form (k^{×})^{m}, and so we can take H′ = (\overline{k}^{×})^{m}, which is linearly reductive (see, e.g., [34, Chapter 5, Theorem 36]). Thus, we shall be able to apply (4.4) in arbitrary characteristic.

Furthermore, we can always reduce the situation of (4.1) to a case in which H is replaced by a closed subgroup of a torus. While this process (explained in the next paragraph) may change the H-strata, it has no effect on the validity of the Dixmier-Moeglin Equivalence, and so we will obtain the equivalence whenever the hypotheses of (4.1) are satisfied.

4.6. Assuming – as we can – that the action of H on A is faithful, there is an embedding of H into the torus D = (k^{×})^{n} such that the closure of H also acts via automorphisms of A. To start, observe that the ordered monomials y_{1}^{m_{1}}y_{2}^{m_{2}} ⋯ y_{n}^{m_{n}} form a k-basis for A. Hence, there is a faithful and rational action of D on A by vector space automorphisms, where each ordered monomial y_{1}^{m_{1}}y_{2}^{m_{2}} ⋯ y_{n}^{m_{n}} is an eigenvector whose eigenvalue is the character (α_{1}, …, α_{n}) ↦ α_{1}^{m_{1}}α_{2}^{m_{2}} ⋯ α_{n}^{m_{n}}. We identify H and D with their images in the group of vector space automorphisms of A, and we set E = D ∩ Aut_{k} A. Then H ⊆ E.

Observe, for a, b ∈ A, that the set {ϕ ∈ D | ϕ(ab) = ϕ(a)ϕ(b)} is Zariski-closed in D. Therefore E is a closed subgroup of D, acting rationally on A by k-algebra automorphisms. Further, E equals the group of k-rational points of an appropriate k-affine closed subgroup E′ ⊆ (\overline{k}^{×})^{n}, namely (\overline{k}^{×})^{n} ∩ Aut\overline{k}(A ⊗_{k} \overline{k}), and E′ is (linearly) reductive [ibid]. Since H ⊆ E, we may replace H by E, thus obtaining the extra hypotheses of (4.4). Hence, we have proved the following theorem:

4.7 Theorem. Let A be an iterated skew polynomial ring over the field k, equipped with a group H acting via k-algebra automorphisms, as in (4.1). Then A satisfies the Dixmier-Moeglin Equivalence. □

5. Applications

Assume throughout this section that k is an infinite field. We discuss various quantized algebras for which the Dixmier-Moeglin Equivalence can be obtained via (4.4).

5.1 Quantum Affine Space. Let A = O_{q}(k^{n}) be the multiparameter coordinate ring of quantum affine n-space over k, where q = (q_{ij}) is a multiplicatively antisymmetric n×n matrix over k; that is, q_{ii} = 1 and q_{ji} = q_{ij}^{−1} for all i, j. The k-algebra A is generated by elements y_{1}, …, y_{n} subject only to the relations y_{i}y_{j} = q_{ij}y_{j}y_{i} for i, j = 1, …, n. There is a natural action of the torus H = (k^{×})^{n} by k-algebra automorphisms on A such that (α_{1}, …, α_{n}).y_{i} = α_{i}y_{i} for (α_{1}, …, α_{n}) ∈ H and all i. It is already known that the Dixmier-Moeglin Equivalence holds in A [13, 2.5], and it follows from [13, 2.3, 2.11] that the primitive ideals of A are precisely the prime ideals maximal in their H-strata. In case k contains a non-root of unity, these results can also be obtained from (4.4); we leave the details to the reader.

5.2 Quantum Symplectic Space. (i) Next, let A = O_{q}(𝔰𝔭 k^{2n}) be the one-parameter coordinate ring of quantum symplectic 2n-space over k as in [42, Definition 14] or [32, 1.1], where q is a nonzero element of k. (To our knowledge, precise relations for multiparameter quantum symplectic spaces have not appeared in the literature, although such algebras surely exist.) The algebra A is generated by elements y_{1}, …, y_{2n} satisfying relations originally worked out in [42, p. 210]. We give the equivalent, simpler set of relations found by Musson [32, 1.1] (cf. [35, 1.1]):

y_{i}y_{j} = qy_{j}y_{i}  (i < j; j ≠ i′)
y_{i}y_{i′} = q^{2}y_{i′}y_{i} + (q^{2} − 1) ∑_{l=1}^{i−1} q^{l−i}y_{l}y_{l′}  (i ≤ n),

where i′ = 2n + 1 − i.

(ii) For each n-tuple α ∈ (k^{×})^{n}, there is a k-algebra automorphism ω(α) of A such that ω(α)(y_{i}) = α_{i}y_{i} and ω(α)(y_{i′}) = α_{i}^{−1}y_{i′}, for all i = 1, …, n. In particular, this gives a faithful rational action of the torus H = (k^{×})^{n} on A by k-algebra automorphisms.

(iii) Musson has shown that A can be presented as an iterated skew polynomial ring of the form

k[y_{1}][y_{1′}; τ′_{1}][y_{2}; τ_{2}][y_{2′}; τ′_{2}, δ′_{2}] ⋯ [y_{n}; τ_{n}][y_{n′}; τ′_{n}, δ′_{n}],

where the k-algebra automorphisms τ_{i}, τ′_{i} and the k-linear τ′_{i}-derivations δ′_{i} are determined as follows:

\begin{aligned}τ_{i}(y_{j}) & = \{ \begin{matrix}q^{−1}y_{j} \\ qy_{j} \end{matrix} \\ τ′_{i}(y_{j}) & = \{ \begin{matrix}q^{−1}y_{j} \\ qy_{j} \\ q^{−2}y_{j} \end{matrix} \\ δ′_{i}(y_{j}) & = \{ \begin{matrix}0 \\ (q^{−2} − 1) ∑_{l=1}^{i−1} q^{l−i}y_{l}y_{l′} \end{matrix} \end{aligned}  \begin{matrix}(j < i) \\ (j > i′) \\ (j < i) \\ (j > i′) \\ (j = i) \\ (j ≠ i) \\ (j = i) \end{matrix}

[32, 1.2] (cf. [35, 1.10]). Here we have interchanged q and q^{−1} relative to [32, 1.2] due to our conventions for skew polynomial rings. Observe that τ_{i} and τ′_{i} coincide with the restrictions to the subalgebras k⟨y_{1}, y_{1′}, …, y_{i−1}, y_{(i−1)′}⟩ and k⟨y_{1}, y_{1′}, …, y_{i−1}, y_{(i−1)′}, y_{i}⟩ of automorphisms of the form

h_{i} = ω(q^{−1}, q^{−1}, …, q^{−1})
h′_{i} = ω(q^{−1}, q^{−1}, …, q^{−1}, q^{−2}, 1, 1, …, 1).

With these choices, the h_{i}-eigenvalue of y_{i} is q^{−1}, and the h′_{i}-eigenvalue of y_{i′} is q^{2}.

In view of (5.2), we may now conclude the following from (4.2) and (4.4):

5.3 Theorem. Let A = O_{q}(𝔰𝔭 k^{2n}) as in (5.2i), and let the torus H = (k^{×})^{n} act on A as in (5.2ii). Assume that q is not a root of unity.

(i) The H-prime ideals of A are all completely prime, and there are at most 2^{2n} of them. (ii) The Dixmier-Moeglin Equivalence holds for A, and its primitive ideals are precisely

the prime ideals maximal within their H-strata. □

5.4 Quantum Euclidean Space. Let A = O_{q}(𝔬 k^{n}) be the one-parameter coordinate ring of quantum Euclidean n-space as in [42, Definition 12], with generators y_{1}, …, y_{n}. (In case n is odd, we must require that q has a square root in k.) As in the case of quantum symplectic space, a simpler set of relations is given in [32, 2.1, 2.2; 36, 1.1], and an iterated skew polynomial ring presentation is given in [32, 2.3; 36, 1.2]; we do not write out the details here.

Let m = ⌊n/2⌋ denote the integer part of n/2. For α ∈ H = (k^{×})^{m}, there is a kalgebra automorphism ω(α) of A such that ω(α)(y_{i}) = α_{i}y_{i} and ω(α)(y_{i′}) = α_{i}^{−1}y_{i′} for i = 1, …, m, and such that ω(y_{m+1}) = y_{m+1} if n is odd. (Here i′ = n+1−i.) This gives a rational action of H on A, and one checks easily that the automorphisms appearing in the iterated skew polynomial presentation of A coincide with restrictions of automorphisms ω(α) satisfying (4.1c). Hence, we obtain the following analog of (5.3) from (4.2) and (4.4):

5.5 Theorem. Let A = O_{q}(𝔬 k^{n}) as in (5.4), and let the torus H = (k^{×})^{⌊n/2⌋} act on A as in (5.4). Assume that q is not a root of unity.

(i) The H-prime ideals of A are all completely prime, and there are at most 2^{n} of them. (ii) The Dixmier-Moeglin Equivalence holds for A, and its primitive ideals are precisely

the prime ideals maximal within their H-strata. □

5.6 Quantum Matrices. (i) Let A = O_{λ,p}(M_{n}(k)) be the multiparameter coordinate ring of quantum n × n matrices over k, as studied in [2, 27, 41, 44]. Here, p = (p_{ij}) is a multiplicatively antisymmetric n × n matrix over k, and λ is a nonzero element of k not equal to −1. The k-algebra A is generated by variables y_{11}, y_{12}, …, y_{nn} subject only to the following relations:

y_{ℓm}y_{ij} = \{ \begin{matrix}p_{ℓi}p_{jm}y_{ij}y_{ℓm} + (λ − 1)p_{ℓi}y_{im}y_{ℓj} & when ℓ > i and m > j \\ λp_{ℓi}p_{jm}y_{ij}y_{ℓm} & when ℓ > i and m ≤ j \\ p_{jm}y_{ij}y_{ℓm} & when ℓ = i and m > j. \end{matrix}

(ii) Given any n-tuples α = (α_{1}, …, α_{n}) and β = (β_{1}, …, β_{n}) in (k^{×})^{n}, there are kalgebra automorphisms θ_{l}(α) and θ_{r}(β) of A such that θ_{l}(α)(y_{ij}) = α_{i}y_{ij} and θ_{r}(β)(y_{ij}) = β_{j}y_{ij} for all i, j. Set H = (k^{×})^{n} ×(k^{×})^{n}. There is an action of H on A defined by mapping H → Aut_{k} A according to the rule (α, β) ↦ θ_{l}(α)θ_{r}(β). This action is clearly rational.

(iii) Apply the lexicographic ordering to the double indices 11, 12, …, nn. If the variables y_{ij} are adjoined in this order, A can be presented as an iterated skew polynomial ring

k[y_{11}][y_{12}; τ_{12}] ⋯ [y_{ij}; τ_{ij}, δ_{ij}] ⋯ [y_{nn}; τ_{nn}, δ_{nn}],

as shown in [2, pp. 890-891]. Choose ℓm ∈ {11, …, nn}. Let R_{lm} denote the k-subalgebra of A generated by {y_{ij} | ij < ℓm}, and let S_{ℓm} denote the subalgebra generated by R_{ℓm} ∪ {y_{ℓm}}. Then S_{ℓm} = R_{ℓm}[y_{ℓm}; τ_{ℓm}, δ_{ℓm}], where τ_{ℓm} and δ_{ℓm} are a k-algebra automorphism and a k-linear τ_{ℓm}-derivation of R_{ℓm} determined by their actions on the generators of R_{ℓm} as follows:

τ_{ℓm}(y_{ij}) = \{ \begin{matrix}p_{ℓi}p_{jm}y_{ij} \\ λp_{ℓi}p_{jm}y_{ij} \end{matrix}_{δ_{ℓm}(y_{ij}) = \{ \begin{matrix}(λ − 1)p_{ℓi}y_{im}y_{ℓj} \\ 0 \end{matrix}}

when ℓ ≥ i and m > j when ℓ > i and m ≤ j when ℓ > i and m > j otherwise.

Observe that τ_{ℓm} coincides with the restriction to R_{ℓm} of the automorphism

h_{ℓm} = θ_{l}(p_{ℓ1}, …, p_{ℓn})θ_{r}(p_{1m}, …, p_{m−1,m}, λp_{mm}, λp_{m+1,m}, …, λp_{nm}),

and that the h_{ℓm}-eigenvalue of y_{ℓm} is λ.

5.7. Let A and H be as in (5.6).

(i) It follows from (4.2), when λ is not a root of unity, that the H-prime ideals of A are completely prime, and that there are no more than 2^{n^{2}} of them. (For instance, when n = 2 there are 14.)

(ii) Note that (i) holds in some cases when the prime ideals of A are not all completely prime: If n = 2 and p_{12} = p_{21} = −1, then A/⟨y_{21}, y_{22}⟩ ≅ k⟨y_{11}, y_{12}⟩/⟨y_{11}y_{12} + y_{12}y_{11}⟩. Hence, ⟨y_{21}, y_{22}, y_{11}^{2} − 1⟩ is a prime ideal of A that is not completely prime.

(iii) There is a ℤ×ℤ-grading on A arising from the double indexing of the variables y_{ij}, and it is straightforward to verify that all of the H-prime ideals of A are homogeneous with respect to this grading. Now assume that k is algebraically closed. Then H is a divisible group, and so the only finite H-orbits in Spec A are singletons. It therefore follows directly from (2.1iv), in the algebraically closed case, that every H-prime ideal is prime.

Combining (4.4) and (5.7i), we now obtain:

5.8 Theorem. Let A = O_{λ,p}(M_{n}(k)) as in (5.6i), assuming that λ is not a root of unity. Let the torus H = (k^{×})^{n} × (k^{×})^{n} act on A as in (5.6ii). Then the Dixmier-Moeglin Equivalence is satisfied by A, and its primitive ideals are precisely the prime ideals maximal within their H-strata. □

5.9 Remarks. (i) The multiparameter quantum group O_{λ,p}(GL_{n}(k)) is obtained by inverting the quantum determinant of O_{λ,p}(M_{n}(k)), which is a normal (but not necessarily central) element of O_{λ,p}(M_{n}(k)); see, e.g., [2, Theorem 3]. Also, the action of H extends to O_{λ,p}(GL_{n}(k)), since the quantum determinant is an H-eigenvector. It is now easy to see that the conclusions of (5.8), again assuming that λ is not a root of unity, extend to O_{λ,p}(GL_{n}(k)).

(ii) The Dixmier-Moeglin Equivalence also holds for A = O_{λ,p}(M_{n}(k)) when λ is a root of unity – see [13, 3.2]. This case cannot be handled with the methods used above, since A can have infinitely many H-prime ideals. For example, let n = 2 and let q be a primitive t-th root of unity (in k) for some odd integer t > 1; then take λ = q^{−2} and p = ( \begin{matrix}1 & q^{−1} \\ q & 1 \end{matrix} ). (Then A reduces to the standard one-parameter case, often denoted O_{q}(M_{2}(k)).) In this case, y_{ij}^{t} is central for all i, j [37, 7.2.1]. Now consider the H-ideals J_{α} = ⟨y_{11}^{t}y_{22}^{t}−αy_{12}^{t}y_{21}^{t}⟩, for α ∈ k. Any prime ideal P containing two different J_{α}’s must contain both y_{11}^{t}y_{22}^{t} and y_{12}^{t}y_{21}^{t}. Since y_{12} and y_{21} are normal in A, and since y_{11} and y_{22} are normal modulo either ⟨y_{12}⟩ or ⟨y_{21}⟩, it follows that P must contain one of y_{12}, y_{21} and one of y_{11}, y_{22}; hence, P has height at least 2. On the other hand, any prime minimal over a J_{α} has height 1 by the Principal Ideal Theorem for normal elements [28, 4.1.11]. Therefore no prime ideal of A can be minimal over two different J_{α}’s. Consequently, if we choose an H-prime ideal P_{α} minimal over J_{α} for each α, we obtain infinitely many distinct H-prime ideals in A.

5.10 Quantum Weyl Algebras. (i) Let A = A_{n}^{Q,Γ}(k) be the multiparameter quantized Weyl algebra over k as in [26, 1, 5]. Here, Q = (q_{1}, …, q_{n}) ∈ (k^{×})^{n}, and Γ = (γ_{ij}) is a multiplicatively antisymmetric n × n matrix over k. The algebra A is generated by elements x_{1}, y_{1}, …, x_{n}, y_{n} subject only to the following relations:

\{ \begin{matrix}y_{i}y_{j} = γ_{ij}y_{j}y_{i} & (all i, j) \\ x_{i}x_{j} = q_{i}γ_{ij}x_{j}x_{i} & (i < j) \\ x_{i}y_{j} = γ_{ji}y_{j}x_{i} & (i < j) \\ x_{i}y_{j} = q_{j}γ_{ji}y_{j}x_{i} & (i > j) \\ x_{j}y_{j} = 1 + q_{j}y_{j}x_{j} + ∑_{m<j}(q_{m} − 1)y_{m}x_{m} & (all j) \end{matrix}

(ii) For each α ∈ H = (k^{×})^{n}, there is a k-algebra automorphism ω(α) of A such that ω(α)(x_{i}) = α_{i}x_{i} and ω(α)(y_{i}) = α_{i}^{−1}y_{i}, for all i. In particular, this gives a faithful rational action of H on A by k-algebra automorphisms.

(iii) For ℓ = 1, …, n, let S_{2ℓ−1} denote the k-subalgebra of A generated by y_{1}, x_{1}, …,y_{ℓ−1}, x_{ℓ−1},y_{ℓ}, and let S_{2ℓ} denote the subalgebra generated by y_{1}, x_{1}, …, y_{ℓ}, x_{ℓ}. We can write A as an iterated skew polynomial ring,

A = k[y_{1}][x_{1}; τ_{2}, δ_{2}][y_{2}; τ_{3}][x_{2}; τ_{4}, δ_{4}] ⋯ [y_{n}; τ_{2n−1}][x_{n}; τ_{2n}, δ_{2n}],

where the τ_{ℓ} and δ_{ℓ} are k-algebra automorphisms and k-linear τ_{ℓ}-derivations determined by their actions on generators as follows:

\{ \begin{matrix}τ_{2ℓ−1}(y_{i}) = γ_{ℓi}y_{i} & (i < ℓ) \\ τ_{2ℓ−1}(x_{i}) = γ_{iℓ}x_{i} & (i < ℓ) \\ τ_{2ℓ}(y_{i}) = q_{i}γ_{iℓ}y_{i} & (i ≤ ℓ) \\ τ_{2ℓ}(x_{i}) = q_{i}^{−1}γ_{ℓi}x_{i} & (i < ℓ) \\ δ_{2ℓ}(y_{i}) = 0 & (i < ℓ) \\ δ_{2ℓ}(y_{ℓ}) = 1 + ∑_{m<ℓ}(q_{m} − 1)y_{m}x_{m} \\ δ_{2ℓ}(x_{i}) = 0 & (i < ℓ) \end{matrix}

(cf. [22, 2.1, 2.8]). Observe that τ_{2ℓ} coincides with the restriction to S_{2ℓ−1} of the automorphism

h_{2ℓ} = ω(q_{1}^{−1}γ_{ℓ1}, …, q_{ℓ−1}^{−1}γ_{ℓ,ℓ−1}, q_{ℓ}^{−1}, 1, …, 1),

and that τ_{2ℓ−1} coincides with the restriction to S_{2ℓ−2} of

h_{2ℓ−1} = ω(γ_{1ℓ}, …, γ_{ℓ−1,ℓ}, q_{ℓ}^{−1}, 1, …, 1).

With these choices, the h_{2ℓ}-eigenvalue of x_{ℓ} is q_{ℓ}^{−1}, and the h_{2ℓ−1}-eigenvalue of y_{ℓ} is q_{ℓ}.

In view of (5.10), we may now conclude the following from (4.2) and (4.4):

5.11 Theorem. Let A = A_{n}^{Q,Γ}(k) as in (5.10i), and let the torus H = (k^{×})^{n} act on A as in (5.10ii). Assume that q_{1}, …, q_{n} are not roots of unity.

(i) The H-prime ideals of A are all completely prime, and there are at most 2^{2n} of them. (ii) The Dixmier-Moeglin Equivalence holds for A, and its primitive ideals are precisely

the prime ideals maximal within their H-strata. □

6. Strata Under Torus Actions: The Completely Prime Case

In the previous section we considered certain algebras, equipped with actions by tori H, for which all of the H-prime ideals are completely prime. In this section we present a detailed description of the strata occurring in such cases.

6.1. Let A be a noetherian algebra over an infinite field k (of arbitrary characteristic), and let H = (k^{×})^{r} be a torus, of (finite) rank r over k, acting rationally on A by k-algebra automorphisms. (For example, A and H can be assumed to be as described in (5.1), (5.2), (5.4), (5.6), or (5.10).) Let Ĥ denote the group of rational characters of H, that is, the set of algebraic group morphisms H → k^{×} under the operation of pointwise multiplication. Since k is infinite, Ĥ is free abelian of rank r, with a basis consisting of the component projections (k^{×})^{r} → k^{×}.

Because H acts rationally, A is spanned by its H-eigenvectors [34, Ch. 5, Corollary to Theorem 36], and the corresponding eigenvalues lie in Ĥ. Thus A = ⨁_{x∈Ĥ} A_{x}, where A_{x} denotes the x-eigenspace of A. Further, A_{x}A_{y} ⊆ A_{xy} for x, y ∈ Ĥ, and so A is graded by the group Ĥ. (Conversely, any grading of A by ℤ^{r} arises from a rational action of (k^{×})^{r} on A, as noted in [39, p. 784].)

In order to analyze the stratum of Spec A corresponding to a particular H-prime ideal J (not yet assumed to be completely prime), we first pass to the factor algebra A/J, which is again graded by Ĥ. We then try to simplify the situation by passing to an appropriate localization. Unfortunately, a general “graded Goldie Theorem” – an Ore localization with respect to the homogeneous regular elements, resulting in a graded-semisimple ring – is not available (cf. [33, Part C, I.1.1]). However, for the case in which J is completely prime, we need only a “graded Ore Theorem”, which is quite easy to prove. (As mentioned above, the H-prime ideals of the algebras in (5.1), (5.2), (5.4), (5.6), and (5.10) are all completely prime.)

6.2 Lemma. Let R be a right Ore domain, graded by a group G, and let E denote the set of nonzero homogeneous elements of R. Then E is a right denominator set in R, the localization R[E^{−1}] is graded by G, and every nonzero homogeneous element of R[E^{−1}] is invertible.

Proof. (The reader is referred to [14] for definitions of unexplained terms.) To verify that E is a right denominator set, only the right Ore condition needs to be checked, since the remaining condition (right reversibility) is trivially true in a domain. To prove that E is right Ore, one easily sees that it suffices to show that rE ∩ eR ≠ ∅ for all e ∈ E and all homogeneous elements r ∈ R (cf. [33, Part A, I.6.1]). Since R is a right Ore domain, there exist s, t ∈ R such that s ≠ 0 and rs = et. Suppose r ∈ R_{x} and e ∈ R_{y}, for some x, y ∈ G, and suppose s = s_{1} + ⋯ + s_{n}, with s_{i} ∈ R_{z_{i}} for some distinct z_{i} ∈ G. Then t = t_{1} + ⋯ + t_{n}, for some t_{i} ∈ R_{y^{−1}xz_{i}}, such that rs_{i} = et_{i}. Since some s_{i} ≠ 0 (whence s_{i} ∈ E), this verifies the desired common multiple condition.

Having verified that E is a right denominator set, we obtain the localization R[E^{−1}], and we observe that there is a well-defined G-grading on R[E^{−1}] such that re^{−1} has degree xy^{−1} for all r ∈ R_{x} and e ∈ E ∩ R_{y} (cf. [33, Part A, I.6.2]). It is clear that every nonzero homogeneous element of R[E^{−1}] is invertible. □

6.3. Let R be a ring graded by a (multiplicative) group G, such that every nonzero homogeneous element of R is invertible. One says in this situation that R is a G-graded division ring [33, Part A, I.4]. Note that the identity component R_{1} is then a division ring. Also note, for x, y ∈ G, that R_{x}R_{y} = R_{x}r_{y} for all nonzero r_{y} ∈ R_{y}; in particular, each R_{y} is either zero or a 1-dimensional vector space over R_{1}. Lastly, R is said to be strongly graded provided R_{x}R_{y} = R_{xy}, for all x, y ∈ G.

Lemma. Assume that G is abelian.

(a) The center Z(R) is a homogeneous subring of R, strongly graded by the subgroup G_{Z} = {x ∈ G | Z(R) ∩ R_{x} ≠ 0} of G.

(b) The ring R is a free Z(R)-module, in which Z(R) is a direct summand.

(c) Suppose that G_{Z} is free abelian of finite rank, with basis {g_{1}, …, g_{n}}. Choose a nonzero element z_{j} ∈ Z(R) ∩ R_{g_{j}} for each j. Then Z(R) = (Z(R) ∩ R_{1})[z_{1}^{±1}, …, z_{n}^{±1}], a Laurent polynomial ring over the field Z(R) ∩ R_{1}.

Proof. Part (a) is obvious (cf. [33, Part A, I.4.5(3)]), and allows us to write Z(R) = ⨁_{x∈G_{Z}} Z(R)_{x}, where Z(R)_{x} = Z(R) ∩ R_{x}.

(b) Note that the set G′ = {x ∈ G | R_{x} ≠ 0} is a subgroup of G. After replacing G by G′, we may assume that R_{x} ≠ 0 for all x ∈ G. (In particular, R is now strongly graded by G.)

Observe that S = ⨁_{x∈G_{Z}} R_{x} is a homogeneous subalgebra of R containing Z(R) and strongly graded by G_{Z}. Choose a transversal T for G_{Z} in G, with 1 ∈ T. For t ∈ T choose a nonzero element r_{t} ∈ R_{t}, with r_{1} = 1. Since the multiplication map G_{Z} × T → G is bijective,

R = ⨁_{x∈G_{Z}} ⨁_{t∈T} R_{xt} = ⨁_{x∈G_{Z}} ⨁_{t∈T} R_{x}r_{t} = ⨁_{t∈T} Sr_{t}.

Thus R is a free left S-module with basis {r_{t} | t ∈ T }, and since this basis includes r_{1} = 1, we also see that S is a left S-module direct summand of R. Therefore it suffices to show that S is a free Z(R)-module in which Z(R) is a direct summand.

Choose nonzero elements z_{x} ∈ Z(R)_{x} for x ∈ G_{Z}. Since Z(R)_{1} is a field, we can choose a basis B for S_{1} as a vector space over Z(R)_{1}, with 1 ∈ B. Hence,

S = ⨁_{x∈G_{Z}} S_{1}z_{x} = ⨁_{β∈B} ⨁_{x∈G_{Z}} Z(R)_{1}βz_{x} = ⨁_{β∈B} Z(R)β;

that is, S is a free Z(R)-module with basis B. Since 1 ∈ B, this also shows that Z(R) is a direct summand of S, as desired.

(c) Since the monomials z_{1}^{i_{1}}z_{2}^{i_{2}} ⋯ z_{n}^{i_{n}} are homogeneous with distinct degrees g_{1}^{i_{1}}g_{2}^{i_{2}} ⋯ g_{n}^{i_{n}}, it is clear that z_{1}, …, z_{n} are algebraically independent over Z(R)_{1}. For each x ∈ G_{Z}, there exist unique m_{1}, …, m_{n} ∈ ℤ such that x = g_{1}^{m_{1}}g_{2}^{m_{2}} ⋯ g_{n}^{m_{n}}, and z_{x} = z_{1}^{m_{1}}z_{2}^{m_{2}} ⋯ z_{n}^{m_{n}} is a nonzero element of Z(R)_{x}. Therefore Z(R) = ⨁_{x∈G_{Z}} Z(R)_{1}z_{x} = Z(R)_{1}[z_{1}^{±1}, …, z_{n}^{±1}]. □

6.4 Proposition. Let G be an abelian group and R a G-graded division ring. (a) If I is an ideal of R, then I = R(I ∩ Z(R)).

(b) If J is an ideal of Z(R), then J = (RJ) ∩ Z(R).

Proof. Part (b) is immediate from (6.3.b).

(a) Set J = I ∩ Z(R), and suppose that I ≠ RJ. Choose an element r ∈ I \ RJ with support {x_{1}, …, x_{n}} of minimal cardinality. Then write r = r_{1} +⋯+r_{n} for some nonzero elements r_{i} ∈ R_{x_{i}}.

First, suppose there exists a nonzero element s ∈ I whose support is properly contained in {x_{1}, …, x_{n}}. After possibly renumbering, we may assume that the support of s is contained in {x_{1}, …, x_{n−1}} and includes x_{1}. Write s = s_{1} + ⋯ + s_{n−1}, with each s_{i} ∈ R_{x_{i}} and s_{1} ≠ 0. Now consider the element t = r_{1}^{−1}r − s_{1}^{−1}s, and observe that t is an element in I with support contained in {x_{1}^{−1}x_{2}, …, x_{1}^{−1}x_{n}}. By the minimality of n, both s and t must lie in RJ. But then r = r_{1}t + r_{1}s_{1}^{−1}s ∈ RJ, contradicting our assumptions.

Thus there cannot be any nonzero element in I whose support is properly contained in {x_{1}, …, x_{n}}.

Now set r′ = r_{1}^{−1}r, which is an element of I with support {1, x_{1}^{−1}x_{2}, …, x_{1}^{−1}x_{n}} and identity component 1. Given any nonzero homogeneous element u ∈ R, observe that u^{−1}r′u has the same support as r′ (because G is abelian), and its identity component is also 1. Hence, the difference u^{−1}r′u − r′ has support contained in {x_{1}^{−1}x_{2}, …, x_{1}^{−1}x_{n}}, and so the element v = r_{1}(u^{−1}r′u − r′) is an element of I with support contained in {x_{2}, …, x_{n}}. By the previous paragraph, v = 0, whence u^{−1}r′u = r′. Since u was an arbitrary nonzero homogeneous element of R, this last equality implies that r′ ∈ Z(R). Hence, r = r_{1}r′ ∈ RJ, another contradiction.

Therefore I = RJ. □

6.5 Corollary. If G is an abelian group and R a G-graded division ring, then contraction and extension provide:

(a) Mutually inverse isomorphisms between the lattices of ideals of R and Z(R);

(b) Mutually inverse homeomorphisms between Spec R and Spec Z(R). □

We now return to the situation of (6.1) and apply (6.2)–(6.5), obtaining the following description of the strata:

6.6 Theorem. Let A be a noetherian algebra over an infinite field k, and suppose that a torus H = (k^{×})^{r} acts rationally on A by k-algebra automorphisms. Let J be a completely prime H-invariant ideal of A, and let

S_{J} = {P ∈ Spec A | (P : H) = J}

be the corresponding H-stratum of Spec A. Then there exists an Ore set E_{J} in the algebra A/J such that

(a) The localization map A → A/J → A_{J} = (A/J)[E_{J}^{−1}] induces a homeomorphism of S_{J} onto Spec A_{J}.

(b) Contraction and extension induce mutually inverse homeomorphisms between Spec A_{J} and Spec Z(A_{J}).

(c) Z(A_{J}) is a commutative Laurent polynomial ring over an extension field of k, in r or fewer indeterminates.

Proof. We may clearly assume that J = 0. As in (6.1), let Ĥ denote the group of rational characters of H, and grade A by Ĥ. Let E_{J} denote the set of nonzero homogeneous elements of A with respect to this grading. Then by (6.2), E_{J} is an Ore set in A and the localization A_{J} = A[E_{J}^{−1}] is an Ĥ-graded division ring. Observe that the prime ideals of A disjoint from E_{J} are precisely the prime ideals that contain no nonzero H-eigenvectors. The set of such prime ideals is precisely S_{J}, and therefore part (a) follows from standard localization theory (e.g., [14, 9.22], [28, 2.1.16vii]). Part (b) is given by (6.5), and part (c) follows from (6.3). □

6.7. The above description of the H-strata has been noted for certain quantized algebras in [4, 4.5, 5.1–7]. In view of (5.3), (5.5), (5.7) and (5.11), this description also holds for the H-strata of the algebras O_{q}(𝔰𝔭 k^{2n}), O_{q}(𝔬 k^{n}), O_{λ,p}(M_{n}(k)) and A_{n}^{Q,Γ}(k) discussed in (5.2), (5.4), (5.6) and (5.10). We emphasize that these results do not require k to be algebraically closed, as one might expect if one analyzed the maximal portion of an H-stratum using the Moeglin-Rentschler-Vonessen transitivity theorem (2.6). Moreover, a transitivity result suitable for our purposes follows easily from (6.2)–(6.5), as we now show.

6.8 Theorem. Let A, H, J, and S_{J} be as described in (6.6).

(a) Every rational ideal in S_{J} is maximal in S_{J}.

(b) If k is algebraically closed, then H acts transitively on S_{J} ∩ Rat A.

Proof. We may obviously assume that J = 0 and that S_{J} ∩Rat A is nonempty. Now A is a noetherian domain, graded by the character group Ĥ, which is free abelian of rank r. Let E denote the set of nonzero homogeneous elements of A. By (6.2), E is an Ore set in A, and A[E^{−1}] is an Ĥ-graded division ring. Note that the action of H on A extends naturally to a rational action by k-algebra automorphisms on A[E^{−1}], and that the H-grading on A[E^{−1}] corresponding to this action coincides with the one obtained from the localization process. By (6.6a), the localization map A → A[E^{−1}] induces a bijection from S_{J} onto Spec A[E^{−1}].

Consider a prime ideal P ∈ S_{J}. Since the Ore set E is disjoint from P, its image in A/P consists of regular elements (e.g., [14, 9.21]). Hence,

Fract(A[E^{−1}]/P A[E^{−1}]) ≅ Fract(A/P),

and so P A[E^{−1}] is rational if and only if P is rational. Thus the bijection of S_{J} onto Spec A[E^{−1}] via localization restricts to a bijection of S_{J}∩Rat A onto Rat A[E^{−1}]. Therefore we may replace A by A[E^{−1}], that is, we may assume that A is an Ĥ-graded division ring and that S_{J} = Spec A.

Since Ĥ is free abelian of finite rank, so is the subgroup

G_{Z} = {x ∈ Ĥ | Z(A)_{x} ≠ 0}.

Choose a basis {g_{1}, …, g_{n}} for G_{Z}, and choose a nonzero element z_{j} ∈ Z(A)_{g_{j}} for j = 1, …, n. By (6.3.c), Z(A) is a Laurent polynomial ring of the form Z(A)_{1}[z_{1}^{±1}, …, z_{n}^{±1}].

(a) Let P ∈ Rat A, and set Q = P ∩ Z(A). Then the domain Z(A)/Q embeds in the center of Fract(A/P), which is algebraic over k by our assumption on P. Since Z(A) is a Laurent polynomial ring over the field Z(A)_{1}, it follows that Z(A)_{1} is algebraic over k and that Z(A)/Q is a field. In particular, Q ∈ Max Z(A), and thus P ∈ Max A by (6.5).

(b) Let P_{1}, P_{2} ∈ Rat A, and set Q_{i} = P_{i} ∩ Z(A). Since k is now assumed to be algebraically closed, the observations of the preceding paragraph show that Z(A)_{1} = k and that each Q_{i} = ∑_{j=1}^{n} Z(A)(z_{j} − α_{ij}) for some α_{ij} ∈ k^{×}. Thus by (6.4), each P_{i} = ∑_{j=1}^{n} A(z_{j} − α_{ij}). Recall that each z_{j} ∈ Z(A)_{g_{j}} and that the g_{j} are linearly independent characters of H. Hence, there exists h ∈ H such that g_{j}(h) = α_{1j}α_{2j}^{−1} for all j [18, 16.2C]. Consequently,

h(z_{j} − α_{1j}) = g_{j}(h)z_{j} − α_{1j} = α_{1j}α_{2j}^{−1}(z_{j} − α_{2j})

for all j, from which we conclude that h(P_{1}) = P_{2}. □

We conclude by noting that (6.8) yields a direct proof of the special case of (2.12) corresponding to our present situation.

6.9 Corollary. Let A and H be as in (6.1), and assume that A satisfies the Nullstellensatz over k. Assume further that H -Spec A is finite, and that all H-prime ideals of A are completely prime. Then A satisfies the Dixmier-Moeglin Equivalence, and the primitive ideals of A are precisely the prime ideals maximal within their H-strata.

Proof. By (6.8a), the rational ideals of A are maximal within their H-strata. The corollary thus follows from (2.1iii) and (2.2ii). □

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Department of Mathematics, University of California, Santa Barbara, CA 93106 24 K. R. GOODEARL AND E. S. LETZTER

E-mail address: goodearl@math.ucsb.edu Department of Mathematics, Texas A&M University, College Station, TX 77843 E-mail address: letzter@math.tamu.edu