4 December 99

QUANTIZED PRIMITIVE IDEAL SPACES AS

QUOTIENTS OF AFFINE ALGEBRAIC VARIETIES

K. R. Goodearl

Abstract. Given an affine algebraic variety V and a quantization O_{q}(V) of its coordinate ring, it is conjectured that the primitive ideal space of O_{q}(V) can be expressed as a topological quotient of V. Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological quotient maps are given in the case of quantum toric varieties.

Introduction

A major theme in the subject of quantum groups is the philosophy that in the passage from a classical coordinate ring to a quantized analog, the classical geometry is replaced by structures that should be treated as ‘noncommutative geometry’. Indeed, much work has been invested into the development of theories of noncommutative differential geometry and noncommutative algebraic geometry. We would like to pose the question whether these theories are entirely noncommutative, or whether traces of classical geometry are to be found in the noncommutative geometry. This rather vague question can, of course, be focused in any number of different directions. We discuss one particular direction here, which was developed in joint work with E. S. Letzter [6]; it concerns situations in which quantized analogs of classical varieties contain certain quotients of these varieties.

To take an ideal-theoretic perspective on the question posed above, recall the way in which an affine algebraic variety V is captured in its coordinate ring O(V): the space of maximal ideals, max O(V), is homeomorphic to V, under their respective Zariski topologies. In the passage from commutative to noncommutative algebras, the natural analog of a maximal ideal space is a primitive ideal space – this point of view is taken partly on practical grounds, because noncommutative rings often have too few maximal ideals for various purposes, but also in order to reflect the ideal theory connected with the study of irreducible representations. (Recall that the primitive ideals of an algebra A are precisely the annihilators of the irreducible A-modules.) Thus, given a quantized version of O(V), say O_{q}(V), the natural analog of V is the primitive ideal space prim O_{q}(V). The aspect of the general problem which we wish to discuss here, then, is that of finding classical geometric structure in prim O_{q}(V), and relating it to the structure of V.

We report below on some success in expressing prim O_{q}(V) as a quotient of V with respect to the respective Zariski topologies. Modulo a minor technical assumption, this is

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16D30, 16D60, 16P40, 16S36, 17B37.

This research was partially supported by NATO Collaborative Research Grant 960250 and National Science Foundation research grants DMS-9622876, DMS-9970159.

done in three situations – when V is an algebraic torus, a full affine space, or (generalizing both of those cases) an affine toric variety. We discuss these cases in Sections 2, 3, and 4, respectively; we sketch some of the methods and indicate relationships among the three cases. In particular, in the last section we establish precise formulas for topological quotient maps in the case of quantum toric varieties, maps were only given an existence proof in [6].

Recall that a map ϕ : X → Y between topological spaces is a topological quotient map provided ϕ is surjective and the topology on Y coincides with the quotient topology induced by ϕ, that is, a subset C ⊆ Y is closed in Y precisely when ϕ^{−1}(C) is closed in X. In that case, Y is completely determined (as a topological space) by the topology on X together with the partition of X into fibres of ϕ.

Throughout the paper, we fix an algebraically closed base field k. All the varieties we discuss will be affine algebraic varieties over k.

1. Quantum semisimple groups

We follow the usual practice in writing ‘quantum semisimple groups’ as an abbreviation for ‘quantized coordinate rings of semisimple groups’. Thus, ‘quantum SL_{n}’ refers to any quantization of the coordinate ring O(SL_{n}(k)). We begin by displaying the primitive ideal space of the most basic example, the standard single parameter quantization of SL_{2}.

1.1. Example. Consider the algebra O_{q}(SL_{2}(k)), where q ∈ k^{×} is a non-root of unity. This is the k-algebra with generators X_{11}, X_{12}, X_{21}, X_{22} satisfying the following relations:

\begin{aligned}X_{11}X_{12} & = qX_{12}X_{11} & X_{11}X_{21} & = qX_{21}X_{11} & X_{12}X_{21} & = X_{21}X_{12} \\ X_{21}X_{22} & = qX_{22}X_{21} & X_{12}X_{22} & = qX_{22}X_{12} \end{aligned}

X_{11}X_{22} − X_{22}X_{11} = (q − q^{−1})X_{12}X_{21}  X_{11}X_{22} − qX_{12}X_{21} = 1

Since q is not a root of unity, the primitive ideal space of O_{q}(SL_{2}(k)) is not large, and can be completely described as follows:

⟨X_{11} − α, X_{12}, X_{21}, X_{22} − α^{−1}⟩ (α ∈ k^{×}) ⟨X_{12}⟩  ⟨X_{21}⟩ ⟨X_{12} − βX_{21}⟩ (β ∈ k^{×})

With the Zariski topology, this space seems more like a scheme than an affine variety, since it has many non-closed points. When taken apart as pictured, however, it can be viewed as a disjoint union of four classical varieties: two single points, and two punctured affine lines. Taken as a whole, prim O_{q}(SL_{2}(k)) can be related to the variety SL_{2}(k) via the following map:

( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) ↦ ⟨X_{11} − α, X_{12}, X_{21}, X_{22} − α^{−1}⟩  (α ∈ k^{×})
( \begin{matrix}α & 0 \\ γ & α^{−1} \end{matrix} ) ↦ ⟨X_{12}⟩  (α, γ ∈ k^{×})
( \begin{matrix}α & β \\ 0 & α^{−1} \end{matrix} ) ↦ ⟨X_{21}⟩  (α, β ∈ k^{×})
( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ↦ ⟨X_{12} − βγ^{−1}X_{21}⟩  (α, δ ∈ k; β, γ ∈ k^{×})
(αδ − βγ = 1)

We leave as an exercise for the reader to show that the map SL_{2}(k) ↠ prim O_{q}(SL_{2}(k)) given above is Zariski continuous; in fact, it is a topological quotient map.

1.2. Let G be a connected semisimple algebraic group over ℂ, and let q ∈ ℂ^{×} be a non-root of unity. Quantized coordinate rings of G have been defined in both single parameter and multiparameter versions, which we denote O_{q}(G) and O_{q,p}(G) respectively. We shall not recall the definitions here, but refer the reader to [8], [9], [10], [12], [13], or [4]. The first classification results for primitive ideals in this context were proved in the single parameter case:

Theorem. [Hodges-Levasseur, Joseph] There exists a bijection

prim O_{q}(G) ⟷ { symplectic leaves in G },

where the symplectic leaves are computed relative to the associated Poisson structure on G arising from the quantization.

Proof. This was proved for the cases G = SL_{3}(ℂ) and G = SL_{n}(ℂ) by Hodges and Levasseur in [8, Theorem 4.4.1] and [9, Theorem 4.2], and then for arbitrary G by Joseph [12, Theorem 9.2], [13, Theorems 10.3.7, 10.3.8]. □

Let us rewrite this result to bring in the points of G more directly: There exists a surjection G ↠ prim O_{q}(G) whose fibres are precisely the symplectic leaves in G. In the multiparameter case, the situation is slightly more complicated, as follows.

1.3. Theorem. [Hodges-Levasseur-Toro] There exists a surjection G ↠ prim O_{q,p}(G), but the fibres are symplectic leaves only for certain choices of p.

Proof. The first statement follows from the results in [10, Section 4]; for the second, see [10, Theorems 1.8, 4.18]. □

1.4. We can fill in a bit more detail about these surjections by bringing in a group of symmetries. Let H be a maximal torus in G, acting on O_{q,p}(G) by ‘winding automorphisms’ (cf. [13, (1.3.4)]). For example, if G = SL_{2}(ℂ) and H = \{( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) | α ∈ ℂ^{×}\}, then H

acts on O_{q}(G) so that ( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) . ( \begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ) = ( \begin{matrix}αX_{11} & αX_{12} \\ α^{−1}X_{21} & α^{−1}X_{22} \end{matrix} ). In this case, the induced

action of H on prim O_{q}(G) has four orbits, which are the sets displayed in Example 1.1. It follows from the analyses in [8, 9, 12, 13, 10] that • The number of H-orbits in prim O_{q,p}(G) is finite.

• Each of these H-orbits is Zariski homeomorphic to H modulo the relevant stabilizer.

• The preimage of each H-orbit, under the surjection in Theorem 1.3, is a locally closed subset of G.

These properties hint strongly that there is some topological connection between G and prim O_{q,p}(G).

1.5. Conjecture. There exists an H-equivariant surjection G ↠ prim O_{q,p}(G) which is a topological quotient map (with respect to the Zariski topologies).

If this conjecture were verified, then it together with knowledge of the fibres of the map would completely determine prim O_{q,p}(G) as a topological space.

In the case of O_{q}(SL_{2}(ℂ)), the conjecture can be readily verified ‘by hand’, as already suggested above. There is a further piece of evidence in the SL_{3} case: Brown and the author have constructed an H-equivariant topological quotient map B^{+} ↠ prim O_{q}(B^{+}) where B^{+} is the upper triangular Borel subgroup of SL_{3}(ℂ) [Work in progress].

The major difficulty in the way of the conjecture is that no good global description of the Zariski topology on prim O_{q,p}(G) is known, only the Zariski topologies on the separate H-orbits.

In order to try to obtain a better feel for the problem, let us widen the focus:

1.6. Conjecture. If O_{q}(V) is a quantized coordinate ring of an affine algebraic variety (or group) V, there exists a topological quotient map V ↠ prim O_{q}(V), equivariant with respect to an appropriate group of automorphisms.

Here O_{q}(V) is meant to stand for any quantization of O(V), a phrase which is only slightly less vague than the symbolism, since no definition of a “quantization” of a coordinate ring exists. For descriptions of many commonly studied classes of examples, see [4].

In the remainder of the paper, we discuss some cases where the extended conjecture has been verified.

2. Quantum tori

The simplest case in which to examine our conjecture is that of a quantization of the coordinate ring of an algebraic torus. The parameters required can be arranged as a matrix q = (q_{ij}) ∈ M_{n}(k^{×}) which is multiplicatively antisymmetric, that is, q_{ii} = 1 and q_{ji} = q_{ij}^{−1} for all i, j. (These conditions are assumed in order to avoid degeneracies in the resulting algebra.) The quantum torus over k with respect to these parameters is the algebra

O_{q}((k^{×})^{n}) := k⟨x_{1}^{±1}, …, x_{n}^{±1} | x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i, j⟩.

The structure of this algebra has been analyzed in various cases by many people, among them McConnell and Pettit [15], De Concini, Kac, and Procesi [3], Hodges [7], Vancliff [16], Brown and the author [2], Letzter and the author [5], and Ingalls [11]. In particular, it is known that prim O_{q}((k^{×})^{n}) and spec O_{q}((k^{×})^{n}) are homeomorphic to the maximal and prime spectra, respectively, of the center Z(O_{q}((k^{×})^{n})), via contraction and extension, and that the center is a Laurent polynomial ring over k (e.g., [5, Lemma 1.2, Corollary 1.5]).

2.1. Theorem. [6, Theorem 3.11] For any multiplicatively antisymmetric matrix q ∈ M_{n}(k^{×}), there exist topological quotient maps

(k^{×})^{n} ↠ prim O_{q}((k^{×})^{n})
spec k[y_{1}^{±1}, …, y_{n}^{±1}] ↠ spec O_{q}((k^{×})^{n}),

equivariant with respect to the natural actions of the torus (k^{×})^{n}. □

This case, in many ways, works out too smoothly to really put Conjecture 1.6 to the test. In that regard, the case of quantum affine spaces is much more interesting. Before moving on to that case, however, we describe the form of the maps in Theorem 2.1, since this will preview essential aspects of the quantum affine space case. We proceed by rewriting quantum tori as twisted group algebras, as follows.

2.2. Fix a quantum torus A = O_{q}((k^{×})^{n}). Set Γ = ℤ^{n}, and define σ : Γ × Γ → k^{×} by the rule

σ(α, β) = ∏_{i,j=1}^{n} q_{ij}^{α_{i}β_{j}}.

Then σ is an alternating bicharacter on Γ, and it determines the commutation rules in A, since x^{α}x^{β} = σ(α, β)x^{β}x^{α} for all α, β ∈ Γ, where we have used the standard multi-index notation x^{α} = x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ x_{n}^{α_{n}}. Recall that the radical of σ is the subgroup

rad(σ) = {α ∈ Γ | σ(α, −) ≡ 1}

of Γ. It plays the following role: Z(A) = k[x^{α} | α ∈ rad(σ)] (cf. [5, Lemma 1.2] or [11, Proposition 5.2]).

Now choose a 2-cocycle c : Γ×Γ → k^{×} such that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ. (This can be done in many ways – see [1, Proposition 1, p. 888] or [6, (3.2)], for instance.) We may identify A with the twisted group algebra k^{c}Γ, that is, the k-algebra with a basis {x_{α} | α ∈ Γ} such that

x_{α}x_{β} = c(α, β)x_{α+β}

for α, β ∈ Γ.

Let us write the group algebra kΓ in terms of a basis {y_{α} | α ∈ Γ}, where y_{α}y_{β} = y_{α+β} for α, β ∈ Γ. There is a k-linear (vector space) isomorphism

Φ_{c} : A → kΓ

such that Φ_{c}(x_{α}) = y_{α} for α ∈ Γ. While this map is not multiplicative, it satisfies Φ_{c}(x_{α}x_{β}) = c(α, β)Φ_{c}(x_{α})Φ_{c}(x_{β}) for α, β ∈ Γ (cf. [6, (3.5)]).

The final ingredient needed to describe the topological quotient maps in the current situation is the torus H = (k^{×})^{n} and its natural actions on A and kΓ by k-algebra automorphisms. The induced actions of H on the prime and primitive spectra of A and kΓ will also be required. We identify H with Hom(Γ, k^{×}), which allows us to write the above actions as

h.x_{α} = h(α)x_{α}  and  h.y_{α} = h(α)y_{α}

for h ∈ H and α ∈ Γ. Observe that the map Φ_{c} is H-equivariant. The identification H = Hom(Γ, k^{×}) provides a pairing H × Γ → k^{×}, and we use ^{⊥} to denote orthogonals with respect to this pairing. In particular,

S^{⊥} = {h ∈ H | ker h ⊇ S}

for S ⊆ Γ.

2.3. It is convenient to have a compact notation for the intersection of the orbit of an ideal under a group of automorphisms. If P is an ideal of a ring B, and T is a group acting on B by ring automorphisms, we write

(P : T) = ⋂_{t∈T} t(P).

This ideal can also be described as the largest T -invariant ideal of B contained in P. The same notation can also be applied to any subset of B.

2.4. Theorem. [6, Theorem 3.11] Let A = O_{q}((k^{×})^{n}), and fix Γ, σ, c, Φ_{c}, H as above. Assume that c ≡ 1 on rad(σ) × Γ. (Such a choice for c is always possible; cf. [6, Lemma 3.12].) Then the rule P ↦ Φ_{c}^{−1}(P : rad(σ)^{⊥}), for prime ideals P of kΓ, defines Hequivariant topological quotient maps

spec kΓ ↠ spec A  and  max kΓ ↠ prim A.

The fibres of the second map are exactly the rad(σ)^{⊥}-orbits in max kΓ. □

2.5. As the formula in Theorem 2.4 indicates, the given maps are compositions of two maps, which themselves have nice properties. To express this in more detail, write S = rad(σ), and recall that an S^{⊥}-prime ideal of kΓ is any proper S^{⊥}-invariant ideal Q such that whenever Q contains a product of S^{⊥}-invariant ideals, it must contain one of the factors. Denote by S^{⊥}- spec kΓ the set of S^{⊥}-prime ideals of kΓ, and observe that this set supports a Zariski topology (defined in the obvious way). The first map in Theorem 2.4 can be factored in the form

spec kΓ ↠ S^{⊥}- spec kΓ ⟶^{≈} spec A,

where the map P ↦ (P : S^{⊥}) from spec kΓ to S^{⊥}- spec kΓ is a topological quotient map, and the map Q ↦ Φ_{c}^{−1}(Q) from S^{⊥}- spec kΓ to spec A is a homeomorphism. That P ↦ (P : S^{⊥}) is a topological quotient map is an easy exercise (cf. [6, Proposition 1.7] for a generalization); the bulk of the work in proving Theorem 2.4 goes into establishing the homeomorphism S^{⊥}- spec kΓ ≈ spec A (cf. [6, Proposition 3.10]). The key point is that the hypothesis on c ensures that Φ_{c} is ‘central-semilinear’, that is, Φ_{c}(za) = Φ_{c}(z)Φ_{c}(a) for z ∈ Z(A) and a ∈ A [6, Lemma 3.5].

The factorization above respects maximal and primitive ideals in the following way. Let S^{⊥}- max kΓ denote the set of maximal proper S^{⊥}-invariant ideals of kΓ; this is a subset of S^{⊥}- spec kΓ, which we equip with the relative topology. Then we have the factorization

max kΓ ↠ S^{⊥}- max kΓ ⟶^{≈} prim A,

where P ↦ (P : S^{⊥}) gives a topological quotient map from max kΓ to S^{⊥}- max kΓ, and Q ↦ Φ_{c}^{−1}(Q) gives a homeomorphism from S^{⊥}- max kΓ to prim A (cf. [6, Propositions 3.9, 3.10]).

3. Quantum affine spaces

Let q ∈ M_{n}(k^{×}) again be a multiplicatively antisymmetric matrix. The corresponding quantum affine space over k is the algebra

O_{q}(k^{×}) := k⟨x_{1}, …, x_{n} | x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i, j⟩.

3.1. Denote the algebra O_{q}(k^{×}) by A for purposes of discussion. The torus H = (k^{×})^{n} again acts naturally as k-algebra automorphisms on A. The primitive spectrum of A consists of 2^{n} H-orbits, which we denote prim_{w} A, indexed by the subsets w ⊆ {1, …, n} [5, Theorem 2.3]. Here

prim_{w} A = {P ∈ prim A | x_{i} ∈ P ⟺ i ∈ w}.

Similarly, spec A is a disjoint union of 2^{n} subsets

spec_{w} A = {P ∈ spec A | x_{i} ∈ P ⟺ i ∈ w}.

Each spec_{w} A is homeomorphic, via localization, to the prime spectrum of the quantum torus

A_{w} = (A/⟨x_{i} | i ∈ w⟩)[x_{j}^{−1} | j ∉ w],

and likewise prim_{w} A is homeomorphic to prim A_{w} (cf. [5, Theorem 2.3]).

The partition prim A = ⨆_{w} prim_{w} A is a quantum analog of the standard stratification of affine n-space by its H-orbits. To emphasize the parallel, we label the H-orbits in k^{n} as

(k^{n})_{w} = {(α_{1}, …, α_{n}) ∈ k^{n} | α_{i} = 0 ⟺ i ∈ w},

for w ⊆ {1, …, n}. It follows from Theorem 2.1 that there are H-equivariant topological quotient maps

(k^{n})_{w} ↠ prim A_{w} ≈ prim_{w} A

for all w. The problem is then to patch these maps together coherently. Modulo a minor technical assumption, that can be done:

3.2. Theorem. [6, Theorem 4.11] Let q = (q_{ij}) ∈ M_{n}(k^{×}) be a multiplicatively antisymmetric matrix. Assume that either −1 ∉ ⟨q_{ij}⟩ or char k = 2. Then there exist topological quotient maps

k^{n} ↠ prim O_{q}(k^{n})
spec k[y_{1}, …, y_{n}] ↠ spec O_{q}(k^{n}),

equivariant with respect to the natural actions of the torus (k^{×})^{n}. □

Here ⟨q_{ij}⟩ denotes the subgroup of k^{×} generated by the q_{ij}.

To describe the maps in the above theorem, and identify the fibres of the first, we use a setup analogous to that in the quantum torus case. In particular, we write O_{q}(k^{n}) as a twisted semigroup algebra with respect to a suitable cocycle.

3.3. Fix a quantum affine space A = O_{q}(k^{n}). Set Γ = ℤ^{n} and Γ^{+} = (ℤ^{+})^{n}, and define σ : Γ × Γ → k^{×} as in (2.2). As in the previous case, σ determines the commutation rules in A.

Next, choose a 2-cocycle c : Γ×Γ → k^{×} such that c(α, β)c(β, α)^{−1} = σ(α, β) for α, β ∈ Γ. We identify A with the twisted semigroup algebra k^{c}Γ^{+} with a basis {x_{α} | α ∈ Γ^{+}}, and we write the corresponding semigroup algebra R = kΓ^{+} in terms of a basis {y_{α} | α ∈ Γ^{+}}. There are partitions spec R = ⨆_{w} spec_{w} R and max R = ⨆_{w} max_{w} R analogous to those for spec A and prim A.

Again as before, identify the torus H = (k^{×})^{n} with Hom(Γ, k^{×}), so that the natural actions of H on A and kΓ^{+} are given by

h.x_{α} = h(α)x_{α}  and  h.y_{α} = h(α)y_{α}.

3.4. It is convenient to identify the localizations A_{w} with subalgebras of the twisted group algebra k^{c}Γ, and to identify the corresponding localizations R_{w} of R with subalgebras of the group algebra kΓ. This is done as follows:

A_{w} = ∑_{α∈Γ_{w}} kx_{α}  and  R_{w} = ∑_{α∈Γ_{w}} ky_{α} = kΓ_{w},

where Γ_{w} = {α ∈ Γ | α_{i} = 0 for i ∈ w}. (Here we have extended the x_{α} and the y_{α} to Γindexed bases for k^{c}Γ and kΓ.) As in (2.2), there is an H-equivariant k-linear isomorphism Φ = Φ_{c} : k^{c}Γ → kΓ such that Φ(x_{α}) = y_{α} for α ∈ Γ. This map restricts to H-equivariant k-linear isomorphisms from A onto R and from A_{w} onto R_{w} for each w.

Next, let σ_{w} and c_{w} denote the restrictions of σ and c to Γ_{w}. Then the identification above can be restated as A_{w} = k^{c_{w}}Γ_{w}. We shall need the radical of σ_{w}, and we emphasize that this is a subgroup of Γ_{w}. There is a natural pairing between Γ_{w} and a quotient group of H, but it is more useful to take orthogonals with respect to the pairing of H and Γ; thus, we write

rad(σ_{w})^{⊥} = {h ∈ H | ker h ⊇ rad(σ_{w})}.

3.5. Theorem. [6, Theorem 4.11] Let A = O_{q}(k^{n}), and assume that either −1 ∉ ⟨q_{ij}⟩ or char k = 2. Fix Γ, σ, c, Φ, H, σ_{w} as above. Assume that c is an alternating bicharacter on Γ with c^{2} = σ and such that c(α, β) = 1 whenever σ(α, β) = 1. (Such a choice for c is possible by [6, Lemma 4.2].) Then there are H-equivariant topological quotient maps

spec kΓ^{+} ↠ spec A  and  max kΓ^{+} ↠ prim A

such that P ↦ Φ^{−1}(P : rad(σ_{w})^{⊥}) for P ∈ spec_{w} kΓ^{+}. The fibres of the second map over points in prim_{w} A are exactly the rad(σ_{w})^{⊥}-orbits within max_{w} kΓ^{+}. □

While the precise formula given for the maps in Theorem 3.5 is particularly useful for computations, it may round out the picture to give a global formula (independent of w), as follows.

3.6. Lemma. Let ϕ : spec kΓ^{+} ↠ spec A be the topological quotient map given in Theorem 3.5. For all P ∈ spec kΓ^{+}, the prime ideal ϕ(P) ∈ spec A equals the largest ideal of A contained in the set Φ^{−1}(P).

Proof. We shall need the generators y_{i} = y_{ϵ_{i}} ∈ kΓ^{+} and x_{i} = x_{ϵ_{i}} ∈ A, where ϵ_{1}, …, ϵ_{n} is the standard basis for Γ.

Let P ∈ spec kΓ^{+}; then P ∈ spec_{w} kΓ^{+} for some w. First suppose that P ∈ max_{w} kΓ^{+}. Then ϕ(P) ∈ prim_{w} A, and so ϕ(P) is a maximal element of spec_{w} A [5, Theorem 2.3]. Now ϕ(P) contains the ideal J_{w} := ⟨x_{i} | i ∈ w⟩. On the other hand, the multiplicative set generated by {y_{j} | j ∉ w} in kΓ^{+} is disjoint from P, and Φ^{−1} sends elements of this set to scalar multiples of elements in the multiplicative set X_{w} generated by {x_{j} | j ∉ w} in A. Hence, Φ^{−1}(P) is disjoint from X_{w}.

Let I be the largest ideal of A which is contained in Φ^{−1}(P). Then J_{w} ⊆ ϕ(P) ⊆ I and I is disjoint from X_{w}. Hence, I/J_{w} induces a proper ideal, call it IA_{w}, in the localization A_{w} = (A/J_{w})[X_{w}^{−1}]. Let M be a maximal ideal of A_{w} containing IA_{w}, and let Q be the inverse image of M under the localization map A → A/J_{w} → A_{w}. Then Q ∈ spec_{w} A and ϕ(P) ⊆ I ⊆ Q. Since ϕ(P) is maximal in spec_{w} A, we obtain ϕ(P) = Q and thus ϕ(P) = I. Therefore the lemma holds for P ∈ max_{w} kΓ^{+}.

Now consider an arbitrary P ∈ spec_{w} kΓ^{+}. Since kΓ^{+} is a commutative affine k-algebra, P is an intersection of maximal ideals. It follows easily, as in [2, Proposition 1.3(a)], that P = ⋂_{r} P_{r} for some maximal ideals P_{r} ∈ max_{w} kΓ^{+}. (Namely, P = ⋂_{v⊆{1,...,n}} Q_{v} where each Q_{v} is an intersection of maximal ideals from max_{v} kΓ^{+}. Observe that Q_{v} = kΓ^{+} when v ⊉ w, and that Q_{v} ⊋ P when v ⊋ w. Consequently, P = Q_{w}.) Since the set functions (− : rad(σ_{w})^{⊥}) and Φ^{−1} preserve intersections, we see that ϕ(P) = ⋂_{r} ϕ(P_{r}) and Φ^{−1}(P) = ⋂_{r} Φ^{−1}(P_{r}). By the previous paragraph, each ϕ(P_{r}) is the largest ideal of A contained in Φ^{−1}(P_{r}). Therefore ϕ(P) is the largest ideal of A contained in Φ^{−1}(P). □

3.7. The maps in Theorem 3.5 have analogous factorizations to those in (2.5), which we write

spec kΓ^{+} ↠ G -spec kΓ^{+} ⟶^{≈} spec A
max kΓ^{+} ↠ G -max kΓ^{+} ⟶^{≈} prim A.

Here G stands for the indexed family of groups {S_{w}^{⊥} | w ⊆ {1, …, n}} where S_{w} = rad(σ_{w}), while

G -spec kΓ^{+} = ⨆_{w⊆{1,...,n}}{(P : S_{w}^{⊥}) | P ∈ spec_{w} kΓ^{+}}
G -max kΓ^{+} = ⨆_{w⊆{1,...,n}}{(P : S_{w}^{⊥}) | P ∈ max_{w} kΓ^{+}}.

Closed sets for Zariski topologies on G -spec kΓ^{+} and G -max kΓ^{+} are defined in the usual way; a compatibility condition on the groups in G ensures that the result actually is a topology [6, (2.4), (4.9)]. A fairly general piece of commutative algebra, developed in [6, Section 2], provides the topological quotient maps from spec kΓ^{+} and max kΓ^{+} onto G -spec kΓ^{+} and G -max kΓ^{+}. The homeomorphisms of G -spec kΓ^{+} and G -max kΓ^{+} onto spec A and prim A are constructed by patching together homeomorphisms of S_{w}^{⊥}- spec kΓ_{w} and S_{w}^{⊥}- max kΓ_{w} onto spec A_{w} and prim A_{w} obtained as in (2.5).

3.8. Example. To illustrate the form of these topological quotients, we give the basic example in dimension 3, where various differences between the classical and quantum cases appear. Recall that the single parameter quantum affine spaces, relative to a scalar q ∈ k^{×}, are the algebras

O_{q}(k^{n}) = k⟨x_{1}, …, x_{n} | x_{i}x_{j} = qx_{j}x_{i} for i < j}.

(a) Assume first that q is not a root of unity, and let p be a square root of q in k. In this case, the topological quotient map

k^{3} ≈ max k(ℤ^{+})^{3} ↠ prim O_{q}(k^{3})

described in Theorem 3.5 can be computed as follows [6, (5.2)], where the scalars λ_{i} are all assumed to be nonzero:

(0, 0, 0) ↦ ⟨x_{1}, x_{2}, x_{3}⟩  (λ_{1}, λ_{2}, 0) ↦ ⟨x_{3}⟩
(λ_{1}, 0, 0) ↦ ⟨x_{1} − λ_{1}, x_{2}, x_{3}⟩  (λ_{1}, 0, λ_{3}) ↦ ⟨x_{2}⟩
(0, λ_{2}, 0) ↦ ⟨x_{1}, x_{2} − λ_{2}, x_{3}⟩  (0, λ_{2}, λ_{3}) ↦ ⟨x_{1}⟩
(0, 0, λ_{3}) ↦ ⟨x_{1}, x_{2}, x_{3} − λ_{3}⟩  (λ_{1}, λ_{2}, λ_{3}) ↦ ⟨λ_{2}x_{1}x_{3} − pλ_{1}λ_{3}x_{2}⟩

The right column demonstrates both the compression effect of the quotient process and the fact that O_{q}(k^{3}) has ‘many fewer’ primitive ideals than O(k^{3}). Note the factor p in the last term. Without this factor, we obtain only a surjective map, which is not Zariski continuous.

(b) For contrast, consider the case where q is a primitive t-th root of unity for some odd t > 1, and take p = q^{(t+1)/2}. Then the topological quotient map k^{3} ↠ prim O_{q}(k^{3}) has the form below [6, (5.3)]:

(0, 0, 0) ↦ ⟨x_{1}, x_{2}, x_{3}⟩  (λ_{1}, λ_{2}, 0) ↦ ⟨x_{1}^{t} − λ_{1}^{t}, x_{2}^{t} − λ_{2}^{t}, x_{3}⟩
(λ_{1}, 0, 0) ↦ ⟨x_{1} − λ_{1}, x_{2}, x_{3}⟩  (λ_{1}, 0, λ_{3}) ↦ ⟨x_{1}^{t} − λ_{1}^{t}, x_{2}, x_{3}^{t} − λ_{3}^{t}⟩
(0, λ_{2}, 0) ↦ ⟨x_{1}, x_{2} − λ_{2}, x_{3}⟩  (0, λ_{2}, λ_{3}) ↦ ⟨x_{1}, x_{2}^{t} − λ_{2}^{t}, x_{3}^{t} − λ_{3}^{t}⟩
(0, 0, λ_{3}) ↦ ⟨x_{1}, x_{2}, x_{3} − λ_{3}⟩
(λ_{1}, λ_{2}, λ_{3}) ↦ ⟨x_{1}^{t} − λ_{1}^{t}, x_{2}^{t} − λ_{2}^{t}, x_{3}^{t} − λ_{3}^{t}, λ_{2}x_{1}x_{3} − pλ_{1}λ_{3}x_{2}⟩

In this case, prim O_{q}(k^{3}) is more ‘classical’, in that all its points are closed.

4. Quantum toric varieties and cocycle twists

In this section, we discuss quantizations of toric varieties [11] and extend Theorem 3.5 to that setting. There are several ways in which (affine) toric varieties may be defined. For our purposes, the simplest definition is a variety on which a torus acts (morphically) with a dense orbit. Some equivalent conditions are given in the following theorem.

4.1. Theorem. [14, Satz 5, p. 105] Let H be an algebraic torus, acting morphically on an irreducible affine variety V. The following conditions are equivalent:

(a) There are only finitely many H-orbits in V.

(b) Some H-orbit is dense in V.

(c) The fixed field k(V)^{H} = k.

(d) All H-eigenspaces in k(V) are 1-dimensional. □

For comparison, we state a ring-theoretic version of this theorem. Since it will not be needed below, however, we leave the proof to the reader.

4.2. Theorem. Let R be a commutative affine domain over k, and let H be an algebraic torus acting rationally on R by k-algebra automorphisms. Then the following conditions are equivalent:

(a) There are only finitely many H-orbits in max R.

(b) Some H-orbit is dense in max R.

(c) (Fract R)^{H} = k, where Fract R is the quotient field of R.

(d) All H-eigenspaces in R are 1-dimensional.

(e) There are only finitely many H-invariant prime ideals in R. □

In the noncommutative setting, consider a prime, noetherian, affine k-algebra R, let Fract R be the Goldie quotient ring of R, and replace the maximal ideal space by prim R. The conditions of Theorem 4.2 are no longer equivalent, since prim R may well have a dense point (if 0 is a primitive ideal). It is thus natural to focus on the stronger conditions, (c) and (d).

Following Ingalls [11], we define a quantum (affine) toric variety to be an affine domain over k equipped with a rational action of an algebraic torus H by k-algebra automorphisms, such that the H-eigenspaces are 1-dimensional. If we fix H and require the action to be faithful, such algebras can be classified by elements of ∧^{2}H together with finitely generated submonoids that generate the character group Ĥ [11, Theorem 2.6].

Quantum toric varieties may also be presented as ‘cocycle twists’ (see below) of coordinate rings of classical toric varieties, analogous to the way quantum tori and quantum affine spaces are twisted versions of (Laurent) polynomial rings. Furthermore, quantum toric varieties can be written as factor algebras of quantum affine spaces, which shows, in particular, that they are noetherian. It also allows us to apply Theorem 3.2 and express their primitive spectra as topological quotients of classical varieties. For this purpose, a presentation as a cocycle twist of a commutative affine algebra suffices; one-dimensionality of eigenspaces is not needed. Hence, we can work with a somewhat larger class of algebras than quantum toric varieties.

4.3. Suppose that R is a k-algebra graded by an abelian group G, and that c : G × G → k^{×} is a 2-cocycle. The underlying G-graded vector space of R can be equipped with a new multiplication ∗, where r ∗ s = c(α, β)rs for all homogeneous elements r ∈ R_{α} and s ∈ R_{β}. This new multiplication is associative, and gives R a new structure as G-graded kalgebra, called the twist of R by c (see [1, Section 3] for details). The following alternative presentation is helpful in keeping computations straight. Let R′ be an isomorphic graded vector space copy of R, equipped with a G-graded k-linear isomorphism R → R′ denoted r ↦ r′. Then R′ becomes a G-graded k-algebra such that r′s′ = c(α, β)(rs)′ for r ∈ R_{α} and s ∈ R_{β}. The twist map r ↦ r′ then gives an isomorphism of the twist of R by c onto R′.

Now suppose that R is a commutative affine k-algebra, that G is torsionfree, and that either −1 is not in the subgroup of k^{×} generated by the image of c or char k = 2. Then there exist topological quotient maps

spec R ↠ spec R′  and  max R ↠ prim R′

[6, Theorem 6.3]. We shall show that these maps can be given by formulas analogous to those in Theorem 3.5.

It is convenient to immediately make a reduction to a special choice of c. First, we can replace G by any finitely generated subgroup containing the support of R. Then, as in the proof of [6, Theorem 6.3], there exists an alternating bicharacter d on G, satisfying the same hypotheses as c, such that R′ is isomorphic to the twist of R by d. Thus, there is no loss of generality in assuming that c is an alternating bicharacter, and we shall describe the topological quotient maps above under this assumption.

The torsionfreeness hypothesis on G is only needed for the reduction just indicated. Hence, we drop this assumption in the discussion to follow.

4.4. To recap, we are now assuming that R is a commutative affine k-algebra graded by an abelian group G, that c : G × G → k^{×} is an alternating bicharacter, and that either −1 ∉ ⟨im c⟩ or char k = 2. Let A = R′ be the twist of R by c.

Choose homogeneous k-algebra generators r_{1}, …, r_{n} for R, and set a_{i} = r′_{i} ∈ A for i = 1, …, n. We use these elements to define sets spec_{w} R, spec_{w} A, max_{w} R, prim_{w} A for w ⊆ {1, …, n} along the same lines as in (3.1). Thus

spec_{w} R = {P ∈ spec R | r_{i} ∈ P ⟺ i ∈ w}
spec_{w} A = {P ∈ spec A | a_{i} ∈ P ⟺ i ∈ w},

while max_{w} R = (max R) ∩ (spec_{w} R) and prim_{w} A = (prim A) ∩ (spec_{w} A). Of course, some of these sets may be empty.

Set H = Hom(G, k^{×}), an abelian group under pointwise multiplication. Because R and A are G-graded k-algebras, H acts on them by k-algebra automorphisms such that h.r = h(x)r and h.a = h(x)a for h ∈ H, r ∈ R_{x}, a ∈ A_{x}. Note that the sets spec_{w} R and spec_{w} A are invariant under the induced H-actions, since the elements r_{i} and a_{i} are H-eigenvectors.

For i = 1, …, n, let δ_{i} ∈ G denote the degree of r_{i} and a_{i}. For w ⊆ {1, …, n}, set G_{w} = ∑_{i∉w} ℤδ_{i} and let c_{w} denote the restriction of c to G_{w}. By definition of H, we have a pairing H × G → k^{×}, and we use ^{⊥} to denote orthogonals with respect to this pairing. In particular, for each w we can define rad(c_{w})^{⊥}, which is a subgroup of H.

We can now state the following more precise version of [6, Theorem 6.3]:

4.5. Theorem. Let R be a commutative affine k-algebra, graded by an abelian group G. Let c : G × G → k^{×} be an alternating bicharacter, and let A = R′ be the twist of R by c. Assume that either −1 ∉ ⟨im c⟩ or char k = 2.

With notation as in (4.4), there exist H-equivariant topological quotient maps

ϕ_{s} : spec R ↠ spec A  and  ϕ_{m} : max R ↠ prim A

such that P ↦ (P : rad(c_{w})^{⊥})′ for P ∈ spec_{w} R. Alternatively, ϕ_{s}(P) is the largest ideal of A contained in P′. Moreover, the fibres of ϕ_{m} over points in prim_{w} A are precisely the rad(c_{w})^{⊥}-orbits in max_{w} R.

Note that the second description of ϕ_{s} shows that this map depends only on R, G, and c, that is, it is independent of the choice of homogeneous generators r_{i} ∈ R as in (4.4).

The proof of Theorem 4.5 will be distributed over the following three subsections.

4.6. Let Φ : A → R be the inverse of the twist map; this is a G-graded k-linear isomorphism. In particular, X′ = Φ^{−1}(X) for subsets X ⊆ R.

Set Γ = ℤ^{n} and Γ^{+} = (ℤ^{+})^{n}, and let ρ : Γ → G be the group homomorphism given by the rule ρ(α_{1}, …, α_{n}) = α_{1}δ_{1} + ⋯ + α_{n}δ_{n}. Set c̃ = c ∘ (ρ × ρ), which is an alternating bicharacter on Γ, as is σ = c̃^{2}. For α = (α_{1}, …, α_{n}) ∈ Γ^{+}, define

r^{α} = r_{1}^{α_{1}}r_{2}^{α_{2}} ⋯ r_{n}^{α_{n}} ∈ R_{ρ(α)}  and  a_{α} = (r^{α})′ ∈ A_{ρ(α)}.

Then a_{α}a_{β} = c̃(α, β)a_{α+β} for α, β ∈ Γ^{+}.

Next, set R̃ = kΓ^{+} and Ã = kc̃Γ^{+}, expressed with k-bases {y_{α}} and {x_{α}} such that y_{α}y_{β} = y_{α+β} and x_{α}x_{β} = c̃(α, β)x_{α+β} for α, β ∈ Γ^{+}. Let Φ̃ : Ã → R̃ be the inverse of the twist map; this is a Γ-graded k-linear isomorphism such that Φ̃(x_{α}) = y_{α} for α ∈ Γ^{+}. There exist surjective k-algebra maps τ : R̃ → R and π : Ã → A such that τ(y_{α}) = r^{α} and π(x_{α}) = a_{α} for α ∈ Γ^{+}. Thus, we obtain a commutative diagram

[figure: R̃; τ; Ã; Φ̃; π; R; A; Φ]

Let H̃ = Hom(Γ, k^{×}), acting via k-algebra automorphisms on Ã and R̃ in the usual way. For w ⊆ {1, …, n}, define spec_{w} Ã, spec_{w} R̃, Γ_{w}, Ã_{w}, R̃_{w}, σ_{w}, c̃_{w} as in (3.1), (3.3), (3.4). (To define spec_{w} Ã and spec_{w} R̃, for instance, use the elements x_{ϵ_{i}} ∈ Ã and y_{ϵ_{i}} ∈ R̃, where ϵ_{1}, …, ϵ_{n} denotes the standard basis for Γ.) Because of our hypotheses on c, the group ⟨im c̃⟩ contains no elements of order 2, and hence rad(σ_{w}) = rad(c̃_{w}) for all w. We set S̃_{w} = rad(c̃_{w}).

Now Theorem 3.5 gives us H̃-equivariant topological quotient maps

ϕ̃_{s} : spec R̃ ↠ spec Ã and  ϕ̃_{m} : max R̃ ↠ prim Ã

such that P ↦ Φ̃^{−1}(P : S̃_{w}^{⊥}) for P ∈ spec_{w} R̃. Moreover, the fibres of ϕ̃_{m} over points in prim_{w} Ã are the S̃_{w}^{⊥}-orbits in max_{w} R̃.

Set V = {P ∈ spec R̃ | P ⊇ ker τ} and W = {P ∈ spec Ã | P ⊇ ker π}. The proof of [6, Theorem 6.3] shows that ϕ̃_{s} and ϕ̃_{m} restrict to topological quotient maps V ↠ W and V ∩ max R̃ ↠ W ∩ prim Ã. Therefore there are topological quotient maps ϕ_{s} and ϕ_{m} such that the following diagrams commute:

[figure: spec R̃; ϕ̃_{s}; spec Ã; max R̃; ϕ̃_{m}; prim Ã; V; ⊆; W; ⊆; V ∩ max R̃; ⊆; W ∩ prim Ã; ⊆; spec R; τ^{−1}; ϕ_{s}; spec A; π^{−1}; max R; τ^{−1}; ϕ_{m}; prim A; π^{−1}]

It remains to show that ϕ_{s} and ϕ_{m} have the properties announced in the statement of Theorem 4.5.

Given any P ∈ spec R, we have π^{−1}(ϕ_{s}(P)) = ϕ̃_{s}(τ^{−1}(P)), which by Lemma 3.6 equals the largest ideal of Ã contained in Φ̃^{−1}τ^{−1}(P). If I is an ideal of A contained in P′ = Φ^{−1}(P), then π^{−1}(I) is an ideal of Ã contained in π^{−1}Φ^{−1}(P) = Φ̃^{−1}τ^{−1}(P). Hence, π^{−1}(I) ⊆ π^{−1}(ϕ_{s}(P)), and so I ⊆ ϕ_{s}(P). This shows that ϕ_{s}(P) is the largest ideal of A contained in P′.

The following lemma records some facts and conditions that we shall need to finish the proof.

4.7. Lemma. Let w ⊆ {1, …, n}, and set S_{w} = rad(c_{w}).

(a) ker τ ⊆ (τ^{−1}(P) : S̃_{w}^{⊥}) ⊆ h(τ^{−1}(P)) for P ∈ spec_{w} R and h ∈ S̃_{w}^{⊥}.

(b) (h.(−)) ∘ τ = τ ∘ ((hρ).(−)) for h ∈ H.

(c) ρ^{−1}(S_{w}) ∩ Γ_{w} = S̃_{w} and so ρ(S̃_{w}) = S_{w}.

(d) (hρ)(τ^{−1}(P)) = τ^{−1}(h(P)) for P ∈ spec_{w} R and h ∈ S_{w}^{⊥}.

(e) S̃_{w}^{⊥} = ρ^{∗}(S_{w}^{⊥})Γ_{w}^{⊥}, where ρ^{∗} : H → H̃ denotes the homomorphism given by composition with ρ.

(f) For P ∈ spec_{w} R, the set {τ^{−1}(h(P)) | h ∈ S_{w}^{⊥}} equals the S̃_{w}^{⊥}-orbit of τ^{−1}(P).

Proof. (a) Note that τ^{−1}(P) ∈ spec_{w} R̃ and ϕ̃_{s}(τ^{−1}(P)) ∈ W. Thus Φ̃^{−1}(τ^{−1}(P) : S̃_{w}^{⊥}) ⊇ ker π, and so (τ^{−1}(P) : S̃_{w}^{⊥}) ⊇ ker τ. The remaining inclusion is clear.

(b) This follows from the observation that

h.τ(y_{α}) = h.r^{α} = hρ(α)r^{α} = τ(hρ(α)y_{α}) = τ((hρ).y_{α})

for h ∈ H and α ∈ Γ^{+}.

(c) Observe that ρ(Γ_{w}) = G_{w}. Since S_{w} ⊆ G_{w} and S̃_{w} ⊆ Γ_{w}, the second part of the claim will follow from the first. Now consider an arbitrary element γ ∈ Γ_{w}. Then ρ(γ) ∈ S_{w} if and only if c(ρ(γ), −) ≡ 1 on G_{w} = ρ(Γ_{w}), if and only if c̃(γ, −) ≡ 1 on Γ_{w}, if and only if γ ∈ S̃_{w}.

(d) It is clear from part (b) that τ(hρ)(τ^{−1}(P)) = h(P). Since h ∈ S_{w}^{⊥} and ρ(S̃_{w}) = S_{w} (part (c)), we have hρ ∈ S̃_{w}^{⊥}, whence ker τ ⊆ (hρ)(τ^{−1}(P)) (part (a)). Part (d) follows.

(e) Since S̃_{w} ⊆ Γ_{w}, it is clear that Γ_{w}^{⊥} ⊆ S̃_{w}^{⊥}. As in the proof of part (d), it is also clear that ρ^{∗}(S_{w}^{⊥}) ⊆ S̃_{w}^{⊥}.

Now consider h′ ∈ S̃_{w}^{⊥}; thus S̃_{w} ⊆ ker h′. Hence, h′ induces a homomorphism h_{1} : Γ/S̃_{w} → k^{×}. In view of part (c), the restriction of ρ to Γ_{w} induces an isomorphism ρ_{w} : Γ_{w}/S̃_{w} → G_{w}/S_{w}. Let h_{2} : G_{w} → k^{×} denote the composition

G_{w} ⟶^{quo} G_{w}/S_{w} ⟶^{ρ_{w}^{−1}} Γ_{w}/S̃_{w} ⟶^{⊆} Γ/S̃_{w} ⟶^{h_{1}} k^{×},

and observe that the restrictions of h_{2}ρ and h′ to Γ_{w} coincide. Since k is algebraically closed, k^{×} is a divisible group, and hence k^{×} is injective in the category of abelian groups. Consequently, h_{2} can be extended to a homomorphism h_{3} ∈ H. On one hand, h_{3}ρ and h′ agree on Γ_{w}, whence h′(h_{3}ρ)^{−1} ∈ Γ_{w}^{⊥}. On the other hand, S_{w} ⊆ ker h_{2} ⊆ ker h_{3}, whence h_{3} ∈ S_{w}^{⊥}. Therefore h′ ∈ ρ^{∗}(S_{w}^{⊥})Γ_{w}^{⊥}, and part (e) is proved.

(f) If h ∈ S_{w}^{⊥}, then hρ ∈ S̃_{w}^{⊥} and we see from part (d) that τ^{−1}(h(P)) lies in the S̃_{w}^{⊥}-orbit of τ^{−1}(P).

Conversely, consider h′ ∈ S̃_{w}^{⊥}. By part (e), h′ = (hρ)h_{0} for some h ∈ S_{w}^{⊥} and h_{0} ∈ Γ_{w}^{⊥}. We claim that h_{0} leaves τ^{−1}(P) invariant; it will then follow from part (d) that h′(τ^{−1}(P)) = τ^{−1}(h(P)).

Since P ∈ spec_{w} R, we see that τ^{−1}(P) contains the ideal J_{w} = ⟨y_{ϵ_{i}} | i ∈ w⟩, where ϵ_{1}, …, ϵ_{n} denotes the standard basis for Γ. Note that J_{w} is invariant under H. Since R̃/J_{w} is spanned by the cosets y_{α} + J_{w} for α ∈ Γ_{w} ∩ Γ^{+}, we see that the induced action of Γ_{w}^{⊥} on R̃/J_{w} is trivial. Thus τ^{−1}(P) is indeed invariant under h_{0}, which establishes part (f). □

4.8. (Proof of Theorem 4.5) Let w ⊆ {1, …, n}, and consider P ∈ spec_{w} R. In view of Lemma 4.7(a) and the commutative diagrams in (4.6), we see that

ϕ_{s}(P) = πϕ̃_{s}(τ^{−1}(P)) = πΦ̃^{−1}(τ^{−1}(P) : S̃_{w}^{⊥})
= Φ^{−1}τ(τ^{−1}(P) : S̃_{w}^{⊥}) = (τ(τ^{−1}(P) : S̃_{w}^{⊥}))′.

On the other hand, Lemma 4.7(f) implies that

(τ^{−1}(P) : S̃_{w}^{⊥}) = ⋂_{h∈S_{w}^{⊥}} τ^{−1}(h(P)) = τ^{−1}(P : S_{w}^{⊥}),

whence τ(τ^{−1}(P) : S̃_{w}^{⊥}) = (P : S_{w}^{⊥}). Therefore ϕ_{s}(P) = (P : S_{w}^{⊥})′.

Since H is abelian, the set functions (− : S_{w}^{⊥}) are H-equivariant, as is the twist map (−)′. Hence, it follows from the formula just proved that ϕ_{s} is H-equivariant.

Finally, note that the fibre of ϕ_{m} over any point of prim_{w} A is contained in max_{w} R. Consider P, P′ ∈ max_{w} R. Taking account of (4.6), we see that ϕ_{m}(P) = ϕ_{m}(P′) if and only if ϕ̃_{m}(τ^{−1}(P)) = ϕ̃_{m}(τ^{−1}(P′)), if and only if τ^{−1}(P) and τ^{−1}(P′) lie in the same S̃_{w}^{⊥}-orbit. By Lemma 4.7(f), this occurs if and only if τ^{−1}(P′) = τ^{−1}(h(P)) for some h ∈ S_{w}^{⊥}, and thus if and only if P and P′ lie in the same S_{w}^{⊥}-orbit. □

Once the precise form of the maps in Theorem 4.5 is given, we can easily construct topological quotient maps between different cocycle twists of R, as follows.

4.9. Corollary. Let R be a commutative affine k-algebra, graded by an abelian group G. Let c_{1}, c_{2} : G × G → k^{×} be alternating bicharacters, and let A_{i} be the twist of R by c_{i}. Assume that either −1 ∉ ⟨im c_{1}⟩ ∪ ⟨im c_{2}⟩ or char k = 2. Set H = Hom(G, k^{×}), and let ϕ_{i} : spec R ↠ spec A_{i} be the H-equivariant topological quotient map given in Theorem 4.5.

Assume that c_{2}(x, y) = 1 ⟹ c_{1}(x, y) = 1, for any x, y ∈ G. Then there exists an H-equivariant topological quotient map ϕ : spec A_{1} ↠ spec A_{2} such that the following diagram commutes:

[figure: spec A_{1}; ϕ; spec R; ϕ_{1}; ϕ_{2}; spec A_{2}]

Proof. Set up notation for A_{i} and ϕ_{i} as in (4.4), and abbreviate (c_{i})_{w} by c_{iw}. Let Φ_{i} : A_{i} → R be the inverse of the twist map. Then ϕ_{i} is given by the rule

ϕ_{i}(P) = Φ_{i}^{−1}(P : rad(c_{iw})^{⊥}) for P ∈ spec_{w} R.

Our hypotheses on c_{1}, c_{2} imply that rad(c_{2w}) ⊆ rad(c_{1w}) for all w, whence rad(c_{2w})^{⊥} ⊇ rad(c_{1w})^{⊥} and so

(P : rad(c_{2w})^{⊥}) = ((P : rad(c_{1w})^{⊥}) : rad(c_{2w})^{⊥})

for all P ∈ spec_{w} R. Hence, we can define a map ϕ : spec A_{1} → spec A_{2} such that

ϕ(Q) = Φ_{2}^{−1}(Φ_{1}(Q) : rad(c_{2w})^{⊥})

for Q ∈ spec_{w} A_{1}. It is clear that ϕϕ_{1} = ϕ_{2}. Since ϕ_{1} and ϕ_{2} are H-equivariant topological quotient maps, so is ϕ. □

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Department of Mathematics, University of California, Santa Barbara, CA 93106

E-mail address: goodearl@math.ucsb.edu