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009 ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA

MARTIN LORENZ

ABSTRACT. LetG be an affine algebraic group and letR be an associative algebra with a rational actionof G by algebra automorphisms. We study the inducedG-action on the setSpec R of all prime idealsof R, viewed as a topological space with the Jacobson-Zariski topology, and on the subspaceRat R ⊆

Spec R consisting of all rational ideals ofR. Here, a prime idealP of R is said to be rational if theextended centroidC(R/P ) is equal to the base field. Our results generalize work of Mœglin & Rentschlerand Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs.

The mapP 7→T

g∈G g.P gives a surjection fromSpec R onto the setG-Spec R of all G-primeideals ofR. The fibres of this map yield the so-calledG-stratification ofSpec R which has played acentral role in the recent investigation of algebraic quantum groups, in particular in the work of Goodearland Letzter. We describe theG-strata ofSpec R in terms of certain commutative spectra. Furthermore, weshow that if a rational idealP is locally closed inSpec R then the orbitG.P is locally closed inRat R.This generalizes a standard result onG-varieties. Finally, we discuss the situation whereG-Spec R is afinite set.

INTRODUCTION

0.1. This article continues our investigation [15] of the actionof an affine algebraic groupG on anarbitrary associative algebraR. Our focus will now be on some topological aspects of the inducedaction on the setSpecR of all prime ideals ofR, the main themes being local closedness ofG-orbitsin SpecR and the stratification ofSpecR by means of suitable commutative spectra. The stratificationin question plays a central role in the theory of algebraic quantum groups; see Brown and Goodearl[7] for a panoramic view of this area. Our goal here is to develop the principal results in a context thatis free of the standard finiteness conditions, noetherianness or the Goldie property, that underlie thepioneering works of Mœglin and Rentschler [19], [20], [21],[22] and Vonessen [27], [28].

Throughout, we work over an algebraically closed base fieldk and we assume that the action ofGonR is rational; the definition will be recalled in§3.1. The action will generally be written as

G×R→ R , (g, r) 7→ g.r .

0.2. The setSpecR carries the familiar Jacobson-Zariski topology; see§1.1 for details. Since theG-action onR sends prime ideals to prime ideals, it induces an action ofG by homeomorphisms onSpecR. In the following,G\ SpecR will denote the set of allG-orbits in SpecR. We will alsoconsider the setG-SpecR consisting of allG-prime ideals ofR. Recall that a properG-stable idealIof R is calledG-prime if AB ⊆ I for G-stable idealsA andB of R implies thatA ⊆ I orB ⊆ I.

There are surjective maps

SpecRcan.−→ G\ SpecR , P 7→ G.P = {g.P | g ∈ G} (1)

γ : SpecR −→ G-SpecR , P 7→ P:G =⋂

g∈G

g.P . (2)

2000Mathematics Subject Classification.Primary 16W22; Secondary 16W35, 17B37, 20G42.Key words and phrases.algebraic group, rational action, orbit, prime ideal, rational ideal, primitive ideal, Jacobson-Zariski

topology, locally closed subset, stratification.Research of the author supported in part by NSA Grant H98230-07-1-0008.

1

2 MARTIN LORENZ

See [15, Proposition 8] for surjectivity ofγ. We will giveG\ SpecR andG-SpecR the final topologiesfor these maps: closed subsets are those whose preimage inSpecR is closed [5, I.2.4]. Since (2) factorsthrough (1), we obtain a surjection

G\ SpecR −→ G-SpecR , G.P 7→ P:G . (3)

This map is continuous and closed; see§1.3.

0.3. As in [15], we will be particularly concerned with the subsetsRatR ⊆ SpecR andG-RatR ⊆G-SpecR consisting of all rational andG-rational ideals ofR, respectively. Recall that rationality andG-rationality is defined in in terms of the extended centroidC( . ) of the corresponding factor algebra[15]. Specifically,P ∈ SpecR is said to berational if C(R/P ) = k, andI ∈ G-SpecR isG-rationalif the G-invariantsC(R/I)G ⊆ C(R/I) coincide withk. For the definition and basic properties ofthe extended centroid, the reader is referred to [15]. Here,we just recall thatC(R/P ) andC(R/I)G

always are extension fields ofk, for anyP ∈ SpecR and anyI ∈ G-SpecR. The extended centroid ofa semiprime noetherian (or Goldie) algebra is identical to the center of the classical ring of quotients.In the context of enveloping algebras of Lie algebras and related noetherian algebras, the fieldC(R/P )is commonly called theheart (cœur, Herz) of the primeP (e.g., [10], [3], [4]). We will follow thistradition here.

The setsRatR andG-RatR will be viewed as topological spaces with the topologies that areinduced fromSpecR andG-SpecR: closed subsets ofRatR are the intersections of closed subsets ofSpecR with RatR, and similarly forG-RatR [5, I.3.1]. TheG-action onSpecR stabilizesRatR.Hence we may consider the setG\RatR ⊆ G\ SpecR consisting of allG-orbits inRatR. We endowG\RatR with the topology that is induced fromG\ SpecR; this turns out to be indentical to the finaltopology for the canonical surjectionRatR −→ G\RatR [5, III.2.4, Prop. 10]. By [15, Theorem 1],the surjection (3) restricts to a bijection

G\RatRbij.−→ G-RatR . (4)

This map is in fact a homeomorphism; see§1.5.

0.4. The following diagram summarizes the various topological spaces under consideration and theirrelations to each other.

SpecR

can.

||||xxxxxxxxxxxxxxxxxxxx

γ

"" ""EEEEEEEEEE

EEEEEEEEEE

RatR

yyyy

yyyy

yy

||||yyyy

yyyy

yy

EEEE

EEEE

EE

"" ""EEEEE

EEEEE

� ?

OO

G\ SpecR // // G-SpecR

G\RatR ∼=//

� ?

OO

G-RatR� ?

OO

Here,։ indicates a surjection whose target space carries the final topology,→ indicates an inclusionwhose source has the induced topology, and∼= is the homeomorphism (4).

0.5. The technical core of the article is Theorem 9 which describes theγ-fibre over a givenI ∈G-SpecR. This fibre will be denoted by

SpecI R = {P ∈ SpecR | P:G = I}

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 3

as in [7]. The partition

SpecR =⊔

I∈G-Spec R

SpecI R (5)

is called theG-stratificationof SpecR in [7, II.2]. In the special case whereR is noetherian andG isan algebraic torus, a description of theG-strataSpecI R in terms of suitable commutative spectra wasgiven in [7, II.2.13], based on work of Goodearl and Letzter [11]. For generalR andG, the intersection

RatI R = SpecI R ∩ RatR

was treated in [15, Theorem 22]. Our proof of Theorem 9, to be given in Section 3, elaborates on theone in [15].

AssumingG to be connected for simplicity, we put

TI = C(R/I)⊗k k(G) ,

wherek(G) denotes the field of rational functions onG. The algebraTI is a tensor product of twocommutative fields andTI has no zero divisors. The givenG-action onR and the right regularG-action onk(G) naturally give rise to an action ofG onTI . LettingSpecG(TI) denote the collection ofall G-stable primes ofTI , Theorem 9 establishes a bijection

c : SpecI Rbij.−→ SpecG(TI)

which is very well behaved: the mapc is equivariant with respect to suitableG-actions, it is an orderisomorphism for inclusion, and it allows one to control hearts and rationality. For the precise formula-tion of Theorem 9, we refer to Section 3.

0.6. Theorem 9 and the tools developed for its proof will be used inSection 4 to investigate localclosedness of rational ideals. Recall that a subsetA of an arbitrary topological spaceX is said to belocally closedif A is closed in some neighborhood ofA in X . This is equivalent toA being open in itstopological closureA in X or, alternatively,A being an intersection of an open and a closed subset ofX [5, I.3.3]. A pointx ∈ X is locally closed if{x} is locally closed. ForX = SpecR, this amountsto the following familiar condition: a prime idealP is locally closed inSpecR if and only if P isdistinct from the intersection of all primes ofR that properly containP . A similar formulation holdsforX = G-SpecR ; see§1.4. We remark that “locally closed inG-SpecR” is referred to as “G-locallyclosed” in [21] and [28].

The second main result of this article is the following theorem which will be proved in Section 4.Earlier versions assuming additional finiteness hypotheses are due to Mœglin and Rentschler [19,Theoreme 3.8], [21, Theoreme 3] and to Vonessen [28, Theorem 2.6].

Theorem 1. The following are equivalent for a rational idealP ofR:

(a) P is locally closed inSpecR;(b) γ(P ) = P:G is locally closed inG-SpecR.

Theorem 1 in conjunction with (4) has the following useful consequence. The corollary belowextends [28, Cor. 2.7] and a standard result onG-varieties [14, Satz II.2.2].

Corollary 2. If P ∈ RatR is locally closed inSpecR then theG-orbit G.P is open in its closure inRatR.

Proof. The pointP :G ∈ G-SpecR is locally closed by Theorem 1. Applying the easy fact [5, I.3.3]that preimages of locally closed sets under continuous mapsare again locally closed tof : RatR →

SpecRγ→ G-SpecR, we conclude from (4) thatf−1(P:G) = G.P is locally closed inRatR. �

4 MARTIN LORENZ

0.7. In order to put Theorem 1 and Corollary 2 into perspective, wemention that rational ideals areoftentimes locally closed inSpecR. In fact, for many important classes of algebrasR, rational idealsare identical with the locally closed points ofSpecR. Specifically, we will say that the algebraRsatisfies theNullstellensatzif the following two conditions are satisfied:

(i) every prime ideal ofR is an intersection of primitive ideals, and(ii) Z (EndR V ) = k holds for every simpleR-moduleV .

Recall that an ideal ofR is said to be (right) primitive if it is the annihilator of a simple (right)R-module. Hypothesis (i) is known as theJacobson propertywhile versions of (ii) are referred to asthe endomorphism property[18] or theweak Nullstellensatz[23], [15]. The Nullstellensatz is quitecommon. It is guaranteed to hold, for example, ifk is uncountable and the algebraR is noetherian andcountably generated [18, Corollary 9.1.8], [7, II.7.16]. The Nullstellensatz also holds for any affinePI-algebraR [26, Chap. 6]. For many other classes of algebras satisfyingthe Nullstellensatz, see [18,Chapter 9] or [7, II.7].

If R satisfies the Nullstellensatz then the following implications hold for all primes ofR:

locally closed ⇒ primitive ⇒ rational.

Here, the first implication is an immediate consequence of (i) while the second follows from (ii); see[15, Prop. 6]. The algebraR is said to satisfy theDixmier-Mœglin equivalenceif all three properties areequivalent for primes ofR. Standard examples of algebras satisfying the Dixmier-Mœglin equivalenceinclude affine PI-algebras, whose rational ideals are in fact maximal [24], and enveloping algebras offinite-dimensional Lie algebras; see [25, 1.9] forchark = 0. (In positive characteristics, envelopingalgebras are affine PI.) More recently, the Dixmier-Mœglin equivalence has been shown to hold fornumerous quantum groups; see [7, II.8] for an overview.

Note that the validity of the Nullstellensatz and the Dixmier-Mœglin equivalence are intrinsic toR.However,G-actions can be useful tools in verifying the latter. Indeed, assuming the Nullstellensatz forR, Theorem 1 implies that the Dixmier-Mœglin equivalence is equivalent toP :G being locally closedin G-SpecR for everyP ∈ RatR. This condition is surely satisfied wheneverG-SpecR is in factfinite.

0.8. The final Section 5 briefly addresses the question as to whenG-SpecR is a finite set. Besidesbeing of interest in connection with the Dixmier-Mœglin equivalence (§0.7), this is obviously relevantfor theG-stratification (5); see also [7, Problem II.10.6]. Restricting ourselves to algebrasR satisfyingthe Nullstellensatz, we show in Proposition 14 that finiteness ofG-SpecR is equivalent to the followingthree conditions:

(i) the ascending chain condition holds forG-stable semiprime ideals ofR,(ii) R satisfies the Dixmier-Mœglin equivalence, and(iii) G-RatR = G-SpecR.

Several versions of Proposition 14 for noetherian algebrasR can be found in [7, II.8], where a profusionof algebras is exhibited for whichG-SpecR is known to be finite.

Note that (i) above is no trouble for the standard classes of algebras, even in the strengthened formwhich ignoresG-stability. Indeed, noetherian algebras trivially satisfy the ascending chain conditionfor all semiprime ideals, and so do all affine PI-algebras; see [26, 6.3.36’]. Moreover, as was out-lined in §0.7, the Dixmier-Mœglin equivalence (ii) has been established for a wide variety of algebras.Therefore, in many situations of interest, Proposition 14 says in essence that finiteness ofG-SpecRis tantamount to the equalityG-RatR = G-SpecR. This is also the only condition where theG-action properly enters the picture. The article concludes with some simple examples of torus actionssatisfying (iii). Further work is needed on how to assure thevalidity of (iii) under reasonably generalcircumstances.

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 5

0.9. This article owes a great deal to the ground breaking investigations of Mœglin & Rentschlerand Vonessen. The statements of our main results as well as the basic strategies employed in theirproofs have roots in the aforementioned articles of these authors. We have made an effort to renderour presentation reasonably self-contained while also indicating the original sources at the appropriatepoints in the text. The reader interested in the details of Sections 3 and 4 may wish to have a copy of[28] at hand in addition to [15].

Notations. Our terminology and notation follows [15]. The notations and hypotheses introduced in theforegoing will remain in effect throughout the paper. In particular, we will work over an algebraicallyclosed base fieldk. Furthermore,G will be an affine algebraick-group andR will be an associativek-algebra (with1) on whichG acts rationally byk-algebra automorphisms. For simplicity,⊗k will bewritten as⊗. Finally, for any idealI E R, the largestG-stable ideal ofR that is contained inI will bedenoted by

I:G =⋂

g∈G

g.I .

1. TOPOLOGICAL PRELIMINARIES

1.1. Recall that the closed sets of theJacobson-Zariski topologyonSpecR are exactly the subsets ofthe form

V(I) = {P ∈ SpecR | P ⊇ I}

whereI ⊆ R. The topological closure of a subsetA ⊆ SpecR is given by

A = V(I(A)) where I(A) =⋂

P∈A

P . (6)

The easily checked equalitiesI(V(I(A))) = I(A) andV(I(V(I))) = V(I) show that the operatorsV andI yield inverse bijections between the collection of all closed subsets ofSpecR on one side andthe collection of all semiprime ideals ofR (i.e., ideals ofR that are intersections of prime ideals) onthe other. Thus, we have an inclusion reversing 1-1 correspondence

{closed subsetsA ⊆ SpecR

}1-1←→

{semiprime ideals

I E R

}. (7)

Note that the equalityI(V(I(A))) = I(A) can also be stated as

I(A) = I(A) . (8)

1.2. The action ofG commutes with the operatorsV andI: g.V(I) = V(g.I) andg.I(A) = I(g.A)holds for allg ∈ G and allI ⊆ R, A ⊆ SpecR. In particular, the elements ofG act by homeomor-phisms onSpecR. Furthermore:

If g.A ⊆ A for a closed subsetA ⊆ SpecR andg ∈ G theng.A = A. (9)

In view of the correspondence (7), this amounts to saying that g.I ⊇ I forcesg.I = I for semiprimeidealsI E R. But this follows from the fact that theG-action onR is locally finite [15, 3.1]: Ifr ∈ I satisfiesg.r /∈ I then choose a finite-dimensionalG-stable subspaceV ⊆ R with r ∈ V to getI ∩ V $ g.I ∩ V $ g2.I ∩ V $ . . . , which is impossible.

Finally, consider theG-orbitG.P of a pointP ∈ SpecR. SinceI(G.P ) = P:G, equation (6) showsthat the closure ofG.P in SpecR is given by

G.P = V(P:G) = {Q ∈ SpecR | Q ⊇ P:G}

and (8) givesI(G.P ) = I(G.P ) = P:G .

6 MARTIN LORENZ

Thus, the correspondence (7) restricts to an inclusion reversing bijection

{ G-orbit closures inSpecR }1-1←→ G-SpecR

∈ ∈ (10)

G.P ←→ P:G .

1.3. As was mentioned in the Introduction, the spacesG\ SpecR andG-SpecR inherit their topologyfrom SpecR via the surjections in (1) and (2):

SpecR

π=can.

xxxxrrrrrrrrrrγ

%% %%LLLLLLLLLL

G\ SpecR G-SpecR

with π(P ) = G.P andγ(P ) = P :G. Closed sets inG\ SpecR and inG-SpecR are exactly thosesubsets whose preimage inSpecR is closed. In both cases, preimages are alsoG-stable, and hencethey have the formV(I) for someG-stable semiprime idealI E R.

Let C be a closed subset ofG-SpecR and writeγ−1(C) = V(I) as above. SinceP ⊇ I isequivalent toP:G ⊇ I for P ∈ SpecR, we obtain

C = γ(V(I)) = {P:G | P ∈ SpecR,P ⊇ I}

= {P:G | P ∈ SpecR,P:G ⊇ I}

= {J ∈ G-SpecR | J ⊇ I} .

(11)

Conversely, ifC = {J ∈ G-SpecR | J ⊇ I} for someG-stable semiprime idealI E R thenγ−1(C) = V(I) is closed inSpecR and soC is closed inG-SpecR. Thus, the closed subsets ofG-SpecR are exactly those of the form (11). The partial order onG-SpecR that is given by inclusioncan be expressed in terms of orbit closures by (10): forP,Q ∈ SpecR, we have

Q:G ⊇ P:G ⇐⇒ G.Q ⊆ G.P ⇐⇒ Q ∈ G.P . (12)

The mapγ factors throughπ; so we have a map

γ′ : G\ SpecR→ G-SpecR

with γ = γ′ ◦ π as in (3). Sinceγ−1(C) = π−1(γ′−1(C)) holds for anyC ⊆ G-SpecR, the mapγ′

is certainly continuous. Moreover, ifB ⊆ G\ SpecR is closed thenA = π−1(B) ⊆ SpecR satisfiesγ(A) = γ′(B) andA = V(I) for someG-stable semiprime idealI E R. Hence,γ′(B) = γ(V(I)) isclosed inG-SpecR by (11). This shows thatγ′ is a closed map.

1.4. Recall that a subsetA of an arbitrary topological spaceX is locally closedif and only ifA \A =

A ∩A∁ is a closed subset ofX . By (6), a prime idealP of R is locally closed inSpecR if and only ifV(P ) \ {P} = {Q ∈ SpecR | Q % P} is a closed subset ofSpecR, which in turn is equivalent tothe condition

P $ I(V(P ) \ {P}) =⋂

Q∈SpecRQ%P

Q . (13)

Similarly, (11) implies that aG-prime idealI of R is locally closed inG-SpecR if and only if

I $⋂

J∈G-SpecRJ%I

J . (14)

Lemma 3. LetP ∈ SpecR and letN be a normal subgroup ofG having finite index inG. ThenP:Gis locally closed inG-SpecR if and only ifP:N is locally closed inN -SpecR.

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 7

Proof. For brevity, putX = N -SpecR, Y = G-SpecR andH = G/N . Thus,H is a finite groupacting by homeomorphisms onX . From (3) we obtain a continuous surjectionX → Y , I 7→ I :H ,whose fibres are easily seen to be theH-orbits inX . Thus, we obtain a continuous bijectionH\X →Y . From the description (11) of the closed sets inX andY , we further see that this bijection is closed,and hence it is a homeomorphism. Therefore, the imageI :H of a givenI ∈ X is locally closed inY if and only if the orbitH.I is locally closed inX ; see [5, I.5.3, Cor. of Prop. 7]. Finally, withdenoting topological closure inX , one easily checks that

H.I \H.I =⋃

h∈H

h.({I} \ {I}

)

and, consequently,(H.I \H.I

)∩ {I} = {I} \ {I}. Thus,H.I is locally closed if and only ifI is

locally closed, which proves the lemma. �

1.5. We now turn to the spaceRatR of rational ideals ofR and the associated spacesG\RatR andG-RatR with the induced topologies fromSpecR, G\ SpecR andG-SpecR, respectively; see§0.3.Restricting the mapsπ = can. andγ′ in §1.3, we obtain a commutative triangle of continuous maps

RatRπrat=can.

yyyyssssssssssγrat

%% %%KKKKKKKKKK

G\RatRγ′

rat // G-RatR

The mapγ′rat, identical with (4), is bijective. Furthermore, ifB ⊆ G\RatR is closed then, as in§1.3,we haveγ′rat(B) = {J ∈ G-RatR | J ⊇ I} for someG-stable semiprime idealI E R; soγ′rat(B) isclosed inG-RatR. This shows thatγ′rat is a homeomorphism. For general reasons, the quotient mapπrat is open and the topology onG\RatR is identical to the final topology forπrat [5, III.2.4, Lemme2 and Prop. 10]. By virtue of the homeomorphismγ′rat, the same holds forG-RatR andγrat.

We remark that injectivity ofγ′rat and (12) imply that

Q:G % P:G ⇐⇒ Q ∈ G.P \G.P

holds forP,Q ∈ RatR. Here, can be taken to be the closure inSpecR or in RatR. Focusing onthe latter interpretation, we easily conclude that

P:G is locally closed inG-RatR ⇐⇒ G.P is open in its closure inRatR.

Alternatively, in view of the homeomorphismγ′rat, local closedness ofP:G ∈ G-RatR is equivalent tolocal closedness of the pointG.P ∈ G\RatR, which in turn is equivalent to local closedness of thepreimageG.P = π−1

rat (G.P ) ⊆ RatR; see [5, I.5.3, Cor. of Prop. 7].

2. RING THEORETICAL PRELIMINARIES

2.1. The extended centroid of a ringU will be denoted by

C(U) .

By definition,C(U) is the center of the Amitsur-Martindale ring of quotients ofU . We briefly recallsome basic definitions and facts. For details, the reader is referred to [15].

The ringU is said to becentrally closedif C(U) ⊆ U . If U is semiprime then

U = UC(U)

is a centrally closed semiprime subring of the Amitsur-Martindale ring of quotients ofU ; it is calledthe central closureof U . Furthermore, ifU is prime thenU is prime as well andC(U) is a field.Consequently, for anyP ∈ SpecU , we have a commutative fieldC(U/P ).

8 MARTIN LORENZ

If Γ is any group acting by automorphisms onU then the action ofΓ extends uniquely to an actionon the Amitsur-Martindale ring of quotients ofU , and henceΓ acts onC(U); see [15, 2.3]. IfI is aΓ-prime ideal ofU then the ring ofΓ-invariantsC(U/I)Γ is a field; see [15, Prop. 9].

2.2. A ring homomorphismϕ : U → V is calledcentralizingif the ringV is generated by the imageϕ(U) and its centralizer,CV (ϕ(U)) = {v ∈ V | vϕ(u) = ϕ(u)v ∀u ∈ U}; see [15, 1.5]. The lemmabelow is a special case of [15, Lemma 4] and it can also be foundin an earlier unpublished preprint ofGeorge Bergman [1, Lemma 1].

Lemma 4. Let ϕ : U → V be a centralizing embedding of prime rings. Thenϕ extends uniquelyto a homomorphismϕ : U → V between the central closures ofU andV . The extensionϕ is againinjective and centralizing. In particular,ϕ(C(U)) ⊆ C(V ).

Proof. If I is a nonzero ideal ofU thenϕ(I)V = V ϕ(I) is a nonzero ideal ofV . Henceϕ(I) has zeroleft and right annihilator inV . The existence of the desired extensionϕ now follows from [15, Lemma4], sinceCϕ = C(U) holds in the notation of that result. Uniqueness ofϕ as well as injectivity and thecentralizing property are immediate from [15, Prop. 2(ii)(iii)]. For example, in order to guarantee thatϕ is centralizing, it suffices to check thatCV (ϕ(U)) centralizesϕ(C(U)). To prove this, letq ∈ C(U)andv ∈ CV (ϕ(U)) be given. By [15, Prop. 2(ii)] there is a nonzero idealI E U so thatqI ⊆ U . Foru ∈ I, one computes

ϕ(q)vϕ(u) = ϕ(q)ϕ(u)v = ϕ(qu)v = vϕ(qu) = vϕ(q)ϕ(u) .

This shows that[v, ϕ(q)]ϕ(I) = 0 and [15, Prop. 2(iii)] further implies that[v, ϕ(q)] = 0. �

The lemma implies in particular that every automorphism of aprime ringU extends uniquely to thecentral closureU . (A more general fact was already mentioned above.) We will generally use the samenotation for the extended automorphism ofU .

2.3. The essence of the next result goes back to Martindale et al. [16], [9].

Proposition 5. LetU be a centrally closed prime ring and letV be anyC-algebra, whereC = C(U).Then there are bijections

{P ∈ Spec(U ⊗C V ) | P ∩ U = 0}1-1←→ SpecV

P 7−→ P ∩ V

U ⊗C p ←− [ p .

These bijections are inverse to each other and they are equivariant with respect to all automorphismsofU ⊗C V that stabilize bothU andV . Furthermore, hearts are preserved:

C((U ⊗C V )/P ) ∼= C(V/P ∩ V ) .

Proof. The extensionV → U ⊗C V is centralizing. Therefore, contractionP 7→ P ∩ V is a well-defined mapSpec(U ⊗C V )→ SpecV which clearly has the stated equivariance property.

If V is a prime ring then so isU ⊗C V ; this follows from the fact that every nonzero ideal ofU ⊗C V contains a nonzero element of the formu⊗ v with u ∈ U andv ∈ V ; see [15, Lemma 3(a)]or [9, Theorem 3.8(1)]. By [17, Cor. 2.5] we also know thatC(U ⊗C V ) = C(V ). Consequently,p 7→ U ⊗C p = (U ⊗C V )p gives a mapSpecV → {P ∈ Spec(U ⊗C V ) | P ∩ U = 0} whichpreserves hearts.

Finally, by [15, Lemma 3(c)], the above maps are inverse to each other. �

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 9

2.4. LetU be a prime ring. IfU is an algebra over some commutative fieldF then the central closureU is anF -algebra as well, becauseZ(U) ⊆ C(U) = Z(U).

Proposition 6. LetU andV be algebras over some commutative fieldF , withU prime. Then there isa bijection

{P ∈ Spec(U ⊗F V ) | P ∩ U = 0} −→ {P ∈ Spec(U ⊗F V ) | P ∩ U = 0}

P 7−→ P ∩ (U ⊗F V ) .

This bijection and its inverse are inclusion preserving andequivariant with respect to all automor-phisms ofU ⊗F V that stabilize bothU andU ⊗F V . Furthermore, hearts are preserved under thisbijection.

Proof. Since the extensionU⊗F V → U⊗F V is centralizing, the contraction mapP 7→ P∩(U⊗F V )sends primes to primes, and hence it yields a well-defined mapbetween the sets in the proposition. Thismap is clearly inclusion preserving and equivariant as stated.

For surjectivity, letP be a prime ofU ⊗F V such thatP ∩ U = 0. The canonical map

ϕ : U → U ⊗F V ։ W = (U ⊗F V )/P

is a centralizing embedding of prime rings. By Lemma 4, thereis a unique extension to central closures,

ϕ : U −→ W .

The image of the canonical mapψ : V → U⊗F V ։ W centralizesϕ(U), and hence it also centralizesϕ(U); see the proof of Lemma 4. Therefore,ϕ andψ yield a ring homomorphism

ϕV : U ⊗F V −→ W .

Put P = Ker ϕV . SinceϕV extends the canonical mapU ⊗F V ։ W = (U ⊗F V )/P , we haveP ∩ (U ⊗F V ) = P and

W ⊆ Im ϕV = (U ⊗F V )/P ⊆ W . (15)

In particular, every nonzero ideal ofIm ϕV has a nonzero intersection withW , and henceIm ϕV is aprime ring andP is a prime ideal. SinceP ∩ U = P ∩ U = 0, it follows thatP ∩ U = 0. This provessurjectivity. Furthermore, since the inclusions in (15) are centralizing inclusions of prime rings, theyyield inclusions of extended centroids,C(W ) ⊆ C((U ⊗F V )/P ) ⊆ C(W ) = C(W ) by Lemma 4.Therefore,P has the same heart asP .

To prove injectivity, letP and P ′ be primes ofU ⊗F V , both disjoint fromU \ {0}, such thatP ∩ (U ⊗F V ) ⊆ P ′ ∩ (U ⊗F V ). We claim thatP ⊆ P ′. Indeed, it follows from [15, Prop. 2(ii)]that, for anyq ∈ U ⊗F V , there is a nonzero idealI of U such thatqI ⊆ U ⊗F V . For q ∈ P , weconclude thatqI ⊆ P ′. SinceI(U ⊗F V ) is an ideal ofU ⊗F V that is not contained inP ′, we musthaveq ∈ P ′. Therefore,P ⊆ P ′ as claimed. Injectivity follows and we also obtain that the inversebijection preserves inclusions. This completes the proof. �

We will apply the equivariance property of the above bijection to automorphisms of the formα⊗F β

with α ∈ AutF -alg(U), extended uniquely toU as in§2.2, andβ ∈ AutF -alg(V ).

2.5. The following two technical results have been extracted from Mœglin and Rentschler [19, 3.4-3.6]; see also Vonessen [28, proof of Prop. 8.12]. As above,F denotes a commutative field. If a groupΓ acts on a ringU thenU is called aΓ-ring; similarly for fields. AΓ-ring U is calledΓ-simple ifUhas noΓ-stable ideals other than0 andU .

10 MARTIN LORENZ

Lemma 7. LetF ⊆ L ⊆ K beΓ-fields. Assume thatK = FractA for someΓ-stableF -subalgebraAwhich isΓ-simple and affine (finitely generated). ThenL = FractB for someΓ-stableF -subalgebraB which isΓ-simple and generated by finitely manyΓ-orbits.

Proof. We need to construct aΓ-stableF -subalgebraB ⊆ L satisfying

(i) L = FractB,(ii) B is generated asF -algebra by finitely manyΓ-orbits, and(iii) B is Γ-simple.

Note thatL is a finitely generated field extension ofF , becauseK is. Fix a finite setX0 ⊆ L of fieldgenerators and letB0 ⊆ L denote theF -subalgebra that is generated by

⋃x∈X0

Γ.x. ThenB0 certainlysatisfies (i) and (ii). We will show that the intersection of all nonzeroΓ-stable semiprime ideals ofB0 isnonzero. Consider the algebraB′ = B0A ⊆ K; this algebra isΓ-stable and affine overB0. By genericflatness [8, 2.6.3], there exists some nonzerot ∈ B0 so thatB′[t−1] is free overB0[t

−1]. We claim thatif b0 is anyΓ-stable semiprime ideal ofB0 not containingt thenb0 must be zero. Indeed,b0B0[t

−1]is a proper ideal ofB0[t

−1], and henceb0B′[t−1] is a proper ideal ofB′[t−1]. Therefore,b0A∩A is a

proper ideal ofA which is clearlyΓ-stable. SinceA is Γ-simple, we must haveb0A ∩A = 0. Finally,sinceb0 ⊆ K = FractA, we conclude thatb0 = 0 as desired. Thus,t belongs to every nonzeroΓ-stable semiprime ideal ofB0. Now letB be theF -subalgebra ofL that is generated byB0 and theΓ-orbit of t−1. Clearly,B still satisfies (i) and (ii). Moreover, ifb is any nonzeroΓ-stable semiprimeideal ofB thenb ∩ B0 is aΓ-stable semiprime ideal ofB0 which is nonzero, becauseL = FractB0.Hencet ∈ b and sob = B. This implies thatB is Γ-simple which completes the construction ofB. �

The lemma above will only be used in the proof of the following“lying over” result which will becrucial later on. For a givenΓ-ringU , we let

SpecΓ U

denote the collection of allΓ-stable primes ofU . These primes are certainlyΓ-prime, but the converseneed not hold in general.

Proposition 8. LetU be a primeF -algebra and letΓ be a group acting byF -algebra automorphismsonU . Suppose that there is aΓ-equivariant embeddingC(U) → K, whereK is a Γ-field such thatK = FractA for someΓ-stable affineF -subalgebraA which isΓ-simple.

Then there exists a nonzero idealD E U such that, for everyP ∈ SpecΓ U not containingD, thereis a P ∈ SpecΓ U satisfyingP ∩ U = P .

Proof. Applying Lemma 7 to the given embeddingF ⊆ C = C(U) → K we obtain aΓ-stableF -subalgebraB ⊆ C such thatC = FractB, B is Γ-simple andB is generated asF -algebra by finitelymanyΓ-orbits. Fix a finite subsetX ⊆ B such thatB is generated by

⋃x∈X G.x and let

D = {u ∈ U | xu ∈ U for all x ∈ X} ;

this is a nonzero ideal ofU by [15, Prop. 2(ii)]. In order to show thatD has the desired property, wefirst make the following

Claim. SupposeP ∈ SpecΓ U does not containD. Then, givenw1, . . . , wn ∈ UB ⊆ U , there existsan idealI E U with I * P and such thatwiI ⊆ U for all i.

To see this, note that every element ofUB is a finite sum of terms of the form

w = u(g1.x1) · · · (gr.xr)

with u ∈ U , gj ∈ G andxj ∈ X . The idealJ =(⋂r

j=1 gj .D)r

of U satisfies

wJ ⊆ ug1.(x1D) · · · gr.(xrD) ⊆ U .

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 11

Moreover,J * P , because otherwisegj .D ⊆ P for somej and soD ⊆ P contrary to our hypothesis.Now write the givenwi as above, collect all occuringgj in the (finite) subsetE ⊆ G and lets be the

largest occuringr. Then the idealI =(⋂

g∈E g.D)s

does what is required.

Next, we show thatPB ∩ U = P .

Indeed, for anyu ∈ PB ∩ U , the above claim yields an idealI E U with I * P and such thatuI ⊆ P . SinceP is prime, we must haveu ∈ P which proves the above equality. Now choose anidealQ E UB which contains the idealPB and is maximal with respect to the conditionQ ∩ U = P

(Zorn’s Lemma). It is routine to check thatQ is prime. Therefore,Q = Q:Γ is at least aΓ-prime idealof UB satisfyingQ ∩ U = P . We show thatQ is in fact prime. Letw1, w2 ∈ UB be given such thatw1UBw2 ⊆ Q. ChoosingI as in the claim forw1, w2, we havew1IUw2I ⊆ U ∩ Q = P . SinceP isprime, we conclude thatwjI ⊆ P for j = 1 or 2. Hence,wjIB ⊆ Q =

⋂g∈Γ g.Q. Since eachg.Q

is prime inUB andIB is an ideal ofUB not contained ing.Q, we obtain thatwj ∈⋂

g∈Γ g.Q = Q.

This shows thatQ is indeed prime.Finally, U is the (central) localization ofUB at the nonzero elements ofB. Furthermore,Q∩B = 0,

sinceQ ∩ B is a properΓ-stable ideal ofB. It follows thatP = QC is a prime ideal ofU which isclearlyΓ-stable and satisfiesP ∩ UB = Q. Consequently,P ∩ U = Q ∩ U = P , thereby completingthe proof. �

3. DESCRIPTION OFG-STRATA

3.1. We now return to the setting of§0.1. TheG-action onR will be denoted by

ρ = ρR : G −→ Autk-alg(R) (16)

when it needs to be explicitly referred to; so

g.r = ρ(g)(r) .

Recall from [15, 3.1, 3.4] that rationality of the action ofG onR is equivalent to the existence of ak-algebra map

∆R : R −→ R⊗ k[G] , r 7→∑

i

ri ⊗ fi

such thatg.r =

∑

i

rifi(g)

holds for allg ∈ G andr ∈ R. Here,k[G] denotes the Hopf algebra of regular functions onG, asusual. Thek[G]-linear extension of the map∆R is an automorphism ofk[G]-algebras which will alsobe denoted by∆R:

∆R : R⊗ k[G]∼−→ R⊗ k[G] ; (17)

see [15, 3.4].

3.2. As in [15], the right and leftregular representationsof G will be denoted by

ρr, ρℓ : G→ Autk-alg(k[G]) ;

they are defined by(ρr(x)f) (y) = f(yx) and(ρℓ(x)f) (y) = f(x−1y) for x, y ∈ G.The groupG acts (rationally) on thek-algebraR ⊗ k[G] by means of the mapsIdR⊗ρr,ℓ and

ρ⊗ ρr,ℓ. The following intertwining formulas hold for allg ∈ G:

∆R ◦ (ρ⊗ ρr) (g) = (IdR⊗ρr) (g) ◦∆R (18)

12 MARTIN LORENZ

and∆R ◦ (IdR⊗ρℓ) (g) = (ρ⊗ ρℓ) (g) ◦∆R , (19)

where∆R is the automorphism (17); see [15, 3.3].In the following, k(G) = Fractk[G]

will denote the algebrak(G) of rational functions onG; this is the full ring of fractions ofk[G], andk(G) is also equal to the direct product of the rational function fields of the irreducible components ofG [2, AG 8.1]. The aboveG-actions extend uniquely tok(G) and toR ⊗ k(G). We will use the samenotations as above for the extended actions. The intertwining formulas (18) and (19) remain valid inthis setting, with∆R replaced by its unique extension to ak(G)-algebra automorphism

∆R : R ⊗ k(G)∼−→ R⊗ k(G) . (20)

3.3. We are now ready to describe theG-stratum

SpecI R = {P ∈ SpecR | P:G = I}

over a givenI ∈ G-SpecR. For simplicity, we will assume thatG is connected; sok(G) is a fieldwhich is unirational overk [2, 18.2]. Furthermore,I is a prime ideal ofR by [15, Prop. 19(a)] andsoC(R/I) is a field as well. The groupG acts onR/I by means of the mapρ in (16). This action

extends uniquely to an action on the central closureR/I, and hence we also have aG-action on

C(R/I) = Z(R/I). Denoting the latter two actions byρ again, we obtainG-actionsρ ⊗ ρr on

R/I ⊗ k(G) and onC(R/I)⊗ k(G). As in §2.5, we will write

SpecG(T ) = {p ∈ SpecT | p is (ρ⊗ ρr)(G)-stable}

for T = C(R/I)⊗k(G) orT = R/I⊗k(G). The groupG also acts on both algebrasT via Id⊗ρℓ andthe latter action commutes withρ⊗ ρr. Hence,G acts onSpecG(T ) throughId⊗ρℓ. We are primarilyinterested in the first of these algebras,

TI = C(R/I)⊗ k(G) ,

a commutative domain and a tensor product of two fields.

Theorem 9. For a givenI ∈ G-SpecR, letTI = C(R/I)⊗ k(G). There is a bijection

c : SpecI Rbij.−→ SpecG(TI)

having the following properties, forP, P ′ ∈ SpecI R andg ∈ G:

(a) G-equivariance:c(g.P ) = (Id⊗ρℓ(g))(c(P )) ;(b) inclusions:P ⊆ P ′ ⇐⇒ c(P ) ⊆ c(P ′) ;(c) hearts:there is an isomorphism ofk(G)-fieldsΨP : C(TI/c(P ))

∼−→ C ((R/P )⊗ k(G)) sat-

isfying

ΨP ◦ (ρ⊗ ρr)(g) = (IdR/P ⊗ρr)(g) ◦ΨP ,

Ψg.P ◦ (Id⊗ρℓ)(g) = (ρ⊗ ρℓ)(g) ◦ΨP ;

(d) rationality:P is rational if and only ifTI/c(P ) ∼= k(G).

Note thatC(TI/c(P )) in (c) is just the classical field of fractions of the commutative domainTI/c(P ). Furthermore, in the second identity in (c), we have

(Id⊗ρℓ)(g) : C(TI/c(P ))∼−→ C(TI/c(g.P ))

(ρ⊗ ρℓ)(g) : C ((R/P )⊗ k(G))∼−→ C ((R/g.P )⊗ k(G))

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 13

in the obvious way. For (d), recall from (4) that there existsa rationalP ∈ SpecI R if and only if I isG-rational.

Proof. ReplacingR byR/I, we may assume thatI = 0. Our goal is to establish a bijection betweenSpec0R = {P ∈ SpecR | P :G = 0} andSpecG(T0). For (a), this bijection needs to be equivariantfor theG-action byρ onR and byId⊗ρℓ onT0.

As was pointed out above,R is a prime ring. For brevity, we will put

C = C(R) and K = k(G) .

Thus,C andK are fields andT0 = C ⊗ K. Let R = RC denote the central closure ofR; so R is acentrally closed prime ring and

R⊗K = R⊗C T0 .

By Proposition 5,SpecT0 is in bijection with the set of all primesQ ∈ Spec(R ⊗ K) such thatQ ∩ R = 0. This bijection is equivariant with respect to the subgroups(ρ⊗ ρr)(G) and(Id⊗ρℓ)(G)

of Autk-alg(R ⊗ K), because these subgroups stabilize bothR andT0. Therefore, the bijection inProposition 5 restricts to a bijection

SpecG(T0)1-1←→ {Q ∈ SpecG(R ⊗K) | Q ∩ R = 0}

which is equivariant for theG-action byId⊗ρℓ on T0 and onR ⊗ K. Furthermore, Proposition 6

gives a bijection{Q ∈ Spec(R ⊗ K) | Q ∩ R = 0}1-1←→ {Q ∈ Spec(R ⊗ K) | Q ∩ R = 0} and

this bijection is equivariant with respect to both(ρ⊗ ρr)(G) and(Id⊗ρℓ)(G). As above, we obtain abijection

{Q ∈ SpecG(R⊗K) | Q ∩ R = 0}1-1←→ {Q ∈ SpecG(R⊗K) | Q ∩R = 0}

which is equivariant for theG-action byId⊗ρℓ onR⊗K and onR⊗K. Hence it suffices to constructa bijection

d : Spec0R −→ {Q ∈ SpecG(R⊗K) | Q ∩R = 0} (21)

which is equivariant for theG-action byρ onR and byId⊗ρℓ onR⊗K.For a givenP ∈ Spec0R, consider the homomorphism ofK-algebras

ϕP : R⊗K∆R−→ R⊗K

can.։ SP = (R/P )⊗K , (22)

where∆R is the automorphism (20) and the second map is theK-linear extension of the canonicalepimorphismR ։ R/P . The algebraSP is prime, sinceK is unirational overk. Therefore,

d(P ) = KerϕP = ∆−1R (P ⊗K) (23)

is a prime ideal ofR⊗K. From [15, Lemma 17], we infer thatP:G = d(P )∩R, and sod(P )∩R = 0.Furthermore,d(P ) clearly determinesP . Hence, the mapP 7→ d(P ) yields an injection ofSpec0Rinto {Q ∈ Spec(R⊗K) | Q ∩R = 0}.

We now check that this injection isG-equivariant and has image inSpecG(R ⊗K). Note that (18)and (19) imply the following equalities for allg ∈ G:

ϕP ◦ (ρ⊗ ρr)(g) = (IdR/P ⊗ρr)(g) ◦ ϕP , (24)

ϕg.P ◦ (IdR⊗ρℓ)(g) = (ρ⊗ ρℓ)(g) ◦ ϕP . (25)

In (25), we view(ρ ⊗ ρℓ)(g) as an isomorphismSP∼−→ Sg.P in the obvious way. In particular, (24)

shows thatd(P ) is stable under(ρ⊗ ρr)(G) while (25) gives

d(g.P ) = (IdR⊗ρℓ)(g)(P ) ;

so the mapP 7→ d(P ) isG-equivariant.

14 MARTIN LORENZ

For surjectivity ofd and the inverse map, letQ ∈ SpecG(R ⊗K) be given such thatQ ∩ R = 0.Put

P = R ∩∆R(Q) .

ThenP is prime inR, because the extensionR → R ⊗K is centralizing. Furthermore, [15, Lemma17] givesP :G = ∆−1

R (P ⊗ K) ∩ R = Q ∩ R = 0; soP ∈ Spec0 R. We claim thatQ = d(P ) orequivalently,∆R(Q) = P ⊗ K. To see this, note that∆R (Q) is stable under(Id⊗ρr)(G) by (18).Thus, the desired equality∆R(Q) = P ⊗K follows from [6, Cor. to Prop. V.10.6], because the fieldof ρr(G)-invariants inK is k. Thus,d is surjective and the inverse ofd is given by

d−1(Q) = R ∩∆R(Q) . (26)

This finishes the construction of the desiredG-equivariant bijection (21).To summarize, we have constructed a bijection

c : Spec0R −→ SpecG(T0)

with property (a); it arises as the composite of the following bijections:

Spec0Rd //

c=f◦e−1◦d

��

{Q ∈ SpecG(R⊗K) | Q ∩R = 0}

SpecG(T0) {Q ∈ SpecG(R⊗K) | Q ∩ R = 0}

e

OO

foo

(27)

Formulas ford and its inverse are given in (23) and (26), respectively. Theother maps are as follows:

e(Q) = Q ∩ (R⊗K) , (28)

f(Q) = Q ∩ T0 , (29)

f−1(p) = R⊗C p = (R ⊗K)p ; (30)

see Propositions 5 and 6. The mapsd±1, f±1 ande visibly preserve inclusions, and Proposition 6tells us that this also holds fore−1. Property (b) follows. By Propositions 5 and 6, bothe andf alsopreserve hearts. Thus, we have an isomorphism ofK-fields

ΨP : C(T0/c(P ))∼−→ C ((R⊗K)/d(P ))

∼−→ C(SP ) .

The desired identities forΨ are consequences of (24) and (25). This proves property (c).For (d), note that

K ⊆ T0/c(P ) ⊆ C(T0/c(P )) = Fract (T0/c(P )) ∼= C(SP ) (31)

holds for anyP ∈ Spec0R. If P is rational thenC(SP ) = K by [15, Lemma 7], and soT0/c(P ) = K.Conversely, ifT0/c(P ) = K then (31) givesC(SP ) = K. Since there always is aK-embeddingC(R/P )⊗K → C(SP ) by [15, equation (1-2)], we conclude thatC(R/P ) = k; soP is rational. Thisproves (d), and hence the proof of the theorem is complete. �

3.4. Note that Theorem 9(b) and (d) together imply that rational ideals are maximal in theirG-strata.The following result, which generalizes Vonessen [28, Theorem 2.3], is a marginal strengthening ofthis fact. In particular, the groupG is no longer assumed to be connected.

Proposition 10. LetP ∈ RatR and letI E R be any ideal ofR such thatI ⊇ P . If I:G = P:G thenI = P .

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 15

Proof. LetG0 denote the connected component of the identity inG; this is a normal subgroup of finiteindex inG [2, 1.2]. PuttingI0 = I:G0 andP 0 = P:G0, we have

I0 ⊇ P 0 ⊇ P:G = I:G =⋂

x∈G/G0

x.I0 .

SinceP 0 is G0-prime and allx.I0 areG0-stable ideals ofR, we conclude thatI0 ⊇ P 0 ⊇ x.I0 forsomex and (9) further implies thatI0 = P 0. Therefore, we may replaceG byG0, thereby reducingto the case whereG is connected. Furthermore, replacingI by an ideal that is maximal subject to theconditionI:G = P :G, we may also assume thatI is prime; see [15, proof of Proposition 8]. Thus,PandI both belong toSpecP :GR and Theorem 9(b),(d) yields the result. �

Corollary 11. Let I ∈ G-SpecR be locally closed inG-SpecR. Then the maximal members of theG-stratumSpecI R are locally closed inSpecR. In particular, if R satisfies the Nullstellensatz (see§0.7) then the maximal members of theG-stratumSpecI R are exactly the rational ideals inSpecI R.

Proof. Let P ∈ SpecR be maximal in itsG-stratumSpecI R, whereI = P :G. Then, for anyQ ∈ SpecR with Q % P , we haveQ:G % I. SinceI is locally closed, it follows thatI 6=

⋂Q%P Q:G.

HenceP 6=⋂

Q%P Q which proves thatP is locally closed inSpecR.Finally, rational ideals are always maximal in theirG-strata by Proposition 10. In the presence of

the Nullstellensatz, the converse follows from the preceding paragraph. �

3.5. We review some general results of Mœglin and Rentschler [22]and Vonessen [28]. Some of theconstructions below were already used, in a more specialized form, in the proof of Theorem 9. Theaffine algebraic groupG need not be connected here, but we will only use this materialin the connectedcase later on.

Fix a closed subgroupH ≤ G and letk(G)H = k(H\G) ⊆ k(G)

denote the subalgebra of invariants for the left regular action ρℓ

∣∣H

on k(G). Following Mœglin andRentschler [22] we define, for an arbitrary idealI E R,

I♮ = ∆−1R (I ⊗ k(G)) ∩ (R⊗ k(G)H) . (32)

Here ∆R is the automorphism (20) ofR ⊗ k(G). Thus, I♮ is certainly an ideal ofR ⊗ k(G)H .Furthermore:

• If I is semiprime thenI ⊗ k(G) is a semiprime ideal ofR ⊗ k(G), becausek(G) is a directproduct of fields that are unirational overk. Therefore,I♮ is semiprime in this case. ForconnectedG, we also see that ifI is prime thenI♮ is likewise, as in (23).• The groupG acts onR ⊗ k(G)H by means ofρ⊗ ρr. Formula (18) implies thatI♮ is always

stable under this action. Moreover, if the idealI is H-stable then formula (19) implies thatthe ideal∆−1

R (I ⊗ k(G)) of R ⊗ k(G) is stable under the automorphism groupIdR⊗ρℓ(H).Therefore, [28, Lemma 6.3] (or [6, Cor. to Prop. V.10.6] for connectedG) implies that∆−1

R (I⊗k(G)) = I♮ ⊗k(G)H k(G) holds in this case, and hence

I = ∆R

(I♮ ⊗k(G)H k(G)

)∩R . (33)

To summarize, the mapI 7→ I♮ gives an injection of the set of allH-stable semiprime ideals ofR intothe set of all(ρ⊗ ρr)(G)-stable semiprime ideals ofR⊗ k(G)H . In fact:

Proposition 12 (Mœglin and Rentschler, Vonessen). Let H be a closed subgroup ofG. The mapI 7→ I♮ in (32) gives a bijection from the set of allH-stable semiprime ideals ofR to the set of all(ρ ⊗ ρr)(G)-stable semiprime ideals ofR ⊗ k(G)H . This bijection and its inverse, given by(33),preserve inclusions.

16 MARTIN LORENZ

For the complete proof, see [28, Theorem 6.6(a)].

4. PROOF OFTHEOREM 1

4.1. Proof of Theorem 1 (b)⇒ (a). Let P ∈ RatR be given such thatI = P :G is a locally closedpoint ofG-SpecR. Then the preimageγ−1(I) = SpecI R under the continuous map (2) is a locallyclosed subset ofSpecR, and hence so isSpecI R ∩ {P}. By Proposition 10,SpecI R ∩ {P} = {P},which proves (a).

4.2. Proof of Theorem 1 (a) ⇒ (b). Besides making crucial use of Theorem 9, our proof closelyfollows Vonessen [28, Sect. 8] which in turn is based on Mœglin and Rentschler [19, Sect. 3].

4.2.1. Some reductions.First, recall that the connected component of the identity in G is always anormal subgroup of finite index inG. Therefore, Lemma 3 allows us to assume thatG is connectedand we will do so for the remainder of this section.

We are given an idealP ∈ RatR satisfying (13) and our goal is to show thatP :G is distinct fromthe intersection of allG-primes ofR which properly containP :G; see (14). For this, we may clearlyreplaceR byR/P:G. Hence we may assume that

P:G = 0 .

Thus, the algebraR is prime by [15, Proposition 19(a)], and soC(R) is a field. Our goal now is to showthat the intersection of all nonzeroG-primes ofR is nonzero again. Note that this intersection is identi-cal to the intersection of all nonzeroG-stable semiprime ideals ofR, becauseG-stable semiprimes areexactly the intersections ofG-primes. The intersection in question is also identical to the intersectionof all nonzeroG-stable prime ideals ofR, becauseG-primes are the same asG-stable primes ofR by[15, Proposition 19(a)].

4.2.2. The main lemma.Let R = RC(R) denote the central closure ofR. In the lemma below, whichcorresponds to Vonessen [28, Prop. 8.7], we achieve our goalfor R in place ofR:

Lemma 13. The intersection of all nonzeroG-stable semiprime ideals ofR is nonzero.

Proof. Let GP denote the stabilizer ofP in G and recall thatGP is a closed subgroup ofG [13,I.2.12(5)]. SinceP is locally closed,P is distinct from the intersection of allGP -stable semiprimeideals ofR that properly containP . By Proposition 12 withH = GP , we conclude that the idealP ♮ ∈ Spec

(R⊗KGP

)is distinct from the intersection of all(ρ⊗ ρr)(G)-stable semiprime ideals of

R ⊗ KGP that properly containP ♮. Here, we have putK = k(G) andKGP = k(GP \G) denotesthe invariant subfield ofK for the left regular actionρℓ

∣∣GP

as in§3.5. In other words, the intersection

of all nonzero(ρ ⊗ ρr)(G)-stable semiprime ideals of(R ⊗KGP )/P ♮ is nonzero. By Vonessen [28,Lemma 8.6], the lemma will follow if we can show that there is afinite centralizing embedding

R → (R ⊗KGP )/P ♮ (34)

such that theG-action viaρ ⊗ ρr on (R ⊗ KGP )/P ♮ restricts to theG-action onR via the uniqueextension ofρ.

To construct the desired embedding, writeC = C(R) and consider thek-algebra map

ψP : C → T0 = C ⊗K ։ T0/c(P )∼−→ K , (35)

where the first two maps are canonical and the last map is theK-isomorphism in Theorem 9(d); it isgiven by the isomorphismΨP in Theorem 9(c). (We remark that the mapψP is identical to the one

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 17

constructed in [15, Theorem 22].) The identities forΨP in Theorem 9(c) yield the following formulas:

ψP ◦ ρ(g)∣∣C

= ρr(g) ◦ ψP , (36)

ψg.P = ρℓ(g) ◦ ψP . (37)

(See also (a) in the proof of [15, Theorem 22].) Consider the subfield

KP = ImψP ⊆ K .

Equation (36) implies thatKP is aρr(G)-stable subfield ofK. More precisely, identity (37) showsthat

KP ⊆ KGP .

Moreover, ifg /∈ GP thenc(g.P ) 6= c(P ) and henceψg.P 6= ψP . Therefore, we deduce from (37)thatGP = {g ∈ G | ρℓ(g)(x) = x for all x ∈ KP }. Now Vonessen [28, Proposition 4.5] gives thatthe field extensionKGP /KP is finite (and purely inseparable). Thus, the desired embedding (34) willfollow if we can show that there is a(ρ⊗ ρr)(G)-equivariant isomorphism

(R ⊗KGP )/P ♮ ∼= R⊗C KGP ,

wherec⊗1 = 1⊗ψP (c) holds for allc ∈ C in the ring on the right. But the mapR⊗K ։ R⊗CK ∼=

R⊗C (T0/c(P )) has kernel(e−1 ◦d)(P ) in the notation of (27), and it is(ρ⊗ρr)(G)-equivariant. Therestriction of this map toR⊗KGP has imageR⊗CK

GP and kernel(e−1 ◦d)(P )∩(R⊗KGP ) = P ♮.This finishes the proof of the lemma. �

4.2.3. End of proof.We now complete the proof of Theorem 1 by showing that the intersection of allnonzeroG-stable prime ideals ofR is nonzero.

We use theG-equivariant embeddingψP : C → K = k(G); see (35) and (36). Note thatK =FractA whereA = k[G] is aG-stable affine domain overk whose maximal ideals form oneG-orbit.Therefore,A is G-simple. By Proposition 8, there exists an ideal0 6= D E R such that, for everyG-stable primeP of R not containingD, there is aG-stable prime ofR lying overP .

Now letN denote the intersection of all nonzeroG-stable semiprime ideals ofR; soN 6= 0 byLemma 13 and henceN ∩R 6= 0 by [15, Prop. 2(ii)(iii)]. We conclude from the preceding paragraphthat every nonzeroG-stable prime ideal ofR either containsD or else it containsN∩R. Therefore, theintersection of all nonzeroG-stable prime ideals ofR contains the idealD ∩N ∩R which is nonzero,becauseR is prime. This completes the proof of Theorem 1. �

5. FINITENESS OFG-SpecR

5.1. The following finiteness result is an application of Theorem1 and [15, Theorem 1].

Proposition 14. Assume thatR satisfies the Nullstellensatz. Then the following conditions are equiv-alent:

(a) R has finitely manyG-stable semiprime ideals;(b) G-SpecR is finite;(c) G-RatR is finite;(d) G has finitely many orbits inRatR;(e) R satisfies (i) the ascending chain condition forG-stable semiprime ideals,

(ii) the Dixmier-Mœglin equivalence, and(iii) G-RatR = G-SpecR.

If these conditions are satisfied then rational ideals ofR are exactly the primes that are maximal intheirG-strata.

18 MARTIN LORENZ

Proof. The implications (a)⇒ (b)⇒ (c) are trivial and (c)⇔ (d) is clear from (4). Moreover, theNullstellensatz implies that theG-stable semiprime ideals ofR are exactly the intersections ofG-rational ideals. Thus, (c) implies (a), and hence conditions (a) - (d) are all equivalent.

We now show that (a) - (d) imply (e). First, (i) is trivial from(a). For (ii), note that (b) implies thatall points ofG-SpecR are locally closed. Hence all rational ideals ofR are locally closed inSpecR byTheorem 1, proving (ii). Finally, in order to prove (iii), write a givenI ∈ G-SpecR as an intersectionof G-rational ideals. The intersection involves only finitely many members by (c), and so one of themmust be equal toI byG-primeness. Thus,I ∈ G-RatR which takes care of (iii).

To complete the proof of the equivalence of (a) - (e), we will show that (e) implies (b). By a familiarargument, hypothesis (i) allows us to assume that the algebraR isG-prime andG-SpecR/I is finitefor all nonzeroG-stable semiprime idealsI of R. By (iii) and (4), we know thatP :G = 0 holds forsomeP ∈ RatR. SinceP is locally closed inSpecR by (ii), Theorem 1 implies that0 is locallyclosed inG-SpecR, that is,I =

⋂06=J∈G-Spec R J is nonzero; see (14). Therefore,G-SpecR =

{0} ∪G-SpecR/I is finite.Finally, the last assertion is clear from Corollary 11, because all points ofG-SpecR are locally

closed by (b). �

5.2. We now concentrate on the case whereG is an algebraic torus:G ∼= Gnm with Gm = k∗. In

particular,G is connected and soG-SpecR is simply the set of allG-stable primes ofR by [15,Proposition 19(a)]. Moreover, everyG-moduleM has the form

M =⊕

λ∈X(G)

Mλ ,

whereX(G) is the collection of all morphisms of algebraic groupsλ : G→ Gm and

Mλ = {m ∈M | g.m = λ(g)m for all g ∈ G}

is the set ofG-eigenvectors of weightλ in M .

Lemma 15. If dimkRλ ≤ 1 holds for allλ ∈ X(G) thenG-SpecR = G-RatR.

Proof. Let P ∈ G-SpecR be given. The conditiondimkRλ ≤ 1 for all λ ∈ X(G) passes toR/P .Therefore, replacingR byR/P , we may assume thatR is prime and we must show thatC(R)G = k.For a givenq ∈ C(R)G put I = {r ∈ R | qr ∈ R}; this is a nonzeroG-invariant ideal ofR. HenceI =

⊕λ∈X(G) Iλ and so there is a nonzero elementr ∈ Iλ for someλ. Sinceq isG-invariant, we have

qr ∈ Rλ = kr. Therefore,(q − k)r = 0 holds in the central closureR for somek ∈ k. Inasmuch asnonzero elements ofC(R) are units inR, we conclude thatq = k ∈ k, which proves the lemma. �

5.2.1. Example: affine commutative algebras.The following proposition is a standard result onG-varieties [14, II.3.3 Satz 5]. It is also immediate from the foregoing:

Proposition 16. LetR be an affine commutative domain overk and letG be an algebraick-torusacting rationally onR. Then the following are equivalent:

(i) G-SpecR is finite;(ii) (FractR)G = k;(iii) dimkRλ ≤ 1 for all λ ∈ X(G).

Proof. Since affine commutative algebras satisfy the Nullstellensatz, the Dixmier-Mœglin equivalenceand the ascending chain condition for ideals, Proposition 14 tells us that (i) amounts to the equalityG-SpecR = G-RatR. The implication (iii)⇒ (i) therefore follows from Lemma 15. Furthermore, (i)implies that0 ∈ G-RatR; soC(R)G = k. SinceC(R) = FractR, (ii) follows. Finally, if a, b ∈ Rλ

are linearly independent thena/b ∈ FractR is not a scalar. Hence (ii) implies (iii). �

ALGEBRAIC GROUP ACTIONS ON NONCOMMUTATIVE SPECTRA 19

5.2.2. Example: quantum affine toric varieties.Affine domainsR with a rational action by an alge-braic torusG satisfying the conditiondimkRλ ≤ 1 for all λ ∈ X(G) as in Lemma 15 are calledquantum affine toric varietiesin Ingalls [12].

A particular example isquantum affine spaceR = Oq(kn) = k[x1, . . . , xn]. Here,q denotesfamily of parametersqij ∈ k∗ (1 ≤ i < j ≤ n) and the defining relations ofR are

xjxi = qijxixj (i < j) .

The torusG = Gnm acts onR, with α = (α1, . . . , αn) ∈ G acting by

α.xi = αixi

for all i. The standard PBW-basis ofR,

{xm = xm1

1 xm2

2 . . . xmn

n |m = (m1, . . . ,mn) ∈ Zn≥0} ,

consists ofG-eigenvectors:xm ∈ Rλmwith

λm(α) = αm = αm1

1 αm2

2 . . . αmn

n

Therefore, the conditiondimkRλ ≤ 1 for all λ is satisfied. Moreover,R = Oq(kn) is noetherian andsatisfies the Dixmier-Moeglin equivalence as long ask contains a non-root of unity [7, II.8.4].

Any quantum affine toric varietyR is a quotient of someOq(kn) [12, p. 6]. Hence,R is noetherianand satisfies the Dixmier-Moeglin equivalence (in the presence of non-roots of unity). Therefore,G-SpecR is finite by Proposition 14 and Lemma 15.

5.2.3. Example: quantum2 × 2 matrices.Let R = Oq(M2); this is the algebra with generatorsa, b, c, d and defining relations

ab = qba ac = qca bc = cb

bd = qdb cd = qdc ad− da = (q − q−1)bd .

The torusG4m acts onR as in [7, II.1.6(c)], withD = {(α, α, α−1, α−1) | α ∈ k∗} acting trivially.

Thus,G = G4m/D

∼= G3m acts onR:

(α, β, γ).a = βa (α, β, γ).b = γb

(α, β, γ).c = αβc (α, β, γ).d = αγd

This action does not satisfy conditiondimkRλ ≤ 1 for all λ ∈ X(G). Indeed, the PBW-basis{aibjcldm | i, j, l,m ∈ Z≥0} of R consists ofG-eigenvectors:aibjcldm corresponds to the char-acter(α, β, γ) 7→ αl+mβi+lγj+m. Defining

f : Z4 −→ Z3 , (i, j, l,m) 7→ (l +m, i+ l, j +m)

we have, for any givenλ = (λ1, λ2, λ3) ∈ Z3,

dimkRλ = #(f−1(λ) ∩ Z4

≥0

).

In order to determine this number, note thatKer f = Z(1,−1,−1, 1). Hence, we must count thepossiblet ∈ Z so thatzλ + t(1,−1,−1, 1) ∈ Z4

≥0 where we have putzλ = (λ2 − λ1, λ3, λ1, 0). Weobtain the following conditions ont: λ2 − λ1 + t ≥ 0, λ3 − t ≥ 0, λ1 − t ≥ 0, andt ≥ 0. Therefore,

dimkRλ = max {0,min{λ1, λ3} −max{λ1 − λ2, 0}+ 1} (38)

which can be arbitrarily large.Now consider the algebraR = R/(Dq) whereDq = ad− qbc ∈ ZR is the quantum determinant;

see [7, I.1.9]. SinceDq ∈ Rπ with π = (1, 1, 1), we have

dimkRλ = dimkRλ − dimkRλ−π ≤ 1

20 MARTIN LORENZ

by (38). ThereforedimkRλ ≤ 1 for all λ ∈ X(G). Moreover, assumingq to be a non-root of unity,Rsatisfies the Nullstellensatz and the Dixmier-Mœglin equivalence. Indeed, the algebraR is an image ofquantum4-space, sincead ≡ q2da mod Dq. Therefore, we know from Proposition 14 and Lemma 15that there are finitely manyG-primes ofR that containDq. The remainingG-primes correspond toG-SpecOq(GL2), and by [7, Exercise II.2.M], there are only four of these.

Acknowledgment. The author would like to thank Temple University for granting him a researchleave during the Fall Semester 2008 when the work on this article was completed. Thanks are also dueto the referees for valuable comments and suggestions.

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DEPARTMENT OFMATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA , PA 19122E-mail address: lorenz@temple.eduURL: http://www.math.temple.edu/∼lorenz