aa r X i v : . [ m a t h . QA ] A ug CATENARITY IN QUANTUM NILPOTENT ALGEBRAS
K. R. GOODEARL AND S. LAUNOIS
Abstract.
In this paper, it is established that quantum nilpotent algebras (also knownas CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime idealsbetween any two given prime ideals P ( Q have the same length. This is achievedby proving that the prime spectra of these algebras have normal separation, and thenestablishing the mild homological conditions necessary to apply a result of Lenagan andthe first author. The work also recovers the Tauvel height formula for quantum nilpotentalgebras, a result that was first obtained by Lenagan and the authors through a differentapproach. Introduction
The aim of this paper is to study the prime spectra of quantum algebras. More precisely,we focus on the catenary property – that all saturated chains of inclusions of prime idealsbetween any two fixed prime ideals have the same length – for a large class of (quantum)algebras called quantum nilpotent algebras. Examples of these algebras include for in-stance quantum matrices and more generally quantum Schubert cells. Quantum nilpotentalgebras have also appeared in the literature under the name “CGL extensions”, and theirprime spectra have been proved in some cases to be linked to totally nonnegative matrixvarieties; see for instance [9, 10, 19] for more details.A fundamental property of any affine algebraic variety V is that all saturated chains ofinclusions of irreducible subvarieties of V between any two fixed irreducible subvarietieshave the same length. Restated in terms of the coordinate ring O ( V ), this says that theprime spectrum of O ( V ) is catenary.Quantized coordinate rings of affine varieties are expected to enjoy suitable versionsof the properties of their classical counterparts. In particular, it is conjectured that theprime spectra of quantized coordinate rings must be catenary. This conjecture has beenverified for the quantized coordinate rings of many varieties, such as matrix varieties [3],affine spaces and general and special linear groups [12], simple algebraic groups [15, 30],Schubert cells [28], and Grassmannians [21]. Catenarity has also been established formany related quantum algebras, such as uni- and multiparameter quantum symplectic andeuclidean spaces [24, 16], quantized Weyl algebras [12, 24], and twisted quantum Schubertcell algebras [29]. The above references deal with generic quantum algebras, those whosequantum parameters are non-roots of unity. When the quantum parameters are rootsof unity, such algebras satisfy polynomial identities, and catenarity of affine polynomialidentity algebras follows from a result of Schelter [26, Theorem 1]. Mathematics Subject Classification.
Primary 16T20; Secondary16D25, 16P40, 16S36, 20G42.
Key words and phrases.
Catenary, quantum nilpotent algebra, CGL extension, height formula.The research of the first named author was supported by US National Science Foundation grant DMS-1601184. That of the second named author was supported by EPSRC grant EP/N034449/1.
Here we establish catenarity for all members of the broad family of quantum nilpotentalgebras (defined below). These algebras and localizations thereof cover the generic quan-tum algebras mentioned above, except for quantized coordinate rings of simple algebraicgroups and Grassmannians.By a famous result of Gabber, enveloping algebras of finite dimensional solvable Liealgebras are catenary (see, e.g., [5] or a combination of [23, Appendix Al] and [18, Ch.9]). This result was extended to enveloping algebras of finite dimensional solvable Liesuperalgebras by Lenagan [22]. The method of proof involved establishing good homolog-ical properties of the ring, connecting homological properties with growth, and controllinggrowth properties of prime factors by finding normal elements. (A normal element in aring R is an element x such that xR = Rx .) Abstracting these methods, Lenagan and thefirst author gave a set of homological and ring-theoretical conditions that ensure catenarityof an algebra [12, Theorem 7.1]. The method additionally yields the following useful heightformula, first established by Tauvel [27] for enveloping algebras of solvable Lie algebras:ht( P ) + GKdim( R/P ) = GKdim( R ) for all prime ideals P of R. This formula has been proved for many quantum algebras such as the ones mentionedabove, and Lenagan and the present authors recently proved that all quantum nilpotentalgebras satisfy Tauvel’s height formula [11].In order to apply the above methods to an algebra R , a suitable supply of normalelements in prime factor algebras is needed, in the following form. The prime spectrumSpec R must have normal separation , meaning that for any pair of distinct comparableprime ideals P ( Q in R , the factor Q/P contains a nonzero normal element of
R/P .Normal separation was proved by Cauchon for quantum matrices [3] using ring-theoreticaland combinatorial methods. Later, Yakimov established it for quantum Schubert cells [28]using representation theoretical methods. Here we prove it for a larger class of algebrasusing purely ring-theoretical methods.Establishing normal separation for quantum nilpotent algebras requires most of theeffort in the paper, since existing results can be applied to verify the required homologicalproperties.1.1.
Quantum nilpotent algebras.
Let R an iterated skew polynomial algebra of theform(1.1) R = K [ x ][ x ; σ , δ ] · · · [ x N ; σ N , δ N ] , over a field K , where σ j is an automorphism of the K -algebra R j − := K [ x ][ x ; σ , δ ] . . . [ x j − ; σ j − , δ j − ]and δ j is a K -linear σ j -derivation of R j − , for all j ∈ [[2 , N ]]. (When needed, we denote R := K and set R = K [ x ; σ , δ ] with σ := id K , δ := 0.) In particular, R and the R j are noetherian domains. Definition 1.1.
An iterated skew polynomial extension R as in (1.1) is called a quantumnilpotent algebra or a CGL extension [20, Definition 3.1] if it is equipped with a ratio-nal action of a K -torus H = ( K ∗ ) d by K -algebra automorphisms satisfying the followingconditions:(i) The elements x , . . . , x N are H -eigenvectors.(ii) For every j ∈ [[2 , N ]], δ j is a locally nilpotent σ j -derivation of R j − .(iii) For every j ∈ [[1 , N ]], there exists h j ∈ H such that ( h j · ) | R j − = σ j and h j · x j = q j x j for some q j ∈ K ∗ which is not a root of unity. ATENARITY IN QUANTUM NILPOTENT ALGEBRAS 3 (We have omitted the condition σ j δ j = q j δ j σ j from the original definition, as it followsfrom the other conditions; see, e.g., [14, Eq. (3.1); comments, p.694].)The main theorem of the paper is Theorem 1.2. If R is a quantum nilpotent algebra, then Spec R is catenary, and all primequotients of R satisfy Tauvel’s height formula. The key requirement in proving this theorem is normal separation in Spec R . Existenceof suitable normal elements is established by induction on the number of indeterminatesin R . The following two sections are devoted to the induction step, in which normalelements are constructed in certain skew polynomial algebras in one indeterminate andfactor algebras thereof. Normal separation for quantum nilpotent algebras is achieved inSection 4 together with the desired homological properties, and Theorem 1.2 is provedthere.1.2. Notation and conventions.
Throughout, all algebras will be unital algebras over afixed base field K . All the skew polynomial rings we consider will be of the form A [ X ; σ, δ ]where the coefficient ring A is a K -algebra, σ is a K -algebra automorphism of A , and δ is a K -linear left σ -derivation of A . The K -automorphism and K -linearity assumptionsensure that A [ X ; σ, δ ] is a K -algebra, and that it is noetherian if A is noetherian. Theindeterminate X in A [ X ; σ, δ ] skew-commutes with elements a ∈ A as follows: Xa = σ ( a ) X + δ ( a ). 2. A first construction of normal elements
Basic assumptions.
Let A be a noetherian K -algebra domain and R = A [ X ; σ, δ ] askew polynomial extension.Assume throughout this section that • δ is locally nilpotent. • There is an abelian group H acting on R by K -algebra automorphisms such that X is an H -eigenvector and A is H -stable. • There exists h ◦ ∈ H such that ( h ◦ · ) | A = σ and the h ◦ -eigenvalue λ ◦ of X is not aroot of unity.As noted in [14, Eq. (3.1)], σδ = λ ◦ δσ . More generally [14, Eq. (3.2)],(2.1) ( h · ) | A ◦ δ = χ X ( h ) δ ◦ ( h · ) | A ∀ h ∈ H , where χ r : H → K ∗ denotes the H -eigenvalue of an H -eigenvector r ∈ R .2.2. H -ideals. Recall that if C is a ring equipped with an action of a group H by au-tomorphisms, then the H -ideals of C are the (two-sided) ideals of C invariant under the H -action. An H -prime ( ideal ) of C is any proper H -ideal P such that a product I I of H -ideals of C is contained in P only if I or I is contained in P . The ring C is said to be H -simple provided C = 0 and the only H -ideals of C are 0 and C . The latter condition isequivalent to the condition that 0 is the only H -prime of C .2.3. Cauchon extensions.
If in addition to § • Every H -prime of A is completely prime,then R is a Cauchon extension [20, Definition 2.5].
K. R. GOODEARL AND S. LAUNOIS δ is locally nilpotent, the set S := { X n | n ∈ Z ≥ } is a denominator set in R [2, Lemme 2.1]. Set b R := RS − . Since the elements of S are H -eigenvectors, the action of H on R extends uniquely to an action by K -algebra automorphisms on b R .Let θ : A → b R be the Cauchon map defined by(2.2) θ ( a ) = ∞ X l =0 (1 − λ ◦ ) − l ( l )! λ ◦ δ l σ − l ( a ) X − l . (See (2.5) for the definition of ( l )! λ ◦ .) Cauchon established in [2, Propositions 2.1–2.4] that • θ is an injective K -algebra homomorphism. • θ extends uniquely to an injective K -algebra homomorphism A [ Y ; σ ] → b R with θ ( Y ) = X . • Set B := θ ( A ) and T := θ ( A [ Y ; σ ]) ⊆ b R . Then T = B [ X ; α ] where α is the K -algebra automorphism of B defined by α ( θ ( a )) = θ ( σ ( a )). • S is also a denominator set in T , and T S − = S − T = b R .As is noted in [20, p.327], B ∩ R ⊆ A .By [20, Lemma 2.6] (whose proof only uses the assumptions of § θ is H -equivariant.Since the action of σ on A is given by h ◦ , it follows that α = ( h ◦ · ) | B . Lemma 2.1.
Let a ∈ A \ { } and let s ∈ Z ≥ be maximal such that δ s ( a ) = 0 . Then s isminimal such that θ ( a ) X s ∈ R .Proof. Since δ l ( a ) = 0 for l > s , we have θ ( a ) = P sl =0 c l δ l σ − l ( a ) X − l for some c l ∈ K ∗ .Obviously θ ( a ) X s ∈ R .Suppose that s > θ ( a ) X t ∈ R for some t < s . Then θ ( a ) X s − ∈ R , from whichit follows that δ s σ − s ( a ) X − ∈ R . Now δ s σ − s ( a ) ∈ A ∩ RX , whence δ s σ − s ( a ) = 0. But δ s σ − s = λ s ◦ σ − s δ s , so we obtain δ s ( a ) = 0, contradicting our hypotheses. Therefore s isminimal such that θ ( a ) X s ∈ R . (cid:3) The following lemma is excerpted from the proof of [20, Proposition 2.9].
Lemma 2.2.
Let a ∈ A be a normal H -eigenvector, and let s ∈ Z ≥ be maximal such that δ s ( a ) = 0 . Then the element x := θ ( a ) X s is a normal H -eigenvector in R . In particular, xX = η − Xx , where η is the σ -eigenvalue of a .Proof. Since θ is H -equivariant, the element b := θ ( a ) is a normal H -eigenvector in B , andthe h ◦ -eigenvalue of b equals that of a , namely η . By Lemma 2.1, s is minimal such that bX s ∈ R . This places x in R , and clearly x is an H -eigenvector.Since(2.3) Xb = α ( b ) X = h ◦ ( b ) X = ηbX, we see that xX = η − Xx . Moreover, we see that b is also normal in T and in b R . Inparticular, b b R = b Rb is an ideal of b R . But b b R = x b R , and b Rb = b Rx because x = η − s X s b .Thus, I := x b R ∩ R = b Rx ∩ R is an ideal of R . We show that I = Rx = xR , which will prove that x is normal in R .Obviously I contains Rx and xR .Let y ∈ I . Then y ∈ b b R implies yX u ∈ bT = T b for some u ≥
0. Now yX u = cb forsome c ∈ T , and cX v ∈ R for some v ≥
0. From (2.3), we obtain yX u + v + s = cbX v + s = η − v cX v bX s = η − v cX v x ∈ Rx.
Let t ∈ Z ≥ be minimal such that yX t ∈ Rx , and write yX t = rx for some r ∈ R . ATENARITY IN QUANTUM NILPOTENT ALGEBRAS 5
We wish to show that t = 0. Write r = X i ≥ r i X i , y = X i ≥ y i X i , x = X i ≥ x i X i for some r i , y i , x i ∈ A . In case s = 0, we would have x = b = a ∈ A and so x = a = 0. Incase s >
0, we would have x X − + X i ≥ x i X i − = xX − = bX s − / ∈ R by the minimality of s , so again x = 0. Thus, x = 0 in all cases.Observe that X i ≥ y i X i + t = yX t = rx = X i ≥ r i X i bX s = X i ≥ η i r i bX i + s = X i ≥ η i r i xX i = X i,j ≥ η i r i x j X i + j . If t >
0, it would follow that η r x = 0, whence r = 0. Then r = r ′ X for some r ′ ∈ R ,and so yX t = r ′ Xx = ηr ′ xX. But then yX t − = ηr ′ x ∈ Rx , contradicting the minimality of t . Therefore t = 0.Consequently, y = rx , proving that I = Rx .The proof that I = xR is very similar, and is left to the reader. (cid:3) q -skew calculations. Since δσ = λ − ◦ σδ , the pair ( σ, δ ) is a λ − ◦ -skew derivation inthe terminology of [7]. We shall need the following calculations.The q -Leibniz Rules for the λ − ◦ -skew situation [7, Lemma 6.2] say that(2.4) δ n ( ef ) = n X i =0 (cid:18) ni (cid:19) λ − ◦ σ n − i δ i ( e ) δ n − i ( f ) X n e = n X i =0 (cid:18) ni (cid:19) λ − ◦ σ n − i δ i ( e ) X n − i ∀ n ∈ Z ≥ , e, f ∈ A, where the q -binomial coefficients, for q = λ − ◦ , are given by(2.5) (cid:18) ni (cid:19) q = ( n )! q ( i )! q ( n − i )! q , ( m )! q = ( m ) q ( m − q · · · (1) q , ( m ) q = q m − q − . The argument of [25, Lemme 7.2.3.2] yields
Lemma 2.3.
Let C be a K -algebra domain and ( σ, δ ) a q -skew derivation on C , where q ∈ K ∗ is not a root of unity. Suppose c, e ∈ C with δ ( c ) = ce or δ ( c ) = ec . If there issome m ∈ Z ≥ such that δ m ( c ) = δ m ( e ) = 0 , then δ ( c ) = 0 .Proof. We must show that one of c or e is zero. Suppose that c, e = 0, and let s, t ∈ Z ≥ be maximal such that δ s ( c ) , δ t ( e ) = 0. Assume first that δ ( c ) = ce . By the q -Leibniz Rule, δ s + t ( ce ) = s + t X i =0 (cid:18) s + ti (cid:19) q σ s + t − i δ i ( c ) δ s + t − i ( e ) = (cid:18) s + ts (cid:19) q σ t δ s ( c ) δ t ( e ) = 0 , since (cid:0) s + ts (cid:1) q = 0 because q is not a root of unity. But then δ s + t +1 ( c ) = 0, due to theassumption δ ( c ) = ce . This is impossible, since s + t + 1 > s . The assumption δ ( c ) = ec leads to a similar contradiction. (cid:3) K. R. GOODEARL AND S. LAUNOIS Normal elements in Cauchon extensions
Throughout this section, keep the assumptions of §§ R = A [ X ; σ, δ ] is aCauchon extension.3.1. H -primes in Cauchon extensions. By [13, Lemmas 3.2, 3.3, Proposition 3.4 andtheir proofs],(i) Every H -prime of R is completely prime.(ii) Every H -prime of R contracts to a δ -stable H -prime of A .(iii) For any δ -stable H -prime P of A , there are at most two H -primes of R that contractto P in A . There is always at least one, namely P R .We shall also need the observation(iv) If P is a prime (ideal) of A (or R ), then ( P : H ) := T h ∈H ( h · P ) is an H -prime of A (or R ).It follows that(v) If I is an H -ideal of A (or R ), then all primes minimal over I are H -primes.By the usual localization procedures for skew polynomial rings, σ and δ extend uniquelyto an automorphism and a σ -derivation on A ∗ := Fract A , and the skew polynomial algebra R ∗ := A ∗ [ X ; σ, δ ] equals the localization of R with respect to A \ { } . The H -actions on A and R extend uniquely to actions on A ∗ and R ∗ , and ( h ◦ · ) = σ on A ∗ . Hence, exceptfor local nilpotence of δ , the assumptions of §§ A ∗ and R ∗ .Recall that an inner σ -derivation of A ∗ (or A ) is a map of the form a da − σ ( a ) d , forsome fixed d ∈ A ∗ (or d ∈ A ). Such a σ -derivation is denoted δ d . Proposition 3.1.
Assume that R ∗ is not H -simple. (a) There is a unique element d ∈ A ∗ such that δ = δ d on A ∗ and h · d = χ X ( h ) d for all h ∈ H . In particular, X − d is an H -eigenvector with χ X − d = χ X . (b) There is a unique nonzero H -prime in R ∗ , namely ( X − d ) R ∗ = R ∗ ( X − d ) . (c) Let I ∗ be a proper nonzero H -ideal of R ∗ , let n be the minimum degree for nonzeroelements of I ∗ , and let f = X n + cX n − + [lower terms] , with c ∈ A ∗ , be a monic elementof I ∗ with degree n . Then n > and d = ( λ ◦ − − λ n ◦ ) − c .Proof. These follow from [13, Lemma 3.3] and its proof, since A ∗ is H -simple. (cid:3) Whenever R ∗ is not H -simple, we keep the notation d for the element of A ∗ describedin Proposition 3.1(a). Note that R ∗ = A ∗ [ X − d ; σ ] in this case, and that items (i)–(v)above hold for R ∗ and A ∗ . Corollary 3.2. If R ∗ is not H -simple, then ( X − d ) R ∗ ∩ R is the unique nonzero H -primeof R that contracts to in A . Moreover, any H -ideal of R that contracts to in A iscontained in ( X − d ) R ∗ ∩ R .Proof. On one hand, P ∗ := ( X − d ) R ∗ is a nonzero H -prime of R ∗ that contracts to 0 in A ∗ , whence P ∗ ∩ R is a nonzero H -prime of R that contracts to 0 in A . On the other hand,any nonzero H -prime Q of R with Q ∩ A = 0 localizes to a nonzero H -prime QR ∗ of R ∗ ,whence QR ∗ = P ∗ and thus Q = QR ∗ ∩ R = P ∗ ∩ R .Similarly, any H -ideal I of R with I ∩ A = 0 localizes to an H -ideal IR ∗ of R ∗ . Since I is disjoint from A \ { } , we must have IR ∗ = R ∗ , whence there is at least one prime Q ∗ of R ∗ minimal over IR ∗ . Then Q ∗ is an H -prime ( § Q ∗ = P ∗ . Therefore I ⊆ IR ∗ ∩ R ⊆ P ∗ ∩ R . (cid:3) ATENARITY IN QUANTUM NILPOTENT ALGEBRAS 7
Some normal H -eigenvectors.Lemma 3.3. Assume there is a nonzero H -prime P in R with P ∩ A = 0 . Let a ∈ A bea normal H -eigenvector, and s ∈ Z ≥ maximal such that δ s ( a ) = 0 . (a) If s > , then x := θ ( a ) X s is a normal H -eigenvector in R and x ∈ P . Moreover, d = η − ( λ s ◦ − − a − δ ( a ) and δ ( a ) a = ηλ s ◦ aδ ( a ) , where η := χ a ( h ◦ ) . (b) Now assume that a is the leading coefficient of some element of P with degree .Then a + P is normal in R/P . Moreover, if also s = 0 , then δ ≡ and P = XR .Proof. The ideal P localizes to a nonzero H -prime P ∗ of R ∗ such that P ∗ ∩ R = P , and P ∗ = ( X − d ) R ∗ by Proposition 3.1(b).(a) By Lemma 2.2, x is a normal H -eigenvector in R . Now I := Rx is a nonzero H -idealof R , and I ∩ A = 0 because deg x = s >
0. By Corollary 3.2, I ⊆ P , whence x ∈ P .Note that x = aX s + cX s − + [lower terms], where c = (1 − λ ◦ ) − δσ − ( a ) = η − (1 − λ ◦ ) − δ ( a ) . The ideal I localizes to a proper nonzero H -ideal I ∗ := R ∗ x in R ∗ , and s is the minimumdegree for nonzero elements of I ∗ . Since a − x is a monic element of I ∗ with degree s ,Proposition 3.1(c) implies that d = ( λ ◦ − − λ s ◦ ) − a − c = η − ( λ s ◦ − − a − δ ( a ).Observe that δ ( a ) = da − ηad = η − ( λ s ◦ − − (cid:0) a − δ ( a ) a − ηaa − δ ( a ) (cid:1) , whence η ( λ s ◦ − δ ( a ) = a − δ ( a ) a − ηδ ( a ), and therefore ηλ s ◦ δ ( a ) = a − δ ( a ) a .(b) Assume that aX + c ∈ P for some c ∈ A . Then X + a − c is a monic element of P ∗ with degree 1. Since P ∗ is proper, it contains no nonzero elements of degree 0. Hence, weagain apply Proposition 3.1(c), obtaining d = − a − c .If s = 0, then δ ( a ) = 0, whence δ m ( d ) = − η − m a − δ m ( c ) = 0 for some m ∈ Z ≥ . Since δ ( d ) = dd − σ ( d ) d = (1 − λ ◦ ) d , it follows from Lemma 2.3 that δ ( d ) = 0. But 1 − λ ◦ = 0,so we obtain d = 0. Thus δ = δ ≡ P = XR . Moreover, aX = η − Xa , so a is normal in R , whence also a + P is normal in R/P .Finally, assume that s >
0. By part (a), we have − a − c = d = η − ( λ s ◦ − − a − δ ( a ) , whence δ ( a ) = η (1 − λ s ◦ ) c . Since aX + c ∈ P , it follows that Xa = ηaX + η (1 − λ s ◦ ) c ≡ ηaX + η (1 − λ s ◦ )( − aX ) = ηλ s ◦ aX (mod P ) . As a is already normal in A , we conclude that a + P is normal in R/P . (cid:3) Proposition 3.4.
Assume that every nonzero H -prime of A contains a normal H -eigen-vector.If P ( Q are H -primes of R with P ∩ A = 0 , there exists a normal H -eigenvector u of R/P such that u ∈ Q/P .Proof.
Recall that Q ∩ A is a δ -stable H -prime of A .Assume first that P = 0. Then 0 and P are two H -primes of R that contract to 0 in A ,so Q ∩ A = 0 by § P localizes to a nonzero H -prime P ∗ in R ∗ , and P ∗ = R ∗ ( X − d ) by Proposition3.1(b). Writing d = b − c for some b, c ∈ A with b = 0, we have bX − c = b ( X − d ) ∈ P ∗ ∩ R = P . Thus, the H -ideal J := { a ∈ A | aX + e ∈ P for some e ∈ A } is nonzero, as is then J ∩ ( Q ∩ A ) = J ∩ Q . K. R. GOODEARL AND S. LAUNOIS
There exist primes P , . . . , P r in A minimal over J ∩ Q such that P P · · · P r ⊆ J ∩ Q .Since J ∩ Q is an H -ideal, these P i are H -primes of A ( § P i contains a normal H -eigenvector a i , and thus a := a a · · · a r is a normal H -eigenvectorof A that lies in J ∩ Q . Since a is in J , it is the leading coefficient of an element of P of degree 1. By Lemma 3.3(b), the coset u := a + P is a normal H -eigenvector of R/P .Moreover, u ∈ Q/P because a ∈ Q .Now assume that P = 0. If Q ∩ A = 0, then by hypothesis, Q ∩ A contains a normal H -eigenvector a of A . Then δ l ( a ) ∈ Q ∩ A for all l ∈ Z ≥ , whence the element u := θ ( a ) X s lies in Q , where s ∈ Z ≥ is minimal such that θ ( a ) X s ∈ R . By Lemma 2.2, u is a normal H -eigenvector in R .Finally, suppose that Q ∩ A = 0. As above, the H -ideal J := { a ∈ A | aX + e ∈ Q for some e ∈ A } is nonzero. If J = A , then 1 ∈ J , while if J = A , then J contains a product of nonzero H -primes of A . In either case, there is a normal H -eigenvector a of A that lies in J . Let s ∈ Z ≥ be maximal such that δ s ( a ) = 0.If s >
0, then by Lemmas 2.2 and 3.3(a), u := θ ( a ) X s is a normal H -eigenvector of R that lies in Q . On the other hand, if s = 0, Lemma 3.3(b) shows that δ ≡ Q = XR .In this case, u := X is a normal H -eigenvector of R that lies in Q . (cid:3) Carrying normal H -separation from A to R .Definition 3.5. Suppose C is a K -algebra equipped with an H -action by K -algebra auto-morphisms. Following [8, § H - Spec C has normal H -separation providedthat for any proper inclusion P ( Q of H -prime ideals of C , there exists a normal H -eigenvector of C/P which lies in
Q/P .The condition of normal H -separation only requires a suitable supply of H -eigenvectorswhich are normal in appropriate factor rings. It does not require these normal elements tonormalize via actions of elements of H . That requirement leads to the following strongercondition. We say that H - Spec C has H -normal separation if, for any proper inclusion P ( Q of H -prime ideals of C , the ideal Q/P contains a nonzero element u which is H -normal in C/P , meaning that u is normal and there is some h ∈ H such that uc = ( h · c ) u for all c ∈ C/P . Theorem 3.6. If H - Spec A has normal H -separation, then so does H - Spec R .Proof. Let P ( Q be H -primes of R . Then P := P ∩ A is a δ -stable H -prime of A ( § A , R , P , Q by A/P , R/P R , P/P R , Q/P R , respectively.Thus, there is no loss of generality in assuming that P ∩ A = 0.The hypothesis of normal H -separation now implies that every nonzero H -prime of A contains a normal H -eigenvector of A . Therefore, by Proposition 3.4, there exists anormal H -eigenvector u of R/P such that u ∈ Q/P . This verifies normal H -separation in H - Spec R . (cid:3) Question 3.7. If H - Spec A has H -normal separation, does H - Spec R have H -normalseparation? 4. Proof of the main theorem
Observe that if R = K [ x ][ x ; σ , δ ] · · · [ x N ; σ N , δ N ] is a quantum nilpotent algebra,then R j is a Cauchon extension of R j − for all j ∈ [[2 , N ]]. (The complete primeness of H -primes follows from § ATENARITY IN QUANTUM NILPOTENT ALGEBRAS 9
Theorem 4.1. If R is a quantum nilpotent algebra, then Spec R has normal separation.Proof. Write R as in (1.1), and let H be as in Definition 1.1. Obviously H - Spec R hasnormal H -separation. By induction on N , Theorem 3.6 implies that H - Spec R has normal H -separation. Therefore, by [8, Theorem 5.3], Spec R has normal separation. (cid:3) Question 4.2. If R is a quantum nilpotent algebra, does H - Spec R have H -normal sepa-ration?As far as inclusions 0 ( Q of H -primes are concerned, H -normal separation is known tohold provided the torus H is maximal in the sense of [14, § R itself are H -normal by [14, Corollary 5.4].We now address homological properties of a quantum nilpotent algebra R , some ofwhich are obtained by filtering R so that the associated graded ring gr R is a quantumaffine space. Definition 4.3.
A matrix q = ( q ij ) ∈ M N ( K ) is multiplicatively skew-symmetric provided q ii = 1 for all i and q ji = q − ij for all i , j . Given such a matrix, define the algebra O q ( K N ) := K h x , . . . , x N | x i x j = q ij x j x i ∀ i, j ∈ [[1 , N ]] i . The algebra O q ( K N ) is a quantized coordinate ring of the affine space A N , or a quantumaffine space for short. It is trivially a quantum nilpotent algebra. Notation 4.4. If R is a quantum nilpotent algebra as in Definition 1.1, there are scalars λ ji ∈ K ∗ such that σ j ( x i ) = λ ji x i for 1 ≤ i < j ≤ N . These are the below-diagonal entriesof a multiplicatively skew-symmetric matrix λ = ( λ ij ) ∈ M N ( K ). Lemma 4.5.
Let R be an iterated skew polynomial algebra of length N as in (1.1) , andassume there is a multiplicatively skew-symmetric matrix q = ( q ij ) ∈ M N ( K ) such that σ j ( x i ) = q ji x i for ≤ i < j ≤ N . Then there exist an exhaustive, ascending, locally finite K -algebra filtration ( R n ) n ≥ on R and a K -algebra Z ≥ -grading on O q ( K N ) such that (a) R = K . (b) The canonical generators x , . . . , x N of O q ( K N ) are homogeneous with positive de-gree. (c) gr R and O q ( K N ) are isomorphic as graded K -algebras, where the principal symbolsof the x i in R map to the x i in O q ( K N ) .Proof. This is an application of [1, Chapter 2, Corollary 3.3; Chapter 4, Proposition 6.4,Theorem 6.5]. (cid:3)
Proposition 4.6.
Let R = K [ x ][ x ; σ , δ ] · · · [ x N ; σ N , δ N ] be an iterated skew polynomialalgebra as in (1.1) , and assume that σ j ( x i ) ∈ K ∗ x i for ≤ i < j ≤ N . Then R is anAuslander-regular, Cohen-Macaulay algebra of GK -dimension N .Proof. Auslander-regularity and the GK-dimension value follow by induction on N from[4, Theorem 4.2] and [17, Lemma 2.2]. Let R be filtered as in Lemma 4.5, so that R = K and gr R ∼ = O q ( K N ). Then [6, Theorem 3] implies that R is Cohen-Macaulay. (cid:3) Now we have everything in hand to prove the main theorem.
First Proof of Theorem 1.2.
Clearly R is an affine noetherian K -algebra domain. It isAuslander-Gorenstein and Cohen-Macaulay with finite GK-dimension by Proposition 4.6,and Spec R is normally separated by Theorem 4.1. Therefore by [12, Theorem 1.6], Spec R is catenary and Tauvel’s height formula holds in R . Now consider a prime ideal
P/Q in a prime quotient
R/Q of R . Due to catenarity inSpec R , we have ht( P/Q ) = ht( P ) − ht( Q ). Taking account of the height formula for R ,we obtainGKdim (cid:0) ( R/Q ) / ( P/Q ) (cid:1) + ht( P/Q ) = GKdim(
R/P ) + ht( P ) − ht( Q )= GKdim( R ) − ht( Q ) = GKdim( R/Q ) , which verifies the height formula in R/Q . (cid:3) Second Proof of Theorem 1.2.
Catenarity follows from [31, Theorem 0.1], whose hypothe-ses are verified as follows. (1) Normal separation is given by Theorem 4.1. (2) If R isfiltered as in Lemma 4.5, then gr R is graded isomorphic to O q ( K N ), which is clearlynoetherian and connected graded. Moreover, O q ( K N ) has enough normal elements in thesense of [31], since if P is a graded prime ideal of O q ( K N ) with O q ( K N ) /P = K , thensome x j / ∈ P , whence x j + P is a nonzero homogeneous normal element of O q ( K N ) /P withpositive degree.Tauvel’s height formula for R follows from [31, Theorem 2.23] or [11, Theorem 7.1],and then the height formula may be established for prime quotients of R as in the firstproof. (cid:3) References [1] J.L. Bueso, J. G´omez-Torrecillas, and A. Verschoren,
Algorithmic Methods in Non-Commutative Al-gebra. Applications to Quantum Groups , Kluwer (2003) Dordrecht.[2] G. Cauchon,
Effacement des d´erivations et spectres premiers d’alg`ebres quantiques,
J. Algebra (2003), 476–518.[3] G. Cauchon,
Spectre premier de O q ( M n ( k )) . Image canonique et s´eparation normale , J. Algebra (2003), 519–569.[4] E.K. Ekstr¨om, The Auslander condition on graded and filtered Noetherian rings , in Seminaire Dubreil-Malliavin, 1987-1988, Lecture Notes in Mathematics, Vol. 1404 Berlin (1989) Springer, pp. 220–245.[5] O. Gabber,
Equidimensionalit´e de la vari´et´e caract´eristique , Expos´e de O. Gabber r´edig´e par T.Levasseur, Universit´e de Paris VI, (1982).[6] J. G´omez-Torrecillas and F.J. Lobillo,
Auslander-regular and Cohen-Macaulay quantum groups , Alge-bras and Rep. Theory (2004), 35–42.[7] K.R. Goodearl, Prime ideals in skew polynomial rings and quantized Weyl algebras , J. Algebra (1992), 324–377.[8] K.R. Goodearl,
Prime spectra of quantized coordinate rings , in Interactions between Ring Theory andRepresentations of Algebras (Murcia 1998), (F. Van Oystaeyen and M. Saor´ın, eds.), New York (2000),Dekker, pp. 205–237.[9] K.R. Goodearl, S. Launois, and T.H. Lenagan,
Totally nonnegative cells and matrix Poisson varieties ,Advances in Math. (2011), 779–826.[10] K.R. Goodearl, S. Launois, and T.H. Lenagan,
Torus-invariant prime ideals in quantum matrices,totally nonnegative cells and symplectic leaves , Math. Zeitschrift (2011), 29–45.[11] K.R. Goodearl, S. Launois, and T.H. Lenagan,
Tauvel’s height formula for quantum nilpotent algebras ,Communic. in Algebra (2019), 4194–4209.[12] K.R. Goodearl and T.H. Lenagan, Catenarity in quantum algebras , J. Pure Applied Algebra (1996), 123–142.[13] K.R. Goodearl and E.S. Letzter,
The Dixmier–Moeglin equivalence in quantum coordinate rings andquantized Weyl algebras , Trans. Amer. Math. Soc. (2000), 1381–1403.[14] K.R. Goodearl and M.T. Yakimov,
From quantum Ore extensions to quantum tori via noncommutativeUFDs , Adv. Math. (2015), 672–716.[15] K.R. Goodearl and J.J. Zhang,
Homological properties of quantized coordinate rings of semisimplegroups , Proc. Lond. Math. Soc. (3) (2007), 647–671.[16] K.L. Horton, The prime and primitive spectra of multiparameter quantum symplectic and Euclideanspaces , Communic. in Algebra (2003), 2713–2743.[17] C. Huh and C.O. Kim, Gelfand-Kirillov dimension of skew polynomial rings of automorphism type ,Communic. in Algebra (1996), 2317–2323. ATENARITY IN QUANTUM NILPOTENT ALGEBRAS 11 [18] G. Krause and T.H. Lenagan,
Growth of Algebras and Gelfand-Kirillov Dimension , Pitman (1985)Boston.[19] S. Launois, T.H. Lenagan, and B. Nolan,
Total positivity is a quantum phenomenon: the grassmanniancase , arXiv:1906.06199.[20] S. Launois, T.H. Lenagan, and L. Rigal,
Quantum unique factorisation domains , J. London Math.Soc. (2) (2006), 321–340.[21] S. Launois, T.H. Lenagan and L. Rigal, Prime ideals in the quantum grassmannian , Selecta Math.(N.S.) (2008), 697–725.[22] T.H. Lenagan, Enveloping algebras of solvable Lie superalgebras are catenary , Contemp. Math. (1992), 231–236.[23] T. Levasseur and J.T. Stafford,
Rings of differential operators on classical rings of invariants , MemoirsAmer. Math. Soc. No. 412 (1989).[24] S.-Q. Oh,
Catenarity in a class of iterated skew polynomial rings , Communic. in Algebra (1997),37–49.[25] L. Richard, Equivalence rationnelle et homologie de Hochschild pour certaines alg`ebres polynomialesclassiques et quantiques , Th´ese de doctorat, Universit´e Blaise Pascal (Clermont 2), (2002).[26] W. Schelter,
Affine PI rings are catenary , Bull. Amer. Math. Soc. (1977), 1309–1310.[27] P. Tauvel, Sur les quotients premiers de l’alg`ebre enveloppante d’un alg`ebre de Lie r´esoluble , Bull. Soc.Math. France (1978), 177–205.[28] M. Yakimov,
A proof of the Goodearl-Lenagan polynormality conjecture , Int. Math. Res. Notices (2013), 2097–2132.[29] M. Yakimov, Spectra and catenarity of multi-parameter quantum Schubert cells , Glasgow Math. J. (2013), 169–194.[30] M. Yakimov,
On the spectra of quantum groups , Memoirs Amer. Math. Soc. No. 1078, (2014).[31] A. Yekutieli and J.J. Zhang,
Rings with Auslander dualizing complexes , J. Algebra (1999), 1–51.
Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.
E-mail address : [email protected] School of Mathematics, Statistics and Actuarial Science, University of Kent, Canter-bury, Kent, CT2 7FS, UK
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