Generalised quantum determinantal rings are maximal orders
aa r X i v : . [ m a t h . QA ] F e b Generalised quantum determinantalrings are maximal orders
T H Lenagan and L Rigal
Abstract
Generalised quantum determinantal rings are the analogue in quantum matricesof Schubert varieties. Maximal orders are the noncommutative version of integrallyclosed rings. In this paper, we show that generalised quantum determinantal ringsare maximal orders. The cornerstone of the proof is a description of generalisedquantum determinantal rings, up to a localisation, as skew polynomial extensions.
Let K be a field, let m, n be positive integers and let q be a nonzero element of K . Thealgebra of quantum matrices over K , denoted by O q ( M mn ( K )), is a quantum deformation ofthe coordinate ring of the variety of m × n matrices over K . The set of quantum minors Π in O q ( M mn ( K )) carries a natural partial order with respect to which the standard monomialsform a basis over K : more precisely, O q ( M mn ( K )) is a quantum graded algebra with astraightening law on the poset of quantum minors equipped with the standard partialorder. (Precise definitions are given later.)Given a quantum minor γ in O q ( M mn ( K )), one can define a factor ring of O q ( M mn ( K )),denoted by O q ( M mn ( K )) γ , and known as the generalised quantum determinantal ring/factordetermined by γ . The generalised quantum determinantal factors of O q ( M mn ( K )) are theanalogues for O q ( M mn ( K )) of the quantum Schubert varieties in the grassmannian studiedin [8]. The term generalised quantum determinantal ring is used because special instancesof γ determine the quantum determinantal factors where all quantum minors of a givensize are set to be zero. Generalised quantum determinantal rings were shown to be integral1omains in [8, Proposition 4.3], but the question as to whether or not they are maximalorders was left open, see [8, Remark 4.6]. In this work we show that they are indeedmaximal orders. Maximal orders are the noncommutative analogues of normal varieties,or integrally closed rings.Our motivation, here, comes from noncommutative algebraic geometry. The algebrasthat we study are noncommutative analogues of coordinate rings of natural varieties arisingfrom Lie theory and we want to study them as such. This was already the point of viewin the works [6], [7] and [8], where properties of geometric nature of related algebras,expressible either in ring theoretic language (integrity, normality), or homologically (AS-Cohen-Macaulay, AS-Gorenstein properties) were studied. Let K be a field, and let q a nonzero element of K . The algebra of m × n quantum matricesover K , denoted by O q ( M mn ( K )), is the algebra generated over K by mn indeterminates x ij , with 1 ≤ i ≤ m and 1 ≤ j ≤ n , which commute with the elements of K and aresubject to the relations: x ij x il = qx il x ij , for 1 ≤ i ≤ m, and 1 ≤ j < l ≤ n ; x ij x kj = qx kj x ij , for 1 ≤ i < k ≤ m, and 1 ≤ j ≤ n ; x ij x kl = x kl x ij , for 1 ≤ k < i ≤ m, and 1 ≤ j < l ≤ n ; x ij x kl − x kl x ij = ( q − q − ) x il x kj , for 1 ≤ i < k ≤ m, and 1 ≤ j < l ≤ n. It is well-known that O q ( M mn ( K )) is an iterated skew polynomial extension of K withthe x ij added in lexicographic order. An immediate consequence is that O q ( M mn ( K )) is anoetherian domain.When m = n , the quantum determinant D q is defined by; D q := X ( − q ) l ( σ ) x σ (1) . . . x nσ ( n ) , where the sum is over all permutations σ of { , . . . , n } .The quantum determinant is a central element in the algebra O q ( M nn ( K )).Let I and J be t -element subsets of { , . . . , m } and { , . . . , n } , respectively. It is clearfrom the definitions that the subalgebra of O q ( M mn ( K )) generated by those x ij with i ∈ I and j ∈ J is isomorphic in the obvious way to O q ( M tt ( K )). Then the quantum minor [ I | J ] is defined to be the quantum determinant of this subalgebra. (Note that x ij = [ i | j ]2nd [ ∅ | ∅ ] is taken to be 1.) It is immediate that x ij [ I | J ] = [ I | J ] x ij for i ∈ I and j ∈ J ,but quantum minors do not commute with other variables. Nevertheless, several usefulcommutation relations have been developed, and we will use some of them in this article.The set of all quantum minors is denoted by Π. The set Π is equipped with the partialorder ≤ st defined in [7, Section 3.5]. Namely, if [ I | J ] and [ K | L ] are quantum minors with I = { i < · · · < i u } , J = { j < · · · < j u } , K = { k < · · · < k v } and L = { l < · · · < l v } then [ I | J ] ≤ st [ K | L ] ⇐⇒ u ≥ v,i s ≤ k s for 1 ≤ s ≤ v,j s ≤ l s for 1 ≤ s ≤ v. The algebra of quantum matrices, equipped with the partial order ≤ st defined on the setof quantum minors Π, is a quantum graded algebra with a straightening law (abbreviatedQGASL), as defined in [7], see [7, Theorem 3.5.3]. Definition 2.1.
Let γ ∈ Π and set Π γ := { α ∈ Π | α st γ } . Set I γ to be the idealgenerated by Π γ . The generalised quantum determinantal ring O q ( M mn ( K )) γ associatedto γ is the factor algebra O q ( M mn ( K )) /I γ . (We let p : O q ( M mn ( K )) −→ O q ( M mn ( K )) γ be the canonical projection.)The terminology we use arises in the following way. Let γ = [1 , . . . , t − | , . . . , t − γ consists of the s × s quantum minors with s ≥ t , and O q ( M mn ( K )) γ is the factorring obtained by setting all of the t × t quantum minors to be zero: such algebras areknown as quantum determinantal rings, see, for example, [6] Proposition 2.2.
The generalised quantum determinantal ring O q ( M mn ( K )) γ is a QGASLon the natural projection of Π \ Π γ from O q ( M mn ( K )) to O q ( M mn ( K )) γ .Proof. This follows immediately from [7, Theorem 3.5.3 and Corollary 1.2.6].Recall that an element u of a ring R is a normal element if uR = Ru and is regular ifit is a nonzerodivisor. Corollary 2.3.
The image γ of γ in O q ( M mn ( K )) γ is the unique minimal element of p (Π \ Π γ ) . Further, for each τ ≥ st γ , there exists c τ ∈ K , nonzero, such that γ τ = c τ τ γ ,where τ = p ( τ ) . Consequently, γ is a regular normal element of the generalised quantumdeterminantal ring O q ( M mn ( K )) γ .Proof. See the proof of [7, Lemma 1.2.1]. 3
Relations for a subalgebra of quantum matrices
Let γ = [ A | B ] = [ a , . . . , a t | b , . . . , b t ] be a quantum minor in O q ( M mn ( K )), and let c < c < · · · < c n − t be the column indices of O q ( M mn ( K )) that do not occur in γ and r < r < · · · < r m − t be the row indices of O q ( M mn ( K )) that do not occur in γ . We willuse this notation throughout the paper.Denote by S the t × t quantum matrix subalgebra of O q ( M mn ( K )) generated by the x a i b j . Our strategy to show that the generalised determinantal algebra determined by γ is a maximal order will be to show that it is related via localisation to an algebra T which is an iterated Ore extension. The algebra T is a subalgebra of quantum matricesgenerated by S and a family of quantum minors (explicit generators are given below). Inorder to show that T is an iterated Ore extension, we need to do two things. First, weneed to develop suitable commutation relations between the generators; this is done in thissection. Secondly, we need to show that the generators are independent enough to give apresentation as an iterated Ore extension.Let M be the set of t × t quantum minors that are ≥ st γ , and which differ from γ in precisely one entry. Let T be the subalgebra of O q ( M mn ( K )) generated over S by thequantum minors in M . To study T , we need a notation for the quantum minors in M .First, note that each such minor either has the same row set or column set as γ anddiffers from γ in the column set or row set, respectively, by exactly one element. Let R be the set of such quantum minors with the same row set as γ and C be the set of suchquantum minors with the same column set as γ .Note that a quantum minor [ A | B ⊔ { c ( n − t +1) − i }\{ b j } ] is in M precisely when b j Consider the quantum minor γ = [13 | 12] in O q ( M , ( K )). Then S = K [ x , x , x , x ], a quantum matrix subalgebra of O q ( M , ( K )), while R = { m , m } = { [13 | , [13 | } , C = { n } = { [23 | } and M = { m < m < n } = { [13 | < [13 | < [23 | } . Then, T = S [ m , m , n ] . What we are aiming to do amounts to showing, in this example, that T is (isomorphicto) a three step iterated Ore extension of S , with the “variables” m , m , n added inthis order. It then follows that T is a seven step iterated Ore extension of K , and thatGKdim( T ) = 7.For each m kl that is defined let R ( m kl ) be the subalgebra of T generated by S and the m ij that are less than or equal to m kl in the order defined above, and for each n kl that isdefined let R ( n kl ) be the subalgebra of T generated by S , all of the m ij and the n ij that areless than or equal to n kl in the order defined above. Let m − kl be the m ij that immediatelyprecedes m kl in the above order and let n − kl be the n ij that immediately precedes n kl in theabove order. Then, R ( m kl ) is generated over R ( m − kl ) by m kl and R ( n kl ) is generated over R ( n − kl ) by n kl . (If m − kl does not exist set R ( m − kl ) := S , and, similarly, if n − kl does not existthen R ( n − kl ) is generated over S by all of the m ij .)We need to know suitable commutation relations between the members of M andbetween members of M and the x ij in the quantum matrix subalgebra S .First, we check how the m ij , n ij commute with the generators of S . Lemma 3.2. Let x a k b l be a generator for S . Then(i) x a k b l commutes with m ij when l = j , while x a k b j m ij − qm ij x a k b j = b q X s There are several relations to check. We present the proof of the following claim.(The proofs for all other cases are similar, but easier.) Suppose that i < k and j < l andthat m ij and m kl are defined. Then m il and m kj are defined and m ij m kl − m kl m ij = ( q − − q ) m il m kj . Proof of claim . We know that b j < b l , and b l < c ( n − t +1) − k as m kl is defined. Thus, b j < c ( n − t +1) − k and so m kj = [ A | B ⊔ { c ( n − t +1) − k }\{ b j } ] is defined. Also, b l < c ( n − t +1) − k ,as m kl is defined, and c ( n − t +1) − k < c ( n − t +1) − i as i < k . As a consequence, b l < c ( n − t +1) − i and so m il = [ A | B ⊔ { c ( n − t +1) − i }\{ b l } ] is defined.Set A ′ = A \{ a , a } and B ′ := B \{ b j , b l } . Then m ij = [ A ′ ⊔{ a , a } | B ′ ⊔{ c ( n − t +1) − i , b l } ]and m kl = [ A ′ ⊔ { a , a } | B ′ ⊔ { c ( n − t +1) − k , b j } ].6e need a commutation rule for the quantum minors [ a a | b l c ( n − t +1) − i ] and [ a a | b j c ( n − t +1) − k ], where b j < b l < c ( n − t +1) − k < c ( n − t +1) − i . It is easy to verify that, in O q ( M , ( K )), we have [12 | | − [12 | | 24] = ( q − − q )[12 | | O q ( M mn ( K )),[ a a | b l c ( n − t +1) − i ][ a a | b j c ( n − t +1) − k ] − [ a a | b j c ( n − t +1) − k ][ a a | b l c ( n − t +1) − i ]= ( q − − q )[ a a | b j c ( n − t +1) − i ][ a a | b l c ( n − t +1) − k ] . Using the quantum Muir’s Law of extensible minors, see [8, Proposition 1.3] for example,to re-introduce A ′ and B ′ we obtain m ij m kl − m kl m ij = ( q − − q ) m il m kj , as required. O q ( M mn ( K )) The algebra T is a subalgebra of O q ( M mn ( K )). Recall that there is an action of the torus H = ( K ∗ ) m + n on O q ( M mn ( K )) defined on the generators of O q ( M mn ( K )) in the followingway: if h = ( α , . . . , α m ; β . . . , β n ) then h · x ij := α i β j x ij . The generators x ij , m ij , n ij are all eigenvectors for the action of H ; and so it is easy to check that H restricts toautomorphisms of T and the various subalgebras that we are using to build up T as apurported Ore extension. Our aim is to show that the commutation relations developed inSection 3 can be rephrased by using suitable choices of elements h ∈ H . Definition 4.1. Set h m kl := ( α , . . . , α m ; β , . . . , β n ) where (i) α s = 1 when s ∈ A , and α s = q − when s A , and (ii) β c ( n − t +1) − k = q − , β b l = q , and β s = 1 for s ∈ B \{ b l } , with β s = q − for s B ⊔ { c ( n − t +1) − k } .Also, set h n kl := ( α , . . . , α m ; β , . . . , β n ) where (i) α r ( m − t +1) − k = q − , α l = q , and α s = 1for s ∈ A \{ a l } , while α s = q − for s A ⊔ { r ( m − t +1) − k } , and (ii) β s = 1 for s ∈ B and β s = q − for s B .Let’s check the action of the h that we have just defined on relevant generators of T . Lemma 4.2. The following hold.(1) For i ∈ A , j ∈ B , then h m kl ( x ib l ) = qx ib l and h m kl ( x ij ) = x ij when j = b l .(2) For i ∈ A , j ∈ B , then h n kl ( x a l j ) = qx a l j and h n kl ( x ij ) = x ij when i = a l .(3) Suppose that ( i, j ) < ( k, l ) in lexicographic order, then:(a) h m kl ( m ij ) = m ij when i = k and j = l ; b) h m kl ( m ij ) = q − m ij when i = k and j < l or i < k and j = l .(4) Suppose that ( i, j ) < ( k, l ) in lexicographic order, then:(a) h n kl ( n ij ) = n ij when i = k and j = l ;(b) h n kl ( n ij ) = q − n ij when i = k and j < l or i < k and j = l .(5) h n kl ( m ij ) = m ij , for all n kl and m ij .Proof. We prove (1) and (3a). The proofs of all other claims are similar to one of thesetwo.(1) h m kl ( x ij ) = α i β j x ij . Now i ∈ A , so α i = 1 and h m kl ( x ij ) = β j x ij , which is equal to qx ij when j = b l and equal to 1 .x ij otherwise.(3a) Suppose that i = k and j = l . Now, m ij := [ A | B ⊔ { c ( n − t +1) − i }\{ b j } ] with b j < c ( n − t +1) − i , and m kl := [ A | B ⊔ { c ( n − t +1) − k }\{ b l } ] with b l < c ( n − t +1) − k .Let h = h m kl = ( α , . . . , α m ; β . . . , β n ). Then h m kl ( m ij )= h m kl ([ A | B ⊔ { c ( n − t +1) − i }\{ b j } ])= α a . . . α a t β b . . . c β b j . . . β b t β c ( n − t +1) − i [ A | B ⊔ { c ( n − t +1) − i }\{ b j } ]= α a . . . α a t β b . . . c β b j . . . β b t β c ( n − t +1) − i m ij Hence, we need to evaluate λ := α a . . . α a t β b . . . c β b j . . . β b t β c ( n − t +1) − i . From the defi-nition of h m kl we see that each α a i = 1. Also, for s ∈ B \{ b l } we know that β s = 1.Therefore, λ = β b l β c ( n − t +1) − i . We know that β b l = q , so it remains to evaluate β c ( n − t +1) − i .As i = k it follows that c ( n − t +1) − i = c ( n − t +1) − k so that c ( n − t +1) − i B ⊔ { c ( n − t +1) − k } and so β c ( n − t +1) − i = q − . Hence, λ = β b l β c ( n − t +1) − i = qq − = 1.The previous lemma, together with the results obtained in Section 3 are sufficient toestablish the following result. Proposition 4.3. (i) For x ij ∈ S and for any m kl that is defined, m kl x ij − h − m kl ( x ij ) m kl ∈ R ( m − kl ) (ii) For x ij ∈ S and for any n kl that is defined, n kl x ij − h − n kl ( x ij ) n kl ∈ R ( n − kl ) (iii) m kl m ij − h − m kl ( m ij ) m kl ∈ R ( m − kl ) for ( i, j ) < ( k, l ) in lexicographic order(iv) n kl m ij = h − n kl ( m ij ) n kl for all ( i, j ) and ( k, l ) (v) n kl n ij − h − n kl ( n ij ) n kl ∈ R ( n − kl ) for ( i, j ) < ( k, l ) in lexicographic order. 8e can use this proposition to show that at each stage in the construction of T we havean Ore extension. In order to do this, we need to utilise the following result concerningGelfand-Kirillov dimension of extensions of the type considered in the previous result. See[5] for standard properties of Gelfand-Kirillov dimension. Lemma 4.4. Let B be a K -algebra. Suppose A is a finitely generated subalgebra of B that is an integral domain with finite Gelfand-Kirillov dimension and that x is an elementof B such that B is generated by A and x as an algebra. Furthermore, suppose thereexists an automorphism σ of A and finite-dimensional subspace V of A that generates A as an algebra such that σ ( V ) = V . Suppose that xa − σ ( a ) x ∈ A , for each a ∈ A . Then GKdim( B ) ≤ GKdim( A ) + 1 .Also,(i) δ : A −→ A , defined by δ ( a ) := xa − σ ( a ) x , is a σ -derivation of A , and(ii) if C := A [ y ; σ, δ ] , the natural algebra morphism θ : C −→ B such that θ | A = id A and θ ( y ) = x is an isomorphism if only if GKdim( B ) = GKdim( A ) + 1 .Proof. Note that [8, Lemma 2.3] guarantees that GKdim( B ) ≤ GKdim( A ) + 1. As C is a particular example of such a B , we have GKdim( C ) ≤ GKdim( A ) + 1. However,it is well-known that GKdim( C ) ≥ GKdim( A ) + 1 (see [5, p.164]) and so GKdim( C ) =GKdim( A ) + 1.It is routine to check that δ is a σ -derivation of A . The map θ : C −→ B givenby θ ( f ( y )) := f ( x ) is an epimorphism from C to B . If θ is not an isomorphism thenGKdim( B ) ≤ GKdim( C ) − A ), by [5, Proposition 3.15], while if θ is anisomorphism then GKdim( B ) = GKdim( C ) = GKdim( A ) + 1, as required. Corollary 4.5. GKdim( T ) ≤ ( m + n + 1) t − P ti =1 ( a i + b i ) Proof. Recall that T is generated over K by the t elements x a i b j together with those t × t quantum minors that are greater than γ and differ from γ in exactly one entry. Such aminor which excludes a i is given by including a row index which is bigger than a i butnot equal to any of the other a j . There are m − a i − ( t − i ) such indices. Summing over i = 1 , . . . , t , one obtains mt − P a i − t ( t − / 2. There are also nt − P b j − t ( t − / b j , giving a total of ( m + n ) t − P ( a i + b i ) − t ( t − 1) suchquantum minors. Adding in t for the elements x a i b j produces ( m + n + 1) t − P ti =1 ( a i + b i )generators. At each stage that a new generator is introduced, we have an algebra A , anew generator x to generate an algebra B containing A and an automorphism σ with theproperty that xa − σ ( a ) x ∈ A for elements a in a generating set of A as an algebra. As σ is an automorphism, this property extends to all elements of A and so the first part9f Lemma 4.4 is applicable to establish that GKdim( B ) ≤ GKdim( A ) + 1. There are( m + n + 1) t − P ti =1 ( a i + b i ) such extensions building up T from the base field k and sothe required inequality is obtained. Let γ = [ A | B ] = [ a , . . . , a t | b , . . . , b t ] be a quantum minor in O q ( M mn ( K )) and set J γ = O q ( M mn ( K )) γ . Then γ is a regular normal element of J γ , by Corollary 2.3; and sowe can invert γ to obtain the localisation J γ [ γ − ]. Our aim is to show that this localisationis isomorphic to a localisation T [ γ − ] of the algebra T constructed in the previous section.As a consequence, J γ [ γ − ] will be a maximal order. From this we will deduce that J γ is amaximal order.There is a natural homorphism θ from T [ γ − ] to J γ [ γ − ], see below. In order to showthat θ is surjective, we need to employ quantum Laplace expansions; while in order to seethat θ is injective, we need to use Gelfand-Kirillov dimension calculations. The details arein the next few results. Lemma 5.1. The quantum minor γ is a regular normal element in the algebra T . Moreprecisely, γ q • -commutes with each of the generators of T .Proof. A variable x ij is in the generating set for T as an algebra precisely when i ∈ A and j ∈ B , in which case x ij commutes with γ .Recall that m ij = [ A | B ⊔ { c ( n − t +1) − i }\{ b j } ], with b j < c ( n − t +1) − i . A simple applica-tion of the quantum Muir’s law [8, Proposition 1.3] shows that the commutation relationbetween γ = [ A | B ] and m ij is the same as that between x a b j and x a c ( n − t +1) − i , and thisis a q -commutation as these two variables are on the same row of a quantum matrix. Asimilar remark applies to the commutation relations between γ and the n ij with the rolesof rows and columns interchanged.As a consequence of the previous lemma, we can form the localisation T [ γ − ] of T obtained by inverting the powers of γ and the canonical morphism T −→ T [ γ − ] is injective.Gelfand-Kirillov dimension behaves well with respect to this localisation, as we see below. Lemma 5.2. (i) GKdim( T [ γ − ]) = GKdim( T ) ≤ ( m + n + 1) t − P ti =1 ( a i + b i ) .(ii) GKdim( J γ [ γ − ]) = GKdim( J γ ) = ( m + n + 1) t − P ti =1 ( a i + b i ) .(iii) GKdim( T [ γ − ]) ≤ GKdim( J γ [ γ − ]) . roof. (i) Let V be the vector space generated by the generators of T (that is, the x ij , m ij and n ij ). Then the previous lemma shows that γV = V γ . Set W := V + γ − K and notethat W generates T [ γ − ] as an algebra and that W γ ⊆ T . Set Y := W γ + Kγ + K , afinite dimensional vector subspace of T . It is easy to check that W n γ n ⊆ Y n . It followsthat dim( W n ) ≤ dim( Y n ) and so GKdim( T [ γ − ]) ≤ GKdim( T ). As it is obvious thatGKdim( T [ γ − ]) ≥ GKdim( T ), equality follows. The inequality is already established inCorollary 4.5.(ii) For the first equality, a similar proof to that in (i) works, taking the generating subspace V to be generated by the image of Π in J γ , and using Corollary 2.3 instead of Lemma 5.1.For the second equality, see [8, Remark 4.2(iii)].(iii) This is immediate, from (i) and (ii).The inclusion of T in O q ( M mn ( K )) induces a natural homomorphism θ : T −→ J γ [ γ − ]which sends any quantum minor [ I | J ] to its image [ I | J ] in J γ ⊆ J γ [ γ − ]. In particular, θ ( γ ) = γ , and so we may extend θ to a homomorphism (also denoted by θ ) from T [ γ − ] to J γ [ γ − ]. Proposition 5.3. The homomorphism θ : T [ γ − ] −→ J γ [ γ − ] is an isomorphism.Proof. The algebra J γ [ γ − ] is generated over K by γ ± together with the image x rs in J γ of the mn generators x rs of O q ( M mn ( K )). Thus, in order to prove surjectivity, it is enoughto see that the x rs are all in the image of θ . This is obvious for the x rs which have r ∈ A and s ∈ B .Let x rs be a generator with ( r, s ) / ∈ A × B .Suppose first that s B . By [3, A.5. Corollary (b)(i)], for any I ⊆ { , . . . , m } and J ⊆ { , . . . , n } with | J | = | I | + 1 we have X j ∈ J ( − q ) | [1 ,j ) ∩ J | x rj [ I | J \ { j } ] = ( ( − q ) | [1 ,r ) ∩ I | [ I ⊔ { r } | J ] ( r / ∈ I )0 ( r ∈ I ) . If we set I = A and J = B ⊔ { s } then we obtain the following relation in O q ( M mn ( K )):( − q ) • x rs [ A | B ] + t X j =1 ( − q ) • x rb j [ A | B ⊔ { s }\{ b j } ] = ( ( − q ) • [ A ⊔ { r } | B ⊔ { s } ] ( r / ∈ A )0 ( r ∈ A ) . 11o start with, suppose in addition that r ∈ A . Let us look at the image in J γ of the aboverelation. The terms x rb j [ A | B ⊔ { s }\{ b j } ] where s < b j are sent to zero because they arenot greater than or equal to γ . In addition, the image in J γ of the remaining such terms isin the image of θ . It follows from this that, in this case, x rs is in the image of θ . Of course,by a similar argument, exchanging rows and columns, we get that x rs is in the image of θ whenever r / ∈ A and s ∈ B .It remains to deal with the case where r / ∈ A and s / ∈ B . In that case, we have therelation ( − q ) • x rs [ A | B ] + P tj =1 ( − q ) • x rb j [ A | B ⊔ { s }\{ b j } ] = ( − q ) • [ A ⊔ { r } | B ⊔ { s } ] in O q ( M mn ( K )). In this relation, the right hand term is not greater than or equal to γ sinceit is a ( t + 1) × ( t + 1) minor. So, taking the image of this latter relation in J γ , we get, bythe same argument as above, that x rs is in the image of θ since, as we have just proved, x rb j is in the image of θ .This finishes the proof that θ is a surjective map.Hence, GKdim( T [ γ − ]) ≥ GKdim( θ ( T [ γ − ])) = GKdim( J γ [ γ − ]) = GKdim( J γ ). To-gether with Lemma 5.2(iii) this gives GKdim( T [ γ − ]) = GKdim( J γ [ γ − ]).Suppose now that θ is not injective. Then ker( θ ) is a nonzero ideal in the noetheriandomain T [ γ ± ]. Hence, GKdim( θ ( T [ γ ± ])) < GKdim( T [ γ ± ]), by [5, Proposition 3.15].However, this contradicts the fact that these two dimensions are equal, as observed in theprevious paragraph. Thus, θ is injective and so θ is an isomorphism. Corollary 5.4. GKdim( T ) = ( m + n + 1) t − P ti =1 ( a i + b i ) .Proof. This follows immediately from Lemma 5.2. Corollary 5.5. The algebra T is an iterated Ore extension.Proof. The algebra T is constructed from K by adding in the ( m + n + 1) t − P ti =1 ( a i + b i )generators one-by-one. At each stage, the Gelfand-Kirillov dimension can increase by atmost one, by Lemma 4.4, and so must increase by exactly one, as GKdim( T ) = ( m + n +1) t − P ti =1 ( a i + b i ). Thus, each stage is an Ore extension, by Lemma 4.4. Remark 5.6. Corollary 5.5 extends Lemma 6.4 of [1], which asserts that in the commu-tative case (that is when q = 1), O ( M mn ( K )) γ is a localisation of a polynomial ring in( m + n + 1) t − P ti =1 ( a i + b i ) indeterminates. Proposition 5.7. The QGASL O q ( M mn ( K )) γ is an integral domain.Proof. The algebra T [ γ − ] is an integral domain as it is a localisation of T which is asubalgebra of the domain O q ( M mn ( K )). As a consequence, the isomorphism T [ γ − ] ∼ =12 q ( M mn ( K )) γ [ γ − ] of Proposition 5.3 shows that O q ( M mn ( K )) γ [ γ − ] is an integral do-main. As γ is a regular normal element of O q ( M mn ( K )) γ by Corollary 2.3, the naturalmap O q ( M mn ( K )) γ −→ O q ( M mn ( K )) γ [ γ − ] is a monomorphism, and so O q ( M mn ( K )) γ isalso an integral domain. Remark 5.8. The previous result applies to all generalised quantum determinantal rings O q ( M mn ( K )) τ , for any τ ∈ Π. In particular, it applies to the upper neighbours of γ whichare the elements τ ∈ Π \ Π γ with the property that if σ ∈ Π with γ < st σ ≤ st τ then σ = τ .This makes available [8, Proposition 2.2.2] which we use in the proof of our main theorembelow. Theorem 5.9. The generalised quantum determinantal ring O q ( M mn ( K )) γ is a maximalorder.Proof. The algebra T [ γ ± ]) is a localisation of an iterated Ore extension, and so is amaximal order, by [9, V. Proposition 2.5, IV. Proposition 2.1]. Thus, O q ( M mn ( K )) γ [ γ − ] isa maximal order, by the isomorphism established in Proposition 5.3. Hence, [8, Proposition2.2.2] applies to the quantum graded algebra with a straightening law O q ( M mn ( K )) γ (whoseunderlying poset has the single minimal element γ ); so we conclude that O q ( M mn ( K )) γ isa maximal order. Remark 5.10. As pointed out in the introduction, our motivation in the present work isto complete the study, from the point of view of noncommutative algebraic geometry, ofgeneralised quantum determinantal rings. Here is a summary of the results.Let γ ∈ Π. Then, O q ( M mn ( K )) γ is an integral domain and a maximal order in itsdivision ring of fractions, as established in [8] and the present work.Further, O q ( M mn ( K )) γ is AS-Cohen-Macaulay, and it is AS-Gorenstein for any nonzero q in K if and only if it is AS-Gorenstein for q = 1. All this can be shown following thearguments developed in paragraph 4 of [7] (see in particular Theorems 4.2 and 4.3). Noticein addition that necessary and sufficient conditions for O ( M mn ( K )) to be AS-Gorensteinare given in [1, Theorem 8.14]. References [1] W Bruns and U Vetter, Determinantal rings. 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