Computation of point modules of finitely semi-graded rings
aa r X i v : . [ m a t h . R A ] A ug Computation of point modulesof finitely semi-graded rings
Oswaldo Lezama [email protected]
Seminario de ´Algebra Constructiva - SAC Departamento de Matem´aticasUniversidad Nacional de Colombia, Sede Bogot´a
Abstract
In this paper we compute the set of point modules of finitely semi-graded rings. In particular, fromthe parametrization of the point modules for the quantum affine n -space, the set of point modulesfor some important examples of non N -graded quantum algebras is computed. Key words and phrases.
Point modules, point functor, Zariski topology, strongly noetherian algebras,finitely semi-graded rings, skew
P BW extensions.2010
Mathematics Subject Classification.
Primary: 16S38. Secondary: 16W50, 16S80, 16S36.
In algebraic geometry a key role play the point modules and its parametrization. The point modules ofalgebras, and the spaces parameterizing them, were first studied by Artin, Tate and Van den Bergh ([2]) inorder to complete the classification of Artin-Schelter regular algebras of dimension 3. Many commutativeand non-commutative graded algebras have nice parameter spaces of point modules. For example, let K be a field and A = K [ x , . . . , x n ] be the commutative K -algebra of usual polynomials, then there existsa bijective correspondence between the projective space P n and the set of isomorphism classes of pointmodules of A ; this correspondence generalizes to any finitely graded commutative algebra generated indegree 1. Similarly, the point modules of the quantum plane and the Jordan plane are parametrized by P . The study of the point modules and its parametrization has been concentrated in finitely gradedalgebras (see [2], [4], [6], [7], [15], [16]), in the present paper we are interested in the computation ofpoint modules for algebras that are not necessarily N -graded, examples of such algebras are the quantumalgebra U ′ ( so (3 , K )), the dispin algebra U ( osp (1 , W ν ( sl (2 , K )), among manyothers. In [11] were introduced the semi-graded rings and for them was proved a classical theorem ofnon-commutative algebraic geometry, the Serre-Artin-Zhang-Verevkin theorem about non-commutativeprojective schemes (see also [8]). Thus, the semi-graded rings are nice objects for the investigation ofgeometric properties of non-commutative algebras that are not N -graded, and include as a particularcase the class of finitely graded algebras. In [10] was initiated the study the point modules of finitelysemi-graded rings, in the present paper we complete this investigation and we will compute the pointmodules of finitely semi-graded rings generated in degree one. In particular, from the parametrization ofthe point modules for the quantum affine n -space, the set of point modules for some important examplesof non N -graded quantum algebras is computedThe paper is organized as follows: In the first section we recall some basic facts and examples onpoint modules for classical N -graded algebras, we include a complete proof (usually not available in the1iterature) of the parametrization of the point modules for the quantum affine n -space (Example 1.7); wereview the definition and elementary properties of semi-graded rings, in particular, the subclass of finitelysemi-graded rings and modules. As was pointed out above, the finitely semi-graded rings generalize thefinitely graded algebras, so in order present sufficient examples of finitely semi-graded rings not beingnecessarily finitely graded algebras, we recall in the last part of the first section the notion of skew P BW extension defined firstly in [9]. Skew
P BW extensions represent a way of describing many importantnon-commutative algebras not necessarily N -graded, remarkable examples are the operator algebras,algebras of diffusion type, quadratic algebras in 3 variables, the quantum algebra U ′ ( so (3 , K )), the dispinalgebra U ( osp (1 , W ν ( sl (2 , K )), among many others. The second and thethird sections contain the novelty of the paper. The main results are Theorem 2.7, which completes theparametrization started in [10], and Theorem 3.1 where we apply the results of the previous sectionsto compute the point modules of some skew P BW extensions, and hence, the point modules of someexamples of non N -graded quantum algebras. Concrete examples are presented in Examples 3.2 and 3.3. In this first subsection we recall some key facts and examples on point modules for classical N -gradedalgebras that we will use later in the paper (see [2], [4] and [15]). Let K be a field, a K -algebra A is finitely graded if the following conditions hold: (i) A is N -graded, A = L n ≥ A n . (ii) A is connected, i.e., A = K . (iii) A is finitely generated as K -algebra. Let A be a finitely graded K -algebra that is generatedin degree 1. A point module for A is a graded left module M such that M is cyclic, generated in degree0, i.e., there exists an element m ∈ M such that M = Am , and dim K ( M n ) = 1 for all n ≥ A is N -graded and m ∈ M , then M is necessarily N -graded). The collection of isomorphic classes of pointmodules for A is denoted by P ( A ).The following examples show the description (parametrization) of point modules for some well-knownfinitely graded algebras. Example 1.1 ([15]) . Let K be a field and A = K [ x , . . . , x n ] be the commutative K -algebra of usualpolynomials. Then,(i) There exists a bijective correspondence between the projective space P n and the set of isomorphismclasses of point modules of A , P n ←→ P ( A ).(ii) The correspondence in ( i ) generalizes to any finitely graded commutative algebra which is generatedin degree 1. Example 1.2 ([15]) . (i) The point modules of the quantum plane are parametrized by P , i.e., thereexists a bijective correspondence between the projective space P and the collection of isomorphism classesof point modules of the quantum plane A := K { x, y } / h yx − qxy i : P ←→ P ( A ).(ii) For the Jordan plane, A := K { x, y } / h yx − xy − x i , can be proved a similar equivalence: P ←→ P ( A ). Example 1.3 ([15]) . The isomorphism classes of point modules for the free algebra A := K { x , . . . , x n } are in bijective correspondence with N -indexed sequences of points in P n , { ( λ ,i : · · · : λ n,i ) ∈ P n | i ≥ } ,in other words, with the points of the infinite product P n × P n × · · · = Q ∞ i =0 P n (note that an element of Q ∞ i =0 P n is a sequence of the form (( λ ,i : · · · : λ n,i )) i ≥ ). Thus, there exists a bijective correspondencebetween Q ∞ i =0 P n and P ( K { x , . . . , x n } ): Q ∞ i =0 P n ←→ P ( A ).2sing the previous example it is possible to compute the collection of point modules for any finitelypresented algebra. For this we need a definition first, see [15]. Suppose that f ∈ K { x , . . . , x n } is ahomogeneous element of degree m (assuming that deg( x v ) := 1, for all v ). Consider the polynomialring B = K [ y v,u ] in a set of ( n + 1) m commuting variables { y v,u | ≤ v ≤ n, ≤ u ≤ m } . The multilinearization of f is the element of B given by replacing each word w = x v m · · · x v x v occurring in f by y v m ,m · · · y v , y v , (= y v , y v , · · · y v m ,m ). Proposition 1.4 ([15], Proposition 3.5) . Let A = K { x , . . . , x n } / h f , . . . , f r i be a finitely presented K -algebra, where the f l are homogeneous of degree d l ≥ , ≤ l ≤ r . For each f l , let g l be themultilinearization of f l . Then, (i) The isomorphism classes of point modules for A are in bijection with the closed subset X of Q ∞ i =0 P n defined by X := { ( p , p , . . . ) | g l ( p i , p i +1 , . . . , p i + d l − ) = 0 for all ≤ l ≤ r, i ≥ } . (ii) Consider for each m ≥ the closed subset X m := { ( p , p , . . . , p m − ) | g l ( p i , p i +1 , . . . , p i + d l − ) = 0 for all ≤ l ≤ r, ≤ i ≤ m − d l } of Q m − i =0 P n . The canonical projection onto the first m coordinates defines a function φ m : X m +1 → X m .Then X is equal to the inverse limit lim ←− X m of the X m with respect to the functions φ m . In particular,if m is such that φ m is a bijection for all m ≥ m , then the isomorphism classes of point modules of A are in bijective correspondence with the points of X m . For strongly Noetherian algebras there is m such that the functions φ m : X m +1 → X m are bijectivefor all m ≥ m . These algebras were studied by Artin, Small and Zhang in [1], and appears naturallyin the study of point modules in non-commutative algebraic geometry (see [2], [3] and [15]). Let K bea field and let A be a left Noetherian K -algebra, it is said that A is left strongly Noetherian if for anycommutative Noetherian K -algebra C , C ⊗ K A is left Noetherian. Corollary 1.5 ([4], Corollary E4.12) . Let A be a finitely graded strongly noetherian algebra, with presen-tation as in Proposition 1.4. Then there is m such that φ m : X m +1 → X m is bijective for all m ≥ m ,and the point modules of A are in bijective correspondence with the points of X m . Example 1.6.
It is known (see Example 3.8 in [15] and also [7]) that for the multi-parameter quantumaffine 3-space, A := K q [ x , x , x ], defined by x x = q x x , x x = q x x , x x = q x x , m = 2 and X = { ( p , σ ( p )) | p ∈ E } ,where E = ( P if q q = q , { ( x : y : z ) ∈ P | x y z = 0 } if q q = q , (1.1)with σ : E → E bijective. Thus, for A we have the bijective correspondence P ( A ) ←→ X ←→ E . Wewill prove this including all details omitted in the above cited references. Following Proposition 1.4, inthis case we have 3 variables, r = 3 and every generator of A has degree 2. We have to show that thedescription of X is as above and for all m ≥ φ m : X m +1 → X m is bijective.Recall that X = { ( p , p ) ∈ P × P | g l ( p , p ) = 0 for 1 ≤ l ≤ } ,3here p := ( x : y : z ), p := ( x : y : z ) and g , g , g are the multilinearization of the definingrelations of A , i.e., g ( p , p ) = y x − q x y , g ( p , p ) = z x − q x z , g ( p , p ) = z y − q y z .Thus, the relations that define A can be written as y x − q x y = 0, z x − q x z = 0 and z y − q y z = 0,and in a matrix form we have y − q x z − q x z − q y x y z = 0 . (1.2)In order to compute X , let E be the projection of X onto the first copy of P . Let p ∈ P , then p ∈ E if and only if (1.2) has a non trivial solution p if and only if the determinant of the matrix F of thissystem is equal 0, i.e., x y z ( q q − q ) = 0. In addition, observe that rank( F ) = 2. In fact, it isclear that rank( F ) = 0 (contrary, x = y = z = 0, false). Suppose that rank( F ) = 1, and assume forexample that the K -basis of F is the first column, so 0 = z = y = x , a contradiction; in a similarway we get a contradiction assuming that the K -basis of F is the second or the third column. Thus,dim K (ker( F )) = 1 and given p = ( x : y : z ) ∈ E there exists a unique p = ( x : y : z ) ∈ P thatsatisfies (1.2), and hence, we define a function σ : E → P , σ ( p ) := p . (1.3)Therefore, X = { ( p , σ ( p )) | p ∈ E } . σ is injective: For this, note that the defining relations of A can be written also in the following matrixway: − q y x − q z x − q z y x y z = 0;let G be the matrix of this system; as above, rank( G ) = 2 and dim K (ker( G )) = 1, thus, if σ ( p ) = p = σ ( p ′ ), then p = p ′ .From the determinant of F arise two cases. Case 1 : q q = q . In this case every point p = ( x : y : z ) of P is in E , i.e., E = P . Let σ : P → P be the function in (1.3), we have to show that σ is surjective. Since in this case p ∈ E if and only if p ∈ E , we define θ : P → P by θ ( p ) := p with Gp = 0. Observe that σ θ = i P :Indeed, σ ( θ ( p )) = σ ( p ) = p ′ with F p ′ = 0, but Gp = 0 is equivalent to F p = 0, and sincedim K (ker( F )) = 1, then p ′ = p . Case 2 : q q = q . In this case p = ( x : y : z ) ∈ E if and only if x y z = 0, i.e., E = { ( x : y : z ) ∈ P | x y z = 0 } . We denote by σ the function in (1.3). We know that σ isinjective; observe that σ ( E ) ⊆ E : Indeed, let p ∈ E , then σ ( p ) = σ ( p ) = p , but since q q = q and det( G ) = 0 (contrary, x = y = z = 0, false), then x y z = 0, i.e., p ∈ E . Only rest toshow that σ : E → E is surjective. As in the case 1, we define θ : E → E by θ ( p ) := p with Gp = 0, then σ ( θ ( p )) = σ ( p ) = p ′ , with F p ′ = 0, but Gp = 0 is equivalent to F p = 0, and sincedim K (ker( F )) = 1, then p ′ = p .Thus, in both cases we have proved that X = { ( p , σ ( p )) | p ∈ E } ←→ E .To complete the example, we will show that X ←→ X . According to Proposition 1.4, we have to provethat for every m ≥ φ m : X m +1 → X m is a bijection. For m = 2, φ ( p , p , p ) = ( p , p ) ∈ X , so p = σ ( p ), note that p = σ ( p ) since by the definition of X , g l ( p , p ) = 0 for 1 ≤ l ≤
3, so by theproved above ( σ ( p ) , σ ( p )) ∈ X . Thus, X = { ( p , σ ( p ) , σ ( p )) | p ∈ E } and φ is bijective. Byinduction, assume that 4 m = { ( p , σ ( p ) , . . . , σ m − ( p )) | p ∈ E } and X m ←→ X m − ,so X m +1 = { ( p , σ ( p ) , . . . , σ m ( p )) | p ∈ E } since ( σ m − ( p ) , σ m ( p )) ∈ X , and hence X m +1 ←→ X m . Example 1.7.
We will show next that the ideas and results of Example 1.6 can be extended to themulti-parameter quantum affine n -space A := K q [ x , . . . , x n ], with n ≥ m = 2 and X is as in (1.4) below.Recall that in A we have the defining relations x j x i = q ij x i x j , 1 ≤ i < j ≤ n .We have n variables, r = n ( n − generators, every generator of A has degree 2 and X = { ( p , p ) ∈ P n − × P n − | g l ( p , p ) = 0 for 1 ≤ l ≤ r } ,where p := ( x : · · · : x n ), p := ( x : · · · : x n ) and the g l are the multilinearization of the definingrelations of A . Thus, the relations that define A can be written as x j x i − q ij x i x j = 0, 1 ≤ i < j ≤ n ,and in a matrix form we have x − q x · · · x − q x · · · x − q x · · · x n · · · − q n x x − q x · · · x q x · · · x q x · · · x n · · · − q n x ... ... ... ... ... ... ...0 0 0 0 0 · · · x n − q n − ,n x n − x x ......... x n − x n = 0 . The size of the matrix F of this system is n ( n − × n . In order to compute X , let E be the projectionof X onto the first copy of P n − . Let p ∈ P n − , then p ∈ E if and only if the previous system hasa non trivial solution p . This last condition is equivalent to rank( F ) = n −
1: Indeed, it is clear thatrank( F ) ≤ n , but rank( F ) = n (contrary, dim K (ker( F )) = 0 and hence x = · · · x n = 0, false); thus,rank( F ) ≤ n −
1, but there exists i ∈ { , . . . , n } such that x i = 0 and the following n − F arelinearly independent (actually, this is true for every x i = 0): (cid:2) · · · x i · · · − q li x i · · · (cid:3) and (cid:2) · · · · · · x k · · · − q ik x i · · · (cid:3) for all 1 ≤ l < i < k ≤ n , where x i is in the l -position, − q li x i is in the i -position, x k is in the i -positionand − q ik x i is in the k -position. Hence, rank( F ) = n −
1. Conversely, if rank( F ) = n −
1, then the systemhas non trivial solution.Thus, p ∈ E if and only if rank( F ) = n − K (ker( F )) = 1, and hence, given p ∈ E there exists a unique p ∈ P n − that satisfies the above matrix system, so we define a function σ : E → P n − , σ ( p ) := p . Therefore, X = { ( p , σ ( p )) | p ∈ E } . Rewriting the defining relations of A as we did in Example 1.6,we conclude that σ is injective. Moreover, Im ( σ ) = E : In fact, recall that rank( F ) = n − F of size n is equal 0; let F be the set of minors of size n of the matrix F , then E is theprojective variety E = \ f ∈F V ( f ) = V ( I F ) , (1.4)where I F is the ideal of K [ x , . . . , x n ] = K [ x , . . . , x n ] generated by all f ∈ F . Hence, from the rewritingof relations we conclude also that p = σ ( p ) ∈ E and, as in Example 1.6, we define a function θ : E → E such that σθ = i E .Finally, X = { ( p , σ ( p )) | p ∈ E } ←→ E and the proof of X ←→ X is exactly as in the final partof Example 1.6. Remark 1.8. (i) For n = 3, F = { x x x ( q q − q ) } and hence (1.4) extends (1.1). For n = 2, A = K q [ x , x ] is the quantum plane and from Example 1.2 we know that P ( A ) ←→ P , but observethat (1.4) also cover this case since F = ∅ and hence E = V (0) = P .(ii) For the affine algebra A = K [ x , . . . , x n ] all constants q ij are trivial, i.e, q ij = 1 and the generatorsof I F in (1.4) are null, so I F = 0 and E = P n − . This result agrees with Example 1.1.(iii) It is important to recall that there exist quotients of finitely graded algebras A for which P ( A ) = ∅ .In fact, some quotient algebras of the multi-parameter quantum affine n -space have empty set of pointmodules, see [5] and [18]. In this preliminary subsection we recall the definition of semi-graded rings and modules introduced firstlyin [11]. Let B be a ring. We say that B is semi-graded ( SG ) if there exists a collection { B n } n ≥ ofsubgroups B n of the additive group B + such that the following conditions hold:(i) B = L n ≥ B n .(ii) For every m, n ≥ B m B n ⊆ B ⊕ · · · ⊕ B m + n .(iii) 1 ∈ B .The collection { B n } n ≥ is called a semi-graduation of B and we say that the elements of B n are homoge-neous of degree n . Let B and C be semi-graded rings and let f : B → C be a ring homomorphism, we saythat f is homogeneous if f ( B n ) ⊆ C n for every n ≥
0. Let B be a SG ring and let M be a B -module. Wesay that M is a Z -semi-graded, or simply semi-graded, if there exists a collection { M n } n ∈ Z of subgroups M n of the additive group M + such that the following conditions hold:(a) M = L n ∈ Z M n .(b) For every m ≥ n ∈ Z , B m M n ⊆ L k ≤ m + n M k .The collection { M n } n ∈ Z is called a semi-graduation of M and we say that the elements of M n arehomogeneous of degree n . We say that M is positively semi-graded, also called N -semi-graded, if M n = 0for every n <
0. Let f : M → N be an homomorphism of B -modules, where M and N are semi-graded B -modules; we say that f is homogeneous if f ( M n ) ⊆ N n for every n ∈ Z .An important class of semi-graded rings that includes finitely graded algebras is the following (see[11]). Let B be a ring. We say that B is finitely semi-graded ( F SG ) if B satisfies the following conditions:(1) B is SG .(2) There exists finitely many elements x , . . . , x n ∈ B such that the subring generated by B and x , . . . , x n coincides with B .(3) For every n ≥ B n is a free B -module of finite dimension.6oreover, if M is a B -module, we say that M is finitely semi-graded if M is semi-graded, finitelygenerated, and for every n ∈ Z , M n is a free B -module of finite dimension. Remark 1.9. (i) It is clear that any N -graded ring is SG .(ii) Any finitely graded algebra is a F SG ring.From the definitions above we get the following elementary facts.
Proposition 1.10 ([11]) . Let B = L n ≥ B n be a SG ring. Then, (i) B is a subring of B . Moreover, for any n ≥ , B ⊕ · · · ⊕ B n is a B − B -bimodule, as well as B . (ii) B has a standard N -filtration given by F n ( B ) := B ⊕ · · · ⊕ B n . (1.5)(iii) The associated graded ring Gr ( B ) satisfies Gr ( B ) n ∼ = B n , for every n ≥ isomorphism of abelian groups ) . (iv) Let M = L n ∈ Z M n be a semi-graded B -module and N a submodule of M . The following conditionsare equivalent: (a) N is semi-graded. (b) For every z ∈ N , the homogeneous components of z are in N . (c) M/N is semi-graded with semi-graduation given by ( M/N ) n := ( M n + N ) /N , n ∈ Z . Remark 1.11. (i) If B is a F SG ring, then for every n ≥ Gr ( B ) n ∼ = B n as B -modules.(ii) Observe if B is F SG ring, then B B p = B p for every p ≥
0, and if M is finitely semi-graded, then B M n = M n for all n ∈ Z . P BW extensions
In order to present enough examples of
F SG rings not being necessarily finitely graded algebras, we recallin this subsection the notion of skew
P BW extension defined firstly in [9].
Definition 1.12 ([9]) . Let R and A be rings. We say that A is a skew P BW extension of R ( also calleda σ − P BW extension of R ) , if the following conditions hold: (i) R ⊆ A . (ii) There exist finitely many elements x , . . . , x n ∈ A such A is a left R -free module with basis Mon( A ) := { x α = x α · · · x α n n | α = ( α , . . . , α n ) ∈ N n } , with N := { , , , . . . } .The set M on ( A ) is called the set of standard monomials of A . (iii) For every ≤ i ≤ n and r ∈ R − { } , there exists c i,r ∈ R − { } such that x i r − c i,r x i ∈ R. (1.6)7iv) For every ≤ i, j ≤ n , there exists c i,j ∈ R − { } such that x j x i − c i,j x i x j ∈ R + Rx + · · · + Rx n . (1.7) Under these conditions we will write A := σ ( R ) h x , . . . , x n i . Associated to a skew
P BW extension A = σ ( R ) h x , . . . , x n i , there are n injective endomorphisms σ , . . . , σ n of R and σ i -derivations δ i , as the following proposition shows. Proposition 1.13 ([9]) . Let A be a skew P BW extension of R . Then, for every ≤ i ≤ n , there existsan injective ring endomorphism σ i : R → R and a σ i -derivation δ i : R → R such that x i r = σ i ( r ) x i + δ i ( r ) ,for each r ∈ R . Some particular cases of skew
P BW extensions are the following.
Definition 1.14 ([9]) . Let A be a skew P BW extension. (a) A is quasi-commutative if the conditions ( iii ) and ( iv ) in Definition 1.12 are replaced by (iii’) For every ≤ i ≤ n and r ∈ R − { } there exists c i,r ∈ R − { } such that x i r = c i,r x i . (1.8)(iv’) For every ≤ i, j ≤ n there exists c i,j ∈ R − { } such that x j x i = c i,j x i x j . (1.9)(b) A is bijective if σ i is bijective for every ≤ i ≤ n and c i,j is invertible for any ≤ i < j ≤ n . Definition 1.15.
Let A be a skew P BW extension of R with endomorphisms σ i , ≤ i ≤ n , as inProposition 1.13. (i) For α = ( α , . . . , α n ) ∈ N n , σ α := σ α · · · σ α n n , | α | := α + · · · + α n . If β = ( β , . . . , β n ) ∈ N n , then α + β := ( α + β , . . . , α n + β n ) . (ii) For X = x α ∈ M on ( A ) , exp( X ) := α and deg( X ) := | α | . (iii) Let = f ∈ A , t ( f ) is the finite set of terms that conform f , i.e., if f = c X + · · · + c t X t , with X i ∈ M on ( A ) and c i ∈ R − { } , then t ( f ) := { c X , . . . , c t X t } . (iv) Let f be as in (iii) , then deg( f ) := max { deg( X i ) } ti =1 . The next theorems establish some results for skew
P BW extensions that we will use later, for theirproofs see [12], [11] and [13].
Theorem 1.16 ([12]) . Let A be an arbitrary skew P BW extension of the ring R . Then, A is a filteredring with filtration given by F m := ( R, if m = 0 { f ∈ A | deg ( f ) ≤ m } , if m ≥ and the corresponding graded ring Gr ( A ) is a quasi-commutative skew P BW extension of R . Moreover,if A is bijective, then Gr ( A ) is quasi-commutative bijective skew P BW extension of R . heorem 1.17 ([12]) . Let A be a quasi-commutative skew P BW extension of a ring R . Then,1. A is isomorphic to an iterated skew polynomial ring of endomorphism type, i.e., A ∼ = R [ z ; θ ] . . . [ z n ; θ n ] .
2. If A is bijective, then each endomorphism θ i is bijective, ≤ i ≤ n . Theorem 1.18 (Hilbert Basis Theorem, [12]) . Let A be a bijective skew P BW extension of R . If R isa left (right) Noetherian ring then A is also a left (right) Noetherian ring. Theorem 1.19 ([13]) . Let K be a field and let A = σ ( R ) h x , . . . , x n i be a bijective skew P BW extensionof a left strongly Noetherian K -algebra R . Then A is left strongly Noetherian. Theorem 1.20 ([11]) . Any skew
P BW extension A = σ ( R ) h x , . . . , x n i is a F SG ring with semi-graduation A = L k ≥ A k , where A k := R h x α ∈ M on ( A ) | deg( x α ) = k i . Example 1.21.
Many important algebras and rings coming from mathematical physics are particularexamples of skew
P BW extensions, see [12] and [14]: Habitual ring of polynomials in several variables,Weyl algebras, enveloping algebras of finite dimensional Lie algebras, algebra of q -differential operators,many important types of Ore algebras, algebras of diffusion type, additive and multiplicative analoguesof the Weyl algebra, dispin algebra U ( osp (1 , U ′ ( so (3 , K )), Woronowicz algebra W ν ( sl (2 , K )), Manin algebra O q ( M ( K )), coordinate algebra of the quantum group SL q (2), q -Heisenbergalgebra H n ( q ), Hayashi algebra W q ( J ), differential operators on a quantum space D q ( S q ), Witten’s defor-mation of U ( sl (2 , K )), multiparameter Weyl algebra A Q, Γ n ( K ), quantum symplectic space O q ( sp ( K n )),some quadratic algebras in 3 variables, some 3-dimensional skew polynomial algebras, among many others.For a precise definition of any of these rings and algebras see [12] and [14]. F SG rings
In this section we complete the parametrization of point modules of
F SG rings that was initiated in [10].The main result of the present section is Theorem 2.7 below. For completeness, we include the basic factsof parametrization studied in [10], omitting the proofs.
Definition 2.1.
Let B = L n ≥ B n be a F SG ring that is generated in degree . (i) A point module for B is a finitely N -semi-graded B -module M = L n ∈ N M n such that M is cyclic,generated in degree , i.e., there exists an element m ∈ M such that M = Bm , and dim B ( M n ) =1 for all n ≥ . (ii) Two point modules M and M ′ for B are isomorphic if there exists a homogeneous B -isomorphismbetween them. (iii) P ( B ) is the collection of isomorphism classes of point modules for B . The following result is the first step in the construction of the geometric structure for P ( B ). Theorem 2.2.
Let B = L n ≥ B n be a F SG ring generated in degree . Then, P ( B ) has a Zariskitopology generated by finite unions of sets V ( J ) defined by V ( J ) := { M ∈ P ( B ) | Ann ( M ) ⊇ J } , where J ranges the semi-graded left ideals of B . roof. See [10], Theorem 9.
Definition 2.3.
Let B = L n ≥ B n be a F SG ring generated in degree such that B is commutativeand B is a B -algebra. Let S be a commutative B -algebra. A S -point module for B is a N -semi-graded S ⊗ B B -module M which is cyclic, generated in degree , M n is a locally free S -module with rank S ( M n ) = 1 for all n ≥ , and M = S . P ( B ; S ) will denote the set of S -point modules for B . Theorem 2.4.
Let B = L n ≥ B n be a F SG ring generated in degree such that B is commutativeand B is a B -algebra. Let B be the category of commutative B -algebras and let S et be the category ofsets. Then, P defined by B P −→ S etS P ( B ; S ) S → T P ( B ; S ) → P ( B ; T ) , given by M T ⊗ S M is a covariant functor called the point functor for B .Proof. See [10], Theorem 10.
Definition 2.5.
Let B = L n ≥ B n be a F SG ring generated in degree such that B is commutativeand B is a B -algebra. We say that a B -scheme X parametrizes the point modules of B if the pointfunctor P is naturally isomorphic to h X . Taking S = B we get that P ( B ) ⊆ P ( B ; B ). If B = K is a field, then clearly P ( B ) = P ( B ; K ). Theorem 2.6.
Let B = L n ≥ B n be a F SG ring generated in degree such that B = K is a fieldand B is a K -algebra. Let X be a K -scheme that parametrizes P ( B ) . Then, there exists a bijectivecorrespondence between the closed points of X and P ( B ) .Proof. See [10], Theorem 11.
Theorem 2.7.
Let B = L n ≥ B n be a F SG ring generated in degree such that B = K is a field and B is a K -algebra. Then, (i) There is an injective function ( P ( B ) / ∼ ) α −→ P ( Gr ( B )) ,where ∼ is the relation in P ( B ) defined by [ M ] ∼ [ M ′ ] ⇔ Gr ( M ) ∼ = Gr ( M ′ ) ,with [ M ] the class of point modules isomorphic to the point module M . (ii) Let X be a K -scheme that parametrizes P ( Gr ( B )) . Then there is an injective function from ( P ( B ) / ∼ ) to the closed points of X : ( P ( B ) / ∼ ) e α −→ X .Proof. (i) We divide the proof of this part in three steps. Step 1 . Note that Gr ( B ) is a finitely graded K -algebra generated in degree 1: According to Proposition1.10, Gr ( B ) is N -graded with Gr ( B ) n ∼ = B n for all n ≥ K -vector spaces), in particular, Gr ( B ) = K ; if x , . . . , x m ∈ B generate B as K -algebra, then x , . . . , x m ∈ Gr ( B ) generate Gr ( B ) as K -algebra. Step 2 . Let [ M ] ∈ P ( B ), with M = L n ≥ M n , M = Bm , with m ∈ M , and dim K ( M n ) = 1 for all n ≥
0. As in Proposition 1.10, we define 10 r ( M ) := L n ≥ Gr ( M ) n , Gr ( M ) n := M ⊕···⊕ M n M ⊕···⊕ M n − ∼ = M n (isomorphism of K -vector spaces).The structure of Gr ( B )-module for Gr ( M ) is given by bilinearity and the product b p · m n := b p m n ∈ Gr ( M ) p + n .This product is well-defined: If b p = c p , then b p − c p ∈ B ⊕ · · · ⊕ B p − and hence ( b p − c p ) m n ∈ M ⊕ · · · ⊕ M p − n , so in Gr ( M ) p + n we have ( b p − c p ) m n = 0, i.e., b p m n = c p m n ; now, if m n = l n ,then m n − l n ∈ M ⊕ · · · ⊕ M n − , whence b p ( m n − l n ) ∈ M ⊕ · · · ⊕ M p + n − , so in Gr ( M ) p + n we have b p ( m n − l n ) = 0, i.e., b p m n = b p l n . The associative law for this product holds: b q · ( b p · m n ) = b q · ( b p m n ) = b q ( b p m n ) = ( b q b p ) m n = ( c + · · · + c p + q ) m n = c p + q m n = ( b q b p ) · m n .Finally, 1 · m n = m n .Observe that Gr ( M ) = Gr ( B ) · m : Let m n ∈ Gr ( M ) n , with m n ∈ M n , we have m n = bm =( b + · · · + b n ) m = b m + · · · + b n − m + b n m = b n m = b n · m . Step 3 . We define P ( B ) α ′ −→ P ( Gr ( B ))[ M ] [ Gr ( M )]It is clear that α ′ is well-defined, i.e., if M ∼ = M ′ , then Gr ( M ) ∼ = Gr ( M ′ ).Note that the relation ∼ defined in the statement of the theorem is an equivalence relation, let [[ M ]]be the class of [ M ] ∈ P ( B ), then we define( P ( B ) / ∼ ) α −→ P ( Gr ( B ))[[ M ]] α ′ ([ M ]) = [ Gr ( M )] . It is clear that α is a well-defined injective function.(ii) This follows from (i) and applying Theorem 2.6 to Gr ( B ). P BW extensions
In this final section we apply the results of the previous section to compute the point modules of someskew
P BW extensions, and hence, of some important examples of non N -graded quantum algebras. Theorem 3.1.
Let K be a field and A = σ ( K ) h x , . . . , x n i be a bijective skew P BW extension such that A is a K -algebra. Then, there exists m ≥ such that there is an injective function from P ( A ) / ∼ tothe closed points of X m : ( P ( A ) / ∼ ) −→ X m .Proof. From Proposition 1.20, A is a F SG ring generated in degree 1 with A = K , and by hypothesis, A is a K -algebra. Observe that the filtration in Theorem 1.16 coincides with the one in (1.5), thus, fromTheorem 1.17 we get that Gr ( A ) is the n -multiparametric quantum space K q [ x , . . . , x n ] (in A , x i r = rx i ,for all r ∈ K and 1 ≤ i ≤ n ). It is clear that Gr ( A ) is a finitely graded algebra with finite presentation asin Proposition 1.4, moreover, Gr ( A ) is strongly noetherian (Theorem 1.19), so the claimed follows fromCorollary 1.5 and Theorem 2.7.For the exact definition of any of the quantum algebras in the following examples see [12] and [14].11 xample 3.2. The following examples of skew
P BW extensions satisfy the hypothesis of Theorem 3.1,and hence, for them there is m ≥ P ( A ) / ∼ to theclosed points of X m .1. The enveloping algebra of a Lie algebra G of dimension n : In this case Gr ( A ) = K [ x , . . . , x n − ],so m = 1 and X = P n − (Example 1.1). We have a similar situation for the algebra S h of shiftoperators: In this case Gr ( A ) = K [ x , x ], so m = 1 and X = P .2. Quantum algebra U ′ ( so (3 , K )), with q ∈ K − { } : Since in this case Gr ( A ) is a 3-multiparametricquantum affine space, then from Example 1.6 we get that m = 2 and X is as in (1.1). Wehave similar descriptions for the dispin algebra U ( osp (1 , W ν ( sl (2 , K )),where ν ∈ K − { } is not a root of unity, nine types of 3-dimensional skew polynomial algebras,and the multiplicative analogue of the Weyl algebra with 3 variables.3. For the following algebras, Gr ( A ) = K q [ x , . . . , x n ], so from Example 1.7 we conclude that m = 2and X is as in (1.4): The q -Heisenberg algebra, the algebra of linear partial q -dilation operators, thealgebra D for multidimensional discrete linear systems, the algebra of linear partial shift operators. Example 3.3.
The following algebras are not skew
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