Some open problems in the context of skew PBW extensions and semi-graded rings
aa r X i v : . [ m a t h . R A ] S e p Some open problems in the context ofskew
P BW extensions and 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 discuss some open problems of non-commutative algebra and non-commutativealgebraic geometry from the approach of skew
P BW extensions and semi-graded rings. More exactly,we will analyze the isomorphism arising in the investigation of the Gelfand-Kirillov conjecture aboutthe commutation between the center and the total ring of fractions of an Ore domain. The Serre’sconjecture will be discussed for a particular class of skew
P BW extensions. The questions aboutthe noetherianity and the Zariski cancellation property of Artin-Schelter regular algebras will bereformulated for semi-graded rings. Advances for the solution of some of the problems are included.
Key words and phrases.
Gelfand-Kirillov conjecture, Serre’s conjecture, Artin-Schelter regular alge-bras, Zariski cancellation problem, skew
P BW extensions, semi-graded rings.2010
Mathematics Subject Classification.
Primary: 16S36. Secondary: 16U20, 16D40, 16E05, 16E65,16S38, 16S80, 16W70, 16Z05.
Commutative algebra and algebraic geometry are source of many interesting problems of non-commutativealgebra and geometry. For example, the famous theorem of Serre of commutative projective algebraicgeometry states that the category of coherent sheaves over the projective n -space P n is equivalent toa category of noetherian graded modules over a graded commutative polynomial ring. The study ofthis equivalence for non-commutative algebras gave origin to an intensive study of the so-called non-commutative projective schemes associated to non-commutative finitely graded noetherian algebras, andproduced a beautiful result, due to Artin and Zhang, and also proved independently by Verevkin, knownas the non-commutative version of Serre’s theorem. Another famous example that has to be mentionedis the Zariski cancellation problem on the algebra of commutative polynomials K [ x , . . . , x n ] over a field K , this problem asks if given a commutative K -algebra B and an isomorphism K [ x , . . . , x n ][ t ] ∼ = B [ t ],follows that K [ x , . . . , x n ] ∼ = B . This problem has been reformulated for non-commutative algebras andoccupied the attention of many researchers in the last years. In this paper we will present some famousopen problems of non-commutative algebra and non-commutative algebraic geometry, but interpretingthem from the approach of skew P BW extensions and semi-graded rings. For some of these problemswe will include some advances. Skew
P BW extension were define first in [36], many important algebrascoming from mathematical physics are particular examples of skew
P BW extensions: The envelopingalgebra U ( G ), where G is a finite dimensional Lie algebra, the algebra of q -differential operators, thealgebra of shift operators, the additive analogue of the Weyl algebra, the multiplicative analogue of1he Weyl algebra, the quantum algebra U ′ ( so (3 , K )), 3-dimensional skew polynomial algebras, the dispinalgebra, the Woronowicz algebra, the q -Heisenberg algebra, among many others (see [38] for the definitionof these algebras). On the other hand, the semi-graded rings generalize the finitely graded algebras andthe skew PBW extensions, this general class of non-commutative rings was introduced in [37].The problems that we will analyze and formulate from the general approach of skew P BW extensionsand semi-graded rings are: 1) Investigate the Serre-Artin-Zhang-Verevkin theorem (see [10]) for semi-graded rings. 2) Study the Gelfand-Kirillov conjecture ([28]) for bijective skew
P BW extensions overOre domains. Moreover, investigate for them the commutation between the center and the total ringof fractions. 3) Give a matrix-constructive proof of the Quillen-Suslin theorem (see [53], [46] and [55])for a particular class of skew polynomials rings. Include an algorithm that computes a basis of a givenfinitely generated projective module. 4) Investigate if the semi-graded Artin-Schelter regular algebras([9]) are noetherian domains. 5) Investigate the cancellation property (see [13]) for a particular class ofsemi-graded Artin-Schelter regular algebras of global dimension 3.Despite of the paper is a revision of previous works, written with the purpose of reformulating somefamous open problems in a more general context, some new results are included. The first problem wassolved in [37] assuming that the semi-graded rings are domains, in the present paper we will prove animproved result avoiding the restriction about the absence zero divisors. With respect to the problem4), we give an advance showing that a particular class of semi-graded Artin-Schelter regular algebras arenoetherian domains. Moreover, all of examples of semi-graded Artin-Schelter regular algebras exhibitedin the present paper are noetherian domains. The new results of the paper are concentrated in Theorem1.24, Corollary 1.25 and Theorem 4.14.The precise description of the classical famous open problems, and the new reformulation of them inthe context of the present paper, will be given below in each section. We have included basic definitionsand facts about finitely graded algebras, skew
P BW extensions and semi-graded rings, needed for abetter understanding of the paper. If not otherwise noted, all modules are left modules; B will denotea non-commutative ring; Z ( B ) is the center of B ; M s ( B ) be the ring of s × s matrices over B , and GL s ( B ) := { F ∈ M s ( B ) | F is invertible in M s ( B ) } be the general linear group over S ; K will be a field. Our first problem concerns with the generalization of a famous theorem of non-commutative projectivealgebraic geometry over finitely graded algebras. In this subsection we review the theorem and somewell-known basic facts of finitely graded algebras (see [51]).
Definition 1.1.
Let K be a field. It is said that a K -algebra A is finitely graded if the following conditionshold: • A is N -graded: A = L n ≥ A n . • A is connected, i.e., A = K . • A is finitely generated as K -algebra. The most remarkable examples of finitely graded algebras for which non-commutative projectivealgebraic geometry has been developed are the quantum plane, the Jordan plane, the Sklyanin algebraand the multi-parametric quantum affine n -space (see [51]).Let A be a finitely graded K -algebra and M = L n ∈ Z M n be a Z -graded A -module which is finitelygenerated. Then,(i) For every n ∈ Z , dim K M n < ∞ .(ii) The Hilbert series of A is defined by h A ( t ) := P ∞ n =0 (dim K A n ) t n .2iii) The Gelfand-Kirillov dimension of A is defined byGKdim( A ) := sup V lim n →∞ log n dim K V n , (1.1)where V ranges over all frames of A and V n := K h v · · · v n | v i ∈ V i (a frame of A is a finitedimensional K -subspace of A such that 1 ∈ V ; since A is a K -algebra, then K ֒ → A , and hence, K is a frame of A of dimension 1).(iv) A famous theorem of Serre on commutative projective algebraic geometry states that the cate-gory of coherent sheaves over the projective n -space P n is equivalent to a category of noetheriangraded modules over a graded commutative polynomial ring. The study of this equivalence for non-commutative finitely graded noetherian algebras is known as the non-commutative version of Serre’stheorem and is due to Artin, Zhang and Verevkin. In the next numerals we give the ingredientsneeded for the formulation of this theorem (see [10], [57]).(v) Suppose that A is left noetherian. Let gr − A be the abelian category of finitely generated Z -gradedleft A -modules. The abelian category qgr − A is defined in the following way: The objects are thesame as the objects in gr − A , and we let π : gr − A → qgr − A be the identity map on the objects.The morphisms in qgr − A are defined in the following way: Hom qgr − A ( π ( M ) , π ( N )) := lim −→ Hom gr − A ( M ≥ n , N/T ( N )),where the direct limit is taken over maps of abelian groups Hom gr − A ( M ≥ n , N/T ( N )) → Hom gr − A ( M ≥ n +1 , N/T ( N ))induced by the inclusion homomorphism M ≥ n +1 → M ≥ n ; T ( N ) is the torsion submodule of N andan element x ∈ N is torsion if A ≥ n x = 0 for some n ≥
0. The pair (qgr − A , π (A)) is called the non-commutative projective scheme associated to A , and denoted by qgr − A. Thus, qgr − A is aquotient category, qgr − A = gr − A/ tor − A .(vi) ( Serre’s theorem ) Let A be a commutative finitely graded K -algebra generated in degree . Then,there exists an equivalence of categories qgr − A ≃ coh(proj( A )). In particular , qgr − K [ x , . . . , x n ] ≃ coh( P n ).(vii) Suppose that A is left noetherian and let i ≥
0, it is said that A satisfies the χ i condition if forevery finitely generated Z -graded A -module M , dim K ( Ext jA ( K, M )) < ∞ for any j ≤ i ; the algebra A satisfies the χ condition if it satisfies the χ i condition for all i ≥ K is finitely generatedas A -module, then dim K ( Ext jA ( K, M )) = dim K ( Ext jA ( K, M ))).(viii) (
Serre-Artin-Zhang-Verevkin theorem ) If A is left noetherian and satisfies χ , then Γ( π ( A )) ≥ isleft noetherian and there exists an equivalence of categories qgr − A ≃ qgr − Γ( π ( A )) ≥ , (1.2) where Γ( π ( A )) ≥ := L ∞ d =0 Hom qgr − A ( π ( A ) , s d ( π ( A ))) and s is the autoequivalence of qg r − A defined by the shifts of degrees. .2 Skew P BW extensions
The second ingredient needed for setting some of the problems that we will discuss in the present paperare the skew
P BW extensions introduced first in [36]. Skew
P BW extensions are non-commutative ringsof polynomial type and cover many examples of quantum algebras and rings coming from mathematicalphysics: Habitual ring of polynomials in several variables, Weyl algebras, enveloping algebras of finite di-mensional Lie algebras, algebra of q -differential operators, many important types of Ore algebras, algebrasof diffusion type, additive and multiplicative analogues of 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 )), coordinatealgebra of the quantum group SL q (2), q -Heisenberg algebra H n ( q ), Hayashi algebra W q ( J ), differentialoperators on a quantum space D q ( S q ), Witten’s deformation of U ( sl (2 , K )), multiparameter Weyl al-gebra A Q, Γ n ( K ), quantum symplectic space O q ( sp ( K n )), some quadratic algebras in 3 variables, some3-dimensional skew polynomial algebras, particular types of Sklyanin algebras, among many others. Fora precise definition of any of these rings and algebras see [38] Definition 1.2 ([36]) . 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 Mon( 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. (1.3)(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 ∈ R + Rx + · · · + Rx n . (1.4) 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, as the following proposition shows. Proposition 1.3 ([36], Proposition 3) . Let A be a skew P BW extension of R . Then, for every ≤ i ≤ n ,there exist an 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 . From Definition 1.2 (iv), there exists a unique finite set of constants c ij , d ij , a ( k ) ij ∈ R such that x j x i = c ij x i x j + a (1) ij x + · · · + a ( n ) ij x n + d ij , for every 1 ≤ i, j ≤ n. (1.5)A particular case of skew P BW extension is when all derivations δ i are zero. Another interesting caseis when all σ i are bijective and the constants c ij are invertible. Definition 1.4 ([36]) . Let A be a skew P BW extension. A is quasi-commutative if the conditions ( iii ) and ( iv ) in Definition 1.2 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 . (1.6)(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 . (1.7)(b) A is bijective if σ i is bijective for every ≤ i ≤ n and c i,j is invertible for any ≤ i < j ≤ n . Observe that quasi-commutative skew
P BW extensions are N -graded rings, but arbitrary skew P BW extensions are semi-graded rings as we will see in the next subsection. Actually, the main motivationfor constructing the non-commutative algebraic geometry of semi-graded rings is due to arbitrary skew
P BW extensions.Many properties of skew
P BW extensions have been studied in previous works (see [2], [3], [38], [47],[48], [49], [50]). The next theorem establishes two ring theoretic results for skew
P BW extensions.
Theorem 1.5 ([38], [2]) . Let A be a bijective skew P BW extension of a ring R . (i) ( Hilbert Basis Theorem ) If R is a left ( right ) Noetherian ring then A is also left ( right ) Noetherian. (ii) (
Ore’s theorem ) If R is a left Ore domain R . Then A is also a left Ore domain. The third notion needed for setting the problems analyzed in the paper are the semi-graded rings that wewill recall in this subsection. In [37] were proved some properties of them, in particular, it was showedthat graded rings, finitely graded algebras and skew
P BW extensions are particular cases of this type ofnon-commutative rings.
Definition 1.6.
Let B be a ring. We say that B is semi-graded ( SG ) if there exists a collection { B n } n ≥ of subgroups 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 homo-geneous of degree n . Let B and C be semi-graded rings and let f : B → C be a ring homomorphism, wesay that f is homogeneous if f ( B n ) ⊆ C n for every n ≥ . Definition 1.7.
Let B be a SG ring and let M be a B -module. We say that M is a Z -semi-graded, orsimply semi-graded, if there exists a collection { M n } n ∈ Z of subgroups M n of the additive group M + suchthat the following conditions hold: (i) M = L n ∈ Z M n . (ii) For every m ≥ and n ∈ Z , B m M n ⊆ L k ≤ m + n M k .We say that M is positively semi-graded, also called N -semi-graded, if M n = 0 for every n < . Let f : M → N be an homomorphism of B -modules, where M and N are semi-graded B -modules, we saythat f is homogeneous if f ( M n ) ⊆ N n for every n ∈ Z . efinition 1.8. Let B be a SG ring and M be a semi-graded module over B . Let N be a submodule of M , we say that N is a semi-graded submodule of M if N = L n ∈ Z N n . Definition 1.9.
Let B be a ring. We say that B is finitely semi-graded ( F SG ) if B satisfies the followingconditions: (i) B is SG . (ii) There exists finitely many elements x , . . . , x n ∈ B such that the subring generated by B and x , . . . , x n coincides with B . (iii) For every n ≥ , B n is a free B -module of finite dimension.Moreover, if M is a B -module, we say that M is finitely semi-graded if M is semi-graded, finitely gener-ated, and for every n ∈ Z , M n is a free B -module of finite dimension. Remark 1.10.
Observe if B is F SG , 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 .From the definitions above we get the following conclusions. Proposition 1.11 ([37]) . Let B = L n ≥ B n be a SG ring and I be a proper two-sided ideal of B semi-graded as left ideal. 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 . (1.8)(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 . (v) B/I is SG . Proposition 1.12 ([37]) . (i) Any N -graded ring is SG . (ii) Let K be a field. Any finitely graded K -algebra is a F SG ring. (iii)
Any skew
P BW extension is a
F SG ring.
Remark 1.13. (i) In [37] was proved that the previous inclusions are proper.(ii) The class of
F SG rings also includes properly the multiple Ore extensions introduced in [58].6 .4 Problem 1 • Investigate the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings .This problem was partially solved in [37] where it was assumed that the semi-graded left noetherianring B is a domain, however, the Serre-Artin-Zhang-Verevkin theorem for finitely graded algebras doesnot include this restriction. Next we will present the main ingredients of an improved proof where thedomain restriction has been removed.Let B = L n ≥ B n be a SG ring that satisfies the following conditions:(C1) B is left noetherian.(C2) B is left noetherian.(C3) For every n , B n is a finitely generated left B -module.(C4) B ⊂ Z ( B ). Proposition 1.14 ([37], Proposition 5.2) . Let sgr − B be the collection of all finitely generated semi-graded B -modules, then sgr − B is an abelian category where the morphisms are the homogeneous B -homomorphisms. Definition 5.3 in [37] can be improved in the following way.
Definition 1.15.
Let M be an object of sgr − B . (i) For s ≥ , B ≥ s is the least two-sided ideal of B that satisfies the following conditions: (a) B ≥ s contains L p ≥ s B p . (b) B ≥ s is semi-graded as left ideal of B . (ii) An element x ∈ M is torsion if there exist s, n ≥ such that B n ≥ s x = 0 . The set of torsion elementsof M is denoted by T ( M ) . M is torsion if T ( M ) = M and torsion-free if T ( M ) = 0 . Theorem 1.16 ([37], Theorem 5.5) . The collection stor − B of torsion modules forms a Serre subcategoryof sgr − B , and the quotient category qsgr − B := sgr − B/ stor − B is abelian. Remark 1.17.
Recall from the general theory of abelian categories (see [54]) that the quotient functor π : sgr − B → qsgr − B is exact and defined by π ( M ) := M and π ( f ) := f ,with M and M f −→ N in sgr − B and the morphisms in the category qsgr − B are defined by Hom qsgr − B ( M, N ) := lim −→ Hom sgr − B ( M ′ , N/N ′ ) , (1.9)where the direct limit is taken over all M ′ ⊆ M , N ′ ⊆ N in sgr − B with M/M ′ ∈ stor − B and N ′ ∈ stor − B (see [29], [24] and also [54] Proposition 2.13.4). Definition 1.18 ([37], Definition 6.1) . Let M be a semi-graded B -module, M = L n ∈ Z M n . Let i ∈ Z ,the semi-graded module M ( i ) defined by M ( i ) n := M i + n is called a shift of M , i.e., M ( i ) = L n ∈ Z M ( i ) n = L n ∈ Z M i + n . Proposition 1.19 ([37], Proposition 6.3) . Let s : sgr − B → sgr − B defined by M M (1) , M f −→ N M (1) f (1) −−−→ N (1) , f (1)( m ) := f ( m ) , m ∈ M (1) .Then, (i) s is an autoequivalence. (ii) For every d ∈ Z , s d ( M ) = M ( d ) . (iii) s induces an autoequivalence of qsgr − B also denoted by s . Proposition 1.20. sπ = πs .Proof. We have sgr − B π −→ qsgr − B s −→ qsgr − B and sgr − B s −→ sgr − B π −→ qsgr − B ,so, for M in sgr − B , sπ ( M ) = s ( π ( M )) = π ( M )(1) = M (1) and πs ( M ) = π ( M (1)) = M (1); for M f −→ N in sgr − B , sπ ( f ) = s ( f ) = f (1) and πs ( f ) = π ( f (1)) = f (1). Definition 1.21.
Let s be the autoequivalence of qsgr − B defined by the shifts of degrees. We define Γ( π ( B )) ≥ := L ∞ d =0 Hom qsgr − B ( π ( B ) , s d ( π ( B ))) . The domain condition on B in Lemma 6.10 of [37] has been removed in the following simplified version. Lemma 1.22.
Let B be a ring that satisfies (C1)-(C4) . (i) Γ( π ( B )) ≥ is a N -graded ring. (ii) Let B := L ∞ d =0 Hom sgr − B ( B, s d ( B )) . Then, B is a N -graded ring and there exists a ring homo-morphism B → Γ( π ( B )) ≥ . (iii) For any object M of sgr − B Γ( M ) ≥ := L ∞ d =0 Hom sgr − B ( B, s d ( M )) is a graded B -module, and Γ( π ( M )) ≥ := L ∞ d =0 Hom qsgr − B ( π ( B ) , s d ( π ( M ))) is a graded Γ( π ( B )) ≥ -module. (iv) B has the following properties: (a) ( B ) ∼ = B and B satisfies (C2) . (b) B satisfies (C3) . More generally, let N be a finitely generated graded B -module, then everyhomogeneous component of N is finitely generated over ( B ) . (c) B satisfies (C1) . (v) If B satisfies X , then π ( B )) ≥ satisfies (C2) . (b) Γ( π ( B )) ≥ satisfies (C3) . More generally, let N be a finitely generated graded Γ( π ( B )) ≥ -module, then every homogeneous component of N is finitely generated over (Γ( π ( B )) ≥ ) . (c) Γ( π ( B )) ≥ satisfies (C1) .Proof. We include only the proof of the part (v) since the proof of the others are exactly as in [37].(v) We set Γ := Γ( π ( B )) ≥ . Then,(a) Γ satisfies (C2): From (1.9) we have Γ = Hom qsgr − B ( π ( B ) , π ( B )) = Hom qsgr − B ( B, B ) =lim −→ Hom sgr − B ( I ′ , B/N ′ ), where the direct limit is taken over all pairs ( I ′ , N ′ ) in sgr − B , with I ′ , N ′ ⊆ B , B/I ′ ∈ stor − B and N ′ ∈ stor − B . Since π is a covariant functor, we obtain a ring homomorphism(taking in particular I ′ = B and N ′ = 0)( B ) = Hom sgr − B ( B, B ) γ −→ Hom qsgr − B ( B, B ) = Γ γ ( f ) := π ( f ) = f .Since B ∼ = ( B ) (isomorphism defined by α ( x ) = α x , α x ( b ) := bx, x ∈ B , b ∈ B ), then Γ , Γ and B are B -modules. Actually, they are B -algebras: We check this for B , the proof for Γ( π ( B )) ≥ is similar,and from this we get also that Γ is a B -algebra. If f ∈ Hom sgr − B ( B, B ( n )) , g ∈ Hom sgr − B ( B, B ( m )), x ∈ B and b ∈ B , then [ x · ( f ⋆ g )]( b ) = x · ( s n ( g ) ◦ f )( b ) = xg ( n )( f ( b ));[ f ⋆ ( x · g )]( b ) = [ s n ( x · g ) ◦ f ]( b ) = ( x · g )( n )( f ( b )) = xg ( n )( f ( b )).Since B is noetherian, in order to prove that Γ is a noetherian ring, the idea is to show that Γ isfinitely generated as B -module, but since B satisfies X , this follows from Proposition 3.1.3 (3) in [10].Thus, Γ is a commutative noetherian ring, and hence, Γ satisfies (C2).(b) Γ satisfies (C3): Since Γ is graded, Γ d is a Γ -module for every d , but by γ in (a), the idea is toprove that Γ d is finitely generated over B , but again we apply Proposition 3.1.3 (3) in [10].For the second part of (b), let N be a Γ-module generated by a finite set of homogeneous elements x , . . . , x r , with x i ∈ N d i , 1 ≤ i ≤ r . Let x ∈ N d , then there exist f , . . . , f r ∈ Γ such that x = f · x + · · · + f r · x r , from this we can assume that f i ∈ Γ d − d i , but as was observed before, every Γ d − d i is finitely generated as Γ -module, so N d is finitely generated over Γ for every d .(c) Γ satisfies (C1): By (iii), Γ is not only SG but N -graded. The proof of (C1) is exactly as in thepart (iv) of Lemma 6.10 in [37]. Proposition 1.23 ([10], Proposition 2.5) . Let S be a commutative noetherian ring and ρ : C → D be ahomomorphism of N -graded left noetherian S -algebras. If the kernel and cokernel of ρ are right bounded,then D ⊗ C − defines an equivalence of categories qgr − C ≃ qgr − D , where ⊗ denotes the graded tensorproduct. The solution of Problem 1 is as follows.
Theorem 1.24.
Let B be a SG ring that satisfies (C1)-(C4) and assume that B satisfies the condition X , then there exists an equivalence of categories qgr − B ≃ qgr − Γ( π ( B )) ≥ . (1.10) Proof.
The ring homomorphism B ρ −→ Γ( π ( B )) ≥ (1.11) f + · · · + f d π ( f ) + · · · + π ( f d )satisfies the conditions of Proposition 1.23, with S = B , C = B and D = Γ( π ( B )) ≥ . In fact, from theproof of Lemma 1.22 we know that B and Γ( π ( B )) ≥ are N -graded left noetherian B -algebras. Since B X , we can apply the proof of part S10 in Theorem 4.5 in [10] to conclude that thekernel and cokernel of ρ are right bounded.We will see next that our Theorem 1.24 extends the Serre-Artin-Zhang-Verevkin equivalence (1.2). Corollary 1.25.
Let B be a SG ring that satisfies (C1)-(C4) . Then, (i) There is an injective homomorphism of N -graded B -algebras η : B → Gr ( B ) . (ii) If B = K is a field and Gr ( B ) is left noetherian and satisfies X , then B satisfies X and thefollowing equivalences of categories hold: qgr − Gr ( B ) ≃ qgr − B ≃ qgr − Γ( π ( B )) ≥ . (1.12)(iii) If B is finitely graded and satisfies X , then B ∼ = B and the Serre-Artin-Zhang-Verevkin equivalence qgr − B ≃ qgr − Γ( π ( B )) ≥ holds.Proof. (i) η is defined by (see Proposition 1.11) ∞ M d =0 Hom sgr − B ( B, B ( d )) = B η −→ Gr ( B ) = ∞ M d =0 Gr ( B ) d = ∞ M d =0 B ⊕ · · · ⊕ B d B ⊕ · · · ⊕ B d − f + · · · + f d f (1) + · · · + f d (1) , with f i ∈ Hom sgr − B ( B, B ( i )), 0 ≤ i ≤ d . It is clear thet η is additive and η (1) = 1; η is multiplicative: η ( f n ⋆ g m ) = η ( s n ( g m ) ◦ f n ) = ( s n ( g m ) ◦ f n )(1) = s n ( g m )( f n (1)) = g m ( f n (1)) = f n (1) g m (1) = f n (1) g m (1) = η ( f n ) η ( g m ). η is a B -homomorphism: Let x ∈ B and f d ∈ Hom sgr − B ( B, B ( d )), since B ∼ = ( B ) , then η ( x · f d ) = η ( f d ◦ α x ) = ( f d ◦ α x )(1) = f d ( α x (1)) = f d ( x ) = x · f d (1) = x · f d (1) = x · f d (1) = x · η ( f d ). η is injective: If f (1) + · · · + f d (1) = 0, then f k (1) = 0 for every 0 ≤ k ≤ d , therefore f k (1) ∈ ( B ⊕ · · · ⊕ B k − ) ∩ B k since f k (1) ∈ B ( k ) = B k .(ii) Since B and Gr ( B ) are N -graded left noetherian K -algebras ( K a field) and the kernel andcokernel of η are right bounded, we apply Lemma 8.2 in [10] to conclude that B satisfies X . Thus, fromTheorem 1.24 we get the second equivalence of (1.12). Applying Proposition 1.23 to η we obtain the firstequivalence.(iii) We define θ by ∞ M d =0 Hom sgr − B ( B, B ( d )) = B θ −→ B = ∞ M d =0 B d f + · · · + f d f (1) + · · · + f d (1) , with f i ∈ Hom sgr − B ( B, B ( i )), 0 ≤ i ≤ d . As (i), we can prove that θ is an isomorphism of B -algebras.Thus, we get the Serre-Artin-Zhang-Verevkin equivalence qgr − B ≃ qgr − Γ( π ( B )) ≥ . Example 1.26.
In [37] was proved that the following examples of skew
P BW extensions are semi-gradedrings (most of them non N -graded) and satisfy the conditions (C1)-(C4), moreover, in each case, B satisfiesthe condition X , therefore, for these algebras Theorem 1.24 is true. In every example B = K is a field:Enveloping algebra of a Lie K -algebra G of dimension n , U ( G ); the quantum algebra U ′ ( so (3 , K )), with q ∈ K − { } ; the dispin algebra U ( osp (1 , W ν ( sl (2 , K )), where ν ∈ K − { } is not a root of unity; eight types of 3-dimensional skew polynomial algebras.10 Gelfand-Kirillov conjecture
In this section we will review some aspects of the quantum version of the Gelfand-Kirillov conjecture andwe will formulate a related problem in the context of the skew
P BW extensions. We start recalling theclassical conjecture and some well known cases where the conjecture has positive answer. In this section Q ( B ) denotes the total ring of fraction of an Ore (left and right) domain B . Conjecture 2.1 (Gelfand-Kirillov, [28]) . Let G be an algebraic Lie algebra of finite dimension over afield L , with char ( L ) = 0 . Then, there exist integers n, k ≥ such that Q ( U ( G )) ∼ = Q ( A n ( L [ s , . . . , s k ])) , (2.1) where A n ( L [ s , . . . , s k ]) is the general Weyl algebra over L . Recall that G is algebraic if G is the Lie algebra of a linear affine algebraic group. A group G is linearaffine algebraic if G is an affine algebraic variety such that the multiplication and the inversion in G aremorphisms of affine algebraic varieties.Next we recall some examples of algebraic Lie algebras for which the classical conjecture (2.1) holds. Example 2.2. (i) The algebra of all n × n matrices over a field L with char( L ) = 0. The same is truefor the algebra of matrices of null trace ([28], Lemma 7).(ii) A finite dimensional nilpotent Lie algebra over a field L , with char( L ) = 0. Moreover, in this case, Q ( Z ( U ( G ))) ∼ = Z ( Q ( U ( G ))) ([28], Lemma 8).(iii) A finite dimensional solvable algebraic Lie algebra over the field C of complex numbers (see [32],Theorem 3.2)(iv) Any algebraic Lie algebra G over an algebraically closed field L of characteristic zero, withdim( G ) ≤ Remark 2.3. (i) More examples can be found in [14], [23], [33] and [45].(ii) There are examples of non-algebraic Lie algebras for which the conjecture is false. However, otherexamples show that the conjecture holds for some non-algebraic Lie algebras (see [28], Section 8).We are interested in the quantum version of the Gelfand-Kirillov conjecture, i.e., in this case U ( G ) isreplaced for a quantum algebra and the Weyl algebra A n ( L [ s , . . . , s k ]) in (2.1) is replaced by a suitable n -multiparametric quantum affine space K q [ x , . . . , x n ] , as it is shown the following examples. Example 2.4 ([15]) . Let K be a field and B := K [ x ][ x ; σ , δ ] · · · [ x n ; σ n , δ n ] be an iterated skewpolynomial ring with some extra adequate conditions on σ ’s and δ ’s. Then there exits q := [ q i,j ] ∈ M n ( K )with q ii = 1 = q ij q ji , for every 1 ≤ i, j ≤ n , such that Q ( B ) ∼ = Q ( K q [ x , . . . , x n ]). Example 2.5 ([5], Theorem 3.5) . Let A Q, Γ n ( K ) be the multiparameter quantized Weyl algebra (see [38]);in particular, consider the case when there exists a parameter q ∈ K ∗ that is non root of unity, such thatevery parameter in Q = [ q , . . . , q n ] and Γ = [ γ ij ] is a power of q , and in addition, q i = 1, 1 ≤ i ≤ n .Under these conditions, there exits q := [ q ij ] ∈ M n ( K ) with q ii = 1 = q ij q ji , and q ij is a power of q ,1 ≤ i, j ≤ n , such that Q ( A Q, Γ n ( K )) ∼ = Q ( K q [ x , . . . , x n ]), Z ( Q ( K q [ x , . . . , x n ])) = K . Example 2.6 ([5], Theorem 2.15) . Let U + q ( sl m ) be the quantum enveloping algebra of the Lie algebraof strictly superior triangular matrices of size m × m over a field K .(i) If m = 2 n + 1, then Q ( U + q ( sl m )) ∼ = Q (K q [ x , . . . , x n ]),11here K := Q ( Z ( U + q ( sl m ))) and q := [ q ij ] ∈ M n ( K ), with q ii = 1 = q ij q ji , and q ij is a power of q for every 1 ≤ i, j ≤ n .(ii) If m = 2 n , then Q ( U + q ( sl m )) ∼ = Q (K q [ x , . . . , x n ( n − ]),where K := Q ( Z ( U + q ( sl m ))) and q := [ q ij ] ∈ M n ( n − ( K ), with q ii = 1 = q ij q ji , and q ij is a powerof q for every 1 ≤ i, j ≤ n ( n − Q ( Z ( U + q ( sl m ))) ∼ = Z ( Q ( U + q ( sl m ))). • Study the Gelfand-Kirillov conjecture for bijective skew
P BW extensions over Ore domains. More-over, investigate for them the isomorphism Q ( Z ( A )) ∼ = Z ( Q ( A )).The ring Q k,n q ,σ ( R ) of skew quantum polynomials over R , also denoted by R q ,σ [ x ± , . . . , x ± k , x k +1 , . . . , x n ],conforms a particular type of quasi-commutative bijective skew P BW extension. In [2] was proved thefollowing partial solution of Problem 2.
Theorem 2.7 ([2], Corollary 5.1) . If R is an Ore domain ( left and right ) , then Q ( Q k,n q ,σ ( R )) ∼ = Q ( Q q ,σ [ x , . . . , x n ]) , with Q := Q ( R ) . (2.2)The precise definition of R q ,σ [ x ± , . . . , x ± k , x k +1 , . . . , x n ] and Q q ,σ [ x , . . . , x n ] can be found in [2].Observe that the n -multiparametric quantum affine space K q [ x , . . . , x n ] was replaced in (2.2) by the n -multiparametric skew quantum space Q q ,σ [ x , . . . , x n ]. Given a ring, it is an interesting problem to investigate if the finitely generated projective modules over itare free. This problem becomes very important after the formulation in 1955 of the famous Serre’s problemabout the freeness of finitely generated projective modules over the polynomial ring K [ x , . . . , x n ], K afield (see [6], [7], [11], [34]). The Serre’s problem was solved positively, and independently, by Quillen inUSA, and by Suslin in Leningrad, USSR (St. Petersburg, Russia) in 1976 ([46], [55]).For arbitrary skew P BW extensions the problem has negative answer, for example, let A := R [ x ; σ ]be the skew polynomial ring over R := K [ y ], where K is a field and σ ( y ) := y + 1, from [43], 12.2.11 wecan conclude that there exist finitely generated projective modules over A that are not free. However,in [8] was proved that if K is a field, A := K [ x , . . . , x n ; σ ], σ is an automorphism of K of finite order, x i x j = x j x i and x i r = σ ( r ) x i , for every r ∈ K and 1 ≤ i, j ≤ n , then every finitely generated projectivemodule over A is free. The proof in [8] of this theorem ( Quillen-Suslin theorem ) is not construcitve, i.e.,the proof shows the existence of a basis for every finitely generated projective module M over A , but anexplicit basis of M was not constructed. • Give a matrix-constructive proof of the Quillen-Suslin theorem for A := K [ x , . . . , x n ; σ ] . Includean algorithm that computes a basis of a given finitely generated projective A -module.
12n [21] was solved the previous problem for the case of only one variable, in this particular situationweaker hypotheses can be set, namely, K is a division ring, σ is not necessarily of finite order and a nontrivial σ -derivation δ of K can be added. Theorem 3.1 ([21], Theorem 2.1) . Let K be a division ring and A := K [ x ; σ, δ ] , with σ bijective. Then A is a PF -ring, i.e., every finitely generated projective module over A is free. The proof of the previous theorem given in [21] is not only matrix-constructive, but also some al-gorithms that compute the basis of a given finitely generated projective A -module are exhibited andimplemented in Maple . Next we review the main ingredients of the proof and include an illustrativeexample of algorithms.
Proposition 3.2 ([27]) . Let B be a ring. B is a PF -ring if and only if for every s ≥ , given anidempotent matrix F ∈ M s ( B ) , there exists an invertible matrix U ∈ GL s ( B ) such that U F U − = (cid:20) I r (cid:21) , (3.1) where r = dim ( h F i ) , ≤ r ≤ s , and h F i represents the free left B -module M generated by the rows of F . Moreover, the final r rows of U form a basis of M . The matrix-constructive proof of Theorem 3.1 consists in constructing explicitly the matrix U inProposition 3.2 from the entries of the given idempotent matrix F (recall that a left module M over aring B is finitely generated projective if and only if M is the left B -module generated by the rows ofa idempotent matrix over B ). In [21] were designed two algorithms for calculating the basis of a givenfinitely generated projective module: a constructive simplified version and a more complete computationalversion over a field. The computational version was implemented using Maple r Algorithm for the Quillen-Suslin theorem:Constructive versionINPUT : A := K [ x, σ, δ ]; F ∈ M s ( A ) an idempotent matrix. OUTPUT : Matrices U , U − and a basis X of h F i , where UF U − = (cid:20) I r (cid:21) and r = dim ( h F i ) . (3.2) INITIALIZATION : F := F . FOR k from 1 to n − DO
1. Follow the reduction procedures (B1) and (B2) in the proof of Theorem3.1 in order to compute matrices U ′ k , U ′− k and F k +1 such that U ′ k F k U ′− k = (cid:20) α k F k +1 (cid:21) , where α k ∈ { , } . U k := (cid:20) I k − U ′ k (cid:21) U k − ; compute U − k .3. By permutation matrices modify U n − . RETURN U := U n − , U − satisfying (3.2), and a basis X of h F i . xample 3.3. ([21], Example 3.4) Let A := K [ x, σ, δ ], K := Q ( t ), σ ( f ( t )) := f ( qt ) and δ ( f ( t )) := f ( qt ) − f ( t ) t ( q − , where q ∈ K − { , } . We consider the idempotent matrix F := [ F (1) F (2) F (3) F (4) ] ∈ M ( A ),with F ( i ) the ith column of F and a ∈ Q , where F (1) = − t qx ( − ta + 2 t ) x − a + 2 tx + 2 − ,F (2) = − tx + 2 − t qx + ( ta − t ) x + 2 a − − tx − tx + 2 ,F (3) = − tx − (cid:0) − t qa + 3 t q (cid:1) x + (cid:0) a t − ta + 8 t (cid:1) x + 2 a − a + 1 t qx + ( − ta + 4 t ) x − a + 2( ta − t ) x + 2 a − ,F (4) = − t q x + (cid:0) − q t − t q (cid:1) x − tx + 2 − t q x + (cid:0) − q t − t q (cid:1) x + ( − ta + t ) x − a + 2 tx + 2 t qx + 2 tx − . Applying the algorithms we obtain U = tx + 1 0 t qx + 2 tx − t qx + 3 tx − tx − − ta + 2 t ) x − a + 2 − t qx − tx + 2 tx − t qx + a − t qx + 2 tx −
11 0 tx tx + 1 , U − tx − − tx − a − − tx + a − − t qx ta − t ) x + 2 a − t q x − ( − q + a − t qx − ta + 3 t ) x + 1 − − − tx − t qx tx tx + 2 − t qx − tx + 1 , U F U − = , Therefore, r = 2 and the last two rows of U conform a basis X = { x , x } , of h F i , x = ( tx − , , t qx + a − , t qx + 2 tx − x = (1 , , tx, tx + 1). Remark 3.4.
For n ≥
2, the commutativity of K in Problem 3 is necessary, in [34] (p. 36) is provedthat if B is a division ring, then B [ x, y ] is not PF . Artin-Schelter regular algebras (shortly denoted as AS ) were introduced by Michael Artin and WilliamSchelter in [9]. In non-commutative algebraic geometry these algebras play the role of K [ x , . . . , x n ] incommutative algebraic geometry, thus, in particular, the noetherian AS algebras satisfy the condition X of Subsection 1.1 (see Theorem 12.6 in [54]), and hence, for them the Serre-Artin-Zhang-Verevkintheorem holds. Nowadays AS algebras are intensively investigated, in the next subsection we present thedefinition and some properties and open problems on these algebras.14 .1 Artin-Schelter regular algebras Definition 4.1 ([9]) . Let K be a field and A be a finitely graded K -algebra. It says that A is an Artin-Schelter regular algebra ( AS ) if (i) gld( A ) = d < ∞ . (ii) GKdim( A ) < ∞ . (iii) Ext iA ( A K, A A ) ∼ = ( if i = dK ( l ) A if i = d for some shift l ∈ Z . Remark 4.2. (i) In [51] is showed that the third condition is equivalent to
Ext iA ( K A , A A ) ∼ = ( if i = d A K ( l ) if i = d. In addition, since K ∼ = A/A ≥ is finitely generated as left A -module, then we can replace Ext iA ( A K, A A )by Ext iA ( A K, A A ). The same is true for Ext iA ( K A , A A ).(ii) Some key examples of Artin-Schelter regular algebras are: Let A = K [ x , . . . , x n ] be the usualcommutative polynomial ring in n variables over the field K , with any weights deg( x i ) = d i ≥ ≤ i ≤ n , then A is AS ; if A is a commutative AS algebra, then A is isomorphic to a weightedcommutative polynomial ring; the Jordan and quantum planes are AS of dimension 2; more generally,the n -multiparametric quantum affine space K q [ x , . . . , x n ] is AS . AS algebras of dimension ≤ AS algebras of dimension 4 or 5.(iii) Artin-Schelter open problems : In [9] were formulated the following problems:1. Any AS algebra is noetherian?2. Any AS algebra is a domain?There are some advances with respect to the previous open problems: Both questions above have positiveanswer for any AS algebra of dimension ≤
3. In addition, the same is true for all known concrete examplesof AS algebras of higher dimension. If A is AS noetherian and its dimension is ≤
4, then A is a domain. AS algebras are N -graded and connected, recently Gaddis in [25] and [26] introduces the technique of homogenization and the notion of essentially regular algebras in order the study of the Artin-Scheltercondition for non N -graded algebras. On the other hand, with the purpose of giving new examplesof AS algebras, in [41] and [42] are defined the Z s -graded Artin-Schelter regular algebras and there inhave been proved some results for the classification of AS algebras of dimension 5 with two generators.The semi-graded Artin-Schelter regular algebras that we will introduce in the present section is a newdifferent approach to this problem, and extend the classical notion of Artin-Schelter regular algebradefined originally in [9]. In a forthcoming paper we will generalize some classical well-known results onArtin-Schelter regular algebras to the semi-graded case. Definition 4.3.
Let K be a field and B be a K -algebra. We say that B is a left semi-graded Artin-Schelterregular algebra ( SAS ) if the following conditions hold: (i) B is a F SG ring with semi-graduation B = L p ≥ B p . B is connected, i.e., B = K . (iii) lgld( B ) := d < ∞ . (iv) Ext iB ( B ( B/B ≥ ) , B B ) ∼ = ( if i = d ( B/B ≥ ) B if i = d Remark 4.4. (i) If K ∩ B ≥ = 0, then B/B ≥ = 0; if K ∩ B ≥ = 0 then B/B ≥ ∼ = K , where thisis an isomorphism of K -algebras induced by the canonical projection ǫ : B → K (recall that B ≥ issemi-graded). Moreover, B ( B/B ≥ ) ∼ = B K and ( B/B ≥ ) B ∼ = K B .(ii) Our definition extends (except for the GKdim) the classical notion of Artin-Schelter algebra sincein such case B/B ≥ ∼ = K , lgld( B ) = pd( B K ) = pd( K B ) = rgld( B ) = gld( B ) and Ext iB ( K, B ) =
Ext iB ( K, B ) (see [51]). Thus, every Artin-Schelter regular algebra is
SAS . • Is any
SAS algebra with finite GKdim a left noetherian domain?We will present next some examples of
SAS algebras that are not Artin-Schelter, in every example,the algebra is a left-right noetherian domain.For the matrix representation of homomorphisms of left modules in this section we will use the leftrow notation and for right modules the right column notation. In order to compute free resolutions wewill apply Theorem 19 of [35] that has been implemented in the library
SPBWE.lib developed in
Maple by W. Fajardo in [19] (see also [20]). This library contains the packages
SPBWETools , SPBWERings and
SPBWEGrobner with utilities to define and perform calculations with skew
P BW extensions.
Example 4.5.
The Weyl algebra A ( K ) is not Artin-Schelter, but it is a SAS algebra. In fact, thecanonical semi-graduation of A ( K ) is A ( K ) = K ⊕ K h x, y i ⊕ K h x , xy, y i ⊕ · · · Recall that gld( A ( K )) ≤ A ( K ) ≥ = A ( K ), so A ( K ) /A ( K ) ≥ = 0. Hence, thecondition (iv) in Definition 4.3 trivially holds. This argument can be applied also to the general Weylalgebra A n ( K ) and the generalized Weyl algebra B n ( K ). Example 4.6.
The quantum deformation A q ( K ) of the Weyl algebra is not Artin-Schelter, but it is a SAS algebra. Indeed, recall that A q ( K ) is the K -algebra defined by the relation yx = qxy + 1, with q ∈ K − { } ( A q ( K ) coincides with the additive analog of the Weyl algebra in two variables, as well as,with the linear algebra of q -differential operators in two variables). The semi-graduation of A q ( K ) is asin the previous example, moreover gld( A q ( K )) ≤ A q ( K ) ≥ = A q ( K ), so A q ( K ) /A q ( K ) ≥ = 0 andthe condition (iv) in Definition 4.3 holds. Example 4.7.
Consider the algebra J := K h x, y i / h yx − xy + y + 1 i , according to Corollary 2.3.14 in[25], gld( J ) = 2. The semi-graduation of J is as in Example 4.5, so ( J ) ≥ = J . Thus, J is SAS but is not Artin-Schelter.
Example 4.8.
Now we consider the dispin algebra U ( osp (1 , B (see Example 1.26). Recall that B is defined by the relations x x − x x = x , x x + x x = x , x x − x x = x .Since B is not finitely graded, then B is not Artin-Schelter. We will show that B is SAS . We know thatgld( B ) = 3 (see [38]); the semi-graduation of B is given by B = K ⊕ K h x , x , x i ⊕ K h x , x x , x x , x , x x , x i ⊕ · · · B ≥ = ⊕ p ≥ B p and this ideal coincides with the two-sided ideal of B generated by x , x , x ;moreover, B ( B/B ≥ ) ∼ = B K , where the structure of left B -module for K is given by the canonicalprojection ǫ : B → K (the same is true for the right structure, ( B/B ≥ ) B ∼ = K B ). With SPBWE we getthe following free resolution of B K : → B φ = h − x x x i −−−−−−−−−−−−−−−→ B φ = x − x x − x x − x −−−−−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−→ B ǫ −→ K → . Now we apply
Hom B ( − , B B ) and we get the complex of right B -modules → Hom B ( K, B ) ǫ ∗ −−→ Hom B ( B, B ) φ ∗ −−→ Hom B ( B , B ) φ ∗ −−→ Hom B ( B , B ) φ ∗ −−→ Hom B ( B, B ) → . Note that
Hom B ( K, B ) = 0: In fact, let α ∈ Hom B ( K, B ) and α (1) := b ∈ B , then α ( x
1) = α (0) =0 = x α (1) = x b , so b = 0 (recall that B is a domain) and from this α ( k ) = 0 for every k ∈ K , i.e., α = 0. Moreover, from the isomorphisms of right B -modules Hom B ( B, B ) ∼ = B and Hom B ( B , B ) ∼ = B we obtain the complex0 → B φ ∗ = x x x −−−−−−→ B φ ∗ = x − x x − x x − x −−−−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x x i −−−−−−−−−−−−−−→ B → . So,
Ext B ( K, B ) =
Hom B ( K, B ) = 0,
Ext B ( K, B ) = 0 =
Ext B ( K, B ) and
Ext B ( K, B ) =
B/Im ( φ ∗ ) = B/B ≥ ∼ = K B . This shows that B is SAS . Example 4.9.
The next examples are similar to the previous, in every case gld( B ) = 3 and B is SAS .The free resolutions have been computed with
SPBWE .(a) Consider the universal enveloping algebra of the Lie algebra sl (2 , K ), B := U ( sl (2 , K )). B is the K -algebra generated by the variables x, y, z subject to the relations[ x, y ] = z, [ x, z ] = − x, [ y, z ] = 2 y. The free resolutions are: → B φ = h − z y − x i −−−−−−−−−−−−−→ B φ = y − x z − − x z + 2 − y −−−−−−−−−−−−−−−−−−−→ B φ = xyz −−−−−→ B ǫ −→ K → . → B φ ∗ = xyz −−−−−−→ B φ ∗ = y − x z − − x z + 2 − y −−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − z y − x i −−−−−−−−−−−−−→ B → . (b) Now let B := U ( so (3 , K )) be the K -algebra generated by the variables x, y, z subject to the relations[ x, y ] = z, [ x, z ] = − y, [ y, z ] = x. In this case we have → B φ = h − z y − x i −−−−−−−−−−−−−→ B φ = y − x z − − x z − y −−−−−−−−−−−−−−→ B φ = xyz −−−−−→ B ǫ −→ K → , → B φ ∗ = xyz −−−−−−→ B φ ∗ = y − x z − − x z − y −−−−−−−−−−−−−−→ B φ ∗ = h − z y − x i −−−−−−−−−−−−−→ B → . (c) Quantum algebra B := U ′ ( so (3 , K )), with q ∈ K − { } : x x − qx x = − q / x , x x − q − x x = q − / x , x x − qx x = − q / x .In this case the free resolutions are: → B φ = h − x x − x i −−−−−−−−−−−−−−−−→ B φ = x − qx q / qx − q / − x q / x − qx −−−−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−→ B φ ∗ = x − qx q / qx − q / − x q / x − qx −−−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x − x i −−−−−−−−−−−−−−−−→ B → . (d) Woronowicz algebra B := W ν ( sl (2 , K )), where ν ∈ K − { } is not a root of unity: x x − ν x x = (1 + ν ) x , x x − ν x x = νx , x x − ν x x = (1 + ν ) x .The free resolutions are: → B φ h − ν x ν x − x i −−−−−−−−−−−−−−−−−−−−−−−−→ B φ ν x − x νν x ν − x x − ( ν − ν x −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ B φ x x x −−−−−−−−→ B ǫ −→ K → , → B φ ∗ x x x −−−−−−−−→ B φ ∗ ν x − x νν x ν − x x − ( ν − ν x −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ B φ ∗ h − ν x ν x − x i −−−−−−−−−−−−−−−−−−−−−−−−→ B → . In the following example we study the semi-graded Artin-Schelter condition for the eight types of3-dimensional skew polynomial algebras considered in Example 1.26, five of them are
SAS and the otherthree are not.
Example 4.10.
The first type coincides with the dispin algebra taking β = −
1, and the fifth typecorresponds to U ( so (3 , K )), thus they are SAS . For the other six types we compute next the freeresolutions with
SPBWE : x x − x x = 0 , x x − βx x = x , x x − x x = 0 ( SAS ): → B φ = h − x x − βx i −−−−−−−−−−−−−−−−−→ B φ = x − x x − − βx x − x −−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−→ B φ ∗ = x − x x − − βx x − x −−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x − βx i −−−−−−−−−−−−−−−−−→ B → . x − x x = x , x x − βx x = 0 , x x − x x = x ( SAS ): → B φ = h − x x − βx i −−−−−−−−−−−−−−−−−−→ B φ = x + 1 − x x − βx x − x + 1 −−−−−−−−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−−→ B φ ∗ = x + 1 − x x − βx x − x + 1 −−−−−−−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x − βx i −−−−−−−−−−−−−−−−−−→ B → . x x − x x = x , x x − βx x = 0 , x x − x x = 0 (not SAS ) : → B φ = h − x x − − βx i −−−−−−−−−−−−−−−−−−−−−→ B φ = x − x x − βx x − x + 1 −−−−−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−−→ B φ ∗ = x − x x − βx x − x + 1 −−−−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x − − βx i −−−−−−−−−−−−−−−−−−−−−→ B → . x x − x x = 0 , x x − x x = 0 , x x − x x = x ( SAS ): → B φ = h − x x − x i −−−−−−−−−−−−−−−−−→ B φ = x − x x − x x − x −−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−−→ B φ ∗ = x − x x − x x − x −−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x x − x i −−−−−−−−−−−−−−−−−→ B → . x x − x x = − x , x x − x x = x + x , x x − x x = 0 (not SAS ) : → B φ = h − x + 2 x − x i −−−−−−−−−−−−−−−−−−−−→ B φ = x − x x − − − x x − − x −−−−−−−−−−−−−−−−−−−−−−→ B φ = x x x −−−−−−−→ B ǫ −→ K → , → B φ ∗ = x x x −−−−−−−→ B φ ∗ = x − x x − − − x x − − x −−−−−−−−−−−−−−−−−−−−−−→ B φ ∗ = h − x + 2 x − x i −−−−−−−−−−−−−−−−−−−−→ B → . x x − x x = x , x x − x x = x , x x − x x = 0 (not SAS ):19 → B φ (cid:2) − x x − − x − (cid:3) −−−−−−−−−−−−−−−−−−−−−−−−−−→ B φ x − x x − x − x − x −−−−−−−−−−−−−−−−−−−−−−−→ B φ x x x −−−−−−−−→ B ǫ −→ K → , → B φ ∗ x x x −−−−−−−−→ B φ ∗ x − x x − x − x − x −−−−−−−−−−−−−−−−−−−−−−−−→ B φ ∗ (cid:2) − x x − − x − (cid:3) −−−−−−−−−−−−−−−−−−−−−−−−−−→ B → . Next we present an algebra essentially regular in the sense of Gaddis (see [25] and [26]) but not
SAS .A N -filtered algebra A is essentially regular if and only if Gr ( A ) is AS (see Proposition 2.3.7 in [25]). Example 4.11.
Consider the algebra U := K { x, y } / h yx − xy + y i , by Corollary 2.3.14 in [25], U isessentially regular of global dimension 2. Clearly U is not Artin-Schelter, actually we will show that U isnot SAS . The semi-graduation of U is as in Example 4.5, so U ≥ = ⊕ p ≥ U p and this ideal coincides withthe two-sided ideal of U generated by x and y . Observe that U ( U / U ≥ ) ∼ = U K , where the structure of left U -module for K is given by the canonical projection ǫ : U → K (the same is true for the right structure,( U / U ≥ ) U ∼ = K U ). The following sequence is a free resolution of U K :0 → U φ = h y − x i −−−−−−−−−−−→ U φ = xy −−−−−→ U ǫ −→ K → . This statement can be proved using
SPBWE or simply by hand. In fact, ǫ is clearly surjective; φ is injectivesince if φ ( u ) = 0 for u ∈ U , then u (cid:2) y − x (cid:3) = 0, so u = 0 since U is a domain. Im ( φ ) = ker( ǫ ) = U ≥ since (cid:2) u v (cid:3) (cid:20) xy (cid:21) = ux + vy , with u, v ∈ U . Im ( φ ) ⊆ ker( φ ) since φ φ = 0: (cid:2) y − x (cid:3) (cid:20) xy (cid:21) = yx + (1 − x ) y = 0.Now, ker( φ ) ⊆ Im ( φ ): In fact, let (cid:2) u v (cid:3) ∈ ker( φ ), then ux + vy = 0; let u = u + u x + u y + u x + u xy + u y + · · · , v = v + v x + v y + v x + v xy + v y + · · · ,from ux + vy = 0 we conclude that all terms of u involving only x ′ s have coefficient equals zero, i.e., u = py for some p ∈ U , hence vy = − pyx = − p ( xy − y ) = p (1 − x ) y , but since A is a domain, v = p (1 − x ),whence, (cid:2) u v (cid:3) = p (cid:2) y − x (cid:3) ∈ Im ( φ ).Now we apply Hom U ( − , U U ) and we get the complex of right U -modules0 → Hom U ( K, U ) ǫ ∗ −→ Hom U ( U , U ) φ ∗ −→ Hom U ( U , U ) φ ∗ −→ Hom U ( U , U ) → . As in Example 4.8,
Hom U ( K, U ) = 0, Hom U ( U , U ) ∼ = U and Hom U ( U , U ) ∼ = U , so we obtain thecomplex 0 → U φ ∗ = xy −−−−−−→ U φ ∗ = h y − x i −−−−−−−−−−−→ U → , φ ∗ is injective. Moreover, Im ( φ ∗ ) = ker( φ ∗ ): Indeed, it is clear that Im ( φ ∗ ) ⊆ ker( φ ∗ ); let (cid:20) uv (cid:21) ∈ ker( φ ∗ ), then yu + (1 − x ) v = 0, from this by a direct computation we get that v = q ′ y for some q ′ ∈ U ,but again by a direct computation it is easy to show that given a polynomial q ′ there exists q ∈ U suchthat q ′ y = yq , whence v = yq for some q ∈ U . Hence, yu + (1 − x ) yq = 0 implies y ( u − xq ) = 0, so u = xq .Therefore, ker( φ ∗ ) ⊆ Im ( φ ∗ ) and we have proved the claimed equality.Thus, Ext U ( K, U ) = Hom U ( K, U ) = 0, Ext U ( K, U ) = 0 and Ext U ( K, U ) = U /Im ( φ ∗ ) ≇ K U .In fact, suppose there exists a right U -module isomorphism U /Im ( φ ∗ ) α −→ K U , let α (1) := λ , then α (1) · (1 − x ) = λ · (1 − x ), so α (1 · (1 − x )) = λ , i.e., α (0) = 0 = λ , whence α = 0, a contradiction. Weconclude that U is not SAS .Now we will show an algebra that is
SAS but is not essentially regular.
Example 4.12.
Let S be the algebra of Theorem 4.0.7 in [25] defined by B := K { x, y } / h yx − i . B has a semi-graduation as in Example 4.5. It is known that gld( B ) = 1 (see [25], Proposition 4.1.1 andalso [12]) and it is clear that B/B ≥ = 0. Thus, the condition (iv) in Definition 4.3 trivially holds and B is SAS . Now observe that Gr ( B ) ∼ = R yx , where R yx is defined in [25] by R yx := K { x, y } / h yx i .According to Corollary 2.3.14 in [25], B is not essentially regular.We conclude the list of examples with an algebra that is not essentially regular neither SAS . Example 4.13.
Let B := R yx . Observe that B is a finitely graded algebra with graduation as in inExample 4.5; by a direct computation we get the following exact sequences0 → B φ = h y i −−−−−−−−→ B φ = xy −−−−−→ B ǫ −→ K → , → B φ ∗ = xy −−−−−−→ B φ ∗ = h y i −−−−−−−−→ B → . According to [25], gld( B ) = 2, but note that B/yB ≇ B K . In fact, suppose there exists a right B -moduleisomorphism B/yB α −→ K B , let α (1) := λ , then α (1) · x = λ · x , so α ( x ) = 0, i.e., x = 0 but clearly x / ∈ yB .This says that B is not SAS neither essentially regular.The previous examples induce the following general result.
Theorem 4.14.
Let K be a field and A := σ ( R ) h x , . . . , x n i be a bijective skew P BW extension thatsatisfies the following conditions: (i) R and A are K -algebras. (ii) lgld( R ) < ∞ . (iii) R is a F SG ring with semi-graduation R = L p ≥ R p . (iv) R is connected, i.e., R = K . (v) For ≤ i ≤ n , σ i , δ i in Proposition 1.3 are homogeneous, and there exist i, j such that the parameter d ij in (1 . satifies d ij ∈ K − { } . hen, B is a SAS algebra.Proof.
First note that A is F SG and connected with semi-graduation A := K , A p := K h R q x α | q + | α | = p i for p ≥ x α as in Definition 1.2 and | α | := α + · · · + α n . Observe that if r , . . . , r m generate R as K -algebra,then r , . . . , r m , x , . . . , x n generate A as K -algebra. Moreover, dim K A p < ∞ for every p ≥ A ) < ∞ (see [38]). The condition (v) in the statement of the theorem says that A/A ≥ = 0, so the condition (iv) in Definition 4.3 trivially holds. Remark 4.15.
From the general homological properties of skew
P BW extensions we know that if R isa noetherian domain, then A is a noetherian domain ([38]). This agrees with the question of Problem 4. The Zariski cancellation problem (ZCP) arises in commutative algebra and can be formulated in thefollowing way: Let K be a field, A := K [ t , . . . , t n ] be the commutative algebra of usual polynomials and B be a commutative K -algebra, if A [ t ] ∼ = B [ t ], then A ∼ = B ?The ZCP has very interesting connections with some famous classical problems: The Automorphismproblem, the Dixmier conjecture, the Jacobian conjecture, among some others, see a discussion in [18].Recently, the ZCP has been considered for non-commutative algebras, in [13] Bell and Zhang studied theZariski cancellation problem for some non-commutative Artin-Schelter regular algebras; other works onthis problem are [16], [17], [40], [56]. In this section we will formulate a problem related to the Zariskicancellation problem in the context of SAS algebras.Despite of the most important results about the cancellation problem are for algebras over fields (seefor example Theorem 5.4), the general recent formulation of this problem is for R -algebras, where R is anarbitrary commutative domain (Definition 5.1). Thus, the general definition below includes the algebrasover fields as well as the particular case of Z -algebras, i.e., arbitrary rings. Definition 5.1 ([13]) . Let R be a conmutative domain and let A be a R -algebra. (i) A is cancellative if for every R -algebra B , A [ t ] ∼ = B [ t ] ⇒ A ∼ = B . (ii) A is strongly cancellative if for any d ≥ and every R -algebra B , A [ t , . . . , t d ] ∼ = B [ t , . . . , t d ] ⇒ A ∼ = B . (iii) A is universally cancellative if for any R -flat finitely generated commutative domain S such that S/I ∼ = R for some ideal I of S , and any R -algebra BA ⊗ S ∼ = B ⊗ S ⇒ A ∼ = B ,where the tensor product is over R . emark 5.2. (i) All isomorphisms in the previous definition are isomorphisms of R -algebras, and hence,these notions depend on the ring R .(ii) Observe that the commutative Zariski cancellation problem asks if the K -algebra of polynomials K [ t , . . . , t n ] is cancellative. Abhyankar-Eakin-Heinzer in [1] proved that K [ t ] is cancellative (actually,they proved that every commutative finitely generated domain of Gelfand-Kirillov dimension one is can-cellative, see Corollary 3.4 in [1]); Fujita in [22] and Miyanishi-Sugie in [44] proved that if charK = 0,then K [ t , t ] is cancellative; if charK >
0, Russell in [52] proved that K [ t , t ] is cancellative. Recently,in 2014, Gupta proved that if n ≥ charK > K [ t , . . . , t n ] is not cancellative (see [30], [31]).The ZCP problem for K [ t , . . . , t n ] remains open for n ≥ charK = 0. Proposition 5.3 ([13], Remark 1.2) . For any R -algebra A ,universally cancellative ⇒ strongly cancellative ⇒ cancellative. For the investigation of the ZCP problem for a given non-commutative algebra A have been usedsome subalgebras of A . The first one is the center. Theorem 5.4 ([13], Proposition 1.3) . Let K be a field and A be a K -algebra. If Z ( A ) = K , then A isuniversally cancellative, and hence, cancellative. In [39] was computed the center of many K -algebras interpreted as skew P BW extensions, some ofthem with trivial center, and hence, such algebras are cancellative.Another subalgebra involved in the study of the ZCP problem is the Makar-Limanov invariant.
Definition 5.5.
Let A be a R -algebra. (i) Der( A ) denotes the collection of R -derivations of A and LND( A ) the collection of locally nilpotent R -derivations of A ; δ ∈ Der( A ) is locally nilpotent if given a ∈ A there exists n ≥ such that δ n ( a ) = 0 . (ii) The Makar-Limanov invariant of A is defined to be ML( A ) := \ δ ∈ LND( A ) ker( δ ) . Theorem 5.6 ([13], Theorems 3.3 and 3.6) . Let A be a R -algebra that is a finitely generated domain offinite Gelfand-Kirillov dimension. (i) If ML( A [ t ]) = A , then A is cancellative. (ii) If charR = 0 and ML( A ) = A , then A is cancellative. The effective computation of ML( A ) is in general a difficult task, therefore other strategies andtechniques have been introduced in order to investigate the cancellation property for non-commutativealgebras. For example, setting algebraic condition on A , it is interesting to know if A becomes cancellative.In this direction recently have been proved the following results. Theorem 5.7 ([40], Theorem 4.1) . Let A be a R -algebra. If A is left ( or right ) artinian, then A isstrongly cancellative, and hence, cancellative. Theorem 5.8 ([56], Theorem 0.1) . Let A be a K -algebra, K a field with charK = 0 . If A is a noetherian ( left and right ) AS algebra generated in degree with gld( A ) = 3 and A is not P I , then A is cancellative. Observe that if in the previous theorem we could remove the condition not
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