Existence  of critical modules of GK-dimension 2  over elliptic algebras

Ajitabh, Kaushal

doi:10.1090/s0002-9939-00-05322-3

Cited by 4 publications

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“…, where D (n e ,n o ) are smooth quasi-projective varieties of dimension 2(n e − (n e − n o ) 2 ). If in addition A is of generic type A then the varieties D (n e ,n o ) are affine.…”

Section: Theorem 13 Assume K Is Uncountable Let a Be An Elliptic Cmentioning

confidence: 99%

“…Anthony Henderson pointed out to us this number is given by the number of partitions of n e − (n e − n o ) 2 . Alternatively, see [16].…”

Section: Moreover If a Is Elliptic For Which σ Has Infinite Order Thimentioning

confidence: 99%

“…The Hilbert series of the Veronese subalgebra A (2) = k ⊕ A 2 ⊕ A 4 ⊕ · · · of A is the same as the Hilbert series of the commutative ring k[x 0 , x 1 , x 2 , x 3 ]/(x 0 x 1 − x 2 x 3 ) which is the homogeneous coordinate ring of a quadratic surface (quadric) in P 3 . Since [33] Tails(A) ∼ = Tails(A (2) ) and cd X = 2 we therefore think of Proj A as a quantum quadric, a noncommutative analogue of the quadric surface P 1 × P 1 .…”

Section: Definition 23 [3]mentioning

confidence: 99%

“…Since [33] Tails(A) ∼ = Tails(A (2) ) and cd X = 2 we therefore think of Proj A as a quantum quadric, a noncommutative analogue of the quadric surface P 1 × P 1 . See also [24].…”

Section: Definition 23 [3]mentioning

confidence: 99%

“…• j : E ∼ = P 2 if A is quadratic, respectively j : E ∼ = P 1 × P 1 if A is cubic; or • j : E → P 2 is an embedding of a divisor E of degree three if A is quadratic, respectively j : E → P 1 × P 1 where E is a divisor of bidegree (2,2) if A is cubic and σ ∈ Aut(E). In the first case we say A is linear, otherwise A is called elliptic.…”

Section: Introductionmentioning

confidence: 99%

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Ideals of cubic algebras and an invariant ring of the Weyl algebra

Naeghel

Marconnet

2007

Journal of Algebra

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We classify reflexive graded right ideals, up to isomorphism and shift, of generic cubic three-dimensional Artin-Schelter regular algebras. We also determine the possible Hilbert functions of these ideals. These results are obtained by using similar methods as for quadratic Artin-Schelter algebras [K. De Naeghel, M. Van den Bergh, Ideal classes of three-dimensional Sklyanin algebras, J. Algebra 276 (2) (2004) 515-551; K. De Naeghel, M. Van den Bergh, Ideal classes of three dimensional Artin-Schelter regular algebras, J. Algebra 283 (1) (2005) 399-429]. In particular our results apply to the enveloping algebra of the Heisenberg-Lie algebra from which we deduce a classification of right ideals of the invariant ring A ϕ 1 of the first Weyl algebra A 1 = k x, y /(xy − yx − 1) under the automorphism ϕ(x) = −x, ϕ(y) = −y.

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“…, where D (n e ,n o ) are smooth quasi-projective varieties of dimension 2(n e − (n e − n o ) 2 ). If in addition A is of generic type A then the varieties D (n e ,n o ) are affine.…”

Section: Theorem 13 Assume K Is Uncountable Let a Be An Elliptic Cmentioning

confidence: 99%

“…Anthony Henderson pointed out to us this number is given by the number of partitions of n e − (n e − n o ) 2 . Alternatively, see [16].…”

Section: Moreover If a Is Elliptic For Which σ Has Infinite Order Thimentioning

confidence: 99%

Section: Definition 23 [3]mentioning

confidence: 99%

“…Since [33] Tails(A) ∼ = Tails(A (2) ) and cd X = 2 we therefore think of Proj A as a quantum quadric, a noncommutative analogue of the quadric surface P 1 × P 1 . See also [24].…”

Section: Definition 23 [3]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ideals of cubic algebras and an invariant ring of the Weyl algebra

Naeghel

Marconnet

2007

Journal of Algebra

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Blowup subalgebras of the Sklyanin algebra

Rogalski

2011

Advances in Mathematics

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We describe some interesting graded rings which are generated by degree-3 elements inside the Sklyanin algebra S, and prove that they have many good properties. Geometrically, these rings R correspond to blowups of the Sklyanin P 2 at 7 or fewer points. We show that the rings R are exactly those degree-3-generated subrings of S which are maximal orders in the quotient ring of the 3-Veronese of S.The author was partially supported by NSF grants DMS-0600834 and DMS-0900981. subalgebras of T . Now g ∈ T 1 and T /T g ∼ = B(E, L 3 , σ 3 ). Let x indicate the image of x ∈ S under the quotient map S → S/Sg. Given any effective Weil divisor D on E with 0 ≤ deg D ≤ 7, we let. In other words, V (D) consists exactly of those global sections of L 3 which vanish along the divisor D. We define R(D) = k V (D) ⊆ T . One goal of the paper is to prove that the rings R(D) have many nice properties (with the notable exception of finite global dimension). We summarize these in the following theorem. The definitions of the properties involved are reviewed in Section 2.(2) (Theorems 6.3, 6.7) R is strongly noetherian, satisfies the Artin-Zhang χ conditions, is Auslander-Gorenstein and Cohen Macaulay. The noncommutative projective scheme proj-R has cohomological dimension 2. R is a maximal order.(3) (Theorem 10.4) There is a homogeneous ideal I of R with GK R/I ≤ 1 which is essentially minimal among such ideals in the following sense: given a homogeneous ideal J of R with GK R/J ≤ 1, then J ⊇ I ≥n for some n ≥ 0.We make a comment on the significance of part (3) of the theorem, which is actually the part on which we expend the most effort in the paper. The generic Sklyanin algebra S and its Veronese ring T are known to have no homogeneous factor rings of GK-dimension 1. The rings R(D) sometimes do, so the point of part(3) above is that such factor rings can be strongly controlled. This fact is needed in the proof of our main result, which can be found in Section 10. It classifies degree 1-generated orders in T , as follows.Theorem 1.2. Let V ⊆ T 1 and let A = k V ⊆ T . Assume that Q gr (A) = Q gr (T ).(1) There is a unique effective divisor D on E with 0 ≤ deg D ≤ 7 such that A ⊆ R(D) with A and R(D) equivalent orders. In particular, the rings R(D) are exactly the degree-1 generated subalgebras of T which are maximal orders in Q gr (T ).(2) If A is noetherian, then in part (1) A ⊆ R(D) is a finite ring extension. If g ∈ A, then A is indeed noetherian.When g ∈ A in the preceding theorem, then it can happen that A is not noetherian and A ⊆ R(D) is not a finite ring extension; see Section 11. Also, similar methods can be used to analyze the subalgebras of S generated in degree 1, which was in fact the original project we attempted. Since dim k S 1 = 3, the only interesting degree-1-generated subalgebras are A = k V , where V ⊆ S 1 with dim k V = 2. We show in Theorem 12.2 below that for such an A, either the 3-Veronese ring A (3) is equal to the ring R(D) for a divisor D of degree 3, or else A equals S in all large degrees.

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Hilbert series of modules of GK-dimension two over elliptic algebras

Naeghel

2007

Journal of Algebra

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We characterize the Hilbert functions and minimal resolutions of (critical) Cohen-Macaulay graded right modules of Gelfand-Kirillov dimension two over generic quadratic and cubic three-dimensional ArtinSchelter regular algebras. See also [Y. Berest, G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an appendix by Michel Van den Bergh), Int. Math. Res. Not. 2002 (26) (2002) 1347-1396; L. Le Bruyn, Moduli spaces for right ideals of the Weyl algebra, J. Algebra 172 (1995) 32-48; T.A. Nevins, J.T. Stafford, Sklyanin algebras and Hilbert schemes of points, math.AG/0310045, 2003. [8,14,15]].

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Existence of critical modules of GK-dimension 2 over elliptic algebras

Cited by 4 publications

References 4 publications

Ideals of cubic algebras and an invariant ring of the Weyl algebra

Ideals of cubic algebras and an invariant ring of the Weyl algebra

Blowup subalgebras of the Sklyanin algebra

Hilbert series of modules of GK-dimension two over elliptic algebras

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