The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Bell, Jason P.; Rogalski, D.; Sierra, Susan J.

doi:10.1007/s11856-010-0111-0

Cited by 31 publications

(34 citation statements)

References 31 publications

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“…Let Q = Ann(M ) and so Q = 0 by Lemma 4.7. If P is the largest homogeneous prime ideal contained in Q, then P corresponds to a reduced closed subscheme of F ⊂ E and B/P ∼ = B(F, O F (1), σ| F ) ( [5], Lemma 3.3). These subschemes are well-understood in this case, and so either F corresponds to a subscheme in the twisted homogeneous coordinate ring associated to gr(A), or else F is fixed pointwise by σ| F , in which case B/P is commutative.…”

Section: H(a) = T (W )/(R) and Gr(a) = T (V )/(S)mentioning

confidence: 99%

PBW deformations of Artin–Schelter regular algebras

Gaddis

2016

J. Algebra Appl.

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We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.

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Section: H(a) = T (W )/(R) and Gr(a) = T (V )/(S)mentioning

confidence: 99%

PBW deformations of Artin–Schelter regular algebras

Gaddis

2016

J. Algebra Appl.

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“…It turns out that the phenomena exhibited by the above theorems is not specific to differential-algebraic geometry. A difference-algebraic analogue was proved in 2010 (independently) by Bell-Rogalski-Sierra [6] and Cantat [10]. We will say more about that in §7 below.…”

Section: Jouanalou-type Theoremsmentioning

confidence: 85%

“…orbit is a φ-invariant subvariety that is not the union of irreducible φ-invariant subvarieties. The following Jouanolou-type theorem is a special case of Theorem 3.4 that appeared originally in [6] and [10], independently.…”

Section: An Aside On the Difference Casementioning

confidence: 94%

“…This is a finitely generated free abelian group, and the action is said to be quasi-unipotent if all of its eigenvalues are roots of unity. In [17] it is shown that quasi-unipotence is equivalent to the existence of a "φ-ample divisor on X", which was the condition isolated in [6] as relevant to higher codimension analogues of Theorem 7.1. Conjecture 8.5 of [6] can therefore be expressed as follows:…”

Section: An Aside On the Difference Casementioning

confidence: 99%

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Model theory and Kähler geometry

Moosa

Pillay

2008

Model Theory With Applications to Algebra and Analysis

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We survey and explain some recent work at the intersection of model theory and bimeromorphic geometry (classification of compact complex manifolds). Included here are the essential saturation of the many-sorted structure C of Kähler manifolds, the conjectural role of hyperkähler manifolds in the description of strongly minimal sets in C, and Campana's work on the isotriviality of hyperkähler families and its connection with the nonmultidimensionality conjecture.

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“…, y d,j ) ∈ C d such that Equation (2) holds. So the cardinality of T q is the same as the set of solutions to Equations (2) in C n+md with x q = 1. It is this latter set that we count.…”

Section: Proof We Writementioning

confidence: 99%

Poisson algebras via model theory and differential-algebraic geometry

Bell¹,

Launois²,

Sánchez³

et al. 2017

J. Eur. Math. Soc.

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Abstract. Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

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The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Cited by 31 publications

References 31 publications

PBW deformations of Artin–Schelter regular algebras

PBW deformations of Artin–Schelter regular algebras

Model theory and Kähler geometry

Poisson algebras via model theory and differential-algebraic geometry

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