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177
Noncommutative geometry based on commutator expansions
 J. Reine Angew. Math
, 1998
"... Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas ..."
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Cited by 36 (0 self)
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Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas
Serre duality for noncommutative projective schemes
 Proc. Amer. Math. Soc
, 1997
"... Abstract. We prove the Serre duality theorem for the noncommutative projective scheme proj A when A is a graded noetherian PI ring or a graded noetherian ASGorenstein ring. Let k be a eld and let A = L i0 Ai be an Ngraded right noetherian kalgebra. In this paper we always assume that A is locall ..."
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Cited by 30 (8 self)
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Abstract. We prove the Serre duality theorem for the noncommutative projective scheme proj A when A is a graded noetherian PI ring or a graded noetherian ASGorenstein ring. Let k be a eld and let A = L i0 Ai be an Ngraded right noetherian kalgebra. In this paper we always assume that A is locally nite in the sense that each Ai is
Noncommutative twotori with real multiplication as noncommutative projective varieties
 J. Geom. Phys
"... Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate a ..."
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Cited by 26 (4 self)
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Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative twotorus with real multiplication. These results are based on the techniques of [14] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.
Ideal classes of the Weyl algebra and noncommutative projective geometry
, 2001
"... Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le B ..."
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Cited by 26 (3 self)
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Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.
Marginal and relevant deformations of N=4 field theories and noncommutative moduli spaces of vacua
, 2000
"... We study marginal and relevant supersymmetric deformations of the N = 4 superYangMills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as noncommutative spaces. The construction of these spaces relies on the representation t ..."
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Cited by 24 (3 self)
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We study marginal and relevant supersymmetric deformations of the N = 4 superYangMills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as noncommutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from Fterm constraints. These field theories are dual to superstring theories propagating on deformations of the AdS5×S 5 geometry. We study Dbranes propagating in these vacua and introduce the appropriate notion of algebraic geometry for noncommutative spaces. The resulting moduli spaces of Dbranes have several novel features. In particular, they may be interpreted as symmetric products of noncommutative spaces. We show how mirror symmetry between these deformed geometries and orbifold theories follows from Tduality. Many features of the dual closed string theory may be identified within the noncommutative algebra. In particular, we make progress towards understanding the Ktheory necessary for backgrounds where the NeveuSchwarz antisymmetric tensor of the string is turned on, and we shed light on some aspects of discrete
Criteria for σampleness
 J. Amer. Math. Soc
"... Abstract. In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a σample divisor, where σ is an automorphism of a projective scheme X. Many open questions regarding σample divisors ha ..."
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Cited by 23 (1 self)
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Abstract. In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a σample divisor, where σ is an automorphism of a projective scheme X. Many open questions regarding σample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be σample. As a consequence, we show right and left σampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GKdimension. We also characterize which automorphisms σ yield a σample divisor. 1.
Regular algebras of dimension 4 and their A∞Extalgebras
 Duke Math. J
"... ABSTRACT. We construct four families of ArtinSchelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of ArtinSchelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noethe ..."
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Cited by 22 (10 self)
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ABSTRACT. We construct four families of ArtinSchelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of ArtinSchelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and CohenMacaulay. One of the main tools is Keller’s highermultiplication theorem on A∞Extalgebras.
Dualizing complexes and perverse sheaves on noncommutative ringed schemes
, 2002
"... Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to sh ..."
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Cited by 21 (9 self)
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Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to show that a rigid dualizing complex exists on every affine open set in X. In the geometric part of the proof we construct a perverse tstructure on the derived category of bimodules, and this allows us to glue the affine rigid dualizing complexes to
Sklyanin algebras and Hilbert schemes of points
"... We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective ..."
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Cited by 20 (2 self)
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We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2. The generic noncommutative plane corresponds to the Sklyanin algebra S = Skl(E, σ) constructed from an automorphism σ of infinite order on an elliptic curve E ⊂ P 2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1 − n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P² \ E.