Non-commutative $\mathbb{P}^{1}$-bundles over commutative schemes

Bergh, Michel Van den

doi:10.1090/s0002-9947-2012-05469-9

Cited by 35 publications

(99 citation statements)

References 29 publications

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“…An important instance constitutes the basis for noncommutative projective geometry, in which to a sufficiently nice Z-graded algebra A, one associates a category QGr(A) of "quasi-coherent graded modules", obtained as the quotient of the graded modules by the torsion modules. This is motivated by Serre's well-known result that, for A commutative, we have QGr(A) ∼ = Qch(Proj(A)) (see [12,45,49,50] for more details and generalizations).…”

Section: Motivationmentioning

confidence: 99%

Non-commutative deformations and quasi-coherent modules

Van

Liu

Lowen

2016

Sel. Math. New Ser.

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We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the GerstenhaberSchack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have Qch(O) ∼ = Qch(X )). Under a natural identification of GS cohomology of O and

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Section: Motivationmentioning

confidence: 99%

Non-commutative deformations and quasi-coherent modules

Van

Liu

Lowen

2016

Sel. Math. New Ser.

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“…In this section, following [16], we define the noncommutative symmetric algebra of certain R − S-bimodules. Some of the exposition is adapted from [10, Section 2].…”

Section: Duality and Admissible Bimodulesmentioning

confidence: 99%

“…Next, we recall the definition of a noncommutative version of P 1 due to M. Van den Bergh [16]. If K and L are finite extensions of k and N is a K − L-bimodule of finite dimension as both a K-module and an L-module, then one can form the Z-algebra S nc (N ), the noncommutative symmetric algebra of N .…”

Section: Introductionmentioning

confidence: 99%

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Witt’s Theorem for Noncommutative Conic Curves

Nyman

2015

Appl Categor Struct

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“…The key observation is that many interesting categories, such as categories of twisted sheaves on global quotient stacks, can be viewed as suitably defined categories of modules over kernel algebras. Note that a very similar setup involving algebra objects in the monoidal category of sheaf bimodules is used in [50] to define noncommutative P 1 -bundles.…”

Section: Introductionmentioning

confidence: 99%

Kernel algebras and generalized Fourier–Mukai transforms

Polishchuk¹

2011

J. Noncommut. Geom.

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Abstract. We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as Dmodules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of [44] concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon [33] concerning an analog of the Fourier transform for derived categories of quasi-coherent sheaves on a dual pair of generalized 1-motives.

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Non-commutative $\mathbb{P}^{1}$-bundles over commutative schemes

Cited by 35 publications

References 29 publications

Non-commutative deformations and quasi-coherent modules

Non-commutative deformations and quasi-coherent modules

Witt’s Theorem for Noncommutative Conic Curves

Kernel algebras and generalized Fourier–Mukai transforms

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