Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Polishchuk, Alexander

doi:10.1007/978-3-8348-0352-8_16

Cited by 30 publications

(32 citation statements)

References 21 publications

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“…The author thanks Jianmin Chen for the reference [14] and Guodong Zhou for the reference [1]. The author is very grateful to Zengqiang Lin for pointing out several misprints and to David Ploog for many helpful suggestions.…”

Section: Acknowledgmentsmentioning

confidence: 96%

A note on separable functors and monads with an application to equivariant derived categories

Chen

2015

Abh. Math. Semin. Univ. Hambg.

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For an adjoint pair (F, U ) of functors, we prove that U is a separable functor if and only if the defined monad is separable and the associated comparison functor is an equivalence up to retracts. In this case, under an idempotent completeness condition, the adjoint pair (F, U ) is monadic. This applies to the comparison between the derived category of the category of equivariant objects in an abelian category and the category of equivariant objects in the derived category of the abelian category.

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Section: Acknowledgmentsmentioning

confidence: 96%

A note on separable functors and monads with an application to equivariant derived categories

Chen

2015

Abh. Math. Semin. Univ. Hambg.

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“…In [4] (using work of Polishchuk [15]) we showed that the homomorphism T 3 is an injective map on K 0 (A κ θ ) = Z 8 in the case that θ is irrational -and in the rational case we would have to include Connes' cyclic 2-cocycle that picks out the "label of the trace". Based on the values in Table 2 of [3, (page 37)], 9 the unbounded traces ψ k of Cubic invariant projections take values in the following lattice in the complex plane…”

Section: ρ(E(t)) = E(t) κ(E(t)) = E(t);mentioning

confidence: 99%

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

Walters

2015

Nuclear Physics B

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The Hexic transform ρ of the noncommutative 2-torus A θ is the canonical order 6 automorphism defined by ρ(U ) = V , ρ(V ) = e −πiθ U −1 V , where U , V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU = e 2πiθ UV. The Cubic transform is κ = ρ 2 . These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C * -algebras A ρ θ , A κ θ are noncommutative Z 6 , Z 3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ , a projection e of trace q 2 θ − pq is constructed in A θ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eA θ e contains a Hexic invariant q × q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C ∞ -projections of the continuous field {A t } 0

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“…Then the group Z 2 acts on E by the rule (x : y : z) → (x : −y : z). By a result of Geigle and Lenzing [17, Proposition 4.1 and Example 5.8], see also [33,Corollary 1.4], there is a derived equivalence…”

Section: Non-commutative Curves With Nodal Singularitiesmentioning

confidence: 99%

Tilting on non-commutative rational projective curves

Burban

Drozd

2010

Math. Ann.

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In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy 2 = x 3 + x 2 z.

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Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Cited by 30 publications

References 21 publications

A note on separable functors and monads with an application to equivariant derived categories

A note on separable functors and monads with an application to equivariant derived categories

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

Tilting on non-commutative rational projective curves

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