Rosenberg’s reconstruction theorem

Brandenburg, Martin

doi:10.1016/j.exmath.2017.08.005

Cited by 14 publications

(12 citation statements)

References 14 publications

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“…Similarly, its action on the left hand tensorands of the odd components is the initial action of ε 1 ⊲ y. In conclusion, identifying (6)(7)(8) with M by a * ∼ = k ∼ = c * , a * = c * = 1, y ∈ A 1 acts as ε −1 1 (y) on the even-degree components of a ε 1 (M ) and as ε 1 (y) on the odd-degree components.…”

Section: Whenmentioning

confidence: 83%

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Exotic elliptic algebras of dimension 4

Chirvăsitu

Smith

2017

Advances in Mathematics

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Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A(E, τ ) that depends on this data:where S(E, τ ) is the 4-dimensional Sklyanin algebra associated to (E, τ ), M2(k) is the ring of 2 × 2 matrices over k, and Γ is (Z/2) × (Z/2) acting in a particular way as automorphisms of S and M2(k). The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E[2] on E. Following the ideas and results in papers of Artin-Tate-Van den Bergh, Smith-Stafford, and Levasseur-Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj nc (A). Point modules and fat point modules determine "points" in Proj nc (A). Line modules determine "lines" in Proj nc (A). Line modules for A are in bijection with certain lines in P(A * 1 ) ∼ = P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G(1, 3). Shelton-Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P 5 , and that deg(L) = 20. The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P(A * 1 ). Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.2010 Mathematics Subject Classification. 16E65, 16S38, 16T05, 16W50. Key words and phrases. Sklyanin algebras, elliptic algebras, line modules.2. The 4-dimensional Sklyanin algebras S(E, τ ) 2.1. Definition of the Sklyanin algebra. Always, α 1 , α 2 , α 3 are fixed elements in k − {0, ±1} such that α 1 + α 2 + α 3 + α 1 α 2 α 3 = 0. We often write α = α 1 , β = α 2 , and γ = α 3 . As in [9, §6], we fix a, b, c, i ∈ k such that a 2 = α, b 2 = β, c 2 = γ, and i 2 = −1.

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Section: Whenmentioning

confidence: 83%

“…We can say more. the action of y on the left hand tensorands of the even components in (6)(7)(8) is the initial action of ε −1 1 ⊲ y. Similarly, its action on the left hand tensorands of the odd components is the initial action of ε 1 ⊲ y.…”

Section: Whenmentioning

confidence: 99%

Exotic elliptic algebras of dimension 4

Chirvăsitu

Smith

2017

Advances in Mathematics

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“…In particular,

F

fixes the structure sheaf

scriptO

. It is well known that there is a unique automorphism

θ

double-struckX

such that

F ≃ θ^{*}

, the pullback functor; see [3, Theorem 5.4]. Here, we use the fact that a locally noetherian scheme is quasi‐separated.…”

Section: The Objective Categoriesmentioning

confidence: 99%

“…We have the following commutative diagram

where ‘res’ is the restriction functor, and we identify

coh - U_{i}

with

R_{i}

‐mod. The restriction functor ‘res’ induces the well‐known equivalence between

coh - U_{i}

and the Serre quotient category of

coh - X

by those sheaves supported on the complement of

U_{i}

; compare [3, Example 4.3]. It follows that

{false(θ |_{U_{i}} false)}^{*}

and thus

^{σ_{i}} false(- false)

are iso‐preserving.…”

Section: The Objective Categoriesmentioning

confidence: 99%

Liftable derived equivalences and objective categories

Chen

2020

Bull. London Math. Soc.

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We give two proofs of the following theorem and a partial generalization: if a finite-dimensional algebra A is derived equivalent to a smooth projective scheme, then any derived equivalence between A and another algebra B is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain class of projective schemes is triangle-objective, that is, any triangle autoequivalence on it, which preserves the isomorphism classes of all objects, is necessarily isomorphic to the identity functor.

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“…It is proved in [9] that this is a Grothendieck category. Furthermore X can be recovered from QchpXq [7,10,16].…”

Section: Non-commutative Geometrymentioning

confidence: 99%

Noncommutative versions of some classical birational transformations

Presotto¹,

Bergh²

2016

J. Noncommut. Geom.

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In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct an analogue of the Cremona transform for non-commutative projective planes.

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Rosenberg’s reconstruction theorem

Cited by 14 publications

References 14 publications

Exotic elliptic algebras of dimension 4

Exotic elliptic algebras of dimension 4

Liftable derived equivalences and objective categories

Noncommutative versions of some classical birational transformations

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