## EQUIVALENCES FOR NONCOMMUTATIVE PROJECTIVE SPACES

### Citations

218 |
Fourier-Mukai Transforms in Algebraic Geometry
- Huybrechts
- 2006
(Show Context)
Citation Context ...ctor, a previous result by Bondal and Orlov ([6], [15]) regarding autoequivalences preserving an ample sequence yields the result. A good account of this result can also be found in Huybrechts’ book (=-=[10]-=-). We will now prove (1)⇒(4). Let σ ∈ Σn and m1, ...,mn as in (1). Choose a1i and a 2 i such that mi = a2i a1 i and inductively define aki = ak−2 i a k−1 i . Further define Φ : Bω −→ Bω′ by Φ(α k l ) ... |

176 |
Noncommutative projective schemes
- Artin, Zhang
- 1994
(Show Context)
Citation Context ...c zero and ωij ∈ K ∗, for all i and j. Note that ωijωji = 1. Our objects of study will be the spaces Pn−1ω := Proj(S n ω). These are the noncommutative projective spaces associated to Snω , following =-=[2]-=-. They are pairs (tails(Snω), piS n ω) where tails(S n ω) is the quotient category of finitely generated graded modules over Snω , gr(S n ω) by its subcategory of torsion modules and piS n ω is the pr... |

175 |
Graded algebras of global dimension 3
- Artin, Schelter
- 1987
(Show Context)
Citation Context ...ctor AXX , where X = (X1, ..., Xn) is the vector formed by the generators of Snω . This is because I n ω is generated in degree 2. A similar matrix appears in the original work of Artin and Schelter (=-=[1]-=-). For each x ∈ E1, we need at least a 1-dimensional space of solutions for the equation Axȳ = 0 so that we get a point in the projective space. This happens if and only if the rank of Ax is less or ... |

155 | Representations of associative algebras and coherent sheaves, Izv - Bondal - 1989 |

132 |
Some algebras associated to automorphisms of elliptic curves
- Artin, Tate, et al.
- 1990
(Show Context)
Citation Context ...ions of Snω and S n ω′ are isomorphic. We study birational equivalences between these spaces in section 3. An interesting invariant of such a space is its point variety. We recall the definition from =-=[4]-=-. We start with point modules. Definition 1.1 (Artin, Tate, Van den Bergh). A graded R-module M is said to be a point module if: Supported by FCT - Portugal, research grant SFRH/BD/28268/2006. Special... |

132 |
Reconstruction of a variety from the derived category and groups of autoequivalences
- Bondal, Orlov
(Show Context)
Citation Context ...is clear for 0 ≤ i ≤ n− 1 since the middle equivalence is induced from Φ). Observing that this sequence is ample ([3]), and that it is preserved by the functor, a previous result by Bondal and Orlov (=-=[6]-=-, [15]) regarding autoequivalences preserving an ample sequence yields the result. A good account of this result can also be found in Huybrechts’ book ([10]). We will now prove (1)⇒(4). Let σ ∈ Σn and... |

126 | Equivalences of derived categories and K3 surfaces - Orlov - 1997 |

119 |
Blocks of Tame Representation Type and Related Algebras
- Erdmann
- 1990
(Show Context)
Citation Context ...hat Bω is a basic algebra. Indeed, a progenerator in mod(Bω) such that its endomorphism algebra is Bω′ (thus basic as well) has to be Bω. Thus Morita equivalence in this case implies isomorphism (see =-=[8]-=- for more details). Remark 2.8. Minamoto and Mori have recently showed that the graded Morita equivalence classes within certain families of algebras (Snω in our case) depend only on the isomorphism... |

52 | Twisted graded algebras and equivalences of graded categories
- Zhang
- 1996
(Show Context)
Citation Context ...perscript n will be dropped and assumed to be fixed. 2. Graded equivalences Our target is to classify up to equivalence the categories Gr(Snω). For this we will use Zhang’s theoy of twisting systems (=-=[18]-=-) which we quickly review now. Let R and S be connected N0-graded right Noetherian K-algebras. Definition 2.1 (Zhang). A twisting system is a set τ = {τn : n ∈ N0} of K-linear, degree preserving bijec... |

24 |
Quadratic Algebras Associated with the Union of a Quadric and a Line in P3
- Vancliff
- 1994
(Show Context)
Citation Context ...ety of P n−1 defined by⋂ q(ijk)(ω) 6=1 V (XiXjXk), where V (XiXjXk) is the zero locus of XiXjXk in P n−1. Proof. This proof was first sketched by Hattori ([9]). It also uses some ideas from Vancliff (=-=[17]-=-). For practical purposes we shall consider the generators of Inω rewritten in the form θjiXjXi − θijXiXj where θ is a fixed n × n matrix of parameters in C∗ such that θijθ −1 ji = ωij . Following the... |

20 | Quantum unique factorisation domains,
- Launois, Lenagan, et al.
- 2006
(Show Context)
Citation Context ...tive set of parameters, the only normal elements (i.e., elements x of Sω such that xSω = Sωx) of degree 1 are scalar multiples of X1, X2, ..., Xn. For that purpose we recall the following result from =-=[11]-=-. Lemma 2.7 (Launois, Lenagan, Rigal,[11]). Let R be a prime noetherian ring and suppose that d, s are normal elements of R such that dR is prime and s /∈ dR. Then, there is a unit v ∈ R such that sd ... |

8 | The structure of AS-Gorenstein algebras,
- Minamoto, Mori
- 2011
(Show Context)
Citation Context ...f algebras (Snω in our case) depend only on the isomorphism classes of certain related finite dimensional algebras (Bnω in our case). In that sense, their results generalise the previous proposition (=-=[13]-=-). 3. Birational equivalence The context in which these q-cyclic numbers, defined in the previous section, appear naturally is explored below. They actually concern the birational classification of th... |

7 | Helices on del Pezzo surfaces and tilting Calabi-Yau algebras
- Bridgeland, Stern
(Show Context)
Citation Context ...= ωijα k i α k−1 j , where k ∈ {2, 3} and 1 ≤ i 6= j ≤ 4. It is easy to see that such a strong exceptional sequence forms a basis for the Grothendieck group of the derived category (see, for example, =-=[7]-=-). Therefore, the number of elements in such sequence is preserved via derived equivalence. Lemma 1.5. If Db(tails(Snω)) ∼= Db(tails(Smω′)) then m = n. In particular if the categories themselves are e... |

5 |
Sur les endomorphismes des tores quantiques
- Richard
(Show Context)
Citation Context ...taining Sω as a subalgebra, i.e., Tω = C 〈 X±11 , ..., X ±1 n 〉 / 〈XjXi − ωijXiXj , i, j ∈ {1, ..., n}〉. Richard classified, up to isomorphism, this family of algebras, which are called quantum tori (=-=[16]-=-). Lemma 3.2 (Richard, [16]). (1) Tω ∼= Tω′ if and only if there is a matrix A = (aij) ∈ GLn(Z) such that ω ′ ij = ∏ 1≤k,l≤n ω akiatj kt ; (2) Tω is simple if and only if there is not a nonzero vector... |

1 |
Mirror symmetry for weighted projective spaces and their noncommutative deformations
- Auroux, Katzarkov, et al.
- 2008
(Show Context)
Citation Context ... respective quotient. Hence, Tails(Snω) represents quasicoherent sheaves over Pn−1ω . This follows Artin and Zhang’s ([2]) formulation of noncommutative projective geometry. We follow the approach in =-=[3]-=- to study these categories and we shall keep the notation therein unless stated otherwise. Recall ([2],[14]) that Pn−1ω ∼= Pn−1ω′ if there is an equivalence of categories F between tails(Snω) and tail... |

1 |
Noncommutative projective spaces of quantum affine coordinate rings which are birational but not isomorphic, unpublished
- Hattori
(Show Context)
Citation Context ... 4.2. The point variety of Snω is the subvariety of P n−1 defined by⋂ q(ijk)(ω) 6=1 V (XiXjXk), where V (XiXjXk) is the zero locus of XiXjXk in P n−1. Proof. This proof was first sketched by Hattori (=-=[9]-=-). It also uses some ideas from Vancliff ([17]). For practical purposes we shall consider the generators of Inω rewritten in the form θjiXjXi − θijXiXj where θ is a fixed n × n matrix of parameters in... |

1 |
den Bergh
- Bruyn, Smith, et al.
- 1996
(Show Context)
Citation Context ...ee that (E2)red = (E1)red. Thus, the right projection following the inverse of the left projection can be regarded as an isomorphism whose graph is (Ω2)red. Lemma 4.3 (Le Bruyn, Smith, Van den Bergh, =-=[12]-=-). If (Ω2)red is the graph of an isomorphism between (E1)red and (E2)red then Ω2 is the graph of an isomorphism between E1 and E2. This lemma proves that Ω2 can be regarded as the graph of an automorp... |

1 |
Noncommutative projective spaces and point schemes, Algebras
- Mori
(Show Context)
Citation Context ...nd Zhang’s ([2]) formulation of noncommutative projective geometry. We follow the approach in [3] to study these categories and we shall keep the notation therein unless stated otherwise. Recall ([2],=-=[14]-=-) that Pn−1ω ∼= Pn−1ω′ if there is an equivalence of categories F between tails(Snω) and tails(S n ω′) such that F (O n ω) = O n ω′ . An equivalence at the level of graded modules naturally induces an... |