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## Noncommutative proj and coherent algebras

Venue: | Math. Res. Lett |

Citations: | 11 - 0 self |

### Citations

176 |
Noncommutative projective schemes
- Artin, Zhang
- 1994
(Show Context)
Citation Context ... quotient of the category of coherent modules over the corresponding algebra by the subcategory of finite-dimensional modules. In the Noetherian case a similar result was proved by Artin and Zhang in =-=[2]-=-. Introduction The main result of this paper is a slight generalization of the theorem of Artin and Zhang in [2] characterizing certain class of abelian categories that can be viewed as noncommutative... |

174 |
Faisceaux algébriques cohérents
- Serre
- 1955
(Show Context)
Citation Context ...t category QGRA of the category of graded A-modules by the subcategory of torsion modules. If A is commutative and is generated by a finite number of elements of degree 1 then by the theorem of Serre =-=[10]-=- the category of quasicoherent sheaves on Proj(A) is equivalent to QGRA. Therefore, one would like to think about QGRA as a suitable replacement for the latter category in the case when A is noncommut... |

91 | Noncommutative curves and noncommutative surfaces
- Stafford, Bergh
- 2001
(Show Context)
Citation Context ...egory of (nonnegatively) graded algebras is a subcategory of the category of Z-algebras: to every graded algebra A = ⊕n≥0An one can associate a Z-algebra AZ = ⊕Aij with Aij = Aj−i. As was observed in =-=[11]-=-, sec. 11.1, the AZ-theorem can be extended to the case when an ample sequence of objects does not have the form (σ n (O), n ∈ Z) for some object O and some autoequivalence σ, by working with Z-algebr... |

80 |
Polishchuk, Homological properties of associative algebras: the method of helices
- Bondal, E
- 1994
(Show Context)
Citation Context ... Our generalization is also formulated using the language of Zalgebras. It is worth mentioning that the notion of a coherent Z-algebra arises naturally in the theory of geometric helices developed in =-=[4]-=-. Namely, it was proved in [7] that in the situation when a triangulated category is generated by the geometric helix, certain natural pair of subcategories defines a t-structure iff the Z-algebra ass... |

49 | Categories of holomorphic vector bundles on non-commutative two-tori
- Polishchuk, Schwarz
- 2003
(Show Context)
Citation Context ...ies of holomorphic bundles on noncommutative tori. These categories can also be viewed as hearts of certain non-standard t-structures in derived categories of coherent sheaves on elliptic curves (see =-=[9]-=-). It is very easy to see that none of these categories is Noetherian. More precisely, every non-zero object in these categories is non-Noetherian. On the other hand, in the case when a noncommutative... |

38 |
Twisted homogeneous coordinate rings
- Artin, Bergh
- 1990
(Show Context)
Citation Context ...ue for M = Pi. Now using the fact that every coherent module is a quotient of a module from P one can easily prove the required statement by diagram chasing (one should check surjectivity first - see =-=[1]-=- (3.13)(i),(iii)). It follows that Γ ∗ M ∈ C0 for M ∈ coh b A, so Γ ∗ induces a functor Ψ : cohprojA = cohA/ coh b A → C/C0 such that Φ ◦ Ψ ≃ Id. Furthermore, for every X ∈ C and m ∈ Z there is a cano... |

26 | Noncommutative two-tori with real multiplication as noncommutative projective varieties
- Polishchuk
(Show Context)
Citation Context ...bjects can still be described in terms of the This work was partially supported by NSF grant DMS-0070967. 1corresponding graded algebra. Applications to noncommutative two-tori will be considered in =-=[8]-=-. We exploit the idea going back to Serre’s paper [10] that the correct abelian category replacing the category of finitely generated modules in the non-Noetherian case is the category of coherent mod... |

20 |
On a non-commutative analogue of the category of coherent sheaves on a projective scheme
- Verevkin
- 1992
(Show Context)
Citation Context ...ted by A1 over A0 = k there is an equivalence of categories cohprojA ≃ cohprojA (n) , where A (n) = ⊕i≥0Ain is a Veronese subalgebra of A. The similar result in the Noetherian case is due to Verevkin =-=[12]-=-, Theorem (A-5) (see also [2], Prop. 5.10). From now on we will always assume that the sequence E = (Ei) satisfies the following finiteness condition: for every object X ∈ C one has dimk HomC(Ei, X) <... |

2 |
Gröbner bases and the coherence of monomial associative algebra
- Piontkovskĭı
- 1996
(Show Context)
Citation Context ...gebra and its Veronese subalgebras. We believe that in the noncommutative world many natural construction lead to coherent algebras (but not necessarily Noetherian ones). For example, Piontkovskii in =-=[5]-=- proved coherence of a graded algebra with finite number of generators and a finite number of defining monomial relations. In [6] this result is generalized to a broader class of algebras. Acknowledgm... |

1 |
Noncommutative Gröbner bases, coherence of associative algebras, and divisibility in semigroups
- Piontkovskii
(Show Context)
Citation Context ...as (but not necessarily Noetherian ones). For example, Piontkovskii in [5] proved coherence of a graded algebra with finite number of generators and a finite number of defining monomial relations. In =-=[6]-=- this result is generalized to a broader class of algebras. Acknowledgments. This note is mostly based on a part of the author’s diploma work [7] carried out at Moscow State University in 1993 under t... |

1 |
On coherent algebras, diploma work
- Polishchuk
- 1993
(Show Context)
Citation Context ...rmulated using the language of Zalgebras. It is worth mentioning that the notion of a coherent Z-algebra arises naturally in the theory of geometric helices developed in [4]. Namely, it was proved in =-=[7]-=- that in the situation when a triangulated category is generated by the geometric helix, certain natural pair of subcategories defines a t-structure iff the Z-algebra associated to this helix is coher... |

1 | Koszul duality, JGP 5, no.3 - Beilinson, Ginzburg, et al. - 1988 |