#### DMCA

## 4 NONCOMMUTATIVE GRASSMANIAN OF CODIMENSION TWO HAS COHERENT COORDINATE RING

### Citations

203 | Generators and representability of functors in commutative and noncommutative geometry, Mosc - Bondal, Bergh - 2003 |

178 |
Noncommutative projective schemes
- Artin, Zhang
- 1994
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Citation Context ...ebras play the roles of coordinate rings of noncommutative projective spaces in a version of noncommutative projective geometry [Po, BVdB] which generalizes the well-known approach of Artin and Zhang =-=[AZ]-=-. Namely, suppose that a regular algebra of global dimension d is graded coherent. Consider the quotient category qgrA = cmodA/ torsA of the category cmodA of finitely presented (=graded coherent) rig... |

172 |
Graded algebras of global dimension 3
- Artin, Schelter
- 1987
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Citation Context ...re we are interested in the case d = 3, that is, in the case of noncommutative planes. The famous classification of 3-dimensional regular algebras A of polynomial growth is obtained Artin and Shelter =-=[AS]-=-. Particularly, they have shown that these algebras are Noetherian (hence, coherent). It is not hard to construct also non-Noetherian 3-dimensional regular algebras (e.g., one may follow the approach ... |

153 | Representations of associative algebras and coherent sheaves - Bondal - 1990 |

80 |
Polishchuk, Homological properties of associative algebras: the method of helices
- Bondal, E
- 1994
(Show Context)
Citation Context ...l dimension (say, d) and satisfies the following Gorenstein property: Ext iA(k, k) ∼= { k∗[l] for some l ∈ Z, i = d 0, i 6= d. The same notion of regularity is extend (following Bondal and Polishchuk =-=[BP]-=-) to a slightly more general case of a Z-algebra A, see Subsection 2.1 below. Regular algebras play the roles of coordinate rings of noncommutative projective spaces in a version of noncommutative pro... |

29 | Quadratic Algebras. University Lecture Series, 37 - Polishchuk, Positselski - 2005 |

21 |
Synergy in the theories of Gröbner bases and path algebras
- Farkas, Feustel, et al.
- 1993
(Show Context)
Citation Context ..., where the word w has length at least r, we have u ∗ v = (u ∗ w)s. Note that the above definition is compatible with the standard assumptions of the Gröbner basis theory for ideals in path algebras =-=[FFG]-=-. In this theory, it is assumed that the above set of generators X of a path algebra kQ of a quiver Q consists of two parts, X = V ∪ E, where V is the set of vertices and E is the set of arrows of the... |

13 | Coherent algebras and noncommutative projective lines - Piontkovski |

11 | Deformation theory of objects in homotopy and derived categories III: abelian categories
- Lunts, Orlov
(Show Context)
Citation Context ...r 3-dimensional regular algebra. In contrast to the previous cases, it seems that the approach based on the above lemma fails for this algebra.1 This algebra is introduced by Efimov, Luntz, and Orlov =-=[ELO]-=- under the name Noncommutative Grassmanian (see Subsection 2.2 below for an explicit definition). According to [ELO, Section 7], a noncommutative Grassmanian NGr(m,n) ‘is a true noncommutative moduli ... |

11 | Noncommutative proj and coherent algebras - Polishchuk - 2005 |

9 |
Algebraic structure of Yang-Mills theory, The unity of mathematics, The unity of
- Movshev, Schwarz
- 2006
(Show Context)
Citation Context ...ebra is coherent. However, there are important examples for which the coherence is established. These are the octonion algebra of P. Smith [S], the Yang–Mills algebra introduced by Movshev and Swartz =-=[MS]-=- which is coherent by a theorem of Herscovich [H] (the Yang–Mils algebra introduced by Connes and Dobios–Violette [CDV] is a particular case of it), and 3-Calabi–Yau algebras which are Ore extensions ... |

4 |
Noncommutative Groebner bases, coherence of associative algebras, and divisibility in semigroups, Fundam
- Piontkovski
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Citation Context ...lgebra (in particular, it admits a quadratic Gröbner basis of relations). Note that we do not know if Â is PBW or not. In Proposition 3.5, we prove that A satisfies a property of bounded processing =-=[P01]-=-, that is, the structure of the multiplication of paths in this quiver algebra is essentially depend only on bounded segments of the multipliers. We recall necessary definitions in Subsection 3.3. Usi... |

4 | A 3-Calabi-Yau algebra with G2 symmetry constructed from the octonions, arXiv:1104.3824v1
- Smith
(Show Context)
Citation Context ...research grant 12-01-0134. 1 2 DMITRI PIONTKOVSKI such an algebra is coherent. However, there are important examples for which the coherence is established. These are the octonion algebra of P. Smith =-=[S]-=-, the Yang–Mills algebra introduced by Movshev and Swartz [MS] which is coherent by a theorem of Herscovich [H] (the Yang–Mils algebra introduced by Connes and Dobios–Violette [CDV] is a particular ca... |

2 |
Representations of super Yang-Mills algebras
- Herscovich
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Citation Context ...mples for which the coherence is established. These are the octonion algebra of P. Smith [S], the Yang–Mills algebra introduced by Movshev and Swartz [MS] which is coherent by a theorem of Herscovich =-=[H]-=- (the Yang–Mils algebra introduced by Connes and Dobios–Violette [CDV] is a particular case of it), and 3-Calabi–Yau algebras which are Ore extensions of 2-Calaby–Yau ones by He, Oystaeyen, and Zhang ... |

2 | Linear equations over noncommutative graded rings
- Piontkovski
(Show Context)
Citation Context ...als [P01, Prop. 7]: If a right sided ideal I in A is generated in degreed ≤ d for some d, then its relations are concentrated in degrees ≤ d + 6. It follows that A is universally coherent in terms of =-=[P05]-=-, see also [P05, Prop. 4.10]. Similar linear estimates for the generators of the entries of the minimal projective resolution for each finitely presented A–module follow from [P05, Prop. 4.3]. Referen... |

1 |
Graded Calabi Yau algebras of dimension 3, Journal of pure and applied algebra
- Bocklandt
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Citation Context ...)(m− n), . . . ,OP(kn)(−1),OP(kn)). It follows from [BP] that A is Koszul and Gorenstein of global dimension d+ 1. Note that A is a so-called graded Calabi–Yau algebra of dimension 3 (in the sense of =-=[Bock]-=-). This follows from the same property of the corresponding algebra Â, see Subsection 3.1 below. It is pointed out in [ELO, Remark 8.23] that the description of the derived category of QModA (“quasic... |

1 |
Oystaeyen F., Zhang Y., Graded 3-Calabi-Yau algebras as Ore extensions of 2-Calabi-Yau algebras
- He, Van
- 2013
(Show Context)
Citation Context ... (the Yang–Mils algebra introduced by Connes and Dobios–Violette [CDV] is a particular case of it), and 3-Calabi–Yau algebras which are Ore extensions of 2-Calaby–Yau ones by He, Oystaeyen, and Zhang =-=[HOZ]-=-. In all these cases, the coherence property is proved using the same lemma [P08, Prop. 3.2]. It states that if a non-trivial two-sided ideal I in a graded algebra A is free as a left module and the q... |