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## Structures of AS-Gorenstein Algebras

Venue: | Adv. Math |

Citations: | 8 - 2 self |

### Citations

774 | An introduction to homological algebra, Cambridge studies in advanced mathematics 38 - Weibel - 1994 |

643 | Representation Theory of Artin Algebras - Auslander, Reiten, et al. - 1995 |

522 | Triangulated categories in the representation theory of finite-dimensional algebras - Happel - 1988 |

342 | Noncommutative noetherian rings - McConnell, Robson - 2001 |

176 |
Noncommutative projective schemes
- Artin, Zhang
- 1994
(Show Context)
Citation Context ...ras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. The quotient category TailsA := GrModA/TorsA is called the noncommutative projective scheme associated to A in =-=[2]-=-. Let pi : GrModA → TailsA be the quotient functor. We often denote by M := piM ∈ TailsA for M ∈ GrModA. Note that the autoequivalence M 7→ M(m) preserves (direct limits of) modules finite dimensional... |

84 | Existence theorems for dualizing complexes over noncommutative graded and filtered - Bergh - 1997 |

79 | Homological properties of associative algebras: the method of helices (Russian - Bondal, Polishchuk - 1993 |

69 | Dualizing complexes over noncommutative graded algebras - Yekutieli - 1992 |

52 | Twisted graded algebras and equivalences of graded categories
- Zhang
- 1996
(Show Context)
Citation Context ... as a graded abelian group with new multiplication defined by a ∗ b = τ j(a)b where a ∈ Ai, b ∈ Aj . The map A → A defined by a 7→ τ i(a) for a ∈ Ai gives an isomorphism τ−1A → Aτ as graded rings. By =-=[25]-=-, GrMod τA ∼= GrModAτ ∼= GrModA for any τ ∈ AutA. We will give some relationships between these two notions of twist. Lemma 2.10. Let A be an N-graded ring connected over R, τ ∈ AutA a graded ring aut... |

42 |
den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc
- Bondal, Van
- 1996
(Show Context)
Citation Context ...implies ` ≥ 1. Proposition 3.4 for M = A/A≥m allows us to use results given in [5, Section 4]. Recall that an objectM of D(TailsA) is called compact if HomD(TailsA)(M,−) commutes with direct sums. By =-=[5]-=-, A(i) are compact in D(TailsA) for i ∈ Z, and {A(i) | i ∈ Z} generates D(TailsA), i.e., an object M of D(TailsA) is zero if and only if HomD(TailsA)(A(i),M[p]) = 0 for all i ∈ Z and all p ∈ Z. 23 Pro... |

36 | Growth of graded Noetherian rings - Stephenson, Zhang - 1997 |

35 |
Deformed Calabi-Yau completions
- Keller
(Show Context)
Citation Context ...uasi-Fano algebra of dimension n. Remark 1.4. The preprojective algebra of a quasi-Fano algebra of dimension n was called the (n+1)-Calabi-Yau completion or the derived (n+1)-preprojective algebra in =-=[10]-=-. One of the conditions for an algebra R to be quasi-Fano of dimension n can be thought of as the condition that the derivedm-preprojective algebra of R is a usual algebra (not only a dg-algebra) if a... |

22 | den Bergh, Ideal classes of three dimensional Sklyanin algebras - Naeghel, Van |

16 | Capital Structure
- SC
(Show Context)
Citation Context ...algebra, then the trivial extension A := ∆R is an AS-Gorenstein algebra over R of dimension 0 and of Gorenstein parameter −1. One of the characterizations of an AS-regular algebra over k was given in =-=[11]-=-, namely, a graded algebra A connected over k is AS-regular if and only if the Yoneda Ext-algebra A! := ⊕i∈N ExtiA(k, k) is graded Frobenius. This fact motivates the following result. Proposition 4.20... |

13 | Coherent algebras and noncommutative projective lines
- Piontkovski
(Show Context)
Citation Context ...We now consider AS-regular algebras of dimension 2. As an application of Theorem 4.12, we prove the following theorem. If A is connected over k, then this theorem was already proved by D. Piontkovski =-=[18]-=-. (See also [13, Theorem 7.2].) Theorem 4.16. Every AS-regular algebra A over R of dimension 2 is graded right coherent. Proof. The equivalence Q : GrModA → GrModA[`] tells us that A is graded right c... |

13 |
Non-Noetherian regular rings of dimension 2
- Zhang
- 1998
(Show Context)
Citation Context ...raded right coherent. Thus we conclude that A is graded right coherent. To prove this theorem, Piontkovski used an explicit description of an ASregular algebra over k of dimension 2 due to J.J. Zhang =-=[26]-=-, namely, every ASregular algebra over k of dimension 2 is given by k〈x1, . . . , xp〉/ ( ∑p i=1 σ(xi)xp−i) for some σ ∈ Autkk〈x1, . . . , xp〉. Using representation theory of quivers, we can also repro... |

12 | G-algebras, twistings, and equivalences of graded categories
- Sierra
(Show Context)
Citation Context ...nd B are isomorphic as Z-algebras if there is an algebra isomorphism φ : A → B such that φ(Aij) = Bij for all i, j ∈ Z. For any Z-graded algebra A, we define the Z-algebra A = ⊕i,j∈ZAj−i. We refer to =-=[19]-=- for more details on Z-algebras. As another application, we have the following result (cf. [22]). Theorem 4.17. Let A,A′ be AS-regular algebras over k, and S = ∇A,S′ = ∇A′ their Beilinson algebras. If... |

10 |
duality for generalized Auslander regular algebras. Trends in the representation theory of finite-dimensional algebras
- Mart́ınez-Villa
- 1997
(Show Context)
Citation Context ...n]) ∼= νDRΓm(A)(`)[−n] ∼= ν(Aν(−`)[n])(`)[−n] ∼= (Aν)ν−1 ∼= A in D(GrModAe) by a graded version of Lemma 2.8 (2), hence the second claim. 3.3 Generalized AS-regular Algebras due to Martinez-Villa. In =-=[12]-=-, Martinez-Villa gave another generalization of AS-regular algebra for N-graded algebras. Definition 3.15. A locally finite N-graded algebra A is called generalized ASregular of dimension n if the fol... |

5 | Graded self-injective algebras ”are” trivial extensions
- Chen
(Show Context)
Citation Context ...omorphisms in Proposition 4.4, the top horizontal arrow is the composition and the bottom horizontal arrow is the multiplication. Quasi-Veronese algebras and Beilinson algebras introduced in [16] and =-=[6]-=- will play an important role for the rest of this paper. Definition 4.6. Let A = ⊕i∈ZAi be a Z-graded algebra and r ∈ N+. 1. The r-th Veronese algebra of A is a Z-graded algebra defined by A(r) := ⊕ i... |

5 | Derived categories and tilting - Keller - 2007 |

4 | Artin-Schelter regular algebras and categories - Martinez-Villa, Solberg |

4 |
Ampleness of two-sided tilting complexes, preprint
- Minamoto
(Show Context)
Citation Context ...te global dimension is one of the major projects. The purpose of this paper is to connect these classification problems. The first author of this paper recently introduced a notion of Fano algebra in =-=[15]-=-, which is a nice class of finite dimensional algebras of finite global dimension. For example, every path algebra of finite acyclic quiver of infinite representation type is Fano [15]. In this paper,... |

2 |
Equivalences for noncommutative projective spaces, preprint
- Vitoria
(Show Context)
Citation Context ... = Bij for all i, j ∈ Z. For any Z-graded algebra A, we define the Z-algebra A = ⊕i,j∈ZAj−i. We refer to [19] for more details on Z-algebras. As another application, we have the following result (cf. =-=[22]-=-). Theorem 4.17. Let A,A′ be AS-regular algebras over k, and S = ∇A,S′ = ∇A′ their Beilinson algebras. If A1 6= 0, then the following are equivalent: (1) GrModA ∼= GrModA′. (2) S ∼= S′ as algebras. (3... |