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## DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS

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641 | Commutative Ring Theory, - Matsumura - 1986 |

372 |
Cohen-Macaulay rings, Cambridge
- Bruns, Herzog
- 1998
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Citation Context ...sketches of) proofs, although the results are well-known, and the arguments based on standard techniques in commutative algebra. For unexplained terminology the reader is referred to Bruns and Herzog =-=[12]-=-. Graded-commutative rings. Let R = ⊕ i>0R i be a graded-commutative ring ; thus R is an N-graded ring with the property that rs = (−1)|r||s|sr for any r, s in R. Elements in a graded object are assum... |

252 |
Des Categories derivees. des categories abeliennes. Asterisque v.
- Verdier
- 1996
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Citation Context ...from its graded center; in particular Koszul objects are employed. A triangulated category T is by definition an additive Z-category equipped with a class of exact triangles satisfying various axioms =-=[30]-=-. Here, Z-category simply means that there is a fixed equivalence Σ: T→ T. Given any additive Z-category T = (T,Σ), we introduce some natural finiteness condition for objects of T as follows: Let R = ... |

190 | On differential graded categories,
- Keller
- 2006
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Citation Context ...Theorem 4.2. The following result is a variation on Theorems 4.2 and 4.5 which might be useful in some contexts. The hypothesis on T holds, for example, when it is algebraic, in the sense of Keller =-=[19]-=-. Theorem 4.6. Let T be a triangulated category with functorial mapping cones. If H is a cohomological functor and G a generator for T such that H∗(G) is in noethfl(R), for some ring R acting centrall... |

178 |
The spectrum of an equivariant cohomology ring,
- Quillen
- 1971
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Citation Context ...ring acting centrally on Db(A). One may specialize further to the case where k is a field of characteristic p and A = kG is the group algebra of a finite group G. It follows from a theorem of Quillen =-=[24]-=- that cxkG(k) equals the p-rank of G. Thus, Corollary 5.7 yields the following inequalities ll(kG) ≥ dimDbst(kG) ≥ rankp(G)− 1 . These estimates were first obtained in [23] using different methods. Re... |

163 | Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, Available from https://tspace.library.utoronto.ca/handle/1807/16682
- Buchweitz
- 1987
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Citation Context ...ension of an Artin algebra. Indeed, for a non-semisimple Artin algebra A one has an inequality rep.dimA ≥ dimDbst(A) + 2 , where Dbst(A) is the stable derived category of A, in the sense of Buchweitz =-=[13]-=-. As one application of the preceding result, we bound the representation dimension of A by the Krull dimension of Hochschild cohomology. Corollary 1.2. Let k be an algebraically closed field and A a ... |

111 | Cohomology of finite group schemes over a field.
- Friedlander, Suslin
- 1997
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Citation Context ... inequality holds because thickll(A)(A/r) = Db(A); see [25, Lemma 7.35]. The rest are obtained by combining Proposition 5.6 and Theorem 5.4. Remark 5.8. In view of results of Friedlander and Suslin =-=[18]-=-, the preceding result applies, in particular, to the case when A is a co-commutative Hopf algebra. In this case, one could choose either the Hochschild cohomology of A over k, or the k-algebra Ext∗A(... |

98 |
Representation dimension of Artin algebras
- Auslander
- 1971
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Citation Context ...orphism; we call it a Koszul object of r on X. Let Y be an object in T and set M = Hom ∗ T(X, Y ). Applying Hom ∗ T(−, Y ) to the triangle above yields an exact sequence of R-modules: M[d + 1] ∓r −→ M=-=[1]-=- −→ Hom ∗ T(X /r, Y ) −→ M[d] ±r −→ M[0] . □6 BERGH, IYENGAR, KRAUSE, AND OPPERMANN This gives rise to an exact sequence of graded R-modules (3.2) 0 −→ (M/rM)[1] −→ Hom ∗ T(X /r, Y ) −→ (0 : r)M [d] ... |

93 | Dimensions of triangulated categories,
- Rouquier
- 2008
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Citation Context ...ings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras. 1. Introduction A notion of dimension for a triangulated category was introduced by Rouquier in =-=[25]-=-. Roughly speaking, it corresponds to the minimum number of steps needed to generate the category from one of its objects. Consideration of this invariant has been critical to some recent developments... |

63 | Ideals in triangulated categories: phantoms, ghosts and skeleta - Christensen - 1998 |

60 | Support varieties and Hochschild cohomology ring - Snashall, Solberg |

58 | Local cohomology and support for triangulated categories - Benson, Iyengar, et al. |

50 | Relative homological algebra and purity in triangulated category - Beligiannis |

42 |
den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc
- Bondal, Van
- 1996
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Citation Context ...e representation dimension of the exterior algebra is d+1, thereby obtaining the first example of an algebra with representation dimension more than three. On the other hand, Bondal and Van den Bergh =-=[11]-=- proved that any cohomological finite functor on the bounded derived category of coherent sheaves on a smooth algebraic variety over a field is representable, by establishing that that triangulated ca... |

38 | Support varieties for selfinjective algebras - Erdmann, Holloway, et al. |

31 | Homology of perfect complexes - Avramov, Buchweitz, et al. |

28 | Representation dimension of exterior algebras
- Rouquier
(Show Context)
Citation Context ...on of this invariant has been critical to some recent developments in algebra and geometry: Using the dimension of the stable category of an exterior algebra on a d-dimensional vector space, Rouquier =-=[26]-=- proved that the representation dimension of the exterior algebra is d+1, thereby obtaining the first example of an algebra with representation dimension more than three. On the other hand, Bondal and... |

19 | Global Hochschild (co-)homology of singular spaces - Buchweitz, Flenner |

17 | Homology of Noetherian rings and local - Tate - 1957 |

16 |
Representation dimension of artin algebras, Queen Mary College Math. Notes, Queen Mary
- Auslander
- 1971
(Show Context)
Citation Context ...isomorphism; we call it a Koszul object of r on X. Let Y be an object in T and set M = Hom∗T(X,Y ). Applying Hom ∗ T(−, Y ) to the triangle above yields an exact sequence of R-modules: M [d+ 1] ∓r−→M =-=[1]-=- −→ Hom∗T(X/r, Y ) −→M [d] ±r−→M [0] . 6 BERGH, IYENGAR, KRAUSE, AND OPPERMANN This gives rise to an exact sequence of graded R-modules (3.2) 0 −→ (M/rM)[1] −→ Hom∗T(X/r, Y ) −→ (0 : r)M [d] −→ 0 , wh... |

14 |
Chain maps inducing zero homology maps
- Kelly
- 1965
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Citation Context ...fold extensions of thick 1 T(G). In the literature, the subcategory thicknT(G) has sometimes been denoted 〈G〉n. The next result is contained in [10, Lemma 2.1]. Similar results have appeared in Kelly =-=[20]-=-, Carlsson [15, Proof of Theorem 16], Christensen [16, Theorem 3.5], Beligiannis [7, Corollary 5.5], Rouquier [25, Lemma 4.11], and Avramov, Buchweitz, and Iyengar [3, Proposition 2.9]. Lemma 3.2 (Gho... |

12 | Representations and cohomology II,” Cambridge studies in advanced mathematics 31 - Benson - 1991 |

8 | Representation dimension and finitely generated cohomology
- Bergh
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Citation Context ... of Koszul objects and elementary observations concerning ‘eventually noetherian modules’; this is inspired by the approach in [6]. Another important ingredient is a version of the ‘Ghost Lemma’ from =-=[10]-=-; see Lemma 3.2. Our principal motivation for considering dimensions of triangulated categories is that it provides a way to obtain lower bounds on the representation dimension of an Artin algebra. In... |

7 | Modules with prescribed cohomological support
- Avramov, Iyengar
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Citation Context ...m 1.1 is contained in Theorem 4.2. The proof is based on a systematic use of Koszul objects and elementary observations concerning ‘eventually noetherian modules’; this is inspired by the approach in =-=[6]-=-. Another important ingredient is a version of the ‘Ghost Lemma’ from [10]; see Lemma 3.2. Our principal motivation for considering dimensions of triangulated categories is that it provides a way to o... |

7 |
On the graded centers and block cohomology
- Linckelmann
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Citation Context ...Σn satisfying ηΣ = (−1)nΣη. Composition gives Z∗(T) a structure of a graded-commutative ring; see, for instance, [14, §3], especially Lemma 3.2.1, which explains the signed commutation rule, and also =-=[21]-=-. In what follows, we assume that a graded-commutative ring R acts centrally on T, via a homomorphism R→ Z∗(T). What this amounts to is specifying for each X in T a homomorphism of rings φX : R→ End∗T... |

6 | Class and rank of differential modules
- Avramov, Buchweitz, et al.
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Citation Context ...angle obtained (by suitable suspension) from the one in (3.1). We claim that for each i = 1, . . . , c the following properties hold: (1) HomnT(X, θi) = 0 for n 0; (2) Hom∗T(X/ r, θi) is surjective; =-=(3)-=- HomnT(X/ r,K0) 6= 0 for infinitely many n ≥ 0. Indeed, for each W ∈ T the triangle (3.4) induces an exact sequence Hom∗T(W,Ki) Hom∗T(W,θi)−−−−−−−−→ Hom∗T(W,Ki−1) ±rsi−−→ Hom∗T(W,Ki−1)[s|ri|] of grade... |

5 |
Avramov: Homological asymptotics of modules over local rings. Commutative algebra
- L
- 1987
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Citation Context ...AND OPPERMANN Now fix an object X in T and suppose that for each Y ∈ T there exists an integer n such that the following properties hold: (1) the graded R-module ⊕ i>nHom i T(X,Y ) is noetherian, and =-=(2)-=- the R0-module HomiT(X,Y ) is of finite length for i ≥ n. In this case, Hom∗T(X,Y ) has finite (Krull) dimension over R ev, the subring of R consisting of elements of even degree, which is a commutati... |

4 | On the homology of finite free (Z/2)k-complexes, - Carlsson - 1983 |

3 | A lower bound for the representation dimension of kCnp
- Oppermann
(Show Context)
Citation Context ...s from a theorem of Quillen [24] that cxkG(k) equals the p-rank of G. Thus, Corollary 5.7 yields the following inequalities ll(kG) ≥ dimDbst(kG) ≥ rankp(G)− 1 . These estimates were first obtained in =-=[23]-=- using different methods. Remark 5.9. We should like to note that when A is an Artin k-algebra which is also projective as a k-module, one has a natural first choice for the ring acting centrally on D... |

2 |
Dimension of stable derived categories of local rings
- Avramov, Iyengar
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Citation Context ... of A, and denoted codimA; here edimA is the embedding dimension of A, that is to say, the k-vector space dimension of m/m2. The result below holds also without the hypothesis that A is complete; see =-=[5]-=-. For the definition of a complete intersection ring, see [12, §2.3]. Corollary 5.10. Let A be a commutative local ring, complete with respect to the topology induced by its maximal ideal. When A is c... |