## Realizability Of Modules Over Tate Cohomology (2001)

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Citations: | 18 - 1 self |

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@MISC{Benson01realizabilityof,

author = {David Benson and Henning Krause and Stefan Schwede},

title = {Realizability Of Modules Over Tate Cohomology},

year = {2001}

}

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### Abstract

Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)-module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kG-module M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.

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Citation Context ... spectrum X is a natural Postnikov system (in the technical sense of Definition A.6) in the stable homotopy category. More generally, suppose the triangulated category T is endowed with a t-structure =-=[5]-=-. Then the truncations give a Postnikov system, infinite in both directions, of the form (A.9) X[−1, ∞)[−1] �� + X[0, ∞) �� + X[1, ∞)[1] �� + X[2, ∞)[2] �� △ ��� ��� ... � △ �� ��� △ ���� ... �� �� ��... |

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Citation Context ...The argument for Ω−1X is similar, but dual. (iii) Part (iii) of Proposition 3.4 shows that the realizable E-modules and the realizable maps form a covariantly finite subcategory of Mod-E. Recall from =-=[1]-=- that a subcategory C of Mod-E is covariantly finite if every object X in Mod-E admits a left C-approximation. This is by definition a map X → C with C ∈Csuch that for every map X → C ′ with C ′ ∈Cthe... |

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Citation Context ... of the stable module category. The category StMod-L is a triangulated category, with shift given by the inverse of Ω and where the triangles come from short exact sequences in Mod-L (see for example =-=[24]-=-, or [13, Thm. 5.6] in the case of group algebras). If V is a Z-graded k-module, we write V [n] forthegradedk-module with V [n] i = V n+i .IfVsupports a differential d: V i −→ V i+1 , then the differe... |

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Citation Context ...ity conditions on H-spaces; Kadeishvili showed that the cohomology of any differential graded algebra A has an A∞structure, unique up to (non-canonical) isomorphism, which is A∞-quasi-isomorphic to A =-=[25]-=-. Hence the entire A∞-structure on H ∗ A recovers the quasi-isomorphism type of A. Kadeishvili also showed that specifying an A3-structure on an associative3624 DAVID BENSON, HENNING KRAUSE, AND STEF... |

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Citation Context ...he n-fold shift of N. ThenT (N,N) ∗ is a graded ring and T (N,M) ∗ forms a graded right T (N,N) ∗-module under composition. For general background on derived and triangulated categories, see SGA4 1 2 =-=[18]-=- (Appendix by Verdier), Gelfand and Manin [21], Verdier [40], and Weibel [41]. For background on unbounded complexes, see Spaltenstein [37]. 2. A realizability obstruction in stable module categories ... |

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Citation Context ...yclic, so the projection map Hom • L ( ˆ P∗, ˆ P∗) −→ Hom • L ( ˆ P∗, ˆ P∗)/T is a quasi-isomorphism of dgas. This smaller model is variously attributed to Vogel [23], Mislin [32], Benson and Carlson =-=[9]-=-. It also receives a homomorphism from the endomorphism dga of an ordinary resolution P∗ (as opposed to a complete resolution) of N. Indeed, any graded map f : P [n] → P can be extended trivially to a... |

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Citation Context ... of Rickard [35] shows that any localization of ˆ H ∗ (G, k) is realizable by an idempotent kG-module. More generally, any localization of a realizable module is again realizable. Benson and Gnacadja =-=[10]-=- have given an example of aflat ˆ H ∗ (G, k)-module which is not realizable. This is related to the existence of filtered systems in StMod-kG which do not lift to filtered systems in Mod-kG. For this ... |

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Citation Context ...,N) ∗ . The question of which C-modules are realizable is precisely the question of to what extent Brown representability for homology theories holds; see the paper of Christensen, Keller, and Neeman =-=[15]-=- for a recent discussion. The analogue of Theorem 3.7 for families of compact objects provides an answer to this question. 3.10. Relation to (3, −1)-periodicity for coherent functors. The natural tran... |

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Citation Context ...l Hochschild class to obtain multiplicative information about the cohomology of the total space in a spherical fibration. A generalization of the class γA has been studied by Baues and coauthors [2], =-=[3]-=-, [4]; their class, called the ‘universal Toda bracket’, lies in the (3, −1)-cohomology group of an appropriate homotopy category. For simplicity we restrict our attention to algebras over a field k. ... |

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Citation Context ...ys that the ideal T is acyclic, so the projection map Hom • L ( ˆ P∗, ˆ P∗) −→ Hom • L ( ˆ P∗, ˆ P∗)/T is a quasi-isomorphism of dgas. This smaller model is variously attributed to Vogel [23], Mislin =-=[32]-=-, Benson and Carlson [9]. It also receives a homomorphism from the endomorphism dga of an ordinary resolution P∗ (as opposed to a complete resolution) of N. Indeed, any graded map f : P [n] → P can be... |

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Citation Context ... this just says that the ideal T is acyclic, so the projection map Hom • L ( ˆ P∗, ˆ P∗) −→ Hom • L ( ˆ P∗, ˆ P∗)/T is a quasi-isomorphism of dgas. This smaller model is variously attributed to Vogel =-=[23]-=-, Mislin [32], Benson and Carlson [9]. It also receives a homomorphism from the endomorphism dga of an ordinary resolution P∗ (as opposed to a complete resolution) of N. Indeed, any graded map f : P [... |

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Citation Context ...ulated category T is endowed with a t-structure [5]. Then the truncations give a Postnikov system, infinite in both directions, of the form (A.9) X[−1, ∞)[−1] �� + X[0, ∞) �� + X[1, ∞)[1] �� + X[2, ∞)=-=[2]-=- �� △ ��� ��� ... � △ �� ��� △ ���� ... �� �� ��� �� ��� � �� ��� ��� � �� ��� � R−1 R0 Here X[i, ∞) is the truncation of X below dimension i, andwhereRiis the i-fold shift of an object belonging to t... |

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Citation Context ...is is explained in the context of group rings of arbitrary groups in Benson and Carlson [9], Mislin [32]. But often it is not clear whether it can be calculated using complete resolutions. Kropholler =-=[30]-=- shows that in the context of modules of type FP∞ over group rings of groups in a very large class3648 DAVID BENSON, HENNING KRAUSE, AND STEFAN SCHWEDE denoted LHF, complete cohomology can be calcula... |

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Citation Context ...i.e., there exists a normal subgroup U � G such that the composite is an isomorphism. P incl. −−→ G proj. −−−→ G/U Proof. Condition (ii) is a special case of (i). For the implication (ii) ⇒ (iii) see =-=[39]-=- or [19, Thm. 7.2.6]. The restriction map from the cohomology of G to the cohomology of its Sylow subgroup P is always injective. If (iii) holds, then it is also surjective since the inclusion P → G h... |

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Citation Context ...gs of groups in a very large class3648 DAVID BENSON, HENNING KRAUSE, AND STEFAN SCHWEDE denoted LHF, complete cohomology can be calculated using complete resolutions. See also Cornick and Kropholler =-=[16]-=-, [17], Benson [7], [8]. (iv) A theorem of Rickard [35] shows that any localization of ˆ H ∗ (G, k) is realizable by an idempotent kG-module. More generally, any localization of a realizable module is... |

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Citation Context ...hschild class to obtain multiplicative information about the cohomology of the total space in a spherical fibration. A generalization of the class γA has been studied by Baues and coauthors [2], [3], =-=[4]-=-; their class, called the ‘universal Toda bracket’, lies in the (3, −1)-cohomology group of an appropriate homotopy category. For simplicity we restrict our attention to algebras over a field k. Howev... |

3 |
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Citation Context ...groups in a very large class3648 DAVID BENSON, HENNING KRAUSE, AND STEFAN SCHWEDE denoted LHF, complete cohomology can be calculated using complete resolutions. See also Cornick and Kropholler [16], =-=[17]-=-, Benson [7], [8]. (iv) A theorem of Rickard [35] shows that any localization of ˆ H ∗ (G, k) is realizable by an idempotent kG-module. More generally, any localization of a realizable module is again... |

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Citation Context ...1 Λ [26]. The class γA contains the information about Massey triple products in the cohomology of the dga A; see Remark 5.10. For an introduction to A∞-algebras we recommend [29]. Berrick and Davydov =-=[11]-=- use the canonical Hochschild class to obtain multiplicative information about the cohomology of the total space in a spherical fibration. A generalization of the class γA has been studied by Baues an... |

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