## BIMODULE COMPLEXES VIA STRONG HOMOTOPY ACTIONS

Citations: | 3 - 1 self |

### BibTeX

@MISC{Keller_bimodulecomplexes,

author = {Bernhard Keller},

title = {BIMODULE COMPLEXES VIA STRONG HOMOTOPY ACTIONS},

year = {}

}

### OpenURL

### Abstract

Abstract. We present a new and explicit method for lifting a tilting complex to a bimodule complex. The key ingredient of our method is the notion of a strong homotopy action in the sense of Stasheff. 1.

### Citations

404 | An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge - Weibel - 1994 |

342 | Homological algebra of mirror symmetry
- Kontsevich
- 1994
(Show Context)
Citation Context ...and [3]. The essential new ingredient of our method is the notion of a strong homotopy action (=A1 -action) due to Stashe [11], [12] and Kadeishvili [2], and recently popularized again by Kontsevich [=-=6]-=-, [7]. The present article is self-contained but the interested reader maysnd more information and references on strong homotopy methods in [5]. We will show that if A is projective over k and T is ri... |

319 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ...s f : T → T[−n], n > 0. We also prove a unicity result which improves on [8] and [2]. The essential new ingredient of our method is the notion of a strong homotopy action (=A∞-action) due to Stasheff =-=[10]-=-, [11] and recently popularized again by Kontsevich [6], [5]. The present article is self-contained but the interested reader may find more information on strong homotopy methods in [4]. We will show ... |

135 |
Morita theory for derived categories
- Rickard
- 1989
(Show Context)
Citation Context ...ned and the analogy with the case of module categories seems to break down. Nevertheless, for complexes T satisfying the `Toda condition' HomDB (T ; T [ n]) = 0 ; for all n > 0 ; J. Rickard succeeded =-=[8-=-] in constructing a triangle functor F : D A ! DB taking AA to T , where D A DA denotes the subcategory of right bounded complexes. This construction was at the basis of the proof of his `Morita theo... |

17 |
Triangulated categories and geometry, Course at the Ecole Normale Supérieure
- Kontsevich
- 1998
(Show Context)
Citation Context ...improves on [8] and [2]. The essential new ingredient of our method is the notion of a strong homotopy action (=A∞-action) due to Stasheff [10], [11] and recently popularized again by Kontsevich [6], =-=[5]-=-. The present article is self-contained but the interested reader may find more information on strong homotopy methods in [4]. We will show that if A is projective over k and T is right bounded with p... |

14 |
The category of differential coalgebras and the category of A(∞)-algebras (Russian), Trudy Tbiliss
- Kadeishvili
- 1985
(Show Context)
Citation Context ...) = −m3(a, b, d(x)) − d(m3(a, b, x)) for all a, b ∈ A and x ∈ T. We construct a complex ˜ X as follows: The underlying graded module of ˜ X is The differential is given by A ⊗ A ⊗ A ⊗ T[2] ⊕ A ⊗ A ⊗ T=-=[1]-=- ⊕ A ⊗ T. d(a, b, c, x) = −ab ⊗ c ⊗ x + a ⊗ bc ⊗ x + a ⊗ m3(b, c, x) − a ⊗ b ⊗ m2(c, x) + a ⊗ b ⊗ c ⊗ d(x) , d(a, b, x) = −ab ⊗ x + a ⊗ m2(b, x) − a ⊗ b ⊗ d(x) , d(a, x) = a ⊗ d(x) , where a, b, c ∈ A... |

12 |
A remark on tilting theory and DG
- Keller
- 1993
(Show Context)
Citation Context ...in [8]. Date: September 20, 1999, last modied on January 14, 2000. 1991 Mathematics Subject Classication. Key words and phrases. A-innity algebra, Derived category, Tilting. 1 2 BERNHARD KELLER In [3] and [4], we gave an a priori construction of a lift (up to isomorphism) of T to a complex of bimodules X (under suitablesatness hypotheses). This made it possible to dene F = L(? A X) and to give a ... |

9 |
Homotopy associativity of H-spaces
- Stashe¤
- 1963
(Show Context)
Citation Context ...ained in section 10. We also prove a uniqueness result which improves on [9] and [3]. The essential new ingredient of our method is the notion of a strong homotopy action (=A1 -action) due to Stashe [=-=11]-=-, [12] and Kadeishvili [2], and recently popularized again by Kontsevich [6], [7]. The present article is self-contained but the interested reader maysnd more information and references on strong homo... |

6 | Derived equivalences for group rings - Koenig, Zimmermann - 1998 |

5 |
equivalences as derived functors
- Derived
- 1991
(Show Context)
Citation Context ... DB taking AA to T , where D A DA denotes the subcategory of right bounded complexes. This construction was at the basis of the proof of his `Morita theorem for derived categories'. Later he showed [=-=9]-=- that if F restricts to an equivalence between the bounded derived categories (and suitablesatness hypotheses hold), then after replacing T by an isomorphic object, it is possible to lift the A-action... |

2 |
The category of dierential coalgebras and the category of A(1)-algebras (Russian), Trudy Tbiliss
- Kadeishvili
- 1985
(Show Context)
Citation Context ...have m 2 (ab; x) m 2 (a; m 2 (b; x)) = m 3 (a; b; d(x)) d(m 3 (a; b; x)) for all a; b 2 A and x 2 T . We construct a complex e X as follows: The underlying graded module of e X is AsAsA T [2] AsA T [=-=-=-1] A T : The dierential is given by d(a; b; c; x) = absc x + asbc x + a m 3 (b; c; x) asb m 2 (c; x) + asbsc d(x) ; d(a; b; x) = ab x + a m 2 (b; x) asb d(x) ; d(a; x) = a d(x) ; where a; b; c 2 A, x... |

2 |
associativity of H-spaces
- Homotopy
- 1963
(Show Context)
Citation Context ...in section 10. We also prove a uniqueness result which improves on [9] and [3]. The essential new ingredient of our method is the notion of a strong homotopy action (=A1 -action) due to Stashe [11], [=-=12]-=- and Kadeishvili [2], and recently popularized again by Kontsevich [6], [7]. The present article is self-contained but the interested reader maysnd more information and references on strong homotopy m... |

1 | tensor products for the category of A(1)-algebras and A(1)-modules - Twisted - 1986 |