## Smashing Subcategories And The Telescope Conjecture - An Algebraic Approach (1998)

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Venue: | Invent. Math |

Citations: | 33 - 8 self |

### BibTeX

@ARTICLE{Krause98smashingsubcategories,

author = {Henning Krause},

title = {Smashing Subcategories And The Telescope Conjecture - An Algebraic Approach},

journal = {Invent. Math},

year = {1998},

volume = {139},

pages = {99--133}

}

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### Abstract

. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...

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Citation Context ...ting out a mistake in a preliminary version of this paper. In addition, I am grateful to an anonymous referee for numerous suggestions. 1. Purity 1.1. Pure-exactness. Let C be a triangulated category =-=[26, 27]-=- and suppose that arbitrary coproducts exist in C. An object X in C is called compact if for every family (Y i ) i2I in C the canonical map ` i Hom(X;Y i ) ! Hom(X; ` i Y i ) is an isomorphism. We den... |

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Citation Context ...he canonical map ` i Hom(X;Y i ) ! Hom(X; ` i Y i ) is an isomorphism. We denote by C 0 the full subcategory of compact objects in C and observe that C 0 is a triangulated subcategory of C. Following =-=[20]-=-, the category C is called compactly generated provided that the isomorphism classes of objects in C 0 form a set, and Hom(C;X) = 0 for all C in C 0 implies X = 0 for every object X in C. Examples of ... |

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Citation Context ... Bousfield and Ravenel if I is a class of identity maps. In fact, they showed for the stable homotopy category that every class of compact objects generates a localizing subcategory which is smashing =-=[6, 23]-=-. However, if I is a class of arbitrary maps in C, it is not clear that there exists a localizing subcategory which is generated by I. Our analysis of smashing subcategories is based on the concept of... |

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Citation Context ... full triangulated subcategory B of C is localizing if B is closed under taking coproducts. The quotient category C=B is, by definition, the category of fractions C [\Sigma \Gamma1 ] (in the sense of =-=[10]-=-) with respect to the class \Sigma of maps Y ! Z which admit a triangle X ! Y ! Z ! X[1] with X in B. Thus the correspondingsquotient functor C ! C [\Sigma \Gamma1 ] is the universal functor which inv... |

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Citation Context ...erefore OE = 0 since Hom(OE 0 ; (g ffi f)(X)[\Gamma1]) = 0 by Lemma 3.7. We conclude that e(X) = 0 and therefore Hom(C I ; X) = 0. 3.4. Approximations. We need to recall the following definition from =-=[2]-=-. Let Y be a class of objects in a category C. Then a map X ! Y is a left Y-approximation of X if Y belongs to Y and if the induced map Hom(Y;Y 0 ) ! Hom(X;Y 0 ) is surjective for every Y 0 in Y. For ... |

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Citation Context ... Bousfield and Ravenel if I is a class of identity maps. In fact, they showed for the stable homotopy category that every class of compact objects generates a localizing subcategory which is smashing =-=[6, 23]-=-. However, if I is a class of arbitrary maps in C, it is not clear that there exists a localizing subcategory which is generated by I. Our analysis of smashing subcategories is based on the concept of... |

68 | The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bous¯eld and - Neeman - 1992 |

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Citation Context ...ure-injective modules. The concept of purity has been studied extensively by algebraists. Pure-exactness and pure-injectivity for modules over a ring have been introduced by Cohn [7], and we refer to =-=[13]-=- for a modern treatment of this subject. Let us recall briefly the relevant definitions. Letsbe an associative ring with identity. We consider the category Modsof (right) -modules. A sequence 0 ! X ! ... |

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Citation Context ...tivated by analogous concepts for the category of modules over a ring [7]. In this context one frequently studies the indecomposable pure-injective modules; they form the Ziegler spectrum of the ring =-=[28]-=-. We shall see that the isomorphism classes of indecomposable pure-injective objects in C form a set which we denote by SpC. Theorem C. Let B be a smashing subcategory of C, and let U be the set of ob... |

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Citation Context ...e [15, Lemma 4.1]. A subset U of Sp C is Zariski-open if and only if there exists some class I of maps in C 0 such that U = fX 2 Sp C j Hom(OE; X) = 0 for some OE 2 Ig; see [9, Chap. VI]. We refer to =-=[18]-=- for a detailed discussion of both topologies in the context of modules over a ring. A map OE : X ! Y in C is said to be a pure-injective envelope of X if Y is pure-injective and a composition / ffi O... |

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Citation Context ...osed under taking coproducts since f 0 and f 00 preserve coproducts. Thus B = C by [20, Lemma 3.2], and therefore j : f 0 ! f 00 is an isomorphism. The following consequence generalizes a result from =-=[8]-=-. 12 HENNING KRAUSE Corollary 2.4. Let C be a compactly generated triangulated category and let f : C ! A be a cohomological functor into an abelian AB 5 category A. Suppose also that f preserves copr... |

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Citation Context ... for all compact objectssC. An object X is called pure-injective if every pure monomorphism X ! Y splits. These definitions are motivated by analogous concepts for the category of modules over a ring =-=[7]-=-. In this context one frequently studies the indecomposable pure-injective modules; they form the Ziegler spectrum of the ring [28]. We shall see that the isomorphism classes of indecomposable pure-in... |

17 | A remark on the generalized smashing conjecture
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Citation Context ...of identity maps between compact objects. In this generality, the conjecture is known to be false. In fact, Keller gives an example of a smashing subcategory which contains no non-zero compact object =-=[14]-=-. Despite some efforts of Ravenel [24], the conjecture remains open for the stable homotopy category. The characterization of smashing subcategories leads to a classification in terms of certain ideal... |

17 | Exactly definable categories - Krause - 1998 |

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Citation Context ...+ OE ffi ' I = (fi ffi ff 0 + OE) ffi ' I : Thus oe I factors through ' I , and this finishes the proof. For some further discussion of the relation between pure-injectives in Modsand Modswe refer to =-=[16, 5]-=-. 2. Cohomological and exact functors 2.1. Extending functors. Let C be any triangulated category. We recall the following well-known property of the Yoneda functor h : C ! modC, X 7! Hom(\Gamma; X). ... |

12 | Stable equivalence preserves representation type
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Citation Context ...+ OE ffi ' I = (fi ffi ff 0 + OE) ffi ' I : Thus oe I factors through ' I , and this finishes the proof. For some further discussion of the relation between pure-injectives in Modsand Modswe refer to =-=[16, 5]-=-. 2. Cohomological and exact functors 2.1. Extending functors. Let C be any triangulated category. We recall the following well-known property of the Yoneda functor h : C ! modC, X 7! Hom(\Gamma; X). ... |

9 | Progress report on the telescope conjecture
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Citation Context ...ts. In this generality, the conjecture is known to be false. In fact, Keller gives an example of a smashing subcategory which contains no non-zero compact object [14]. Despite some efforts of Ravenel =-=[24]-=-, the conjecture remains open for the stable homotopy category. The characterization of smashing subcategories leads to a classification in terms of certain ideals which we now explain. We denote by C... |

9 |
Phantom maps and purity in modular representation theory
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Citation Context ...herefore I =s+ I = (s0 + ) I : Thus I factors through I , and thissnishes the proof. For some further discussion of the relation between pure-injectives in Mod and Mod we refer to =-=[16, 5]-=-. 2. Cohomological and exact functors 2.1. Extending functors. Let C be any triangulated category. We recall the following well-known property of the Yoneda functor h : C ! modC, X 7! Hom(−;X). Lemma ... |

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8 |
On a theorem of Brown and Adams, Topology 36
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Citation Context ...preceding theorem is automatically satisfied if there are at most countably many isomorphism classes of maps between compact objects in C; in particular the stable homotopy category has this property =-=[21]-=-. The classification of smashing subcategories has the following consequence. Corollary. A localizing subcategory B of C is smashing if and only if B is generated by a class of maps between compact ob... |

6 | On The Freyd categories of an additive category
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Citation Context ...ying f 0 (Hom(\Gamma; X)) = f(X) for all X in C. Suppose now that C is triangulated. Then we have the following characterization of flat C-modules which has been observed independently by Beligiannis =-=[3]-=-. Lemma 2.7. The following are equivalent for an additive functor M : C op ! Ab: (1) M is a flat C-module; (2) M is a cohomological functor; (3) M is a fp-injective C-module. 14 HENNING KRAUSE Proof. ... |

4 | Flat and coherent functors - Oberst, Rohrl - 1970 |

3 |
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Citation Context ...nting out a mistake in a preliminary version of this paper. In addition, I am grateful to an anonymous referee for numerous suggestions. 1. Purity 1.1. Pure-exactness. LetC be a triangulated category =-=[26,27]-=- and suppose that arbitrary coproducts exist in C. An object X in C is called compact if for every family .Yi/i2I in C the canonical map ‘ i Hom.X;Yi/ ! Hom.X; ‘ i Yi/ is an isomorphism. We denote by ... |

2 |
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Citation Context ... , X 7! Hom(\Gamma; X)j C0 , is fully faithful; (5) C has filtered colimits. This characterization, and indeed a host of other equivalent statements have been obtained independently by Beligiannis in =-=[4]-=-. 3. Localization 3.1. Cohomological ideals. Let C be an additive category. An ideal I in C consists of subgroups I(X; Y ) in Hom(X;Y ) for every pair of objects X;Y in C such that for all OE in I(X; ... |

2 |
Prefaisceaux, in ``Theorie des Topos et CohomologieÂ Etale des Schemas
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Citation Context ... is flat if and only if M is asltered colimit of representable functors [22, Theorem 3.2]. Therefore FlatC is equivalent to the category of ind-objects over C in the sense of Grothendieck and Verdier =-=[11]-=-. In particular, Flat C is a category withsltered colimits, and every functor f : C ! D into a category D withsltered colimits extends uniquely to a functor f 0 : Flat C ! D preservingsltered colimits... |

2 | Exactly de categories - Krause - 1998 |

1 |
A variant of E.H. Brown’s representabilty theorem, Topology 10
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Citation Context ... map ending in X and a universal pure monomorphism starting in X. Theorem D. For every object X in C there exists, up to isomorphism, a unique triangle X 0 ff \Gamma! X fi \Gamma! X 00 fl \Gamma! X 0 =-=[1]-=- having the following properties: (A1) a map OE : Y ! X is a phantom map if and only if OE factors through ff; (A2) every endomorphism OE of X 0 satisfying ff = ff ffi OE is an isomorphism. The same t... |

1 | Rings of quotients, Springer-Verlag - Stenstr��m - 1975 |

1 | The Connection between the K -theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and - Neeman - 1992 |