## Model category structures on chain complexes of sheaves

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Venue: | Trans. Amer. Math. Soc |

Citations: | 30 - 0 self |

### BibTeX

@ARTICLE{Hovey_modelcategory,

author = {Mark Hovey},

title = {Model category structures on chain complexes of sheaves},

journal = {Trans. Amer. Math. Soc},

year = {},

volume = {353},

pages = {2441--2457}

}

### OpenURL

### Abstract

of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently

### Citations

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Citation Context ...een unable to find a published reference for this fact, we include a direct proof of smallness in an appendix, based on the Gabriel-Popescu theorem. For sheaves and schemes, we try to refer mostly to =-=[Har77]-=-, but we occasionally need more advanced results. The author would like to thank Matthew Ando and Amnon Neeman for helpful discussions about sheaves, and Dan Christensen and the referee for useful sug... |

358 |
Homotopical Algebra
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Citation Context ...l that the derived category is a category, in the usual sense of having only a set of maps between any two objects. One excellent way to cope with this problem is Quillen’s theory of model categories =-=[Qui67]-=-. If one can prove that Ch(A) is a model category with the quasiisomorphisms as the weak equivalences, then it follows from Quillen’s theory that the derived category is a category. Furthermore, one h... |

353 | Residues and duality - Hartshorne - 1966 |

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150 |
Model categories, Mathematical Surveys and Monographs 63
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Citation Context ...d spaces. To understand this paper, the reader needs to know some basic facts about model categories, Grothendieck categories, and sheaves. A good introduction to model categories is [DS95]. The book =-=[Hov98]-=- is a more in-depth study, but still starting from scratch. All the terms we need are defined in [Hov98]; we will give specific references as needed. For Grothendieck categories, [Ste75] is a basic re... |

147 | Algebras and modules in monoidal model categories
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Citation Context ...A similar counterexample works if we think of S as the underlying space of Spec Z (p). There is an additional condition that a symmetric monoidal model category might satisfy, called the monoid axiom =-=[SS97]-=-. This axiom guarantees that the monoids in a symmetric monoidal model category, and the modules over a given monoid, themselves form model categories. The monoid axiom asserts that every map in K-cof... |

131 |
Homotopy theories and model categories, Handbook of algebraic topology
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Citation Context ...ary maps of ringed spaces. To understand this paper, the reader needs to know some basic facts about model categories, Grothendieck categories, and sheaves. A good introduction to model categories is =-=[DS95]-=-. The book [Hov98] is a more in-depth study, but still starting from scratch. All the terms we need are defined in [Hov98]; we will give specific references as needed. For Grothendieck categories, [St... |

127 |
Axiomatic stable homotopy theory
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Citation Context ...nsor have to be interpreted in D(QCo(S)), so are really derived versions. This follows from the corresponding fact in O-Mod itself, and the fact that locally free sheaves are flat. In the language of =-=[HPS97]-=-, then, we have proved the following corollary. Corollary 2.6. Suppose S is a finite-dimensional noetherian scheme with enough locally frees. Then the category D(QCo(S)) is a unital algebraic stable h... |

127 |
Resolutions of unbounded complexes
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Citation Context ...uary 24, 2000. 2000 Mathematics Subject Classification. Primary 18F20, 14F05, 18E15, 18E30, 18G35, 55U35. 2441 c○2001 American Mathematical Societys2442 MARK HOVEY Spaltenstein on unbounded complexes =-=[Spa88]-=- and their generalizations in [TLS99] follow immediately from the existence of the injective model structure. However, the categories just mentioned are closed symmetric monoidal under the tensor prod... |

96 | editors. Théorie des intersections et théorème de Riemann-Roch - Berthelot, Grothendieck, et al. - 1971 |

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Sheafifiable homotopy model categories
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Citation Context ...m X to Y in the derived category as chain homotopy classes of chain maps from a cofibrant replacement for X to a fibrant replacement for Y . It is a folk theorem, apparently due to Joyal [Joy84] (see =-=[Bek99]-=-), that Ch(A) is a model category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations whenever A is a Grothendieck abelian category. We call this the injective model structu... |

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Citation Context ...95]. The book [Hov98] is a more in-depth study, but still starting from scratch. All the terms we need are defined in [Hov98]; we will give specific references as needed. For Grothendieck categories, =-=[Ste75]-=- is a basic reference. We also need the fact that every object X in a Grothendieck category A is small, in the sense that A(X, −) commuteswithκ-indexed colimits, for all cardinals κ with sufficiently ... |

22 |
Letter to A. Grothendieck
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Citation Context ...morphisms from X to Y in the derived category as chain homotopy classes of chain maps from a cofibrant replacement for X to a fibrant replacement for Y . It is a folk theorem, apparently due to Joyal =-=[Joy84]-=- (see [Bek99]), that Ch(A) is a model category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations whenever A is a Grothendieck abelian category. We call this the injective ... |

18 |
Salorio, Localization in categories of complexes and unbounded resolutions
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Citation Context ...ct Classification. Primary 18F20, 14F05, 18E15, 18E30, 18G35, 55U35. 2441 c○2001 American Mathematical Societys2442 MARK HOVEY Spaltenstein on unbounded complexes [Spa88] and their generalizations in =-=[TLS99]-=- follow immediately from the existence of the injective model structure. However, the categories just mentioned are closed symmetric monoidal under the tensor product, as are most Grothendieck categor... |

14 | Differential graded homological algebra, preprint - Avramov, Foxby, et al. - 2004 |

8 | Notes on Derived Categories and Derived Functors, preprint - Lipman |

2 | Dierential graded homological algebra, preprint - Avramov, Foxby, et al. - 1997 |

1 |
Derived categories and projective classes, preprint
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Citation Context ...l structure on Ch(A) foraGrothendieck category A by generalizing the usual projective model structure when A = R-Mod for some ring R. Our approach is related to, but not identical with, the method of =-=[Chr98]-=-; the difference is that we need the weak equivalences to be the quasi-isomorphisms, whereas Christensen is willing to relax that hypothesis. Recall from [Hov98, Section 2.3] that the projective model... |

1 |
Cohomology of sheaves, lecture notes
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Citation Context ...mpact, quasi-separated scheme. This result has been rediscovered in various forms by several people (including the author). Moerdijk and Pronk constructed the injective model structure for sheaves in =-=[MP92]-=-. Many of the results of Received by the editors February 24, 2000. 2000 Mathematics Subject Classification. Primary 18F20, 14F05, 18E15, 18E30, 18G35, 55U35. 2441 c○2001 American Mathematical Society... |

1 | Théorie des intersections et théorème de Riemann-Roch - Ferrand, Jouanolou, et al. - 1971 |