## Etale realization on the A¹-homotopy theory of schemes (2001)

Venue: | MATH |

Citations: | 6 - 3 self |

### BibTeX

@ARTICLE{Isaksen01etalerealization,

author = {Daniel C. Isaksen},

title = {Etale realization on the A¹-homotopy theory of schemes},

journal = {MATH},

year = {2001},

pages = {37--63}

}

### OpenURL

### Abstract

We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of pro-simplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the

### Citations

771 |
Algebraic Geometry, Graduate Texts
- Hartshorne
- 1977
(Show Context)
Citation Context ...Limits of Schemes. We first study how finite limits interact with étale maps and separated maps. The results here are not particularly striking, but they do not appear in the standard literature [10] =-=[11]-=- [23] [30]. A technical result about fiber products comes first. The more general claim about arbitrary finite limits follows relatively easily. Lemma 3.1. Consider a diagram of schemes U �� V �� W X ... |

518 | Etale Cohomology
- Milne
- 1980
(Show Context)
Citation Context ...s of Schemes. We first study how finite limits interact with étale maps and separated maps. The results here are not particularly striking, but they do not appear in the standard literature [10] [11] =-=[23]-=- [30]. A technical result about fiber products comes first. The more general claim about arbitrary finite limits follows relatively easily. Lemma 3.1. Consider a diagram of schemes U �� V �� W X � Y s... |

474 |
Prefaiscaux in: Théorie des Topos et Cohomologie Etale des Schémas, Springer L.N.M
- Grothendieck, Verdier
- 1972
(Show Context)
Citation Context ...covers reappear in Section 5. 4.1. Preliminaries on Pro-Categories. We begin with a review of the necessary background on pro-categories. This section contains only standard material on procategories =-=[1]-=- [2] [8] [16]. Definition 4.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hompro-C(X, Y ) = lim s Composition is defined in the natural way. colim HomC(Xt, Ys)... |

364 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...follows. Section 2 begins with a review of simplicial presheaves and their homotopy theory. We assume familiarity with closed model structures. General references on this topic include [13], [14], or =-=[26]-=-. We conform to the conventions of [13] as closely as possible. See also [6] for more details on model structures as applied to simplicial presheaves. Next comes the definition of the étale realizatio... |

359 | Higher algebraic K-theory - Quillen - 1974 |

353 | Homotopy Limits, Completions and Localizations - Bous, Kan - 1972 |

292 |
Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...o handle in the category of pro-spaces. Finally, Section 5 gives the hypercover descent theorem for étale topological types. We make a few final remarks on terminology. We always mean simplicial sets =-=[21]-=- whenever we refer to spaces. We write sSet for the category of simplicial sets. An étale map U → X is any map such that U is a (possibly infinite) disjoint union of schemes U i and each map U i → X i... |

264 | Categories for the Working Mathematician (Graduate Texts - Lane - 1998 |

221 | Model categories
- Hovey
- 1999
(Show Context)
Citation Context ...he paper follows. Section 2 begins with a review of simplicial presheaves and their homotopy theory. We assume familiarity with closed model structures. General references on this topic include [13], =-=[14]-=-, or [26]. We conform to the conventions of [13] as closely as possible. See also [6] for more details on model structures as applied to simplicial presheaves. Next comes the definition of the étale r... |

194 |
Model categories and their localizations
- Hirschhorn
- 2003
(Show Context)
Citation Context ...y theory of pro-spaces. Section 2 begins with a review of simplicial presheaves and their homotopy theory. We assume familiarity with closed model structures. General references on this topic include =-=[Hi]-=-, [Ho], or [Q1]. We conform to the conventions of [Hi] as closely as possible. See also [D] for more details on model structures as applied to simplicial presheaves. The first major result is that the... |

161 |
Mazur Étale homotopy
- Artin, B
- 1969
(Show Context)
Citation Context ...oming is that the analytic realization does not work over fields with positive characteristic. Varieties over abstract fields have no underlying analytic topology. However, the étale topological type =-=[2]-=- [9] is a substitute. In characteristic zero, the étale topological type EtX of a variety X is the pro-finite completion of the underlying analytic space of X. In any characteristic, EtX carries infor... |

123 | Cohomologie Galoisienne - Serre - 1964 |

119 |
Simplicial presheaves
- Jardine
- 1987
(Show Context)
Citation Context ...ory of simplicial presheaves on Sm/S. The notation stands for “spaces over S”. This category has several model structures. Morel and Voevodsky start with the Nisnevich local injective model structure =-=[J]-=-, in which the cofibrations are all monomorphisms and the weak equivalences are detected by Nisnevich sheaves of homotopy groups. They then formally invert the maps X × A 1 → X for every scheme X to o... |

86 | Classifying spaces and spectral sequences - Segal - 1968 |

49 |
Étale homotopy of simplicial schemes
- Friedlander
- 1982
(Show Context)
Citation Context ...g is that the analytic realization does not work over fields with positive characteristic. Varieties over abstract fields have no underlying analytic topology. However, the étale topological type [2] =-=[9]-=- is a substitute. In characteristic zero, the étale topological type EtX of a variety X is the pro-finite completion of the underlying analytic space of X. In any characteristic, EtX carries informati... |

46 |
Cech and Steenrod homotopy theories with applications to geometrio topology
- Edwards, Hastings
- 1976
(Show Context)
Citation Context ...eappear in Section 5. 4.1. Preliminaries on Pro-Categories. We begin with a review of the necessary background on pro-categories. This section contains only standard material on procategories [1] [2] =-=[8]-=- [16]. Definition 4.1. For a category C, the category pro-C has objects all cofiltering diagrams in C, and Hompro-C(X, Y ) = lim s Composition is defined in the natural way. colim HomC(Xt, Ys). t □��... |

39 |
Etude Locale des Schemas et des Morphismes de
- Grothendieck, Dieudonne
- 1964
(Show Context)
Citation Context ...nite Limits of Schemes. We first study how finite limits interact with étale maps and separated maps. The results here are not particularly striking, but they do not appear in the standard literature =-=[EGA]-=- [Ha] [Mi] [T]. Proposition 4.2. Let f : U → X be a map of finite diagrams of schemes such that the map f a : U a → X a is étale (resp., separated) for every a. Then the map limf : limU → limX is étal... |

38 |
The completely decomposed topology on schemes and associated descent spectral sequence in algebraic K-theory. Algebraic K-theory, I: higher
- Nisnevich
- 1973
(Show Context)
Citation Context ...type over S. We consider two Grothendieck topologies on this category. The étale topology has covers consisting of finite collections of étale maps that have surjective images. The Nisnevich topology =-=[N]-=- has covers consisting of finite collections of étale maps {U a → X} that have surjective images and such that for every point x of X, there is a point u of some U a such that the map k(x) → k(u) on r... |

37 | Universal homotopy theories
- Dugger
(Show Context)
Citation Context ...motopy theory. We assume familiarity with closed model structures. General references on this topic include [13], [14], or [26]. We conform to the conventions of [13] as closely as possible. See also =-=[6]-=- for more details on model structures as applied to simplicial presheaves. Next comes the definition of the étale realization functor, and the first major result is that it is homotopy invariant on th... |

36 | Local projective model structures on simplicial presheaves - Blander - 2001 |

33 | Hypercovers and simplicial presheaves
- Dugger, Hollander, et al.
(Show Context)
Citation Context ...l weak equivalences are the weak equivalences in this model structure. The following proposition tells us that our definition of local weak equivalences is the same as the usual one. Proposition 2.4. =-=[7]-=- The local weak equivalences are the same as the topological weak equivalences of [19, § 2]. Remark 2.5. Note that the same kind of T -localization can be applied to the objectwise injective model str... |

30 |
Localization of Model Categories
- Hirschhorn
- 2000
(Show Context)
Citation Context ...s of the paper follows. Section 2 begins with a review of simplicial presheaves and their homotopy theory. We assume familiarity with closed model structures. General references on this topic include =-=[13]-=-, [14], or [26]. We conform to the conventions of [13] as closely as possible. See also [6] for more details on model structures as applied to simplicial presheaves. Next comes the definition of the é... |

30 |
A1-homotopy theory of schemes
- Morel, Voevodsky
- 2001
(Show Context)
Citation Context ...Dugger for useful conversations. 12 DANIEL C. ISAKSEN 1. Introduction In the recent proof of the Milnor conjecture [31], a certain realization functor from the A1-homotopy category of schemes over C =-=[25]-=- to the ordinary homotopy category of spaces plays a useful role. The basic idea is to detect that a certain map in the stable A1-homotopy category is not homotopy trivial by checking that its image i... |

21 | A model structure for the category of pro-simplicial sets
- Isaksen
(Show Context)
Citation Context ...s of the constructions concerning the étale topological type that first appeared in [9]. Section 4 concerns pro-spaces. We review only the bare essentials of pro-spaces and their homotopy theory. See =-=[15]-=- for details. Some results from [16] on calculating colimits of pro-spaces are also necessary. A k-truncated realization functor is a necessary tool because the infinite colimits that are used to cons... |

19 |
Ensembles profinis simpliciaux et interprétation géométrique du foncteur T
- Morel
- 1996
(Show Context)
Citation Context ...on étale cohomology induced by the projection X × A 1 → X. The projection induces an isomorphism in étale cohomology by [23, Cor. VI.4.20]. □ Remark 2.17. It is also possible to use the completion of =-=[24]-=- in order to define a slightly different A 1 -homotopy invariant étale realization functor. See [17] for more details. The next corollary follows from Theorem 2.15 in the same way that Corollary 2.8 f... |

18 |
The Milnor Conjecture, preprint
- Voevodsky
- 1996
(Show Context)
Citation Context ...ir Voevodsky for suggesting the problem. The author also thanks Ben Blander and Dan Dugger for useful conversations. 12 DANIEL C. ISAKSEN 1. Introduction In the recent proof of the Milnor conjecture =-=[31]-=-, a certain realization functor from the A1-homotopy category of schemes over C [25] to the ordinary homotopy category of spaces plays a useful role. The basic idea is to detect that a certain map in ... |

14 |
Universal homotopy theories, Adv
- Dugger
(Show Context)
Citation Context ...motopy theory. We assume familiarity with closed model structures. General references on this topic include [Hi], [Ho], or [Q1]. We conform to the conventions of [Hi] as closely as possible. See also =-=[D]-=- for more details on model structures as applied to simplicial presheaves. The first major result is that the étale realization functor is homotopy invariant on the local projective model structure fo... |

14 | Thomason Algebraic K-theory eventually surjects onto topological K-theory
- Dwyer, Friedlander, et al.
- 1982
(Show Context)
Citation Context ...t to the realization Re(n ↦→ EtUn). The étale hypercover descent theorem is interesting for its own sake, even though our application is to A 1 -homotopy theory. For example, it is closely related to =-=[DFST]-=-. Our work can probably be used to give a more conceptual proof of [DFST, Thm. 9], in which only the properties of the étale topological type are used (andETALE REALIZATION ON THE A 1 -HOMOTOPY THEOR... |

13 |
Introduction to Étale Cohomology
- Tamme
- 1994
(Show Context)
Citation Context ...Schemes. We first study how finite limits interact with étale maps and separated maps. The results here are not particularly striking, but they do not appear in the standard literature [10] [11] [23] =-=[30]-=-. A technical result about fiber products comes first. The more general claim about arbitrary finite limits follows relatively easily. Lemma 3.1. Consider a diagram of schemes U �� V �� W X � Y such t... |

9 | Completions of pro-spaces
- Isaksen
(Show Context)
Citation Context ...A 1 -homotopy invariant functor, it is necessary to complete away from the characteristics of the residues fields of S. We use here a functorial model for Z/p-completion of pro-spaces as described in =-=[17]-=-, where p is a prime not occurring as a characteristic of a residue field of S. Below is a summary of the details of this construction. Let X be a pointed space. Then ˆ X is a tower · · · → (Z/p)2X → ... |

9 | Calculating limits and colimits in pro-categories, Fund - Isaksen |

6 | Duality and pro-spectra
- Christensen, Isaksen
(Show Context)
Citation Context ...n established. We also hope to stabilize our techniques to obtain a functor on stable A1-homotopy theory. Although some progress on the foundations of the homotopy theory of pro-spectra has been made =-=[5]-=- [18], it is not yet clear whether these theories are suitable for the current application. In this paper, the first shortcoming of the analytic realization described above remains unfixed. We plan to... |

6 | Homotopy theories, Mem - Heller - 1988 |

6 |
Approximation filtrante de diagrammes finis par Pro-C
- Meyer
- 1980
(Show Context)
Citation Context ...on is just a natural transformation such that the maps fs represent the element f of lim s colim HomC(Xt, Ys) t ∼ = lim colim s t HomC( ˜ Xt, ˜ Ys). Every map has a level representation [2, App. 3.2] =-=[22]-=-. More generally, suppose given any diagram A → pro-C : a ↦→ X a . A level representation of X is: a cofiltered index category I; a functor ˜ X : A × I → C : (a, s) ↦→ ˜ Xa s ; and isomorphisms X a → ... |

4 |
The pro-Atiyah-Hirzebruch spectral sequence
- Isaksen
(Show Context)
Citation Context ...tablished. We also hope to stabilize our techniques to obtain a functor on stable A1-homotopy theory. Although some progress on the foundations of the homotopy theory of pro-spectra has been made [5] =-=[18]-=-, it is not yet clear whether these theories are suitable for the current application. In this paper, the first shortcoming of the analytic realization described above remains unfixed. We plan to fill... |

2 | Calculating limits and colimits in pro-categories, preprint - Isaksen |