## The K-theory presheaf of spectra (2008)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Jardine08thek-theory,

author = {J. F. Jardine},

title = {The K-theory presheaf of spectra},

year = {2008}

}

### OpenURL

### Abstract

This paper has evolved from notes for a lecture entitled “ Étale K-theory: a modern

### Citations

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Higher algebraic K-theory. I
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Citation Context ... 6 also uses the additivity theorem for K-theory. Recall that there is an exact category Ex(M) whose objects are the short exact sequences E : 0 → M ′ → M → M ′′ → 0 of M. Then the additivity theorem =-=[17]-=-, [25] asserts that the exact functor E ↦→ (M ′ , M ′′ ) induces a weak equivalence s•(Ex(M)) ≃ −→ s•(M) × s•(M). This result implies that the induced maps are weak equivalences for all k ≥ 1. s k •(E... |

189 | Symmetric spectra
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Citation Context ... presheaves. The étale-local stable model structure on the category of presheaves of symmetric spectra on Sch|S (this is the étale-local version of the Hovey-ShipleySmith theory for symmetric spectra =-=[6]-=-) is a little more interesting to describe. The fibrations for the theory are those maps X → Y for which the underlying maps of presheaves of spectra are fibrations for the étale-local model structure... |

175 |
Higher algebraic K-theory of schemes and of derived categories
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Citation Context ...y coincides with ordinary algebraic K-theory in this case. More generally, and in particular if S is not necessarily regular, the comparison map T T K → K with the Thomason-Trobaugh K-theory presheaf =-=[21]-=- (they would write K(S) B for K T T (S)) induces a Nisnevich local weak equivalence Q(K) 0 → Q(K T T ) 0 of associated presheaves of infinite loop spaces. In effect, the maps Kn(T ) = πnK(T ) → πnK T ... |

146 |
Homotopy theory of Γ-spaces, spectra, and bisimplicial sets
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Citation Context ...X means that j is a weak equivalence and Z is fibrant. The fibrant model in the statement of Theorem 6 is chosen in the strict model structure for the category of spectra of Bousfield and Friedlander =-=[1]-=-, [4], in which weak equivalences and fibrations are defined levelwise. There are two main ingredients in the proof of Theorem 6. The first of these is a general construction for simplicial sets. Let ... |

114 |
Simplicial presheaves
- Jardine
- 1987
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Citation Context ...on a (small) Grothendieck site C has the local weak equivalences (or stalkwise weak equivalences, in the presence of a calculus of stalks) for weak equivalences and the monomorphisms for cofibrations =-=[7]-=-. This structure induces a model structure for presheaves of groupoids on C: a map f : G → H of presheaves of groupoids is a weak equivalence (respectively fibration) if the induced map BG → BH is a l... |

86 |
Simplicial homotopy theory, volume 174
- Goerss, Jardine
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Citation Context ...ns that j is a weak equivalence and Z is fibrant. The fibrant model in the statement of Theorem 6 is chosen in the strict model structure for the category of spectra of Bousfield and Friedlander [1], =-=[4]-=-, in which weak equivalences and fibrations are defined levelwise. There are two main ingredients in the proof of Theorem 6. The first of these is a general construction for simplicial sets. Let X be ... |

79 |
Algebraic K-theory of spaces. Algebraic and geometric topology (New
- Waldhausen
- 1983
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Citation Context ...peared before. Everything turns on the following observation: with a little care (in particular, by taking zero objects as base points), one can use Waldhausen’s construction of the K-theory spectrum =-=[25]-=- to functorially associate a symmetric spectrum K(M) to an exact category M. This symmetric spectrum construction takes exact pairings to smash product pairings functorially. Thus, tensor product pair... |

75 | Motivic symmetric spectra
- Jardine
- 2000
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Citation Context ...ups, π∗K/n(T ) = K∗(T, Z/n) in all sections, where the thing on the right is the traditional mod n K-theory of the scheme T . One can topologize K/n, or any other presheaf of (symmetric) spectra [8], =-=[11]-=-, by taking a stably fibrant model with respect to each of the topologies on the big site Sch|S. 15There is a local stable model structure for the étale topology for Sch|S on the category of presheav... |

52 |
A 1 -homotopy theory
- Voevodsky
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Citation Context ...ed colimit of Grassmanians G(n, ∞) ≃ BGLn (motivic weak equivalence) of n-planes in infinite dimensional space [15, 4.3.14]. The level 0 part of the equivalence (16) was originally found by Voevodsky =-=[22]-=-; his proof uses the equivalences in (17). The algebraic K-theory presheaf of spectra K on the site Sm|S satisfies motivic descent when S is regular. In that case, the algebraic K-groups Kn(S), n ≥ 0,... |

45 | The spectral sequence relating algebraic K-theory to motivic cohomology
- Friedlander, Suslin
- 2000
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Citation Context ... all topologized versions of algebraic K-theory. The final trick in this mix of ideas is to use big site vector bundles instead of ordinary vector bundles to construct the K-theory presheaf (see also =-=[2]-=- and [16]). Big site vector bundles restrict functorially (instead of pseudofunctorially) on the big site, and this has the effect of giving a solution to the standard coherence problem that has tradi... |

38 |
Stable homotopy theory of simplicial presheaves
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- 1987
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Citation Context ...y groups, π∗K/n(T ) = K∗(T, Z/n) in all sections, where the thing on the right is the traditional mod n K-theory of the scheme T . One can topologize K/n, or any other presheaf of (symmetric) spectra =-=[8]-=-, [11], by taking a stably fibrant model with respect to each of the topologies on the big site Sch|S. 15There is a local stable model structure for the étale topology for Sch|S on the category of pr... |

33 | A 1 -homotopy theory of schemes - Morel, Voevodsky - 1999 |

32 | The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. www.math.uiuc.edu/K-theory/0280
- Geisser, Levine
- 1998
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Citation Context ...Galois cohomology of a field is generated multiplicatively by elements in dimension 1: Suslin and Voevodsky show that the Bloch-Kato conjecture implies a conjecture of Beilinson and Lichtenbaum [19], =-=[3]-=-, and then the argument is completed with a comparison of motivic descent spectral sequences [14]. The case n = 2 of the Bloch-Kato conjecture is the Milnor conjecture, which was proved by Voevodsky [... |

29 | A Homotopy Theory for Stacks
- Hollander
(Show Context)
Citation Context ... → H of presheaves of groupoids is a weak equivalence (respectively fibration) if the induced map BG → BH is a local weak equivalence (respectively injective fibration) of simplicial presheaves [13], =-=[5]-=-, [12]. Stacks are presheaves of groupoids G which satisfy descent for this model structure: explicitly, G satisfies descent if any fibrant model G → H induces equivalences of groupoids G(U) → H(U) in... |

27 |
Algebraic K-theory and étale cohomology
- Thomason
- 1985
(Show Context)
Citation Context ...ils, in [24]. Formally invert mulitplication by the Bott element β to produce the presheaf of spectra K/n(1/β). This is now easy to do — use the tensor product pairing (8). Thomason’s descent theorem =-=[20]-=-, [9] says that the étale fibrant model induces a stable equivalence K/n(1/β) → F (K/n(1/β)) K/n(1/β)(S) → F (K/n(1/β))(S) for schemes S satisfying the assumptions for the Lichtenbaum-Quillen conjectu... |

23 |
Motivic cohomology with Z/2 coefficients
- Voevodsky
(Show Context)
Citation Context ...], and then the argument is completed with a comparison of motivic descent spectral sequences [14]. The case n = 2 of the Bloch-Kato conjecture is the Milnor conjecture, which was proved by Voevodsky =-=[23]-=-, and a proof of the corresponding case of the Lichtenbaum-Quillen conjecture appears in [18]. A proof of the full Bloch-Kato conjecture is outlined, subject to some missing details, in [24]. Formally... |

19 | Stacks and the homotopy theory of simplicial sheaves, in Equivariant stable homotopy theory and related areas
- Jardine
- 2000
(Show Context)
Citation Context ...of presheaves of groupoids is a weak equivalence (respectively fibration) if the induced map BG → BH is a local weak equivalence (respectively injective fibration) of simplicial presheaves [13], [5], =-=[12]-=-. Stacks are presheaves of groupoids G which satisfy descent for this model structure: explicitly, G satisfies descent if any fibrant model G → H induces equivalences of groupoids G(U) → H(U) in all s... |

18 |
Strong stacks and classifying spaces
- Joyal, Tierney
- 1990
(Show Context)
Citation Context ... f : G → H of presheaves of groupoids is a weak equivalence (respectively fibration) if the induced map BG → BH is a local weak equivalence (respectively injective fibration) of simplicial presheaves =-=[13]-=-, [5], [12]. Stacks are presheaves of groupoids G which satisfy descent for this model structure: explicitly, G satisfies descent if any fibrant model G → H induces equivalences of groupoids G(U) → H(... |

17 |
Generalized Étale Cohomology Theories, volume 146
- Jardine
- 1997
(Show Context)
Citation Context ...R ↦→ Z×BGl(R) + for affine schemes Sp(R), or b) correct in the sense that K0 shows up as a presheaf of stable homotopy groups, but almost notationaly impenetrable — see the fun with pseudofunctors in =-=[9]-=-, for example. 1The first three sections of this paper are intended to repair this difficulty. The basic constructions that are used in this paper are well known, but they are organized here in a way... |

14 | K-theory and motivic cohomology of schemes
- Levine
(Show Context)
Citation Context ...d Voevodsky show that the Bloch-Kato conjecture implies a conjecture of Beilinson and Lichtenbaum [19], [3], and then the argument is completed with a comparison of motivic descent spectral sequences =-=[14]-=-. The case n = 2 of the Bloch-Kato conjecture is the Milnor conjecture, which was proved by Voevodsky [23], and a proof of the corresponding case of the Lichtenbaum-Quillen conjecture appears in [18].... |

13 | On Voevodsky’s algebraic K-theory spectrum, Algebraic Topology
- Panin, Pimenov, et al.
- 2009
(Show Context)
Citation Context ...ologized versions of algebraic K-theory. The final trick in this mix of ideas is to use big site vector bundles instead of ordinary vector bundles to construct the K-theory presheaf (see also [2] and =-=[16]-=-). Big site vector bundles restrict functorially (instead of pseudofunctorially) on the big site, and this has the effect of giving a solution to the standard coherence problem that has traditionally ... |

6 |
The homotopy limit problem for two-primary algebraic K-theory
- Rosenschon, Østvær
- 2005
(Show Context)
Citation Context ... [14]. The case n = 2 of the Bloch-Kato conjecture is the Milnor conjecture, which was proved by Voevodsky [23], and a proof of the corresponding case of the Lichtenbaum-Quillen conjecture appears in =-=[18]-=-. A proof of the full Bloch-Kato conjecture is outlined, subject to some missing details, in [24]. Formally invert mulitplication by the Bott element β to produce the presheaf of spectra K/n(1/β). Thi... |

2 |
On motivic cohomology with Z/ℓ-coefficients
- Voevodsky
- 2003
(Show Context)
Citation Context ...Voevodsky [23], and a proof of the corresponding case of the Lichtenbaum-Quillen conjecture appears in [18]. A proof of the full Bloch-Kato conjecture is outlined, subject to some missing details, in =-=[24]-=-. Formally invert mulitplication by the Bott element β to produce the presheaf of spectra K/n(1/β). This is now easy to do — use the tensor product pairing (8). Thomason’s descent theorem [20], [9] sa... |