## Topological K-theory of Algebraic K-theory Spectra (1999)

Venue: | J. Algebraic K-Theory |

Citations: | 13 - 4 self |

### BibTeX

@ARTICLE{Mitchell99topologicalk-theory,

author = {Stephen A. Mitchell},

title = {Topological K-theory of Algebraic K-theory Spectra},

journal = {J. Algebraic K-Theory},

year = {1999},

volume = {3},

pages = {607--626}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction One of the central problems of algebraic K-theory is to compute the K-groups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = Spec R, then the homology groups of the zero-th space 1 KX are of interest since they are the homology groups of the innite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex K-theory is another important invariant. Let K denote the periodic complex K-theory spectrum, and let ^ K denote its Bouseld `-adic completion

### Citations

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(Show Context)
Citation Context ...ote the complex points of X, with the classical topology. Then the hypotheses (T) hold, and the étale homology groups H∗(X; Zℓ) agree with the singular homology groups H∗(XC; Zℓ), by Artin’s theorem (=-=[4]-=-, Exposé XI). The resulting spectral sequence as in Corollary 5.2 is of limited interest because the hypercohomology spectrum itself can be explicitly identified: 10H · (X; K) ∧ ∼ = F((XC)+, bu) ∧ wh... |

165 | Etale homotopy - Artin, Mazur - 1969 |

121 |
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Citation Context ...eory, ( ˆ K/ℓ) ∗ is essentially the second in an infinite hierarchy of cohomology theories—the Morava K-theories: K(0), K(1), K(2)... Here K(0) is rational cohomology. By work of Hopkins and J. Smith =-=[10]-=-, these theories are “primes” of the ℓ-local stable homotopy category, and together with mod ℓ cohomology as a sort of “prime at infinity” they form a complete list of the distinct primes. Moreover, b... |

86 |
Anneaux locaux henseliens
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Citation Context ...oetherian. (It isn’t clear to the author whether or not this is always true. The local rings are noetherian, since they are filtered colimits of local étale algebras over a local noetherian base; cf. =-=[16]-=-, p. 95.) We have the following fairly general criterion: Let X be a connected noetherian normal scheme with function field F, and let {Fn} denote the cyclotomic tower as usual. Let Xn denote the norm... |

51 |
Etale Homotopy of Simplicial Schemes
- Friedlander
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(Show Context)
Citation Context ... Xét, then ˇ H∗ ét (X; A) ∼ = H∗ ét (X; aA), where a(−) is sheafification. This fact will be used frequently below. An alternative approach would be to use Verdier’s theory of hypercovers, as in [2], =-=[9]-=-. This would obviate the Artin property, and for some purposes would even allow us to replace Xét by a more general site. We will not consider hypercovers here, however, as that approach properly belo... |

46 |
On the K-theory of local fields
- Suslin
- 1984
(Show Context)
Citation Context ... the point is that ℓcompleted algebraic K-theory is already invariant, by [15] and [18]. Thus we have reduced to checking the cases k = C and k = Fp. For k = C, we have a weak equivalence : KC ∧−→bu∧ =-=[19]-=-. Lemma 4.6 The natural map θ top C : ˆ K 0 bu−→Zℓ[[Inj(µ∞C, µ∞C)]] is an isomorphism of Λ ′- modules. Proof: The two Λ ′-modules in the theorem are free of rank one, so it suffices to show θ top C is... |

45 |
On the K–theory of algebraically closed fields
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(Show Context)
Citation Context ...nsions k ⊂ k ′ with k, k ′ algebraically closed. Again, both statements are obvious for F = L. For F = (K ∧ N )0K(−), the point is that ℓcompleted algebraic K-theory is already invariant, by [15] and =-=[18]-=-. Thus we have reduced to checking the cases k = C and k = Fp. For k = C, we have a weak equivalence : KC ∧−→bu∧ [19]. Lemma 4.6 The natural map θ top C : ˆ K 0 bu−→Zℓ[[Inj(µ∞C, µ∞C)]] is an isomorphi... |

44 | Two-primary algebraic K-theory of rings of integers in number "elds
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(Show Context)
Citation Context ...ere is no need for Thomason’s theorem. This brings us to Voevodsky’s proof of the Milnor conjecture, and the subsequent applications to the 2-adic Lichtenbaum-Quillen conjectures by Rognes and Weibel =-=[17]-=-. To make a long story short, combining the affirmed Milnor conjecture with the Bloch-Lichtenbaum spectral sequence results in an étale descent spectral sequence for the 2-adic algebraic K-theory of a... |

43 |
Stable homotopy theory of simplicial presheaves
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Citation Context ... hypercohomology spectrum associated with the algebraic K-theory presheaf. Equivalently, under the hypotheses (T), we can take any globally fibrant model for K in the closed model category of Jardine =-=[11]-=-. Then Thomason’s theorem says that the natural map KX−→H · (X; K) induces a weak equivalence on Bousfield localizations: ˆLKX ∼ = −→ ˆ LH · (X; K) For any spectrum E, the map E−→ ˆ LE induces an isom... |

41 |
Stable homotopy and generalized homology
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(Show Context)
Citation Context ...osition 3.3 ˆ K p Y ∼ = ((K ∧ N )p−1Y ) ∗ . Proof: ˆK p Y ∼ = ˆ K p (N ∧ Y ) ∼ = K p (N ∧ Y ) ∼ = (Kp−1N ∧ Y ) ∗ where the last isomorphism follows from the universal coefficient theorem for K-theory =-=[1]-=-. Now from Proposition 3.2 we have a spectral sequence ˇH p ét (X; (K ∧ N )qK) = H p ét (X; a(K ∧ N )qK) ⇒ (K ∧ N )q−pH · (X; K) where the first equality is by Artin’s theorem. Taking Pontrjagin duals... |

38 |
Algebraic K–theory and étale cohomology, Ann. Sci. École Norm. Sup
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(Show Context)
Citation Context ...culations apply only to the globally fibrant model H · (X; K). 2 Notation and Prerequisites 1. Hypercohomology Spectra. We assume the reader is familiar with either Thomason’s hypercohomology spectra =-=[20]-=-, or better, the closed model category of Jardine [11] [12]. The reader might also find it useful to consult the author’s expository article [14]. We recall from [11] that if Φ = (C, T ) is a Grothend... |

27 | Hypercohomology spectra and Thomason's descent theorem
- Mitchell
- 1996
(Show Context)
Citation Context ...es” of the ℓ-local stable homotopy category, and together with mod ℓ cohomology as a sort of “prime at infinity” they form a complete list of the distinct primes. Moreover, by a theorem of the author =-=[13]-=-, all the Morava K-theories beyond ( ˆ K/ℓ) ∗ vanish on KX. Therefore ˆ K ∗ has a distinguished role in the theory. Even better, ˆ K ∗ KX is accessible to computation, thanks to Thomason’s theorem ver... |

21 | On the joins of Hensel rings - Artin - 1971 |

18 | On the K-theory spectrum of a ring of algebraic integers, K-Theory 14
- Dwyer, Mitchell
- 1998
(Show Context)
Citation Context ...ng the Lichtenbaum-Quillen conjectures for ˆ LKX, the Bousfield localization with respect to ˆ K. The author and Bill Dwyer explicitly computed ˆ K ∗ KX when X is a ring of integers in a number field =-=[7]-=-, a smooth curve over a finite field [8], or a local field [7] (with some exceptions at the prime 2 that were left unresolved; see below for further discussion). In these cases the classical Iwasawa t... |

7 | The generalized Burnside ring and the K-theory of a ring with roots of unity, K-Theory 6
- Dwyer, Friedlander, et al.
- 1992
(Show Context)
Citation Context ...ut localizing. We recall briefly the proof of theorem 4.4. By a theorem of Bousfield [5], ˆ L factors through the zero-th space functor Ω ∞ . But on zero-th spaces, the lift exists without localizing =-=[6]-=-. From the fibre sequence KF ∧ q −→bu∧ ψq −1 −→ bu ∧ we compute ˆ K 0 KFq ∼ = Λ ′ ⊗ Λm−1 Zℓ where Λm−1 = Zℓ[[Γm−1]] and Γm−1 is the unique subgroup of index ℓ m−1 in Γ. Note that, up to units, q = γℓm... |

6 |
Generalized étale cohomology theories, Birkhäuser
- Jardine
- 1997
(Show Context)
Citation Context ...K). 2 Notation and Prerequisites 1. Hypercohomology Spectra. We assume the reader is familiar with either Thomason’s hypercohomology spectra [20], or better, the closed model category of Jardine [11] =-=[12]-=-. The reader might also find it useful to consult the author’s expository article [14]. We recall from [11] that if Φ = (C, T ) is a Grothendieck site with underlying category C and topology T ,the ca... |

6 |
Uniqueness of infinite deloopings for K-theoretic spaces
- Bousfield
- 1987
(Show Context)
Citation Context ...♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✸♣ ♣ ❄ ˆLKFq ✲ ˆLKFq = Remark: If the Lichtenbaum-Quillen conjectures hold, the lift exists without localizing. We recall briefly the proof of theorem 4.4. By a theorem of Bousfield =-=[5]-=-, ˆ L factors through the zero-th space functor Ω ∞ . But on zero-th spaces, the lift exists without localizing [6]. From the fibre sequence KF ∧ q −→bu∧ ψq −1 −→ bu ∧ we compute ˆ K 0 KFq ∼ = Λ ′ ⊗ Λ... |

6 | On the K-theory spectrum of a smooth curve over a finite field , Topology 36
- Dwyer, Mitchell
- 1997
(Show Context)
Citation Context ...∗ KFq, where Fq is the residue field of the ring of integers OF . This latter group is immediately computable from Quillen’s theorem. Details are left to the reader. Smooth curves over a finite field =-=[8]-=-. Let X be a smooth curve over a finite field of characteristic different from ℓ. If X is affine, ˆ K ∗ KX is given by a formula exactly analogous to that of Theorem 6.1. If X is complete, the formula... |

4 | On the K-theory of algebraically closed - Suslin - 1983 |

3 | On the K-theory of local - Suslin - 1984 |

2 | Uniqueness of in deloopings for K-theoretic spaces - Bous - 1987 |

2 |
Higher Algebraic K-theory 1
- Quillen
- 1973
(Show Context)
Citation Context ...rary extensions k ⊂ k ′ with k, k ′ algebraically closed. Again, both statements are obvious for F = L. For F = (K ∧ N )0K(−), the point is that ℓcompleted algebraic K-theory is already invariant, by =-=[15]-=- and [18]. Thus we have reduced to checking the cases k = C and k = Fp. For k = C, we have a weak equivalence : KC ∧−→bu∧ [19]. Lemma 4.6 The natural map θ top C : ˆ K 0 bu−→Zℓ[[Inj(µ∞C, µ∞C)]] is an ... |

1 | On the K-theory spectrum of a smooth curve over a Topology 36 - Dwyer, Mitchell - 1997 |

1 |
Etale Homotopy, Lecture Notes in
- Artin, Mazur
- 1969
(Show Context)
Citation Context ...af on Xét, then ˇ H∗ ét (X; A) ∼ = H∗ ét (X; aA), where a(−) is sheafification. This fact will be used frequently below. An alternative approach would be to use Verdier’s theory of hypercovers, as in =-=[2]-=-, [9]. This would obviate the Artin property, and for some purposes would even allow us to replace Xét by a more general site. We will not consider hypercovers here, however, as that approach properly... |