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Hypercohomology spectra and Thomason’s descent theorem
 IN ALGEBRAIC KTHEORY, FIELDS INSTITUTE COMMUNICATIONS
, 1997
"... The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral se ..."
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The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral sequence of AtiyahHirzebruch type H p q ét (X; Zℓ ()) ⇒ πq−p(KX 2 ℓ) but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the ℓadic integers, and are to be interpreted as zero if q is odd. Throughout this paper, étale cohomology is continuous étale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield ℓadic completion of the spectrum KX, not the naive ℓadic completion of the Kgroups. In a remarkable paper [42], Thomason proved the LichtenbaumQuillen conjectures for a certain localized form of Ktheory socalled “Bottperiodic” Ktheory. The first step was the development of an elaborate theory of hypercohomology spectra H ·(X; E) associated to étale presheaves of spectra E or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally Supported by a grant from the National Science Foundation 1 equipped with a suitable descent spectral sequence, and there is a natural map E(X)−→H · (X; E). In particular, the Ktheory of a scheme X maps to its associated hypercohomology spectrum H · ét(X; K). Fix a prime ℓ, which we will assume is odd in order to simplify the discussion.Let L(−) denote Bousfield localization with respect to complex Ktheory, and let ˆ L(−) denote its ℓadic completion. The main theorem of [42] can be stated as follows. Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull dimension, with sufficiently nice residue fields of characteristic different from ℓ. Then the natural map KX−→H · ét(X; K) induces a weak equivalence
The Generalized Burnside Ring And The KTheory Of A RING WITH ROOTS OF UNITY
, 1992
"... this paper is to extend the results of the third author to this wider context. Theorem 4.1 asserts that after localization at ` the maps ..."
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Cited by 6 (2 self)
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this paper is to extend the results of the third author to this wider context. Theorem 4.1 asserts that after localization at ` the maps
Algebraic Ktheory of the twoadic integers
 J. Pure Appl. Algebra
, 1997
"... Abstract. We compute the twocompleted algebraic Kgroups K∗ ( ˆ Z2) ∧ 2 of the twoadic integers, and determine the homotopy type of the twocompleted algebraic Ktheory spectrum K ( ˆ Z2) ∧ 2. The natural map K(Z) ∧ 2 → K( ˆ Z2) ∧ 2 is shown to induce an isomorphism modulo torsion in degrees 4 ..."
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Abstract. We compute the twocompleted algebraic Kgroups K∗ ( ˆ Z2) ∧ 2 of the twoadic integers, and determine the homotopy type of the twocompleted algebraic Ktheory spectrum K ( ˆ Z2) ∧ 2. The natural map K(Z) ∧ 2 → K( ˆ Z2) ∧ 2 is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.
Weight filtrations in algebraic Ktheory
 In Motives, volume 55 of Proceedings of Symposia in Pure Mathematics
, 1994
"... Abstract. We survey briefly some of the Ktheoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher ..."
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Abstract. We survey briefly some of the Ktheoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic Ktheory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain
Étale descent for real number fields
, 2000
"... Abstract. In this paper we verify the strong Quillen–Lichtenbaum conjecture for integers in real number fields at the prime two. That is, we prove that the Dwyer– Friedlander map from mod two algebraic K–theory to mod two étale topological K–theory is a weak equivalence on zero–connected covers for ..."
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Cited by 3 (2 self)
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Abstract. In this paper we verify the strong Quillen–Lichtenbaum conjecture for integers in real number fields at the prime two. That is, we prove that the Dwyer– Friedlander map from mod two algebraic K–theory to mod two étale topological K–theory is a weak equivalence on zero–connected covers for two–integers in real number fields. The proof is given by comparing two explicit calculations. Contents
THE HOMOTOPY TYPE OF TWOREGULAR KTHEORY
"... Abstract. We identify the 2adic homotopy type of the algebraic Ktheory space for rings of integers in tworegular exceptional number fields. The answer is given in terms of wellknown spaces considered in topological Ktheory. 1. Introduction. Let E be a number field, OE its ring of algebraic inte ..."
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Abstract. We identify the 2adic homotopy type of the algebraic Ktheory space for rings of integers in tworegular exceptional number fields. The answer is given in terms of wellknown spaces considered in topological Ktheory. 1. Introduction. Let E be a number field, OE its ring of algebraic integers, and RE = OE [ 1] the
A topological proof of the vanishing of the product of K3(Z) with K1(Z
 In: Proceedings of the International Conference on Algebraic KTheory, Poznań
, 1995
"... Abstract. This paper gives a short and purely homotopical proof of the triviality of the product map K3(Z) ⊗ K1(Z) → K4(Z). The argument is based on the study of the external product for the homotopy groups of the Ktheory spectrum and the fact that the 4dimensional Hurewicz homomorphism in the a ..."
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Abstract. This paper gives a short and purely homotopical proof of the triviality of the product map K3(Z) ⊗ K1(Z) → K4(Z). The argument is based on the study of the external product for the homotopy groups of the Ktheory spectrum and the fact that the 4dimensional Hurewicz homomorphism in the algebraic Ktheory of Z is an isomorphism. The purpose of this short note is to provide a simple proof of the following assertion concerning the 4th algebraic Kgroup of the ring of integers Z. Theorem. The product map K3(Z) ⊗ K1(Z) − → K4(Z) is trivial. Recall that J. Rognes detected recently a gap in his proof of the vanishing of K4(Z). Consequently, it may be interesting to consider the product of elements of K3(Z) ∼ = Z/48 with elements of K1(Z) ∼ = Z/2 in K4(Z). Notice that the statement of the above theorem could be deduced from Bökstedt’s result on the factorization of the algebraic Ktheory space BGL(Z) + after completion at the prime 2: (BGL(Z)+) ̂ 2 ≃ JK(Z) ̂ 2 × ? (see [B]). However, C. Soulé asked me during the Poznań conference on algebraic Ktheory for a direct proof of the theorem using only topological arguments. It will be an immediate consequence of Propositions 3 and 5 below. Throughout the paper, we shall consider all homology groups with coefficients in Z and use the following notation: for any abelian group G, let us write K(G, n) for the EilenbergMacLane space whose nth homotopy group is G and HG for the EilenbergMacLane spectrum whose 0th homotopy group is G; for a CWcomplex or a spectrum X and an integer n, αn: X → X[n] denotes the nth Postnikov section of X, i.e., αn induces an isomorphism on the ith homotopy group for i ≤ n and πiX[n] = 0 for i> n. Lemma 1. Let S denote the sphere spectrum, X any spectrum, n an integer and m a positive integer. Then the image of the external product ∧ : πnX ⊗ πmS → πn+mX
A Topological Proof Of The Vanishing Of The Product Of ...
"... . This paper gives a short and purely homotopical proof of the triviality of the product map K3 (Z)\Omega K1 (Z) ! K4 (Z) . The argument is based on the study of the external product for the homotopy groups of the Ktheory spectrum and the fact that the 4dimensional Hurewicz homomorphism in the a ..."
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. This paper gives a short and purely homotopical proof of the triviality of the product map K3 (Z)\Omega K1 (Z) ! K4 (Z) . The argument is based on the study of the external product for the homotopy groups of the Ktheory spectrum and the fact that the 4dimensional Hurewicz homomorphism in the algebraic Ktheory of Z is an isomorphism. The purpose of this short note is to provide a simple proof of the following assertion concerning the 4th algebraic Kgroup of the ring of integers Z . Theorem. The product map K 3 (Z)\Omega K 1 (Z) \Gamma! K 4 (Z) is trivial. Recall that J. Rognes detected recently a gap in his proof of the vanishing of K 4 (Z) . Consequently, it may be interesting to consider the product of elements of K 3 (Z) ¸ = Z=48 with elements of K 1 (Z) ¸ = Z=2 in K 4 (Z) . Notice that the statement of the above theorem could be deduced from Bokstedt's result on the factorization of the algebraic Ktheory space BGL(Z) + after completion at the prime 2 : (BGL(Z) ...