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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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Cited by 20 (2 self)
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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 QuillenLichtenbaum Conjecture at the prime 2, preprint
, 1997
"... 2. Foundational results 6 3. Divisibility of motivic cohomology; proof of theorem 3 a) 8 ..."
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Cited by 13 (8 self)
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2. Foundational results 6 3. Divisibility of motivic cohomology; proof of theorem 3 a) 8
Topological Ktheory of Algebraic Ktheory Spectra
 J. Algebraic KTheory
, 1999
"... Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups 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 homotopytype of the spectrum, rather than just the disembodied homotopy groups. ..."
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Cited by 10 (3 self)
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Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups 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 homotopytype of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the Kgroups 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 zeroth 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 Ktheory is another important invariant. Let K denote the periodic complex Ktheory spectrum, and let ^ K denote its Bouseld `adic completion
The smooth Whitehead spectrum of a point at odd regular primes
 TOPOL
, 2003
"... Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(Z[1/p]) at p. Then the pprimary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernelofJ spectrum splits off, and the torsion homotopy of the remainder eq ..."
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Cited by 7 (3 self)
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Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(Z[1/p]) at p. Then the pprimary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernelofJ spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S 1transfer map t: ΣCP ∞ → S. The homotopy of W h(∗) is determined in a range of degrees, and the cohomology of W h(∗) is expressed as an Amodule in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.
Twoprimary algebraic KTheory of pointed spaces
, 2002
"... We compute the mod 2 cohomology of Waldhausen’s algebraic Ktheory spectrum A(∗) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted WhDiff (∗). Using an Adams spectral sequence ..."
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Cited by 6 (3 self)
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We compute the mod 2 cohomology of Waldhausen’s algebraic Ktheory spectrum A(∗) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted WhDiff (∗). Using an Adams spectral sequence we compute the 2primary homotopy groups of these spectra in dimensions ∗ ≤ 18, and up to extensions in dimensions 19 ≤ ∗ ≤ 21. As applications we show that the linearization map L: A(∗) → K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher–Waldhausen map hw: G/O → ΩWh Diff (∗) does not admit a fourfold delooping, and there is a 2complete spectrum map M: Wh Diff (∗) → Σg/o ⊕ which is precisely 9connected. Here g/o ⊕ is a spectrum whose underlying space has the 2complete homotopy type of G/O.
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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Cited by 5 (2 self)
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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.
Class Groups and General Linear Group Cohomology for a Ring of Algebraic Integers
, 1994
"... Note: This is a revised and expanded version of an earlier (and much shorter) preprint with the same title. The main theorems and their proofs are unchanged, but much exposition has been added. Suppose that F is a number field, with ring of integers OF. Let ℓ denote an odd prime and let R = OF [1/ℓ] ..."
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Cited by 3 (2 self)
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Note: This is a revised and expanded version of an earlier (and much shorter) preprint with the same title. The main theorems and their proofs are unchanged, but much exposition has been added. Suppose that F is a number field, with ring of integers OF. Let ℓ denote an odd prime and let R = OF [1/ℓ]. In [3], the author and W.Dwyer gave an explicit conjectural computation of the mod ℓ cohomology of the infinite general linear group GLR. Here is the quickest and simplest statement of the conjecture (all homology and cohomology groups have Z/ℓ coefficients): let U denote the infinite unitary group. Let J ∗ a denote the mod ℓ cohomology of the homotopyfibre of the ℓ ath power map SU→SU. Thus as an algebra, J ∗ a is the tensor product of a certain polynomial algebra P ∗ a and a companion exterior algebra E ∗ a. An explicit description is given in Section 1; here we just remark that the Hopf algebra Pa dual to P ∗ a is H∗BU/(ℓ ath powers). Let P denote the algebra of Steenrod ℓth power operations. Conjecture A. Suppose F contains the ℓth roots of unity µ ℓ. PHopf algebras