On the de Rham–Witt complex in mixed characteristic

Hesselholt, Lars; Madsen, Ib

doi:10.1016/j.ansens.2003.06.001

Cited by 55 publications

(76 citation statements)

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“…The case A r F p can be proved also directly using induction over r (Theorem 2.1 and [HeMa04,(4 Illusie already proved the statement for smooth F p -schemes, in particular, for A r F p .…”

Section: Proof In [Hema04] This Theorem Is Only Proven Formentioning

confidence: 93%

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The generalized de Rham-Witt complex over a field is a complex of zero-cycles

Rülling

2006

J. Algebraic Geom.

153

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Bloch and Esnault defined additive higher Chow groups with modulus m on the level of zero cycles over a field k denoted by CH n ((A 1 k , (m + 1){0}), n − 1), n, m ≥ 1. Bloch and Esnault prove CH n ((A 1 k , 2{0}), n − 1) ∼ = Ω n−1 k/Z . In this paper we generalize their result and prove that the additive Chow groups with higher modulus form a generalized Witt complex over k and are as such isomorphic to the generalized de Rham-Witt complex of Bloch-Deligne-Hesselholt-Illusie-Madsen.

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“…The case A r F p can be proved also directly using induction over r (Theorem 2.1 and [HeMa04,(4 Illusie already proved the statement for smooth F p -schemes, in particular, for A r F p .…”

Section: Proof In [Hema04] This Theorem Is Only Proven Formentioning

confidence: 93%

“…Following Illusie (see [Il79] or [HeMa04]) we give in Section 1 a concrete construction of the de Rham-Witt complex over a Z (p) -algebra, p = 2 a prime. Following Illusie (see [Il79] or [HeMa04]) we give in Section 1 a concrete construction of the de Rham-Witt complex over a Z (p) -algebra, p = 2 a prime.…”

Section: Introductionmentioning

confidence: 99%

The generalized de Rham-Witt complex over a field is a complex of zero-cycles

Rülling

2006

J. Algebraic Geom.

153

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“…The TR-groups are connected by several operators: R, F , V and d. In the ordinary (integer-graded) case, there are maps as follows (see [14] for more details). Inclusion of fixed points induces a map…”

Section: The Fundamental Diagrammentioning

confidence: 99%

“…These spectra are connected by maps R, F , V and d [14], and a homotopy limit over R and F gives us the topological cyclic homology spectrum TC(A; p). Therefore to compute topological cyclic homology, and hence algebraic K -theory in good cases, it is sufficient to understand TR n (A; p) together with R, F : TR n+1 * (A; p) → TR n * (A; p) for each p and n. The homotopy groups of these spectra are denoted TR n q (A; p) = [S q ∧ S 1 /C p n−1 + , T (A)] S 1 .…”

Section: Introductionmentioning

confidence: 99%

RO(S1)-graded TR-groups of
Angeltveit
¹
,
Gerhardt
²

2011
Journal of Pure and Applied Algebra

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Communicated by C.A. Weibel MSC: Primary: 55Q91 Secondary: 19D55 a b s t r a c tWe give an algorithm for calculating the RO(S 1 )-graded TR-groups of F p , completing the calculation started by the second author. We also calculate the RO(S 1 )-graded TR-groups of Z with mod p coefficients and of the Adams summand ℓ of connective complex K -theory with V (1)-coefficients. These calculations are used elsewhere to compute the algebraic Ktheory of certain Z-algebras. * (A) with integral coefficients proves to be too difficult one can instead consider the groups TR n * (A; V ) = π * (T (A) C p n−1 ∧ V ) for a suitable finite complex V . For instance, smashing with the mod p Moore spectrum V (0) = S/p was used in [5] to compute the mod p groups TR n * (Z; V (0)) = TR n * (Z; Z/p) for p ≥ 3. Similarly, smashing with the Smith-Toda complex V (1) = S/(p, v 1 ) was used in [3] to compute TR n * (ℓ; V (1)) for p ≥ 5. Here ℓ is the Adams summand of connective complex K -theory localized at p. In both of these cases, the * refers to an integer grading. We will use this technique of smashing with a finite complex in our computations, which are RO(S 1 )-graded.In this paper we calculate TR n α (F p ), the RO(S 1 )-graded TR-groups of F p , TR n α (Z; V (0)), the RO(S 1 )-graded TR-groups of Z with mod p coefficients, and TR n α (ℓ; V (1)), the RO(S 1 )-graded TR-groups of ℓ with V (1) coefficients. For the last case we Lemma 3.3. In the homotopy orbit to TR spectral sequence, every class in the Tate piece V t * T [−α (n−s) ] hC p s is a permanent cycle, and the image of any differential is contained in the Tate piece. If the short exact sequence in Definition 3.2 splits then all differentials go from a subgroup of the homotopy fixed point piece to a quotient of the Tate piece.Proof. This is a straightforward diagram chase, using the construction of the spectral sequence and Diagram (2.2).We will denote classes in V t * T [−α] hC p n by their name in V * T [−α] tC p n and classes in V h * T [−α] hC p n by their name in V * T [−α] hC p n .

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“…As I mentioned earlier, one of the applications of symmetric spectra [114] was to give the modern formulation of a Quillen model category; for example, as used by Voevodsky. Other important stable homotopy constructions occur in the work of Ib Madsen and his collaborators -including Lars Hesselholt ([106], [107]), Ulrike Tillmann [170] and Michael Weiss -culminating respectively in the calulation of the K-theory of local fields and in the solution of David Mumford's conjecture on the cohomology of moduli spaces [171] (see also [44] for further topology in physics). In addition there are areas which grew out of Ed Witten's construction of elliptic cohomology in the late 1980's.…”

Section: 216mentioning

confidence: 99%

Stable Homotopy Around the Arf-Kervaire Invariant

Snaith¹

2009

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On the de Rham–Witt complex in mixed characteristic

Cited by 55 publications

References 20 publications

The generalized de Rham-Witt complex over a field is a complex of zero-cycles

The generalized de Rham-Witt complex over a field is a complex of zero-cycles

RO(S1)-graded TR-groups of
Angeltveit
¹
,
Gerhardt
²

2011
Journal of Pure and Applied Algebra

Stable Homotopy Around the Arf-Kervaire Invariant

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On the de Rham–Witt complex in mixed characteristic

Cited by 55 publications

References 20 publications

The generalized de Rham-Witt complex over a field is a complex of zero-cycles

The generalized de Rham-Witt complex over a field is a complex of zero-cycles

RO(S1)-graded TR-groups of Angeltveit1, Gerhardt2 2011Journal of Pure and Applied Algebra

Stable Homotopy Around the Arf-Kervaire Invariant

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Product

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About

RO(S1)-graded TR-groups of
Angeltveit
¹
,
Gerhardt
²

2011
Journal of Pure and Applied Algebra