Motivic cohomology over Dedekind rings

Geisser, Thomas

doi:10.1007/s00209-004-0680-x

Cited by 81 publications

(123 citation statements)

References 22 publications

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“…The result with mod p r -coefficients is [9, IV §1] because of Z/p r (n) ∼ = W r Ω n X,log [8]. Finally, rationally motivic cohomology and etale motivic cohomology agree [7,Prop. 3.6], hence the result with rational coefficients is [18,Lemma IV 3.12].…”

Section: Motivic Hodge and De Rham Cohomologymentioning

confidence: 98%

Arithmetic cohomology over finite fields and special values of ζ-functions

Geisser

2006

Duke Math. J.

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Summary. We construct cohomology groups with compact support H i c (Xar, Z(n)) for separated schemes of finite type over a finite field, which generalize Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds, and rational and numerical equivalence agree up to torsion, then the groups H i c (Xar, Z(n)) are finitely generated, form an integral model of ladic cohomology with compact support, and admit a formula for the special values of the ζ-function of X.

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Section: Motivic Hodge and De Rham Cohomologymentioning

confidence: 98%

Arithmetic cohomology over finite fields and special values of ζ-functions

Geisser

2006

Duke Math. J.

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“…As Geisser has shown [30], in the case without log-structure, the isomorphism (1.3) allows to approximate the (continuous) p-adic motivic cohomology (sheaves) of padic varieties by their syntomic cohomology; hence to relate p-adic algebraic cycles to differential forms. This was used to study algebraic cycles in mixed characteristic [54] We hope that the "isomorphism" (1.2) that generalizes (1.3) will allow to extend the above mentioned applications.…”

Section: Syntomic Complexes and P-adic Nearby Cyclesmentioning

confidence: 99%

“…In [62] Tsuji generalized this result to someétale local systems. As Geisser has shown [30], in the case without log-structure, the isomorphism (1.3) allows to approximate the (continuous) p-adic motivic cohomology (sheaves) of padic varieties by their syntomic cohomology; hence to relate p-adic algebraic cycles to differential forms. This was used to study algebraic cycles in mixed characteristic [54], geometric class field theory [41], Beilinson's Tate conjecture [5], variational Hodge conjecture [11], p-adic regulators and special values of p-adic L-functions [58].…”

Section: Introductionmentioning

confidence: 99%

Syntomic complexes and p-adic nearby cycles

Colmez

Nizioł

2016

Invent. math.

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Abstract. We compute syntomic cohomology of semistable affinoids in terms of cohomology of (ϕ, Γ)-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness of etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction. Let O K be a complete discrete valuation ring with fraction field K of characteristic 0 and with perfect residue field k of characteristic p. Let O F = W (k) and F = O F [ 1 p ] so that K is a totally ramified extension of F ; let e = [K : F ] be the absolute ramification index of K. Let O K denote the integral closure of O K in K. Set G K = Gal(K/K), and let ϕ = ϕ W (k) be the absolute Frobenius on W (k). For a log-scheme X over O K , X n will denote its reduction mod p n , X 0 will denote its special fiber.1.1. Statement of the main results.1.1.1. The Fontaine-Messing map. Let X be a fine and saturated log-scheme log-smooth over O K equipped with the log-structure coming from the closed point. Denote by X tr the locus where the log-structure is trivial. This is an open dense subset of the generic fiber of X. For r ≥ 0, let S n (r) X be the (log) syntomic sheaf modulo p n on X 0,ét . It can be thought of as a derived Frobenius and filtration eigenspace of crystalline cohomology or as a relative Fontaine functor. , Kato [36] have constructed period morphisms (i : X 0 ֒→ X, j : X tr ֒→ X) α FM r,n : S n (r) X → i * Rj * Z/p n (r) ′ Xtr , r ≥ 0. from logarithmic syntomic cohomology to logarithmic p-adic nearby cycles. Here we set Z p (r) ′ := 1 p a(r) Z p (r), for r = (p − 1)a(r) + b(r), 0 ≤ b(r) ≤ p − 1. Assume now that X has semistable reduction over O K or is a base change of a scheme with semistable reduction over the ring of integers of a subfield of K. That is, locally, X can be written as Spec(A) for a ring Aétale overIf we put D := {X a+b+1 · · · X d = 0} ⊂ Spec(A) then the log-structure on Spec A is associated to the special fiber and to the divisor D. We have Spec(A) tr = Spec(A K ) \ D K . We prove in this paper that the Fontaine-Messing period map α FM r,n , after a suitable truncation, is essentially a quasi-isomorphism. More precisely, we prove the following theorem. for a universal constant N (not depending on p, X, K, n or r) and a constant c p depending only on p (and d if p = 2).(ii) In general, the kernel and cokernel of this map are annihilated by p N for an int...

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“…Calling this motivic cohomology is justified by Voevodsky's [35] Theorem Observe [6,Section 3] that H i mot is covariant for proper maps (with degree shift) and contravariant for flat maps. The latter applies in particular to the structural morphisms p n : X n → D n .…”

Section: Definition 21 (Compare [6 P 779]) -The Motivic Cohomologmentioning

confidence: 99%

On non-commutative twisting in étale and motivic cohomology

Hornbostel¹,

Kings²

2006

Ann. inst. Fourier

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It is proved that under a technical condition the étale cohomology groupsis a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to H i ( X, Z p (j)). This confirms a consequence of the non-abelian Iwasawa main conjecture. Using the "Bloch-Kato-conjecture" a similar result is proven for motivic cohomology with finite coefficients. 2The goal of this paper is to show that the above twisting map has for j >> 0 indeed finite cokernel assuming the very reasonable condition that the Iwasawa µ-invariant of the number field K vanishes. In the second part of the paper we consider the statement, that the resulting elements are in the image of the regulator from motivic cohomology. Our results in this direction give a weak hint that the elements obtained as twists of units are motivic. Using the "Bloch-Kato-conjecture" for all fields (as announced by Voevodsky), we prove that there is a twisting map for motivic cohomology compatible with the one for étale cohomology under the cycle class map.

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Motivic cohomology over Dedekind rings

Cited by 81 publications

References 22 publications

Arithmetic cohomology over finite fields and special values of ζ-functions

Arithmetic cohomology over finite fields and special values of ζ-functions

Syntomic complexes and p-adic nearby cycles

On non-commutative twisting in étale and motivic cohomology

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