A note onp-adic etale cohomology in the semi-stable reduction case

Hyodo, Osamu

doi:10.1007/bf01388786

Cited by 38 publications

(25 citation statements)

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“…, (s, L s )) with respect to the natural map induced by the structure map Ls) on Yé t are locally free O Y -modules of finite rank and coincide with the modified differential modules ω * Y defined in [Hy1].…”

Section: Boundary Maps On the Sheaves Of P-adic Vanishing Cyclesmentioning

confidence: 95%

p-adic étale Tate twists and arithmetic duality☆

Sato

2007

Annales Scientifiques de l’École Normale Supérieure

126

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In this paper, we define, for arithmetic schemes with semistable reduction, p-adic objects playing the roles of Tate twists inétale topology, and establish their fundamental properties.Résumé: Dans ce papier, nous definissions, pour les schémas arithmétiquesà réduction semistable, des objets p-adiques jouant les rôles de twistsà la Tate en topologieétale, et nousétablissons leurs propriétés fondamentales.

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Section: Boundary Maps On the Sheaves Of P-adic Vanishing Cyclesmentioning

confidence: 95%

p-adic étale Tate twists and arithmetic duality☆

Sato

2007

Annales Scientifiques de l’École Normale Supérieure

126

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“…Via these maps all groups can be filtered compatibly in a way similar to the filtration of the unit group of a local field. Finally, the quotients can be computed explicitly by symbols [12], [34], [40], [62] and they turn out to be isomorphic. This approach to the computation of p-adic nearby cycles goes back to the work of Bloch-Kato [12] who treated the case of good reduction and whose approach was later generalized to semistable reduction by Hyodo [34].…”

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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“…W ω Y is the pro-sheaf of CDGA's consisting of the De Rham-Witt differential forms of Y [6]. It is probably best at this point not to worry about the precise construction and just remember that…”

Section: Brief Reviewmentioning

confidence: 99%

“…Denote by W the ring of Witt vectors of k and by K the fraction field of W . Endow A with the log structure A − 0 →A and let X be a proper smooth connected fine saturated log scheme over A with generic fiber X F and special fiber Y which we assume to be of Cartier type ( [6] We will prove a version of this theorem in the context of the unipotent rational homotopy types defined in [9]. The definitions will be reviewed in the next section, but let us state the main result here.…”

Section: Introductionmentioning

confidence: 99%