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Slopes of FrobeniusX/k proper <strong>and</strong> smoothRΓ(X/W n ) := RΓ((X/W n ) crys , O X/Wn ) = RΓ(X, Ru ∗ O X/Wn )2.1 ⇒ RΓ(X/W n ) = RΓ(X, W n Ω·X )W Ω·X := proj.lim. W nΩ·X ,RΓ(X/W ) := RΓ(X, W Ω·X )2.1 ⇒ RΓ(X/W ) = R lim RΓ(X/W n ) ∈ D(W ) perf(⇒ H m (X/W ) = H m RΓ(X/W ) f. g. over W )34
absolute Frobenius F : X → X givesσ-linear Φ : RΓ(X/W ) → RΓ(X/W )Φ : H m (X/W ) → H m (X/W ) isomorphism mod torsion(i. e. = F -crystal)better : v : W Ω·X → W Ω·X , v|W Ωi X = pd−i−1 V givesΦv = vΦ = p don W Ω·X<strong>and</strong> RΓ(X/W ),(X of pure dim. d)(cf. Berthelot, Berthelot-Ogus)35
Page 1 and 2: p-adic Geometry andHomotopy TheoryL
Page 3 and 4: INTRODUCTIONk perfect, char. p > 0,
Page 5 and 6: • analyzes Frobenius on crystalli
Page 7 and 8: 1975 Deligne proposes alternate con
Page 9 and 10: 1976 - 1978 I. carries out Deligne
Page 11 and 12: ⇒ new construction of W . Ω·X(f
Page 13 and 14: PLAN1. Construction(s) of W . Ω·X
Page 15 and 16: (0) RV = V R(R : W n+1 Ω·X → W
Page 17 and 18: (c) There exist unique operatorsF :
Page 19 and 20: Example : r = 1, A = F p [T ],W Ω
Page 21 and 22: and d is a pd-derivation :dγ k (a)
Page 23 and 24: New constructionsd, F d pd-derivati
Page 25 and 26: • W n Ω 0 X/S = W nO X ,Ω·W n
Page 27 and 28: again :• F : W n Ω i X → W n
Page 29 and 30: • W n Ω·X/S = W nΩ·X /I,I = d
Page 31 and 32: THEOREM 2.1 Assume X/k smooth.Then
Page 33: X = Spec F p [T ],E· = complex of
Page 37 and 38: Other applications• Poincaré dua
Page 39 and 40: th. 2.1 ⇒ : C −n induces W n -l
Page 41 and 42: Katz’s observation : 3.1 ⇒re-co
Page 43 and 44: R : W n+1 Ω i X → W nΩ i X more
Page 45 and 46: • construction works in other con
Page 47 and 48: (X, M)/(s, L) : fine log scheme, lo
Page 49 and 50: Log crystalline (Hyodo-Kato) cohomo
Page 51 and 52: • (for (s, L) = standard log poin
Page 53 and 54: Log de Rham-Witt compleximitate Kat
Page 55 and 56: new feature : ∃ can. homomorphism
Page 57 and 58: comparison th. 2.1 generalizes :Ru
Page 59 and 60: • case X/S smooth : Berthelot-Ogu
Page 61 and 62: Note : (a)∼ −→ A n ⊗ L W nW
Page 63 and 64: (c) if X/S smooth, then :H m (Y/W )
Page 65 and 66: Two main steps :• Use Berthelot-O
Page 67 and 68: ϕ : Ru X1 /R n ∗O → Ru X1 /R n
Page 69 and 70: 5.3 ⇒ 5.1N ≥ log p e, e = [K :
Page 71 and 72: Proof of (ii) :relies on existence
Page 73 and 74: (b) The composition of c nCartier i
Page 75 and 76: Remarks• Ogus (1995) : Simpler pr
Page 77: and construction of comparison mapB

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