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THEOREM 5.4 (Hyodo-Kato)(a) ∃ unique homomorphism of gra<strong>de</strong>d algebrasc n : W n ω ∗ Y → R∗ u Y/Wn ∗O,(a 0 , · · · , a n−1 ) ↦→ ∑ 0≤i≤n−1 p i ã pn−ii ,d(a 0 , · · · , a n−1 ) ↦→ ∑ 0≤i≤n−1 ã pn−i−1i dã i , dlog b ↦→ dlog˜b,(ã i ∈ O Dn (resp. ˜b ∈ M Zn ) lifts a i (resp. b)).72
(b) The composition of c nCartier isom. C nwith the inverse higherR q u Y/(Wn ,W n (L)∗O Cn W n ω q Y c n R q u Y/Wn ∗Ois a ϕ-equivariant section of the can. projection,<strong>and</strong> induces a W n < t >-linear isomorphismh q n : W n < t > ⊗ Wn R q u Y/(Wn ,W n (L)∗O∼ −→ R q u Y/Wn ∗O.73
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p-adic Geometry andHomotopy TheoryL
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INTRODUCTIONk perfect, char. p > 0,
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• analyzes Frobenius on crystalli
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1975 Deligne proposes alternate con
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1976 - 1978 I. carries out Deligne
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⇒ new construction of W . Ω·X(f
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PLAN1. Construction(s) of W . Ω·X
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(0) RV = V R(R : W n+1 Ω·X → W
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(c) There exist unique operatorsF :
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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 and 34: X = Spec F p [T ],E· = complex of
Page 35 and 36: absolute Frobenius F : X → X give
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: Proof of (ii) :relies on existence
Page 75 and 76: Remarks• Ogus (1995) : Simpler pr
Page 77: and construction of comparison mapB
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