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5. THE HYODO-KATO ISOMORPHISMS = Spec A, A complete dvr, char. (0, p),K = Frac(A), k = A/πA residue field, perfectK 0 = Frac(W (k))X/S semi-stable reductionY = X ⊗ k the special fibergoal : for X/S proper, <strong>de</strong>fine (K 0 , ϕ, N)-structure onHdR m (X K/K)using log crystalline cohomology of Y58
• case X/S smooth : Berthelot-Ogus isomorphismK ⊗ W H m (Y/W )∼ −→ H m dR (X K/K)⇒ (K 0 , ϕ)-structure (N = 0)• general case : use log str.M X : can. log str. on X, induced by special fiber(M X = O X ∩ j ∗ O ∗ X K, j : X K ⊂ X)M Y: induced log str. on YL : (N → k, 1 ↦→ 0) a : log str. on Spec k,induced by st<strong>and</strong>ard log str. on S, (N → A, 1 → π) a59
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p-adic Geometry andHomotopy TheoryL
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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 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: comparison th. 2.1 generalizes :Ru
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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