p-adic Geometry and Homotopy Theory - Département de ...

p-adic Geometry andHomotopy TheoryLoen, NorwayAugust 2nd-8th, 20091

Review of de Rham-WittTheoryLuc IllusieUniversité Paris-Sud, Orsay, France2

INTRODUCTIONk perfect, char. p > 0, W = W (k)X/k proper, smoothW Ω·X: the de Rham-Witt complex of X/kan inverse limit of dga W n Ω·Xon X,with W Ω 0 X = W O X, W 1 Ω·X = Ω·X/k3

• calculates crystalline cohomology :H ∗ (X, W Ω·X ) = H∗ (X/W )• relates it to :- Serre’s Witt vector cohomology H ∗ (X, W O X )- Artin-Mazur formal gps Φ q associated with H q (−, G m )(Φ 1 = ̂Pic, Φ 2 = ̂Br, etc. )- Hodge cohomology : H j (X, Ω i X ) 4

• analyzes Frobenius on crystalline cohomologyvia operators F , V : W Ω i X → W Ωi XF V = V F = p,F dV = dslope spectral sequenceE ij1 = Hj (X, W Ω i X ) ⇒ Hi+j (X/W )5

1974 Bloch : for p > 2, dim X < p, constructsinverse sytem of complexes oftypical curves on symbolic part of Quillen’s K groups0 → T C n K 1 → T C n K 2 → · · · → T C n K q+1 → · · ·T C n K q+1 in degree q6

1975 Deligne proposes alternate construction :a universal quotient (L·n ) n≥1 of (Ω·W n (O X )/(Z/p n Z) ) n≥1with operators V : L i n → L i n+1and relations• inspired by a former construction of Lubkin7

• based on explicit description ofsubcomplex of integral forms(see below)• no more K-theory8

1976 - 1978 I. carries out Deligne’s program :• construction of W . Ω·X (any X/F p)• comparison with Bloch’s construction, and with crystallinecohomology (X/k smooth)• global geometric applications (e. g. H ∗ of surfaces)9

1979 K. Kato removes restrictions p > 2 anddim X < p in Bloch’s construction1979 - 1983 I., Raynaud, Nygaard, Ekedahlstudy fine structure of slope spectral sequencediscovery of higher Cartier isomorphisms10

⇒ new construction of W . Ω·X(for X/k smooth)1988 Hyodo-Kato adapt it to log geometry(X/k log smooth, Cartier type)⇒ key tool in construction of Hyodo-Kato isomorphism11

2004 extensions of DRW theory tomixed char. and relative situations• Hesselholt-Madsen : W . Ω·A , A/Z (p) , p > 2• Langer-Zink : W . Ω·A/R , R/Z (p)2007 Olsson : stack-theoretic variants2008 Davis-Langer-Zink : overconvergent variants12

PLAN1. Construction(s) of W . Ω·X2. Comparison with crystalline cohomology3. Higher Cartier isomorphisms4. De Rham-Witt for log schemes5. The Hyodo-Kato isomorphism13

1. CONSTRUCTION(S) OF W . Ω·XX = (X, O X ) ringed topos over F pW n O X : Witt vectors of length n on O XTheorem 1.(a) There exists an inverse system of dgaW n Ω·X = (W nΩ 0 X → W nΩ 1 X → · · · → W nΩ i X → · · · )(n ≥ 1)with operators V : W n Ω i X → W n+1Ω i Xsatisfying14

(0) RV = V R(R : W n+1 Ω·X → W nΩ·X )(1) W n Ω 0 X = W nO X , R = R, V = V(2) V (xdy) = (V x)dV y(3) (V x)d[y] = V (x[y p−1 ]d[y]),y ∈ O X , [y] = (y, 0, · · · , 0)universal for these properties15

(b) The natural mapΩ·W n (O X )/(Z/p n Z) → W nΩ·Xis surjective,an isomorphism for n = 1 :Ω·X/F p= W 1 Ω·X . 16

(c) There exist unique operatorsF : W n Ω i X → W n−1Ω i Xsatisfying :F dV = dF a.F b = F (ab)F a = RF a, a ∈ W n O XF d[a] = [a] p−1 d[a], a ∈ O X , [a] = (a, 0, · · · , 0).17

Proof : (a), (b) : straightforward(c) relies ondescription of W . Ω·A for A = F p[T 1 , · · · , T r ]in terms of the complex E· of integral forms :W n Ω·A = E·/(V n E· + dV n E·), V = pF −1 , F T i = T p iE· ⊂ Ω·C/Q p, C = Q p [T p−∞1 , · · · , T p−∞r ],ω ∈ E i ⇔ ω and dω integral(i. e. coefficients in Z p )18

Example : r = 1, A = F p [T ],W Ω·A := proj.lim. W nΩ·AW A = { ∑ k∈N[1/p] a kT k , a k ∈ Z p , den(k)|a k ∀k,lim k→∞ a k = 0}W Ω 1 A = {∑ k>0,k∈N[1/p] a kT k (dT/T ), a k ∈ Z p ,lim k→∞ den(k).a k = 0}W Ω i A = 0, i > 1. 19

More formulas(1) F : W n Ω i X → W n−1Ω i X satisfiesF V = p, V F = p,V (aF b) = (V a)b, ∀a ∈ W n Ω i X , b ∈ W n+1Ω j X(2) V W n O X ⊂ W n+1 O X has a canonical pd-structure :γ k (V a) = (p k−1 /k!)V a k (k ≥ 1)20

and d is a pd-derivation :dγ k (a) = γ k−1 (a)da, (a ∈ V W n O X )Moreover,F d : W n+1 O X → W n Ω 1 X ,(F d([a] + V b) = [a] p−1 d[a] + db)is also a pd-derivation.21

Structure for X/k smooth0 → gr n W·Ω·X → W n+1Ω·X → W nΩ·X → 0,gr n = (V n + dV n )Ω·Xlocally free of finite type /O X in each degree i(extension of Ω i−1 /Z n by Ω i /B n )22

New constructionsd, F d pd-derivations ⇒ Langer-Zink’s constructionof a relative (pro-F -V -) de Rham-Witt complexW . Ω·X/S(f : X → S a morphism of Z (p) -ringed toposes)• W n Ω·X/S = f −1 W n (O S )-dga• F : W n Ω i X → W n−1Ω i X built in together with 23

V : W n−1 Ω i X → W nΩ i Xsatisfying above formulas(except possibly V F = p)24

• W n Ω 0 X/S = W nO X ,Ω·W n O X /W n O S ,pd → W nΩ·X/S surjective• W . Ω·X/S = W .Ω·X for S = {pt, F p}Applications :• theory of displays (Zink)• generalized Hyodo-Kato isomorphisms (Olsson)Langer-Zink’s construction competes with :25

Hesselholt-Madsen’s constructionof an absolute (pro-F -V -) de Rham-Witt complexW . Ω·X(X → S a Z (p) -ringed topos)• hypothesis : p > 2• W n Ω 0 X = W nO X• W n Ω·X = Z (p) -dga (not a W n(O S )-dga)26

again :• F : W n Ω i X → W n−1Ω i Xbuilt in withV : W n−1 Ω i X → W nΩ i Xsatisfying above formulas (except possibly V F = p)• W . Ω·X = “classical” W .Ω·X for X/{pt, F p}Applications :• relations with K-theory and cyclic homology• construction generalizes to log spaces• relations with p-adic vanishing cycles(fixed points of F ) (Geisser-Hesselholt)27

Comparison LZ - HMcanonical mapW n Ω·X → W nΩ·X/S• surjective, iso if p1 S = 0 and S perfect• not iso for S = {pt, Z p }(W . Ω 1 Z p /Z (p)⊗ Z/p n = 0,W . Ω 1 Z p⊗ Z/p n ⊃ Z ∗ p /Z∗pn p )28

• W n Ω·X/S = W nΩ·X /I,I = dg ideal generated by image of f −1 W n Ω 1 S→W n Ω 1 X(analyzed by Hesselholt for X/S smooth, S noetherian,p1 S nilpotent)Overconvergent de Rham-Witt (Davis-Langer-Zink)gives natural lattices in Berthelot’s rigid cohomologyof X/k smooth (k perfect, char. p)29

2. COMPARISON WITHCRYSTALLINE COHOMOLOGYk perfect field, char(k) = p > 0, W n = W n (k), X/kRecall :• u = u X/Wn : (X/W n ) crys → X canonical morphism(u ∗ (E)(U) = Γ((U/W n ) crys , E))• If ∃ embedding, X ⊂ Z/W n smooth :Ru X/Wn ∗O X/Wn = O D ⊗ Ω·Z/W n,(D = pd-envelope of X ⊂ Z/W n )30

THEOREM 2.1 Assume X/k smooth.Then ∃ can., functorial inverse sytem of isomorphisms(of D(X, W n ))Ru X/Wn ∗O X/Wn∼ −→ Wn Ω·X• compatible with multiplicative structures• compatible with Frobenius(Φ : W n Ω·X → W nΩ·X , Φ|W nΩ i X = pi F ).31

Proof• construction of map :- local case : X → W n X → D → Z,(Z/W n smooth)O D ⊗ Ω·Z/W n→ Ω·W n X/W n ,pd → W nΩ·X-globalization : cohomological descent• map = iso : use local structure of DRW32

X = Spec F p [T ],E· = complex of integral forms ⊂ Ω·Q p [T p−∞ ]/Q pE· = Ω·Z p [T ]/Z p⊕ homotop. triv. complexW n Ω·X = E·/(V n + dV n )E·= Ω·Z/p n [T ]/(Z/p n ) ⊕ acyclic complex 33

Slopes of FrobeniusX/k proper and 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·Xand RΓ(X/W ),(X of pure dim. d)(cf. Berthelot, Berthelot-Ogus)35

Slope spectral sequenceE ij1 = Hj (X, W Ω i X ) ⇒ Hi+j (X/W )• degenerates at E 1 mod p-torsion• K 0 ⊗ H j (X, W Ω i X ) = part of slope ⊂ [i, i + 1[ ofK 0 ⊗ H i+j (X/W )(K 0 = Frac(W )• torsion studied by Nygaard, I-Raynaud, Ekedahl36

Other applications• Poincaré duality, Künneth (Ekedahl)• logarithmic Hodge Witt sheaves W n Ω i X,log ⊂ W nΩ i Xétale loc. generated by dlog[x 1 ] · · · dlog[x i ](Milne, Bloch-Gabber-Kato, Kato, I-Raynaud, Colliot-Thélène, Gros, ...)• cycle and Chern classes in crystalline cohomology(Gros)37

3. HIGHER CARTIER ISOMORPHISMSX/k smoothF n : W 2n Ω i X → W nΩ i Xinduces isomorphismC −n : W n Ω i X∼ −→ H i W n Ω·X ,compatible with products,generalizing standard Cartier isomorphism for n = 1 :C −1 : Ω i X → Hi Ω·X38

th. 2.1 ⇒ : C −n induces W n -linear isomorphism(3.1) W n Ω i X ∼ −→ σ n ∗ H i (X/W n )(H i (X/W n ) = R i u X/Wn ∗O X/Wn )i = 0 : W n O X∼ −→ σn ∗ H 0 (X/W n )(cf. Fontaine-Messing :on X syn )˜WDPn∼−→ O crys n39

X lifted to Z smooth /W n ,(a 0 , · · · , a n−1 ) ∈ W n O X↦→ b pn0 + pbpn−1 1 + · · · + p n−1 b p n−1∈ H 0 (X/W n ) = H 0 dR (Z/W n)b i ∈ O Z ↦→ a ii = 1 : W n Ω 1 ∼X −→ σ∗ n H 1 (X/W n ) = σ∗ n HdR 1 (Z/W n)d(a 0 , · · · , a n−1 ) ↦→ ∑ b pn−i−1i db i(cf. Serre’s map F n d : W n O X → Ω 1 X ) 40

Katz’s observation : 3.1 ⇒re-construction of the de Rham-Witt complex :W n Ω i X := σn ∗ H i (X/W n )d : W n Ω i X → W nΩ i+1X =?R : W n+1 Ω i X → W nΩ i X =?F : W n+1 Ω i X → W nΩ i X =?V : W n Ω i X → W n+1Ω i X =? 41

d = Bockstein = coboundary from0 Ω·Z n /W np n Ω·Z 2n /W 2n Ω·Z n /W n0(X lifted to Z 2n smooth /W 2n , Z n := Z 2n ⊗ W n )F : W n+1 Ω i X → W nΩ i X induced by can. mapH i (X/W n+1 ) → H i (X/W n )V : W n Ω i X → W n+1Ω i X induced byp : H i (X/W n ) → H i (X/W n+1 )42

R : W n+1 Ω i X → W nΩ i X more delicate,uses formal smooth lifting Z/W ,plus lifting F of Frobenius, andϕ = p −i F ∗ on Ω i Z/WRa = b if a = class of ϕb(cf. Mazur-Ogus gauges th. : F ∗ : Ω·Z n→ Ω·Z n ,(i↦→i)quasi-isomorphism)⇒ construction of DRW and subsequent formalism(for X/k smooth) independentof the complex of integral forms for the affine space43

Applications• structure of conjugate spectral sequence (I-Raynaud)(X/k proper and smooth)E ij2 = proj.lim Hi (X, H j (X/W n )) ⇒ H i+j (X/W )- degenerates at E 2 mod p-torsion- K 0 ⊗ proj.lim H i (X, H j (X/W n ))= part of slope ⊂]j − 1, j] in K 0 ⊗ H i+j (X/W )44

• construction works in other contexts :log schemes (Hyodo-Kato)algebraic stacks (Olsson)45

4. DE RHAM-WITT FOR LOG SCHEMESα : L → k : a fine log str. on s = Spec(k)W n (L) : Teichmüller lifting of L to W n (s) :W n (L) = L ⊕ Ker(W n (k) ∗ → k ∗ )L → W n (k) : a ↦→ [α(a)]Example : L = (N → k, 1 ↦→ 0) a (standard log point)W n (L) = (N → W n (k), 1 → 0) a 46

(X, M)/(s, L) : fine log scheme, log smooth and Cartiertype /(s, L)Cartier type (= integral, relative Frobenius exact)ensures the existence of Cartier isomorphism :C −1 : Ω q (X,M) ′ /(s,L)∼−→ F ∗ H q (Ω·(X,M)/(s,L) )(F : (X, M) → ((X, M) ′ = (s, L) × (s,L) (X, M) relativeFrobenius) ,C −1 : a ↦→ F ∗ a, dlog b ↦→ dlog b47

Typical example : (s, L) standard log point,(X, M) of semistable type /(s, L) :étale loc. X = Spec k[t 1 , · · · , t d ]/(t 1 · · · t r ), with chartsk[t 1 , · · · , t d ]/(t 1 · · · t r )k1↦→0N r1↦→(1,··· ,1)(e. g. special fiber of semistable scheme over trait)N48

Log crystalline (Hyodo-Kato) cohomology :H m ((X, M)/(W n , W n (L)) :=H m (((X, M)/(W n , W n (L)) crys , O)= H m (X, Ru (X,M)/(Wn ,W n (L)∗O)((X, M)/(W n , W n (L)) crys : log crystalline site(built with local log pd-thickenings)comes equipped with49

• Frobenius operatorϕ : H m ((X, M)/(W n , W n (L)) → H m ((X, M)/(W n , W n (L))defined by (Frobenius on schemes, p on monoids)ϕ = σ-linear isogeny (Berthelot-Ogus, Hyodo-Kato)∃ σ −1 -linear ψ, ϕψ = ψϕ = p r(r = dim(X) = rk Ω 1 (X,M)/(s,L) ) 50

• (for (s, L) = standard log point)monodromy operatorN : H m ((X, M)/(W n , W n (L)) → H m ((X, M)/(W n , W n (L))N = residue at t = 0 of Gauss-Manin connection onH m ((X, M)/(W n < t >, can)) rel. to W n (with triviallog str.)basic relation :Nϕ = pϕN51

for X/k proper, get (ϕ, N)-module structure onK 0 ⊗ H m ((X, M)/(W, W (L))),whereH m ((X, M)/(W, W (L))) = proj.lim. H m ((X, M)/(W n , W n (L))a f. g. W -module,N is nilpotent52

Log de Rham-Witt compleximitate Katz-I-Raynaud’s re-construction :change notations :type /(k, L)Y = (Y, M) log smooth, CartierW n ω i Y = σn ∗ R i u (Y,M)/(Wn ,W n (L)∗O),d : BocksteinF : given by restriction from W n+1 to W n53

V : given by multiplication by pR : uses Cartier type, variant ofMazur-Ogus gauges th.get pro-complexW . ω Ẏ ,with operators F , V satisfying standard formulas(F V = V F = p, F dV = d, etc.)54

new feature : ∃ can. homomorphismdlog : M → W n ωY 1 ,(a ↦→ dlog ã ∈ H 1 (O D ⊗ Ω·(Z,M Z )/(W n ,W n (L)) ), ã ∈ M Z ↦→a ∈ M,Y ⊂ (Z, M Z )/(W n , W n (L)) log smooth embedding,D = pd-envelope)satisfying55

F dlog = dlog, [α(a)] dlog a = d[α(a)]W n ω 0 Y = W nO Y , andW n ω ∗ Ygenerated as W n(O Y )-algebra bydW n O Y , dlog M gp 56

comparison th. 2.1 generalizes :Ru (Y,M)/(Wn ,W n (L)∗O∼ −→ W n ω ẎRΓ((Y, M)/(W n , W n (L))∼ −→ RΓ(Y, W n ω Ẏ )slope spectral sequence, higher Cartier isom., etc.generalize, too57

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, define (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 standard log str. on S, (N → A, 1 → π) a59

Put A n = A ⊗ Z/p n Z = A ⊗ W W n , S n = Spec A nX n = X ⊗ Z/p n Z = X ⊗ W W n , with induced log str.Consider projective systems (a) and (b) ofD + (X n , A n ) = D + (Y, A n ) :(a) A n ⊗ L W nRu (Y,MY )/(W n ,W n (L))∗O(b) ω·X n /A n:= Ω·(X n ,M X n )/(S n,M S n ) 60

Note : (a)∼ −→ A n ⊗ L W nW n ω·Y/W nTHEOREM 5.1 (Hyodo-Kato)There exists a canonical isomorphismρ π : Q ⊗ (A· ⊗ L W· W . ω·Y/W·)∼ −→ Q ⊗ ω·X·/S·in Q ⊗ proj.sys. D + (Y, A n )(for additive cat. C, Hom Q⊗C = Q ⊗ Hom C ,Q ⊗ K: image of K in Q ⊗ C)61

ρ πu = ρ π exp(log(u)N), u ∈ A ∗ 62COROLLARY 5.2For X/S proper (and semistable), ρ π inducesan isomorphism :ρ π : K ⊗ W H m ((Y, M Y )/(W, W (L))∼ −→ H m dR (X K/K)(gives (K 0 , ϕ, N) structure on H m dR (X K/K))Remarks(a) 5.1 valid for X/S log smooth, Cartier type(b) ρ π depends on π :

(c) if X/S smooth, then :H m (Y/W )∼ −→ H m ((Y, M Y )/(W, W (L)),ρ π = Berthelot-Ogus isomorphism, independent of π.63

Highlights of proof(Rough) idea : (a), (b) come fromF -crystal on W < t > with log pole at t = 0use Frobenius (t ↦→ t p ), an isogeny, contracting thedisc, to connect :(a) = log crys side = fiber at 0,(b) = dR side = general fiber64

Two main steps :• Use Berthelot-Ogus’s method to reduce torigidity th. /W n < t > mod bounded p-torsion• Construction of rigidification(uses de Rham-Witt, lifting of higher Cartier isomorphisms)65

Embed Spec A into Spec W [t], t → π,with log str. N → W [t], 1 → tSpec R n = pd-envelope of Spec A n in Spec W n [t]= pd-envelope of S 1 = Spec A 1 in Spec W n [t]Then :ω·X n /A n= A n ⊗ L R nRu X1 /R n ∗Oand66

ϕ : Ru X1 /R n ∗O → Ru X1 /R n ∗O(F on X 1 , S 1 , ϕt = t p on W n [t])is an isogeny : ∃ψ, ψϕ = ϕψ = p r , r = dim YAs F N : X 1 /S 1 → X 1 /S 1 factorsthrough Y/k for N >> 0,Th. 5.1 follows from rigidity th. :67

THEOREM 5.3 (Hyodo-Kato)(i) There exists a unique homomorphismh : Q⊗(W· < t > ⊗ W·Ru (Y/(W·,W·(L))∗O) → Q⊗Ru (Y/W·)∗Ocompatible with ϕ,and whose composition with natural mapRu (Y/W·)∗O → Ru (Y/(W·,W·(L))∗O= can. projection(ii) h is an isomorphism.68

5.3 ⇒ 5.1N ≥ log p e, e = [K : K 0 ],5.3 ⇒h π = ϕ N hϕ −N : Q ⊗ (R· ⊗ L W· Ru (Y/(W·,W·(L))∗O)∼−→ Q ⊗ Ru X1 /R·∗OandA n ⊗ L R nRu X1 /R n ∗O = ω·X n /A n, ρ π = Q ⊗ (A· ⊗ L R· h π )69

Proof of (i) :use endomorphism ϕ of triangleI n ⊗ L W n Ru (Y/W n )∗O →Ru (Y/Wn )∗O → Ru (Y/(Wn ,W n (L))∗O) →(I n = Ker W n < t >→ W n ) together with :• ϕψ = ψϕ = p r on 3rd term• ϕ i (I n ) ⊂ (p i )!I n⇒ unique ϕ-splitting mod bounded p-torsion70

Proof of (ii) :relies on existence oflifted higher Cartier isomorphismIf (Y, M Y )/(k, L) ⊂ (Z·, M·)/(W·[t], 1 ↦→ t)= compatible system of log smooth embeddingsR q u Y/Wn ∗O = H q (O Dn ⊗ ω·Z n /W n )(D n = pd-envelope of Y in Z n )71

THEOREM 5.4 (Hyodo-Kato)(a) ∃ unique homomorphism of graded 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,and 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

End of proof of (ii) :One shows :H q (h) = Q ⊗ h q· ,with h q·as in 5.4 (uniqueness of ϕ-equivariant sections).Proof of 5.4 :uses explicit presentations of W n ω ∗ Yby generators andrelations.74

Remarks• Ogus (1995) : Simpler proof, variants andgeneralizations of 5.2 for log F -crystalsBut : 5.1, 5.3 crucial for crystalline interpretation ofB + st ⊗ W H m ((Y, M Y )/(W, W (L))75

via Künneth formulas at finite levels :H 0 (S 1 /R n ) ⊗ Rn H m (X 1 /R n )∼ −→ H m (X 1 /R n ),wherelhs = H 0 (S 1 /R n ) ⊗ Wn H m (Y/(W n , W n (L))mod bounded p-torsion by Hyodo-Kato 5.1(X 1 = X 1 ⊗ O K, etc.,can. log. str. on S 1 , X 1 , Y , R n )76

and construction of comparison mapB st ⊗ H m (X K, Q p) ) → B st ⊗ H m ((Y, M Y )/(W, W (L))• Olsson (Astérisque 316, 412 p.) :variants and generalizations of 5.3, 5.4 forcertain algebraic stacks using- Langer-Zink relative de Rham-Witt complex- dictionary :(log scheme) = (scheme over a certain alg. stack)77