6. DE RHAM-WITT COMPLEX AND LOG CRYS- TALLINE COHOMOLOGY

Transcription

1 6. DE RHAM-WITT COMPLEX AND LOG CRYS- TALLINE COHOMOLOGY α : 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 1

2 (X, M)/(s, L) : fine log scheme, log smooth and Cartier type /(s, L) Log crystalline site Crys((X, M)/(W n (s), W n (L)) objects : log DP thickenings (U, T, N, δ), U X étale, i : (U, M) (T, N) exact ( strict) closed immersion, T/W n (s); with DP δ on Ker(O T O U ) compatible with can. DP on pw n (k) 2

3 morphisms : obvious covering families : (U i, T i ) i I (U, T ) : (T i T ) i I étale cover, U i = U T T i 3

4 Log crystalline topos ((X, M)/(W n (s), W n (L)) crys (or (X/W n ) logcrys ) structural sheaf O X/Wn : (U, T ) O T canonical morphism u = u (X,M)/(Wn (s)w n (L)) : (X/W n) logcrys Xét 4

5 Calculation of Ru O X/Wn If i : X Z closed immersion, Z/(W n, W n (L) log smooth DP-envelope j = X D n, D n Z exact closed immersion, s. t. i = composite X D n Z, Kerj = DP-ideal, universal for these properties 5

6 (uses Kato s local exactification : (local) factorization of i : X Y Z, X Y exact closed immersion, Y Z log étale.) Then : Ru O X/Wn = O D Ω Ż/(W n,w n (L)). NB. i : X Z exists only locally (even if X/k projective), general case : cohomological descent

7 Remark Log crystalline site, topos, calculation of Ru O generalize to (X, M) (S, L, I, γ) map of fine log schemes, p nilpotent on S, γ DP on I extendable to X 6

8 back to previous case : Log crystalline (Hyodo-Kato) cohomology : H m ((X, M)/(W n, W n (L)) := H m (((X, M)/(W n, W n (L)) logcrys, O X/Wn ) = H m (X, Ru O X/Wn ) (= H m (X, O D Ω Ż/(W n,w n (L)) ) for i : X Z) comes equipped with 7

9 Frobenius operator ϕ : 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) Cartier type 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) ) 8

10 (for (s, L) = standard log point) monodromy operator N : 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 on H m ((X, M)/(W n < t >, can)) rel. to W n (with trivial log str.) basic relation : Nϕ = pϕn 9

11 for X/k proper, get (ϕ, N)-module structure on K 0 H m ((X, M)/(W, W (L))), where H m ((X, M)/(W, W (L))) = proj.lim. H m ((X, M)/(W n, W n (L)) a f. g. W -module, N is nilpotent 10

12 Log de Rham-Witt complex imitate Katz-I-Raynaud s re-construction : change notations : type /(k, L) Y = (Y, M) log smooth, Cartier W n ω i Y = σn R i u (Y,M)/(Wn,W n (L) O), d : Bockstein F : given by restriction from W n+1 to W n 11

13 V : given by multiplication by p R : uses Cartier type, variant of Mazur-Ogus gauges th. get pro-complex W. ω Ẏ, with operators F, V satisfying standard formulas (F V = V F = p, F dv = d, etc.) 12

14 new feature : can. homomorphism dlog : 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) satisfying 13

15 F dlog = dlog, [α(a)] dlog a = d[α(a)] W n ω 0 Y = W no Y, and W n ωy generated as W n(o Y )-algebra by dw n O Y, dlog M gp 14

16 comparison th. generalizes : Ru (Y,M)/(Wn,W n (L) O W n ω Ẏ RΓ((Y, M)/(W n, W n (L)) RΓ(Y, W n ω Ẏ ) slope spectral sequence, higher Cartier isom., etc. generalize, too 15

17 7. THE HYODO-KATO ISOMORPHISM S = Spec A, A complete dvr, char. (0, p), K = Frac(A), k = A/πA residue field, perfect K 0 = Frac(W (k)) X/S semi-stable reduction Y = X k the special fiber goal : for X/S proper, define (K 0, ϕ, N)-structure on HdR m (X K/K) using log crystalline cohomology of Y 16

18 case X/S smooth : Berthelot-Ogus isomorphism K 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 Y L : (N k, 1 0) a : log str. on s = Spec k, induced by standard log str. on S, (N A, 1 π) a 17

19 Then (s, L) = standard log point, and (Y, M Y ) (s, L) of semi-stable type, in particular, log smooth and Cartier type 18

20 Put A n = A Z/p n Z = A W W n, S n = Spec A n X n = X Z/p n Z = X W W n, with induced log str. M Sn : Y X 1 X n X s S 1 S n S Note : M Sn associated with N A n, 1 π, while W n (L) associated with N W n,

21 Consider projective systems (a) and (b) of D + (X n, A n ) = D + (Y, A n ) : (a) = de Rham-Witt A n L W n Ru (Y,MY )/(W n,w n (L)) O A n L W n W n ω Y/W n (b) = de Rham ω X n /A n := Ω (X n,m X n )/(S n,m S n ) 20

22 THEOREM 7.1 (Hyodo-Kato) There exists a canonical isomorphism ρ 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) 21

23 COROLLARY 7.2 For X/S proper (and semistable), ρ π induces an 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 (1) 7.1 valid for X/S log smooth, Cartier type (2) ρ π depends on π : ρ πu = ρ π exp(log(u)n), u A 22

24 (3) 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. 23

25 Highlights of proof (Rough) idea : (a), (b) come from F -crystal on W < t > with log pole at t = 0 use Frobenius (t t p ), an isogeny, contracting the disc, to connect : (a) = log crys side = fiber at 0, (b) = dr side = general fiber 24

26 Main ingredients : Berthelot-Ogus s method, to reduce to rigidity th. /W n < t > mod bounded p-torsion de Rham-Witt, lifting of higher Cartier isomorphisms to construct rigidification 25

27 Reduction to rigidity th. Embed Spec A into Spec W [t], t π, with log str. N W [t], 1 t Spec R n = dp-envelope of Spec A n in Spec W n [t] = dp-envelope of S 1 = Spec A 1 in Spec W n [t] : A 1 A n R n W n [t] 26

28 X 1 X n S 1 SpecA n SpecR n SpecW n [t] Base change in crystalline cohomology ω X n /A n (= Ru X1 /A n O) = A n L R n Ru X1 /R n O 27

29 ω X n /A n (= Ru X1 /A n O) = A n L R n Ru X1 /R n O recall goal : relate ω X n /A n to A n Wn W n ω Y/k done in 2 steps Step 1 : relate Ru X1 /R n O to Ru Y/Wn <t> O using Frobenius isogeny (Berthelot-Ogus s method) Step 2 : relate Ru Y/Wn <t> O to Ru Y/(Wn,W n (L)) O (= W n ω Y/k ) using Hyodo-Kato rigidity theorem 28

30 Step 1 : relate Ru X1 /R n O to Ru Y/Wn <t> O recall : W n < t > = dp-envelope of W n in W n [t] = dp-envelope of k in W n [t], R n = dp-envelope of A 1 in W n [t] gets diagram (of log schemes), with action of Frobenius (t t p on W n [t]) Y Speck SpecW n < t > X 1 SpecA 1 SpecR n SpecW n [t] 29

31 in particular, gets : ϕ : Ru X1 /R n O Ru X1 /R n O, which is an isogeny ( Cartier type) : ψ, ψϕ = ϕψ = p r, r = dim Y 30

32 For N log p e, e = [K : K 0 ], (πa) pn ) (pa) F N : X 1 /S 1 X 1 /S 1 factors through Y/k : X 1 SpecA 1 SpecR n Y Speck SpecW n < t > X 1 SpecA 1 SpecR n where composite horizontal maps = F N 31

33 ϕ N = isogeny ϕ N : Q ((R, F N ) L W Ru Y/W <t> O) Q Ru X1 /R O completes step 1 : relate Ru X1 /R n O to Ru Y/Wn <t> O 32

34 Step 2 : relate Ru Y/Wn <t> O to Ru Y/(Wn,W n (L)) O (Y, M Y ) (Speck, L) (SpecW n, W n (L)) (SpecW n < t >, can) gives canonical Frobenius equivariant map Ru Y/(W <t>,can) O Ru Y/(W,W (L)) O 33

35 hence ( ) Q Ru Y/(W <t>,can) O Q Ru Y/(W,W (L)) O THEOREM 7.3 (Hyodo-Kato rigidity theorem) (i) (*) admits a unique Frobenius equivariant section s : Q Ru Y/(W,W (L)) O Q Ru Y/(W <t>,can) O (ii) The map h : Q (W < t > W Ru (Y/(W,W (L)) O) Q Ru (Y/W <t>) O defined by s is an isomorphism. 34

36 Recall : ϕ N = isogeny ϕ N : Q ((R, F N ) L W Ru Y/W <t> O) Q Ru X1 /R O Similarly : ϕ N : Q (R L W Ru (Y/(W,W (L)) O) Q ((R, F N ) L W Ru (Y/(W,W (L)) O) 35

37 Then : h π = ϕ N hϕ N : Q (R L W Ru (Y/(W,W (L)) O) Q Ru X1 /R O As A n L R n Ru X1 /R n O = ω X n /A n, ρ π = Q (A L R h π ) : Q W ω Y Q ω X /A (notations h π, ρ π because π used in W [t] A, t π) 36

38 Proof of Hyodo-Kato rigidity th. Proof of (i) : base change in crystalline cohomology : Ru Y/(Wn,W n (L)) O = W n L W n <t> Ru Y/(W n <t>,can) O ϕ-equivariant triangle (I n = Ker W n < t > W n ) I n L W n <t> Ru Y/(W n <t>,can) O Ru Y/Wn <t>,can)) O Ru Y/(Wn,W n (L)) O) 37

39 Idea : powers of ϕ close to zero on left vertex, isogeny on right vertex exists unique ϕ-equivariant section of Q Ru Y/W <t>,can)) O Q Ru Y/(W,W (L)) O) 38

40 More precisely : ϕψ = ψϕ = p r on right vertex ( Cartier type) ϕ i (I n ) (p i )!I n (use DP on I n ) 39

41 Put : B := Ru Y/W <t>,can)) O, C := Ru Y/(W,W (L)) O) ε : B C Key point (Hyodo-Kato) : p 2cr+d kills Ker and Coker of Hom ϕ (C, ε) : Hom ϕ (C, B) Hom ϕ (C, C) where c N, d N such that p rc+1 p c!, p r(i+1) (p i )!p d i 0. (Note : v p (p i!) = (p i 1)/(p 1)) 40

42 Proof of (ii) : relies on existence of lifted higher Cartier isomorphism 41

43 Recall : if (Y, M Y )/(k, L) (Z, M )/(W [t], 1 t) = compatible system of log smooth embeddings, get : R q u Y/Wn <t> O = H q (O Dn ω Z n /W n <t> ) (D n = dp-envelope of Y in Z n ) 42

44 THEOREM 7.4 (Hyodo-Kato) (a) unique homomorphism of graded algebras c n : W n ω Y R u Y/Wn <t> O, (a 0,, a n 1 ) 0 i n 1 p i ã pn i i, in degree 0 d(a 0,, a n 1 ) 0 i n 1 ã pn i 1 i dã i, dlog b dlog b, in degree 1 (ã i O Dn (resp. b M Zn ) lifts a i (resp. b)). 43

45 (b) The composition of c n Cartier isom. C n with the inverse higher R q u Y/(Wn,W n (L) O Cn W n ω q Y c n R q u Y/Wn <t> O is a ϕ-equivariant section of the can. projection, and induces a W n < t >-linear isomorphism h q n : W n < t > Wn R q u Y/(Wn,W n (L) O R q u Y/Wn <t> O. 44

46 Remark : Olsson (Astérisque 316, 4.6.9, 9.4.1é) has shown that c n : W n ωy R u Y/Wn <t> O, is a map of dga (for a Bockstein d on rhs), and recovered h q n = iso. Uses : Langer-Zink de Rham-Witt complex for alg. stacks dictionary (discovered by Lafforgue) : (log scheme) = (scheme over a certain alg. stack) 45

47 End of proof of (ii) : Recall : h : Q (W < t > W Ru (Y/(W,W (L)) O) Q Ru (Y/W <t>) O h q n : W n < t > Wn R q u Y/(Wn,W n (L) O R q u Y/Wn <t> O. One shows : H q (h) = Q h q, (uniqueness of ϕ-equivariant sections) 46

48 Proof of 7.4 : uses explicit presentations of W n ω Y by generators and relations, e. g. as a quotient of ω W n (Y )/(W n,w n (L)) and structure of KerW n+1 ω q Y W nω q Y 47

49 Remarks Ogus (1995) : alternate proof, variants and generalizations of 7.2 for log F -crystals But : 7.1, 7.3 crucial for crystalline interpretation of B st + W H m ((Y, M Y )/(W, W (L)) 48

50 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 ), where lhs = H 0 (S 1 /R n ) Wn H m (Y/(W n, W n (L)) mod bounded p-torsion by Hyodo-Kato 7.1 (X 1 = X 1 O K, etc., can. log. str. on S 1, X 1, Y, R n ) 49

51 and construction of comparison map B st H m (X K, Q p) ) B st H m ((Y, M Y )/(W, W (L)) 50