INTEGRAL p-adic HODGE THEORY, TALK 14 (COMPARISON WITH THE DE RHAMWITT COMPLEX)

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1 INTEGRAL -ADIC HODGE THEORY, TALK 4 (COMPARISON WITH THE DE RHAMWITT COMPLEX) JOAQUIN RODRIGUES JACINTO (NOTES BY JAMES NEWTON). Recollectons and statement of theorem Let K be a erfectod eld of characterstc 0, wth a xed comatble system of -ower roots of unty (ζ r) r, wth rng of ntegers O = O K and resdue eld k. Now we restate the man goal of ths week's talks: Theorem.. Let R be a small formally smooth O-algebra. Then for all r, n 0 we have natural somorhsms: W r Ω n,cont R/O = H n ( W r Ω R ). Remark.2. The mas W r Ω n,cont R/O Hn ( W r Ω R ) wll be constructed usng the unversal roerty of the de RhamWtt comlexes (they dene an ntal object n a sutable category of F -V -rocomlexes). It wll be a formal consequence of ths that the Bocksten derental on H ( W r Ω R ) wll corresond to the derental on W r Ω,cont R/O. We x once for all a framng O T ± R. Recall from last week that we have natural somorhsms: W r Ω R W r Ω roet R RΓ(X, W r Ω X ) where X = Sf(R) and X s the generc bre of X, and where we have by denton W r Ω R := Lη [ζ r ]RΓ cont (Z d, W r (R )), W r Ω roet R := Lη [ζ r ]RΓ(X roet, W r (Ô+ X )), where R = R O T ± O T ±/. We can comute these objects n the derved category by Koszul comlexes; denote the (toologcal) generators of Γ = Z d by γ. Then for a toologcal Γ- module M we have RΓ cont (Γ, M) = K M (γ,, γ d ) =: K M (γ ) Fnally we recall that we have constructed the comlex of A nf -modules AΩ R = Lη µ RΓ(X roet, A nf,x ) = Lη µ RΓ(Γ, A nf (R )) and that we have natural quas-somorhsms AΩ R L A nf, θ r W r (O) = W r Ω R.

2 2 INTEGRAL P -ADIC HODGE THEORY 2. Proof of theorem We are gong to reduce to the case of the Laurent olynomal rng R = O[T ± ]. Then we wll moreover reduce to the case over k and aly the usual Carter somorhsm. We saw n the last talk a rocedue for constructng an F -V -rocomlex out of a commutatve algebra object D n D(A nf ). We let D = RΓ(X roet, A nf,x ), whch comes equed wth a Frobenus ϕ D. Then we dene W n r (D) = H n (Lη µ D L A nf, θ r W r (O)) = H n ( W r Ω R ). By usng the general results from the last lecture, f we check that: Wr n (D) s -torson free, We have W r (O)-algebra mas λ r : W r (R) Wr 0 (D) commutng wth R, F, V, then {W r (D)} r nherts a structure of an F -V -rocomlex and hence, by unversal roerty of the relatve de Rham-Wtt comlexes, we get natural mas λ r : W r Ω R/O H ( W r Ω R ). To check these onts s easy: we saw last week that W n r (D) s -torson free (t follows from the Koszul comlex comutaton) and we can dentfy W 0 r (D) = H 0 ( W r Ω R ) = W r (R), so we take λ r to be the dentty. 2.. Reducton to R = O[R ± ]. We begn by the followng Fact: (a result of Elkk) there s a smooth O-algebra R 0 wth an étale ma O[T ± ] R 0 whch gves our xed ma O T ± R after -adc comleton. Now we consder the followng dagram: W r Ω n O[T ± ]/O W r(o[t ± ]) W r (R 0 ) H n ( W r Ω O T ± ) Wr(O[T ± ]) W r (R 0 ) W r Ω n,cont R 0/O = W rω n,cont R/O H n ( W r Ω R ) The equalty of the bottom-left term follows from the fact that -adc closure commutes wth formng the relatve de Rham-Wtt vectors, as we have seen n the last lecture. To show that the lower horzontal ma s an somorhsm, we wll show that the vertcal and uer horzontal mas are somorhsms after -adc comleton. The uer horzontal ma comes from a varant of the constructon of the mas λ r : W r Ω R/O H ( W r Ω R ) where we relace the restrcted Laurent seres rngs by Laurent olynomal rngs, and wll be exlaned n detal later. The left hand vertcal ma s an somorhsm (after -adcally comletng) by the étale base change roerty of the de RhamWtt comlex, whch says that W r Ω n O[T ± ]/O W r(o[t ± ]) W r (R 0 ) = W r Ω n R 0/O.

3 INTEGRAL -ADIC HODGE THEORY 3 The rght hand vertcal ma s an somorhsm by a smlar roerty for W r Ω R [, Lemma 9.9]. So t remans to show that the uer horzontal ma s an somorhsm (after -adc comleton) The case R = O[T ± ]. Now we let R = O[T ± ], and let D = RΓ(Γ, A nf [U ±/ ]), where we recall that the acton of Γ = Z d on A nf [U ±/ ] s gven by γ U /r = [ε] /r U /r and γ acts as the dentty on owers of U j for j. Remark 2.3. Recall that we have an dentcaton where U /s A nf (O T ±/ ) = A nf U ±/ = [(T /s, T /s+, )]. So now we dene as before W n r (D) = H n (Lη µ D L A nf, θ r W r (O)) and agan we need to check that these modules are -torson free and that we have mas λ r : W r (O[T ± ]) W 0 r (D) commutng wth R, F, V. Once ths s checked, we wll get natural mas λ n r : W r Ω n O[T ± ]/O W n r (D), and we wll show that they are somorhsms. In order to check these, and to comare Wr n (D) wth W r Ω n, we show O[T ± ]/O that we can comute Lη µ D usng a `q-de Rham comlex': Prooston 2.4. (Comare wth Lemma 0.0 n Fred's talk) Lη µ D d = q Ω ( A nf [U ± ]/A nf := Anf [U ± ] A nf [U ± ) ] dlogu, = where the derental n the two term comlex sends U k to [k] q U kdlogu, where [k] q = qk q and we set q = [ε] (dlogu s smly taken as a formal generator). Proof. As for Lemma 0.0 last week. We break u D as a drect sum of eces M a = K Anf ([ε] a(),, [ε] a(d) ) where a runs over functons {,, d} Z[/]. When we aly Lη µ only the terms for a wth mage n Z survve, we calculate the recse values for the survvng terms, and we get the above descrton. We obtan as a consequence a descrton of W n r (D): Lemma 2.5. For all n 0 there are somorhsms of W r (O)-modules Wr n (D) = W r u(s) (O) (d n). a:{,...,d} r Z In artcular, W n r (D) s -torson free. Proof. Ths follows secalsng the recedng Prooston to W r (O) (va the ma θ r, note that θ r ([ε]) = [ζ r]).

4 4 INTEGRAL P -ADIC HODGE THEORY Lemma 2.6. There s a unque W r (O)-algebra ma λ r : W r (O[T ± ]) = W 0 r (D), commutng wth R, F, V, and such that λ r ([T ]) = U. Proof. Ths follows from unravellng the dentcatons whch gve W r (O)[U ±/ ] = W r (O[T ±/ ]) U /s [T /r+s ] W 0 r (D) = H 0 (Lη µ D L A nf, θ r W r (O)) = (W r (O)[U ±/ ]) Γ = W r (O[T ± ]) Now we have vered the necessary roertes, we get an F -V -rocomlex structure on {W r D} r and hence natural mas λ r : W r Ω O[T ± ]/O W r (D) comatble wth everythng (d, F, V, R and the multlcatve structure). It remans to rove: Prooston 2.7. λ n r s an somorhsm for all n 0, r. We know that both sdes have a decomoston as a:{,,d} r Z M a where each M a s a nte drect sum of coes of W r u(a) (O). Frst we clam that the mas λ n r resect these decomostons. Ths s done by extendng the trval acton of Γ = Z d on O[T ± ] (res. the acton of Γ on A nf [U ±/ ] dened above) to Z[/] d by O-lnearty (res. A nf -lnearty) and the formulae γ r T = ζ rt, γ r T j = T j, j (res. γ r U = [ɛ] /r U, γ r U j = U j, j ), checkng that ths acton commutes wth R, F, V and hence wth λ d r, that the nduced acton of γ r on each M a s gven by multlcaton by [ζ a() ], and by usng ths and an sotycal r comonent argument to nally deduce the clam. Usng the above clam, we can easly deduce that t suces to show that each ma λ n r := λ n r Wr(O)W r (k) : W r Ω n O[T ± ]/O W r(o)w r (k) W n r (D) k := W n r (D) Wr(O)W r (k) s an somorhsm. We have already seen that W r Ω n O[T ± ]/O W r(o) W r (k) = W r Ω n k[t ± ]/k. The followng lemma shows that the other term has the same shae. Lemma 2.8. We have somorhsms (we don't clam that ths somorhsm s comatble wth λ n r!) W r (D) k = Wr Ω k[t ± ]/k comatble wth λ 0 r n degree 0. Proof. It follows from Prooston 2.4 that W r (D) k can be comuted by secalsng the q-de Rham comlex to W r (k) (the hgher Tors between W r (D) k and W r (k) over W r (O) vansh). Ths sends ε to, so we get the de Rham cohomology grou W n r (D) k = H n (Ω W r(k)[t ± ]/W r(k) )

5 INTEGRAL -ADIC HODGE THEORY 5 By the classcal Carter somorhsm, ths s gven by the usual de RhamWtt comlex: H n (Ω W r(k)[t ± ]/W ) = W r(k) rω n k[t ± ]/k. We can then check the clam about comatblty wth λ 0 r. To nsh the roof, we can now vew λ r as an endomorhsm of the dga W r Ω k[t ± ]/k, whch s the dentty on the degree 0 graded ece. Snce W r Ω s generated k[t ± ]/k n degree 0, we get by the above lemma λ r s the dentty. Ths comletes the roof of Prooston 2.7, and hence the man theorem. References [] Bhatt, B., Morrow, M. and Scholze, P. Integral -adc Hodge theory, arxv: