POL VAN HOFTEN (NOTES BY JAMES NEWTON)

Transcription

1 INTEGRAL P -ADIC HODGE THEORY, TALK 2 (PERFECTOID RINGS, A nf AND THE PRO-ÉTALE SITE) POL VAN HOFTEN (NOTES BY JAMES NEWTON) 1. Wtt vectors, A nf and ntegral perfectod rngs The frst part of the talk wll cover Wtt vectors, A nf and ntegral perfectod rngs, followng secton 3 of [1] Settng. We fx a prme p, and consder rngs S whch are π-adcally complete and separated wth respect to an element π p. We defne the tlt S = lm S/pS x x p and remark that S = lm S x x p as monods (the usual proof works). Examples. S = Z p, S = F p If S = Ẑp[T ] (the p-adc completon) then S/pS = F p [T ] and S = (F p [T ]) perf = F p. If S = Z cyc p = Z p [µ p ] then S/pS = F p [T 1/p ]/T and S = F p [[T 1/p ]]. Defnton 1.2. A nf (S) = W (S ). Applyng the defnton to the above examples, we get (respectvely): A nf (S) = W (F p ) A nf (S) = W (F p ) A nf (S) = Zp [[T 1/p ]]. We have a map θ : A nf (S) S whch lfts the map S S/pS gven by projectng to the bottom component of the nverse lmt. We can defne θ as θ ]p 0[a = a p 0 where () s the multplcatve map S S gven by consderng S = lm x x p S and projectng onto the frst component. When S s perfectod (defnton s to come), the map θ wll be surjectve (ths wll follow from surjectvty of S S/pS). Next we are gong to defne maps θ r : A nf (S) W r (S) (wth θ 1 = θ). Frst we have some recollectons on Wtt vectors. 1

2 2 INTEGRAL P -ADIC HODGE THEORY 1.3. Wtt vectors. For a rng A, we can defne the (p-typcal) Wtt vectors W (A). As sets we have W (A) = A N. The Wtt vectors come equpped wth rng homomorphsms (the ghost components) for r 0: W (A) ωr A (a 0,..., a r,...) a pr 0 + papr p r a r Truncatng to the frst r entres (a 0,..., a r 1 ), we get the truncated Wtt rng W r (A). We have obvous restrcton maps R : W r+1 (A) W r (A) and also a Wtt vector Frobenus F : W r+1 (A) W r (A). When A has characterstc p, F s smply gven by F (a 0,..., a r ) = (a p 0,..., ap r 1 ), so we can wrte Rφ = φr = F, where φ s the map rasng each component to the pth power. On ghost components, we have ω (F x) = ω +1 (x) for all x W r+1 and r 1. Remark 1.4. We have A nf (S) = lm R W r (S ). Snce R φ = F and S s perfect, the maps φ r : W r (S ) W r (S ) nduce an somorphsm φ : lm W r (S ) lm W r (S ). R Takng Wtt vectors commutes wth nverse lmts so we have lm F W r (S ) = lm lm φ W r (S/pS) = lm φ lm W r (S/pS). Fnally, φ s an automorphsm of lm W r (S/pS) (use φr = Rφ = F ) so we conclude that A nf (S) = lm W r (S/pS). Remark 1.5. The multplcatve map () : S S extends to a multplcatve bjecton S lm S (nverse to the canoncal multplcatve map nduced x x p by S S/pS). At the level of Wtt vectors, the canoncal rng homomorphsm lm W r(s) lm W r (S/pS) s n fact an somorphsm. So we have A nf (S) = lm W r (S). Projectng to the rth component n the nverse lmt, we obtan a rng homomorphsm θ r : A nf (S) W r (S). Defnton 1.6. We set θ r = θ r φ r. Here φ = W (φ), the map nduced by functoralty from the p-power map on S. Lemma 1.7. We have θ r ([x]) = [x (r) ] for x S where we wrte x = (x (0), x (1),...) lm x x p S. We therefore have θ r([x]) = [x (0) ] = [x ] and n partcular θ 1 = θ Integral perfectod rngs. Defnton 1.9. S s (ntegral) perfectod f S s π-adcally complete and: (1) π p p (2) The Frobenus φ : S/pS S/pS s surjectve (3) θ : A nf (S) S has prncpal kernel

3 INTEGRAL P -ADIC HODGE THEORY 3 Examples. Z cyc p Z cyc p T 1/p The p-adc completon of Z p [[X]] Z p [p 1/p,, X 1/p ] If A s an ntegral doman wth p / A, the p-adc completon of an absolute ntegral closure of A s a perfectod rng. Lemma The followng are equvalent: (1) φ : S/pS S/pS s surjectve (2) F : W r+1 (S) W r (S) s surjectve for all r (3) θ r : A nf (S) W r (S) s surjectve for all r Remark If F : W 2 (S) S s surjectve, then part (1) n the above holds. Use the fact that F (a 0, a 1 ) = a p 0 + pa 1. Lemma If π s not a zero dvsor, φ s surjectve and ker(θ) s prncpal then s an somorphsm. φ : S/πS S/π p S Now we consder S = O K, where K s a perfectod feld (the old defnton) wth µ p K and a fxed compatble system of p-power roots of unty (ζ p ). Then we defne ɛ S by ɛ = (1, ζ p, ζ p 2,...) and so [ɛ] A nf (S). Fact: ker(θ) s generated by 1 + [ɛ 1/p ] + + [ɛ 1/p ] p 1. The map θ := lm θ r : A nf W (S) r has kernel generated by µ = [ɛ] 1, and f K s sphercally complete θ s surjectve. To compare wth the more standard perfectod termnology: f S s perfectod and flat over Z p then (S[ 1 π ], S) s a perfectod Huber par the deal defnng the topology on S s (π). 2. The pro-étale ste of an adc space All our adc spaces wll be locally Noetheran adc spaces over Spa(Q p, Z p ), so n partcular they wll be analytc. Defnton 2.1. A map f : X Y of adc spaces s fnte étale f t s affnod and for an open cover of Y by affnods Spa(A, A + ) the pull back Spa(B, B+) of X s fnte étale over Spa(A, A + ).e. A B s fnte étale and B + s the ntegral closure of A + n B. Defnton 2.2. A map of adc spaces f : X Y s étale f x X there s an open neghbourhood x U X and f(u) V open n Y such that f factors as (2.2.0) where s an open mmerson and g s fnte étale. U Remark 2.3. Ths defnton does not work n algebrac geometry! (Note by Pol: I am now unsure of ths) f W V g

4 4 INTEGRAL P -ADIC HODGE THEORY We obtan stes X fét X ét wth coverngs jontly surjectve famles of (fnte) étale maps. Now we want to ntroduce the pro-étale ste. Two advantages of ths wll be: frstly nverse lmts of sheaves wll behave well, so we can compute l-adc cohomology groups as genune cohomology groups of a sheaf Z l = lm Z/l n Z, rather than n nverse lmts of cohomology groups. Secondly, every adc space wll be pro-étalelocally perfectod. The pro-étale ste s sandwched between the pro-categores gven by towers of fnte étale, or étale, maps: pro X fét X proét pro X ét. We don t want to take the whole of pro X ét because t ncludes maps whch are not open (for example, thnk of a tower of dscs of shrnkng rad) Pro-categores and the pro-étale ste. Our ndex categores I are cofltered: ths meams we have the followng two propertes: For every par of objects, j of I we have another object k wth morphsms k and k j. For every par of morphsms f, g : j n I we have an object k and a morphsm h : k such that fh = gh. Now gven a category C, the objects of the pro-category pro C are functors I C from (small) co-fltered categores I. Gven U pro X ét we wrte U = lm U ( vares over objects of the ndex category I). Defnton 2.5. We defne a full subcategory X proét pro X ét by sayng that U s n X proét f U s somorphc to lm U wth (1) U X étale for all (ths s automatc) (2) U U j fnte étale and surjectve for all j. Remark 2.6. We can modfy the second condton n the above defnton by substtutng for all wth for a cofnal system of, and get an equvalent defnton. Defnton 2.7. Gven U = lm U n X proét we defne the topologcal space U = lm U. I Defnton 2.8. (1) Gven a morphsm U V n X proét we say that U V s étale (resp. fnte étale) f there exsts U 0 V 0 an étale (resp. fnte étale) map of adc spaces and V V 0 (the constant pro-object gven by V 0 ) such that U = U 0 V 0 V. (2) We say that U V s pro-étale f U = lm A k (here A k X proét ) wth k A k V étale and A k A k fnte étale surjectve. (3) Fnally, we defne coverngs n X proét to be {U V } such that the U cover V, each map U V s pro-étale, plus a set-theoretc condton (whch s automatc f the nverse lmts are over countable ndex sets). Proposton 2.9. Ths defnton of coverng makes X proét nto a ste n partcular, pro-étale maps are preserved under composton and base change. Pro-étale maps are open (.e. f : U V s open) Gven V U open, we can fnd W U n X proét wth W = V. There s a map of stes ν : X proét X ét

5 INTEGRAL P -ADIC HODGE THEORY 5 (nduced by the functor n the other drecton, takng somethng étale over X to the assocated constant pro-object). In partcular we have functors ν, ν on the categores of abelan sheaves. If F Ab(X ét ) s a sheaf of abelan groups (and X s qcqs) we have H (X ét, F) = H (X proét, ν F). If F Ab(X ét ) the natural adjuncton map F Rν ν F s an somorphsm. Remark If U = lm Spa(A, A + ) then ν F(U) = lm F(Spa(A, A + )) Sheaves on the pro-étale ste. Here are a bunch of sheaves on X proét : O + X (we have O+ X ( lm Spa(A, A + O X, Ô + X = lm O+ X /pn, Ô X = Ô+ X [ 1 p ] = lm O + φ X /p A nf,x = W (Ô+ ) X Ô+ X )) = lm A+ ) Remark In general the sectons of these sheaves are not easy to compute: Ô + X (U) may not equal the p-adc completon of lm A+. But they do behave well on affnod perfectods. Defnton U = lm Spa(A, A + ) s affnod perfectod f s a perfectod Huber par. ( lm A + [1 p ], lm A + ) Example Let X n = Spa(K T ±1/pn, O K T ±1/pn ) for K a perfectod feld. Set X = X 1. Then lm n X n s an affnod perfectod n X proét. References [1] Bhatt, B., Morrow, M. and Scholze, P. Integral p-adc Hodge theory, arxv: