THE CARTIER ISOMORPHISM. These are the detailed notes for a talk I gave at the Kleine AG 1 in April Frobenius

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1 THE CARTIER ISOMORPHISM LARS KINDLER Tese are te detaled notes for a talk I gave at te Klene AG 1 n Aprl Frobenus Defnton 1.1. Let f : S be a morpsm of scemes and p a prme. We say tat S s of caracterstc p, f po S = 0, or equvalently, f te unque morpsm Spec Z factors (also unquely) troug Spec F p Spec Z. Let S be of caracterstc p > 0. Te morpsm F :, gven by te dentty on and by a a p on sectons of O, s called te absolute Frobenus morpsm of. Wrte or (p/s) for te sceme makng te followng dagram cartesan: S F S S. f Denote te projectons and S by W = W /S = (F S ) and f = f (p). Te unque morpsm F /S :, suc tat te dagram F /S F f f W S F S S f (1) Date: February 15, :16. 1 ttp:// 1

2 2 LARS KINDLER commutes s called te relatve Frobenus of f : S. From now on S always denotes a sceme of postve caracterstc p. Remark 1.2. a. We can complete dagram (1) as follows: F F /S f f S W /S F S S F F /S f b. Te morpsms F /S and W /S are omeomorpsms: Te composton F = W /S F /S s a omeomorpsm, W /S = (F S ) s a omeomorpsm, wc can be deduced from dagram (2) and te fact tat absolute Froben are omeomorpsms. In fact, by te dagram F /S W /S and W /S F /S are bot omeomorpsms. Tus W /S and ten F /S are omeomorpsms. c. In fact F /S and W /S are even unversal omeomorpsms: F = W /S F /S s a unversal omeomorpsm and ntegral ( entér ), so F /S s surjectve, ntegral and unversally njectve. Surjectvty and ntegralty are stable under base cange, so F /S s unversally bjectve and closed, ence a unversal omeomorpsm. Example 1.3. Let, S be affne, say = Spec B, S = Spec A, were A s of caracterstc p > 0. Ten = B FA A,.e. ab 1 = b a p, and te relatve Frobenus s gven by a b ab p. If B = A[t], ten A FA A[t] = A[t], W /S s gven by at a p t and F /S by at at p, for a A. Hence te mage of F A[t]/A s A[t p ] A[t] and A[t] s freely generated by t, = 0,..., p 1 over A[t p ]. If te morpsm f s smoot, te relatve Frobenus as very nce propertes. But frst we ll need: Remnder 1.4 (Étale coordnates). Let f : S be a smoot morpsm and q a pont of. Snce Ω 1 /S s locally free, tere s a open affne f (2)

3 THE CARTIER ISOMORPHISM 3 negborood U of q and x 1,..., x n O (U) suc tat x q = 0 for all and suc tat dx 1,..., dx n generate Ω 1 /S U (use Nakayama). Tese sectons defne an S-morpsm : U A n S. Ts morpsm s étale by constructon: Because of smootness ave te exact sequence of O U -modules 0 Ω 1 Z/S Ω 1 U/S Ω 1 U/Z 0, were Z := A n S for brevty. By constructon Ω1 U/Z = 0, so : U Z s smoot and unramfed, ence étale. Te x 1,..., x n are called étale coordnates around q. Proposton 1.5 ([Ill02, Prop. 3.2]). Let S be a sceme of postve caracterstc p and f : S a smoot morpsm of pure relatve dmenson n. Ten te relatve Frobenus F := F /S s fnte flat, and te O -algebra F O s locally free of rank p n. In partcular, f f s étale, ten F s an somorpsm Proof. We wll use 1.4. Frst assume tat f s étale,.e. of relatve dmenson 0. Lookng at dagram (2), we see tat f s étale and tat f = f F /S s étale, so F /S s étale. But F /S s also radcal (tat s, unversally njectve), by Remark 1.2.c.. Ten [SGA1, I.5.1] sows tat F /S s an open mmerson. But snce t also s a omeomorpsm t follows tat F /S s an somorpsm. Next, for Z = A n S, F O Z = F O S [t 1,..., t n ] s generated over O Z by monomals t k tkn n fnte locally free, and ence also flat. wt 0 k p 1, wc sows tat F Z/S s ndeed For te general case, we can assume by 1.4 tat we ave a factorzaton Z g S, were g s te projecton Z = A n S S and s étale. Lookng

4 4 LARS KINDLER at te dagram F /Z F /S FZ Z (F Z/S ) Z F Z/S Z g S Z F S S, we see tat F /S = (F Z/S ) F /Z, so by te above we are done. g 2. Carter Before we get to te man teorem we ave to make a few observatons: Remark 2.1. Let f : S agan be a morpsm of scemes. a. Te morpsms α : F Ω1 /S Ω1 /S and β : F /S Ω1 /S Ω1 /S are trval. In fact, f s s some secton of O, ten α(ds) = β(d(1 s)) = d(s p ) = 0. b. Te dfferental of te complex (F /S ) Ω /S s O -lnear: Agan, locally, a secton of O as te form a FS s, and F d((a FS s)t) = d(sa p t) = sa p dt = (a s) dt, for s a secton of O S, t, a sectons of O. Tus te coomologes, cycles, boundares are O -modules, and Z F Ω /S and H F Ω /S are graded ant-commutatve O - algebras va te exteror product. c. Let Z = A n S. We compute te complex (F Z/S) Ω Z/S. Let K(n) be te complex of F p -vector spaces suc tat K(n) j s generated by t k tkn n dt r1... dt rj, were 0 t p 1 and 0 r 1 <... < r j n. For brevty let F = F Z/S for te moment. We ave seen tat as an O Z -module, F O Z = F O S [t 1,..., t n ] s generated by te monomals t k tkn n wt 0 k p 1, so we can wrte F O Z = O Z Fp K(n) 0.

5 THE CARTIER ISOMORPHISM 5 Smlarly, F Ω 1 Z/S s generated, as an O Z -module, by te forms t k tkn n dt r wt 0 t p 1 and r {1,..., n}, so F Ω Z /S 1 = O Z Fp K(n) 1. Te same calculaton works for F Ω Z/S, so we get an somorpsm of complexes of O Z -modules F Ω Z/S = O Z Fp K(n), Teorem 2.2 (Carter Isomorpsm, [Ill02, Tm. 3.5], [Kat70, Tm. 7.2]). Let S be a sceme of caracterstc p > 0 and f : S a morpsm. Tere exsts a unque omomorpsm of graded O -algebras C 1 = C 1 : Ω /S H ( (F /S ) Ω /Y ), suc tat te followng old: () C0 1 (1) = 1. () C1 1 (d(g 1)) = [gp 1 dg] n H 1 ((F /S ) Ω /S ), for g a secton of O. If f s smoot ten C 1 s an somorpsm. Proof. We follow [Kat70, Tm 7.2]. Let us frst construct te morpsm C 1. Let us frst observe tat to defne a morpsm wt te propertes above, t suffces to defne te morpsm C 1 1 n degree 1. Indeed, snce we want a morpsm of graded O -algebras, C 1 0 can only be defned as above, and snce Ω n /S = n Ω 1 /S, we wll obtan C 1 n wedge-product on H ((F /S ) Ω 1 /S ). as C C1 1, usng te To defne C 1 1, recall tat gvng a morpsm Ω1 /S H1 ((F /S ) Ω /S ) s te same as gvng a S-dervaton O /S H 1 ((F /S ) Ω /S ). As and are omeomorpc va F /S, we can consder O as an abelan seaf on (not an O -module). Ten we can wrte O = O f 1 F S f 1 (O S ). A S-dervaton (.e. a f 1 (O S )-dervaton) O H 1 ((F /S ) Ω /S ), satsfyng property () from above, s ten a f 1 (O S )-lnear map δ : O f 1 F S

6 6 LARS KINDLER f 1 (O S ) H 1 ((F /S ) Ω /S ) satsfyng δ((g s)(g s )) = δ(gg ss ) = (g s)δ(g s ) + (g s )δ(g s) = g p s p δ(g s) + g p s p δ(g s ) = g p δ(gs p s) + g p δ(s p g s ) = g p δ(g ss ) + g p δ(g ss ), (3) and, δ(g 1) = [g p 1 dg]. Now, we defne δ by δ(g s) := sg p 1 dg, and t s easy to ceck tat δ satsfes (3). Ceckng tat t s addtve s a lttle trcker: Frst, t s clear tat δ(g (s + s )) = δ(g s) + δ(g s ). So t remans to ceck tat δ((g + g ) s) δ(g s) δ(g s) = 0 n H 1 ((F /S ) Ω /S ). Observe tat δ((g+g ) s) δ(g s) δ(g s) = s(g+g ) p 1 d(g+g ) sg p 1 dg sg p 1 dg, and tat 1 ( (g + g ) p g p g p) p 1 = p =0 1 p ( ) p g g p Z. Dervaton of ts term gves s ( (g + g ) p 1 d(f + g) g p 1 dg g p 1 dg ) ( p 1 = d s =0 1 p ( ) p )g g p = 0 n H 1 ((F /S ) Ω /S ) as desred. Fnally, f 1 (O S )-lnearty s clear. To prove tat for smoot f : S, te morpsm C 1 s an somorpsm, we frst reduce to te case tat = A n S. Snce te queston s local, 1.4 allows us to factor f nto : A n S and te projecton g : An S S. As before, for brevty, wrte Z = A n S. Snce s étale, te relatve Frobenus F /Z s an somorpsm, so te rectangle F = F /Z (p/z) W/Z (4) Z F Z Z

7 s cartesan. Ts proves tat te square THE CARTIER ISOMORPHISM 7 F /S (p/s) (5) Z F Z/S rel Z (p/s) s also cartesan, were rel s te unque arrow suc tat f (p) = g (p) rel : Consder te followng, slgtly more confusng dagram: W /Z = F/Z (p/z) (p/s) Z (5) rel Z (p/s) Z F Z Te outer rectangle s precsely te one from dagram (4), and te rgt and square s cartesan, as FS S = Z (Z FS S). Te rectangle from dagram (5) s ence cartesan. Snce, F /S, F Z/S are all étale, ts mples tat rel s étale and ence flat. Lookng at (5) agan, we see ([Har77, III.9.5]) tat were Ω Z/S = Ω /S rel (F Z/S) Ω Z/S = (F /S ) Ω Z/S = (F /S ) Ω /S, snce s étale. Now rel s flat, so ts nduces an somorpsm ( ) ( ) rel H (F Z/S ) Ω Z/Y H (F /S ) Ω /S. Snce rel s étale, we also ave rel Ω Z (p/s) /S = Ω (p/s) /S us te commutatve dagram wc fnally gves rel Ω Z (p/s) /S = Ω /S rel C 1 ( ) rel H (F Z/S ) Ω Z/S C 1 = ( ) H (F /S ) Ω /S.

8 8 LARS KINDLER As te constructon of C 1 was functoral,.e. C 1 Z assume tat = Z = A n S. = rel C 1, we may Te rest s rater smple. In 2.1 we computed (F /S ) Ω /S = O (p/s) Fp K(n), f = A n S. As we tensor over te feld F p, we get H ( (F /S )Ω /S) = O (p/s) Fp H (K(n)). In order to prove te teorem we ence need to sow: H 0 (K(n)) = F p. H 1 (K(n)) s te vector space wt bass x p 1 dx, = 1,..., n H (K(n)) = H 1 (K(n)). Next, note tat K(n) s te total complex of K(1) Fp... Fp K(1) (n-factors). Tus, by Künnet s Formula, t suffces to prove tat K(1) as te above propertes, but ts s clear: H 0 (K(1)) = ker(d : 0 <p x F p 0 <p x dxf p ) = F p. H 1 (K(1)) = 0 <p x d x F p / m(d) = x p 1 dxf p. Te trd property s trval, as K(1) j = 0 for j > 1. Corollary 2.3 ([Ill02, Cor. 3.6]). If S s a sceme of caracterstc p > 0 and f : S smoot, ten for any te O (p/s)-modules (F /S ) Ω /S, Z (F /S ) Ω /S, B (F /S ) Ω /S, H ((F /S ) Ω /S ) are locally free of fnte type. Proof. We ave already seen n 1.5 tat te clam s true for (F /S ) Ω /S. By Carter s Teorem, t also follows for H ((F /S ) Ω /S ), snce Ω (p/s) /S s locally free of fnte type. Wtout loss of generalty we can assume tat f can be factored nto : A n S =: Z, and te projecton g : Z S. As n te proof of te teorem we obtan rel (F Z/S) Ω Z/S = (F /S ) Ω /S, and snce (F Z/S ) Ω Z/S = O Z Fp K(n), te clam follows, as te cycles and boundares of ts complex are locally free and rel exact.

9 THE CARTIER ISOMORPHISM 9 Alternatvely, f as dmenson n, ten we know tat te O -module (F /S ) Ω n /S = Zn (F /S ) Ω /S s locally free, and usng te exact sequence 0 B k (F /S ) Ω /S Z k (F /S ) Ω /S H k ((F /S ) Ω /S ) 0 we deduce tat B n (F /S ) Ω /S s also locally free2. Usng descendng nducton, te above exact sequence, and te sort exact sequence 3 0 Z k 1 (F /S ) Ω /S (F /S ) Ω k 1 /S d B k (F /S ) Ω /S 0, te clam follows. Te followng corollary s not relevant for te talk. Corollary 2.4 ([Ogu94, 1.2.5]). Let f : S be smoot and E a quascoerent O (p/s ) -module. Ten E := F /S E s canoncally equpped wt a flat connecton of p-curvature 0, and we ave te Carter somorpsm of graded O (p/s)-algebras E O Ω (p/s) (p/s) /S = (F /S ) H (E Ω /S ), were te rgt and sde s te pus-forward of te coomology of te de Ram complex of (E, ),.e. te complex E Ω /S References E Ω +1 /S. [Har77] R. Hartsorne, Algebrac Geometry, Graduate Texts n Matematcs, vol. 52, [Ill02] Sprnger-Verlag, Luc Illuse, Frobenus and odge degeneraton, SMF/AMS Texts and Monograps, vol. 8, Amercan Matematcal Socety, Provdence, RI, 2002, Translated from te 1996 Frenc orgnal by James Lews and Peters. MR MR (2003g:14009) [Kat70] N. M. Katz, Nlpotent connectons and te monodromy teorem: Applcatons of a result of Turrttn, Inst. Hautes Études Sc. Publ. Mat. (1970), no. 39, MR MR (45 #271) [Ogu94] A. Ogus, F -crystals, Grffts transversalty, and te Hodge decomposton, Astérsque (1994), no. 221, MR MR (95g:14025) 2 If P and P are projectve modules suc tat 0 M P P 0 s exact, ten M s projectve, as te sequence above splts and drect summands of projectve modules are projectve. 3 Recall tat te dfferental d s O -lnear.

10 10 LARS KINDLER [SGA1] A. Grotendeck and M. Raynaud, Revêtements Étales et Groupe Fondamental: Sémnare de Géométre Algébrque de Bos-Mare 1960/61., Lecture Notes n Matematcs, vol. 224, Sprnger-Verlag, 1971.