ÉTALE COHOMOLOGY. Contents

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1 ÉTALE COHOMOLOGY GEUNHO GIM Abstract. Ths note s based on the 3-hour presentaton gven n the student semnar on Wnter We wll bascally follow [MlEC, Chapter I,II,III,V] and [MlLEC, Sectons 1 14]. Contents 1. Introducton Wel Conjecture (Sketch of) Sketch of proof of Wel Conjecture Wel s observaton 3 2. Étale Morphsms 3 3. Grothendeck Topology 5 4. Sheaves on Étale Stes Examples Specal Case : Spectrum of a feld Functors 8 5. Cohomology 9 6. Cohomology of Curves 11 References Introducton 1.1. Wel Conjecture. Let X 0 be a nonsngular projectve varety of dmenson d over F q. Let X = X 0 Fq F q and N m = X(F q m). We defne the zeta functon of X 0 by ) Z(t) = Z(X 0, t) = exp ( t m N m m m=1 Q[[t]] (Note that d dt log Z(X 0, t) = N m t m 1 s the generatng functon for N m.) Then the followng holds. (1) Ratonalty Z(t) Q(t) (2) Functonal ( ) Equaton 1 Z = q d ±q dχ 2 t χ Z(t) t where χ s the Euler characterstc of X 0 (self ntersecton number of X ). Date: Aprl 15,

2 (3) Remann Hypothess Z(t) = P 1(t)P 3 (t) P 2d 1 (t) P 0 (t)p 2 (t) P 2d (t) where P 0 (t) = 1 t, P 2d (t) = 1 q d t and for 1 r 2d 1, we have P r (t) = Z[t] for algebrac numbers α r, satsfyng α r, = q r / 2. (4) Bett Numbers Let β r = deg P r (t), χ = 2d r=0 ( 1) r β r. β r =1 (1 α r, t) If Y s a nonsngular projectve varety over a number feld K, and p s a prme of K, then Y mod p and Y K C have the same bett number (the latter beng dm Q H r (X, Q).) Example 1.1. Let X 0 = P 1, then N m = 1 + q m. We have Z(P 1, t) = exp ( tmm + qm t m ) = m We can easly check (1)-(4) above. 1 (1 t)(1 qt) Example 1.2. Let X 0 = P n, then N m = 1 + q m + + q nm and we have Z(P n, t) = 1 (1 t) (1 q n t) Example 1.3. Let X 0 = E be an ellptc curve. Then we have Z(E, t) = (1 αt)(1 βt) (1 t)(1 qt) where α, β are algebrac numbers satsfyng α = β = q (Sketch of) Sketch of proof of Wel Conjecture. Theorem 1.4 (Lefschetz fxed pont formula). Let X be a nonsngular varety over an algebracally closed feld k. Suppose φ : X X s regular. Then, where the left hand sde s the ntersecton number. (Γ φ X ) = ( 1) r Tr(φ Hr (X et,q l )) Lemma 1.5. Let V be a vector space over k, and φ : V V be an endomorphsm. Suppose P φ (t) = 1 and log P φ (t) = log det(1 φt V ) = (1 c t). Then we have Tr(φ m V ) = c m m=1 Tr(φ m V ) tm m c 1 Proof. Trangulzng gves φ... 0 c d, thus φ m c m c m d 1 1 c t = Now we brefly sketch the proof of some part of Wel s conjecture. Let F : X X be the Frobenus map. Note that N m = X(F q m) = X Fm = (Γ F m X ) = ( 1) r Tr(F m Hr (X et,q l )) 2

3 by the theorem above. (In ths case, each fxed pont has multplcty one.) Also, ( ) Z(X 0, t) = exp = = where P r,l (t) = det(1 Ft Hr (X et,q l )). 2d r=0 2d r=0 exp m ( ( 1) r Tr(F m Hr (X et,q l )) tm r m P r,l (t) ( 1)r+1 Tr(F m Hr (X et,q l )) tm m m Q l (t) Q[[t]] = Q(t) ) ( 1) r Snce F acts on H 0 as the dentty, we have P 0 (t) = 1 t. Also F acts on H 2d by multplcaton by deg F = q d, thus P 2d (t) = 1 q d t. Poncare dualty shows the functonal equaton. Furthermore, we can show that P r,l (t) s ndependent of l and s n Z[t]. Provng Remann hypothess s the hardest part of the proof Wel s observaton. Wel s orgnal observaton was that the conjectures wll follow from sutable cohomology theory as we saw n the above. The cohomology theory from the Zarsk topology s not sutable for several reasons. Frst of all, Zarsk topology s too coarse,.e., there are too few open subsets. Also the cohomology groups H r (X zar, F) vanshes when F s a constant sheaf, or r > dm X. On the other hand, étale cohomology from étale topology (wll be defned later) s sutable n ths case. For example, f X s a varety over C, then we have H r (X et, Λ) = H r (X(C), Λ) for a fnte abelan group Λ where the latter s the sngular cohomology group. Also for X = Spec k for a feld k, étale cohomology concdes wth Galos cohomology. For example, we have H r ((Spec k) et, G m ) = H r (Gal(k sep /k), (k sep ) ). 2. Étale Morphsms From now on, we assume that all rngs are noetheran, and all schemes are locally noetheran. Defnton 2.1. Let A, B be rngs. A morphsm f : A B s called flat f B A : Mod(A) Mod(B) s exact. (Ths s equvalent to say that A f 1 m B m s flat for all maxmal m B.) Let X, Y be schemes. A morphsm φ : Y X s called flat f the nduced map O X,φ(y) O Y,y s flat for all y Y. Proposton 2.2. (1) An open mmerson s flat. (2) Base change, compostons of flat morphsms are flat. (3) If X, Y are varetes and f : Y X s flat, then dm Y x = dm Y dm X f Y x =. (Y x s are called a flat famly.) (4) If f : Y X s flat, of fnte type, and f X, Y are noetheran, then f s open. Proof. (1),(2) From propertes of flat modules. (3) Base change and nducton. (4) See [HarAG, p.266]. Example 2.3. For a fnte separable feld extenson K/k, the morphsm Spec K Spec k s flat. If Z s a proper closed subscheme of a connected scheme X = Spec A, then the ncluson Z X s not flat. (Otherwse, the map s open, thus Z s a connected component of X.) 3

4 Defnton 2.4. Let A, B be rngs. A morphsm f : A B s called unramfed at q Spec B f for p = f 1 (q), we have f (p)b q = qb q and κ(q)/κ(p) s fnte separable. (For local rngs A, B, and a local homomorphsm f : A B, f s unramfed f f (m A )B = m B and κ(m B )/κ(m A ) s fnte separable.) Let X, Y be schemes. A morphsm φ : Y X s called unramfed f φ s of fnte type and O X,φ(y) O Y,y s unramfed for all y Y. Proposton 2.5. (1) Base change, compostons of unramfed morphsms are unramfed. (2) For schemes X, Y, Z and Z g Y f X, f f g s unramfed, then so s g. (3) The followngs are equvalent: (a) f : Y X s unramfed (b) Y x Spec k(x) s unramfed for all x X (c) Y x = Spec k where k /k(x) s a fnte separable extenson for all x X (4) The followngs are equvalent: (a) f : Y X s unramfed (b) Ω Y/X = 0 (c) Y/X : Y Y X Y s an open mmerson. Proof. (1) Clear from (3) (2) There s an exact sequence of O Z -modules: f Ω Y/X Ω Z/X Ω Z/Y 0 Thus ths follows from (4) (3) (a) (b) O Y,y /m x O Y,y = OYx,y (b) (c) For all y Y, k(y)/k s fnte separable. Thus all ponts n Y are closed ponts. Ths means Y s dscrete, and Y = Spec k. (c) (b) Clear from defnton. (4) (a) (b) Reduce to the case when X = Spec A, Y = Spec B for felds A, B. We know that f B/A s separable, then Ω B/A = 0 n ths case. (b) (c) We have Y U Y X Y and Ω Y/X = I/I 2 = 0. I y = 0 for all y Y by closed open mmerson NAK (I y O Y /I y = I y /Iy 2 = 0). Thus I = 0 on open V U contanng Y. So (Y, O Y ) = (V, O V ) as open subschemes of Y X Y. (c) (a) Consder f : Y Spec k for k = k and a closed pont y Y. Y {y} g Y X Y We have Spec O y Spec(O y k O y ) s an open mmerson and O y s local Artnan. Thus we have O y O y = Oy, whch means O y = k and unramfed. Defnton 2.6. Let X, Y be schemes. A morphsm φ : Y X s called étale f t s flat and unramfed. Note 2.7. Let X, Y be varetes over k = k and φ : Y X be a regular map. Then, φ s étale at y Y f and only f dφ y s an somorphsm. (Étale morphsms are sort of local somorphsms n ths sense.) Proposton 2.8. (1) An open mmerson s étale. (2) Base change, compostons of étale morphsms are étale. (3) An étale morphsm s open. 4 Y

5 (4) Consder Z g Y f X for schemes X, Y, Z. If f g s étale and f s unramfed, then g s étale. Proof. (1)-(3) Clear. (4) Consder the followng dagram d Z Γ g Z X Y p 1 p 2 g Z Y f g X f We have Z g Y Γ g g 1. Snce f s unramfed, s an open mmerson. Thus, Γ g s Z X Y Y X Y étale. Snce f g s étale, p 2 s étale (base change). Therefore, g = p 2 Γ g s also étale. 3. Grothendeck Topology Defnton 3.1. Let C be a category and U C. A coverng for U s a set of morphsms (U U) I n C such that (1) for any (U U), V U n C, there s U U V C for each and (U U V V) s a coverng of V (2) for any (U U), (V j U ) j Ij, the composton (V j U U),j s a coverng of U (3) (U d U) and all somorphsms are coverngs of U. The system of coverngs s called a (Grothendeck) topology. A category C wth a topology s called a ste. Example 3.2. (1) X zar (Zarsk ste) Category : open subschemes of X wth open mmersons. Coverng : surjectve famly of open mmersons. (In ths case, we have U U U j = U U j ) Ths corresponds to the usual Zarsk topology. (2) X et (Étale ste) Category : Et/X = {U X, étale} wth X-morphsms. X U X, V X are étale, then the X-morphsm φ s also étale by Proposton 2.8. (3) X Et (Bg étale ste) Category : Sch/X= the category of X-Schemes Coverng : surjectve famly of étale morphsms 5 U φ V Note that f

6 (4) X f l (resp. X Fl ) ((Bg) flat ste) Category : Fl/X (Sch/X) Coverng : surjectve famly of flat morphsms In general, for a class of morphsms E whch contans somorphsms and base change, compostons of morphsms n E, then we can make X E wth the underlyng category Sch/X. Defnton 3.3. Let T 1, T 2 be stes. A functor Cat(T 2 ) Cat(T 2 ) preservng fber products and coverngs s called a contnuous map T 1 T 2. Example 3.4. We have the followng contnuous maps nduced by d X : X Fl X Et X et X zar 4.1. Examples. Defnton 4.1. We defne the followngs: 4. Sheaves on Étale Stes PreSh(X et ) = {F : Et/X Sets( or Ab,... ), contravarant} Sh(X et ) = {F PreSh(X et ) F satsfes (S)} (S) for every U X étale, (U U) an étale coverng of U, we have the followng exact sequence: F(U) In other words, the followng map s exact: 0 F(U) F(U ) F(U U U j ),j F(U ) F(U U U j ),j s (s U ) (s ) (s j U U U j s U U U j ),j Note that we wrte F(U) for F(U X) for smplcty. F(U) does depend on the étale map U X. Proposton 4.2 (Sheaf crteron). A presheaf F PreSh(X et ) s a sheaf f and only f (S) s true for (1) Zarsk coverng (2) (V U) sngle coverng for V, U affne and V U étale. Sketch of proof. Consder a coverng (U U ). The followng commutes. (S) for (U U) : F(U) F(U ) F(U U U j ),j (S) for ( U U) : F(U) F( U ) F ( ( U ) U ( U ) ) Use ths to show (1), (2) are enough. 6

7 Example 4.3. (1) Structure sheaf For U X étale, we defne O Xet (U) = Γ(U, O U ). To check the sheaf condton (S), we need to show that the followng s exact for rngs A, B: for A B étale, fathful. (2) We defne µ n, G a, G m, GL n by 0 A B b 1 b b 1 B A B U µ n (Γ(U, O U )), Γ(U, O U ), Γ(U, O U ), GL n (Γ(U, O U )) (3) Constant sheaves For an abelan group Λ, we defne F Λ : U Λ π 0(U). (4) Let F be a coherent O X -module on X zar. For U φ X étale, we defne F et : U Γ(U, φ F) Note that φ F s also coherent, and (O Xzar ) et = O Xet. Smlary, we can defne F f l. Example 4.4. Let X be a varety over a feld k wth char k = p. We have the followng exact sequences between sheaves on étale ste. (Kummer) 0 µ n G m t t n G m 0 (Artn-Schreer) 0 Z/pZ G a t t p t G a 0 where (n, char k) = 1 and Z/pZ above s a constant sheaf. One can check these at stalks Specal Case : Spectrum of a feld. Let X = Spec K for a feld K. Suppose we have an étale morphsm Y Spec K. Then all affne covers of Y should be of the form Spec( K ) for K /K fnte separable because an étale algebra over a feld K s a product of fnte separable extensons. Thus, we have Y = Spec K for fnte separable K /K. Let G = Gal(K sep /K). Then we have the followng equvalence of categores between Sh(X et ) and G-Mod, the category of dscrete G-modules: where M F = Sh(X et ) G-Mod F M F F M M lm F(Spec K ) and F M (U) = Hom G (Hom X (x, U), M) for a closed pont x X. K /K fnte Remark 4.5. A pont n étale topology means a geometrc pont. For a pont x X, we have the geometrc pont x = Spec k(x) sep X. Ths gves a equvalence of categores Sh(x et ) equv Ab and cohomology H r vanshes for r > 0. But for a non-geometrc pont, ths s not true because we have more than one étale morphsms onto Spec(k(x)). Note that a sheaf on a one-pont space s just one datum, a group asssgned to that pont. Remark 4.6. For a geometrc pont x X, the stalk O X,x s the strct Henselzaton of O X,x by O X,x = lm O U,u = OX,x sh (U,u) étale neghborhood 7

8 4.3. Functors. Proposton 4.7. Sh(X et ) s an abelan category wth enough njectves. We have the followng natural functors Sh(X et ) PreSh(X et ) a functor) s left exact and a (the sheaffcaton functor) s exact. Defnton 4.8. Let π : Y X be a morphsm between schemes. We defne Sh(Y et ) π Sh(X et ) π where (the forgetful as follows. For an étale morphsm U X, the morphsm U X Y Y s also étale. For G Sh(Y et ), we defne (π G)(U) = G(U X Y). For an étale morphsm V Y and for F Sh(X et ), we defne F (V) = lm F(U) where the lmt s over commutatve dagrams U V Y U X étale. We defne π F = a(f ). These functors are adjont by the followng relaton: Hom Yet (π F, G) = Hom Yet (F, G) = Hom Xet (F, π G) Defnton 4.9. Suppose j : U X s an open mmerson and : Z = X \ U X s a closed mmerson. We defne Sh(U et ) j j! Sh(X et ) as follows. j s can be defned by the prevous defnton. For an étale morphsm V φ X and F Sh(U et ), we defne { F(V) f φ(v) U F! (V) = 0 otherwse { Fx f x U And we defne j! F = a(f! ). Note that we have (j! F) x = 0 f x / U Smlary, we have the followng relaton: Hom Xet (j! F, G) = Hom Xet (F!, G) = Hom Uet (F, G U ) We also can defne! as a left adjont functor of.! Sh(Z et ) Sh(X et ) Sh(U et ) Proposton Note that d j F Hom Uet (j F, j F) = Hom Xet (j! j F, F) nduces a morphsm j! j F F, and d F Hom Zet ( F, F) = Hom Xet (F, F) nduces a morphsm F F. We have the followng exact sequence n Sh(X et ). Proof. On stalks, we have 0 j! j F F F 0 { d 0 F x F x 0 0 d 0 0 F x F x 0 8 j! j j f x U f x Z

9 Theorem Let T (X) = {(F 1, F 2, φ) F 1 Sh(Z et ), F 2 Sh(U et ), φ : F 1 j F 2 } where φ s a morphsm n Sh(Z et ). Then, we have an equvalence of categores between Proof. See [MlEC, p. 74]. Sh(X et ) F ( F,j F,φ F : F (j j F)) T (X) 5. Cohomology Defnton 5.1. The functors Γ(X, ), Hom X (F 0, ) : Sh(X et ) Ab are left exact for a fxed F 0 Sh(X et ). Thus we can defne the followng rght derved functors: from Sh(X et ) to Ab. H r (X et, ) := R Γ(X, ) Ext r X(F 0, ) := R Hom X (F 0, ) Remark 5.2. We have H 0 (X et, F) = Γ(X, F) and Ext 0 X (F 0, F) = Hom X (F 0, F). If 0 F F F 0 s exact, then there s a long exact sequence H r (X et, F ) H r (X et, F) H r (X et, F ) H r+1 (X et, F ) and the same for Ext r X. Let Z be a constant sheaf on X. Then we have Hom X (Z, F) F(X) α α X (1) where α X : Z(X) = Z F(X). Therefore, we have Hom X (Z, ) = Γ(X, ) and Ext r X(Z, ) = H r (X et, ). Example 5.3. Suppose X = Spec K for a feld K. We have seen that there s an equvalence of categores Sh(X et ) F M F G-Mod for G = Gal(K sep /K). Snce Γ(X, F) = F(X) = MF G, we have H r ((Spec K) et, F) = H r (Gal(K sep /K), M F ) In partcular, f F = G m, then M F = (K sep ). Many analogues of theorems n sngular cohomology theory hold n étale cohomology. Theorem 5.4. Let X be a scheme. Suppose Z X s closed and let U = X \ Z. For F Sh(X et ), we defne the cohomology of F wth support Z by Γ Z (X, F) = ker(γ(x, F) Γ(U, F)) Ths nduces H r Z(X et, F) as a rght derved functor. Then, there s a long exact sequence H r Z(X et, F) H r (X et, F) H r (U et, F) H r+1 Z (X et, F) Proof. Use the exact sequence 0 j! j Z Z Z 0 and the fact that and Hom X (Z, F) = Γ(X, F). Hom X (j! j Z, F) = Hom U (j Z, j F) = Γ(U, F) Theorem 5.5 (Excson). Let X, X be schemes and π : X X be a morphsm. Let Z X and Z X be closed subschemes. Suppose π Z : Z Z and π(x \ Z ) X \ Z. For F Sh(X et ), we have H r Z(X et, F) H r Z(X et, F X ) for all r. In other words, the cohomology groups wth support Z only depend on a neghborhood of Z n X. 9

10 Proof. Note that π s exact and preserves njectves. Thus, t s enough to show that α on the followng dagram s an somorphsm. 0 Γ Z (X, F X ) Γ(X, F X ) Γ(U, F X ) α 0 Γ Z (X, F) Γ(X, F) Γ(U, F) Use the fact that (X X, U X) s a coverng of X and the sheaf condton to show that α s an somorphsm. We can defne the usual Čech cohomology groups lke n the case of sngular cohomology theory. Let X be a scheme and U = (U X) be an étale coverng. Let P PreSh(X). Defne C r (U, P) = I r+1 P(U 0 U U 1 U U U r ) and defne d r : C r C r+1 by the alternatng sum of restrctons. We defne Ȟ r (U, P) = H r (C (U, P)) and Ȟ r (X, P) = lm Ȟ r (U, P) where the lmt s over U refnements. For F Sh(X et ), we have by the sheaf condton. Ȟ 0 (X, F) = Γ(X, F) = H 0 (X et, F) Remark 5.6. Let X, Y be schemes, and consder a Galos coverng U = (Y X) consstng of a sngle map wth fnte Galos group G. (.e., G acts on Aut X Y and Y X Y = Y.) Suppose for g G F Sh(X et ), we have F( U ) = F(U ). Then, we have Ȟ r (U, F) = H r (G, F(Y)) where the latter s the group cohomology group. Ths s because F(X) F(Y) F(Y X Y) F(Y X Y X Y) F ( ) Y G = F(Y) G F(Y) G 2 corresponds to the cochan complex n group cohomology. Theorem 5.7. Let X be a quas-compact scheme. Suppose that every fnte subset of X s contaned n an open affne subset of X. Then, we have for all r and for all F Sh(X et ). Proof. See [MlLEC, p. 71]. Ȟ r (X, F) = H r (X, F) Note 5.8. On X zar, ths s true for separated X and quas-coherent O X -module F wth affne coverng U. 10

11 Theorem 5.9 (Mayer-Vetors). Let X be a scheme and X = U 0 U 1 be an open cover (n Zarsk sense). Let F Sh(X et ). Then, there s a long exact sequence H r (X, F) H r (U 0, F) H r (U 1, F) H r (U 0 U 1, F) H r+1 (X, F) Proof. Use Čech cohomology wth respect to the coverng {U 0, U 1 }. See [MlLEC, p. 74]. Now we compare cohomologes from dfferent stes on a scheme X. Note that there s a contnuous map between the followng stes nduced by d X. X f l X et X zar Theorem (1) Let F be a quas-coherent O X -module. Then, we have H r (X zar, F) H r (X et, F et ) H r (X f l, F f l ) (2) Let G be a smooth, quas-projectve, commutatve group scheme (e.g., abelan varety) over X. Then, H r (X et, G) H r (X f l, G) (3) Let X be a smooth scheme over C and M be a fnte abelan group. Then, H r (X(C), M) H r (X et, M) In partcular, when r = 0, ths gves a correspondence between connected components. Proof. See [MlLEC, III. 3]. In general, the followng holds. Proposton Let E 1 E 2 be classes of morphsms satsfyng Grothendeck topology condton. Let C 2 /X C 1 /X Sch/X be full subcategores of Sch/X. Defne stes X E = (C /X, E ) for = 1, 2. Consder the contnuous map f : X E1 X E2 nduced by d X and let F Sh(X E1 ). There exsts the Leray spectral exact sequence H r (X E2, R s f F) = H r+s (X E1, F) In partcular, f R s f F = 0 for all s > 0, then we have H r (X E2, f F) H r (X E1, F) Remark The proof of Theorem 5.10 comes down to showng ths partcular case. 6. Cohomology of Curves Let X be a complete connected nonsngular curve over a feld k = k. Suppose (n, char k) = 1 and let g be the genus of X. The followng holds. Γ(X, O X ) f r = 0 Theorem 6.1. (1) H r (X et, G m ) = Pc(X) f r = 1 0 f r > 1 µ n (k) f r = 0 (2) H r (Z/nZ) 2g f r = 1 (X et, µ n ) = Z/nZ f r = 2 0 f r > 2 11

12 Sketch of proof. (1) See [MlLEC, I. 13]. (2) Use the exact sequence and the fact that 0 µ n G m t t n G m 0 Pc 0 (X) n Pc 0 (X) J(k) n J(k) s commutatve where J(k) n J(k) s surjectve wth kernel somorphc to (Z/nZ) 2g. Here J(k) s the Jacoban varety of X. References [ArtGT] M. Artn, Grothendeck Topologes, Lecture Notes, Cambrdge, Mass.: Harvard Unversty, 1962 [HarAG] R. Hartshorne, Algebrac Geometry, Sprnger, 1977 [MlEC] J. S. Mlne, Étale Cohomology, Prnceton Unversty Press, 1980 [MlLEC] J. S. Mlne, Lectures on Étale Cohomology (v2.21), 2013, avalable at 12