p-dimension of henselian elds

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1 p-dimension of henselian elds An application of Ofer Gabber's algebraization technique Fabrice Orgogozo CNRS, École polytechnique (France) El Escorial, / 36

2 Denition (p-cohomological dimension) 1. Let X be a scheme and p a prime number. We say that cd p(x ) N i for all p-torsion étale sheaf F and all integer i > N, we have: H i (Xét, F ) = 0. Alexandre Grothendieck et al., SGA 4, exposé X by Michael Artin 2. Let G be a pronite group. We say that cd p(g) N i for all discrete p-torsion G-module M (with continuous action) we have, for all i > N, we have: H i (G, M) = 0. Jean-Pierre Serre, Cohomologie galoisienne. 2/ 36

3 If k is a eld, (Spec(k))ét = BG k (where G k = Gal(k sep /k)), so cd p(spec k) = cd p(g k ). Si X is an ane scheme of characteristic p > 0, cd p(x ) 1. It comes from the Artin-Schreier exact sequence 0 Z/p Ga Ga 0. 3/ 36

4 Examples Reminder: a C 1 -eld has p-cohomological dimension (for all prime number p) 1. Theorem 1. A nite eld is C 1. b Z 2. If k is algebraically closed, k(t) is C 1 ( Tsen). Frob p G Fp so cd l Fp is exactly 1, for all prime number l. 3. Let A be an henselian, excellent dvr with algebraically closed residue eld. Then Frac(A) is C 1 ( Lang). Excellent: the extension Frac( b A)/Frac(A) is separable. 4/ 36

5 Corollary (of Tsen's result) Let K/k a eld extension of transcendence degree N and p a prime number. Then, cd p(k) N + cd p(k). This is an equality if K/k is of nite type, cd p(k) < + and p 1 k. Corollary (of Lang's result) Let K be a complete discrete valuation eld with perfect residue eld k and p a prime number. Then, we have: cd p(k) 1 + cd p(k). This is an equality if p 1 K. Application: cd p(qp) = 2. 5/ 36

6 If the residue eld k is not perfect, Ω 1 k should be taken into account. via the cohomological symbol ( ). H (K, Z/p( )) Bloch-Kat o K M (K)/p, via the dierential symbol ( ): K M (K) Ω K, {x 1,..., x r } dlog(x 1 ) dlog(x r ). 6/ 36

7 Theorem (Kazuya Kat o (simplied version)) Let A be a henselian excellent discrete valuation ring of mixed characteristic (0, p). Let K, k the corresponding elds. Then: cd p(k) = 1 + dim p(k), where dim p(k) is equal to the p-rank of k, dim k Ω 1 k (= [k : kp ]), or dim k Ω 1 k / 36

8 Denition (of the p-dimension dim p ; rst part) Let κ be a eld of characteristic p > 0 and n N. We dene H n+1 p (κ) as the cokernel of the map (also denoted by "1 C 1 "): where Ω n κ Ω n κ/dω n 1 κ : a db 1 dbn (a a p ) db 1 dbn, b 1 b n b 1 b n and Ω i κ := i^ Ω1 κ/z a κ, b i κ. Characteristic p analogue of H n+1 (Spec(κ)ét, µ n p ). H 1 p(κ) = κ/ (κ). Non zero for Fp. H 2 p(κ) = Br(κ)[p]. 8/ 36

9 Denition (of the p-dimension dim p ; nal part) Let κ be a eld, p a prime number. 1. Assume char.(κ) p. Then dim p(κ) := cd p(κ). 2. Assume char.(κ) = p. i dim p(κ) N [κ : κ p ] p N & H N+1 p (κ ) = 0 κ /κ nite dim p(fp) = 1. Remarks The p-rank is invariant under nite eld extension. Need to consider κ /κ Hp r+1 is a "constant" coecient cohomology theory. (Cf. RΓ(G κ, M) (for various p-torsion G κ-modules M) RΓ(G κ, Z/p) (for nite étale κ /κ).) 9/ 36

10 Theorem (K. Kat o (nal version); analogue of Lang's theorem) Let A be a henselian excellent discrete valuation ring and p a prime number. Then, dim p(k) = 1 + dim p(k). Corollary (Analogue of Tsen's theorem) Let K/k a eld extension of transcendence degree N and p a prime number. Then, dim p(k) N + dim p(k). Proof: use the "classical" formula for cd p and the possibility to make K/k a "residue eld extension" of a characteristic zero dvr extension. Kazuya Kat o. Galois cohomology of complete discrete valuation rings. LNM 967, / 36

11 K. Kat o's conjecture Let A be an integral henselian, excellent (e.g. complete) local ring. Let K be its fraction eld and k its residue eld of characteristic p > 0. Then: dim p(k) = dim(a) + dim p(k). (Here, dim(a) is the Krull dimension.) Theorem (K. Kat o, 1986) Let A be a normal excellent henselian local ring of dimension 2 with residue eld k and fraction eld K. Suppose that k is algebraically closed. Then, for all prime number p char.(k), we have: cd p(k) = 2. 11/ 36

12 Remarks The proof uses the theorem of Merkur'ev-Suslin and resolution of singularities for surfaces. Sh uji Sait o. Arithmetic on two dimensional local rings. Inventiones mathematicæ 85, This has been extended to an arbitrary residue eld by Takako Kuzumaki. 12/ 36

13 In the following, we will K. Kat o's conjecture, namely: Theorem Let A be an integral henselian, excellent local ring. Let K be its fraction eld and k its residue eld of characteristic p > 0. Then: dim p(k) = dim(a) + dim p(k). Remark The equal-characteristic formula is proved rst and used to show the mixed-characteristic formula. 13/ 36

14 Lower bound: dim p (K) dim(a) + dim p (k) Reduction to the normal case We may assume A normal: A ν /A is nite (A is excellent). dim p is invariant by nite extension (when it is nite). The characteristic p case can be shown by using the classical result and the theorem of K. Kat o or, more simply, by using the existence of trace maps on H r+1 p. 14/ 36

15 Lower bound: dim p (K) dim(a) + dim p (k) Induction using K. Kat o's theorem (i.e. dim 1 case) Let p be a height one prime ideal L := Frac A p, B = c A p (complete dvr) and b L := Frac B. Mixed characteristic: GL b G L cd p(k = L) cd p( b L) 1 + dim p(frac A/p) [K. Kat o] 1 + ` dim(a) 1 + dim p(k) [induction]. Equal characteristic: [L : L p ] [ b L : b L p ] (if [L : L p ] is nite) and Hp r+1 (L) Hp r+1 ( b L). 15/ 36

16 Upper bound: dim p (K) dim(a) + dim p (k) Ofer Gabber's algebraization technique Reduction to the complete case (Artin-Popescu; cf. excellency hyp.) nite over "good" ring (i.e. ring of power series). (Proof of) Nagata's Jacobian criterion (equal characteristic) generically étale (and nite) over good ring. Mixed characteristic: use Helmut Epp's result. Ramication locus nite over lower dimensional base (Weierstraÿ) and Renée Elkik's algebraization. relative dimension 1 16/ 36

17 Upper bound: dim p (K) dim(a) + dim p (k) Ofer Gabber's algebraization technique Ofer Gabber. A niteness theorem for non abelian H 1 of excellent schemes. Conférence en l'honneur de Luc Illusie, Orsay, Ofer Gabber. Finiteness theorems for étale cohomology of excellent schemes. Conference in honor of Pierre Deligne, Princeton, Michael Artin. Cohomologie des préschémas excellents d'égales caractéristiques. SGA 4, exposé XIX. 17/ 36

18 Upper bound: dim p (K) dim(a) + dim p (k) Reduction to the complete case Lemma Let A be a local henselian quasi-excellent integral ring, b A its completion and K, b K the respective fraction elds. Then the K-algebra b K is a ltered colimit of K-algebras of nite type with retraction. Denition A ring A is quasi-excellent if it is noetherian and for all x X = Spec(A), the morphism Spec(Ô X,x ) Spec(O X,x ) is regular, for all A /A nite, Reg(Spec(A )) is open. Such a ring is in particular "universally Japanese". For henselian local rings, "excellent"=quasi-excellent. Proof. Immediate corollary of Sorin Popescu's version of M. Artin's approximation theorem. Theorem (S. Popescu; Artin's approximation property) Any nite system of polynomial equations over A has a A-point i it has a ba-point. Corollary Let A as above and F nite presentation functor A Alg Set. Then F(K) F( b K) is an injection. 18/ 36

19 Equicharacteristic upper bound: dim p (K) dim(a) + dim p (k) Around Nagata's Jacobian criterion Theorem (O. Gabber, conf. L. Illusie, lemma 8.1) Let A be an integral local complete noetherian ring of dimension d with residue eld k. There exists a subring A 0 of A, isomorphic to k[[t 1,..., t d ]] such that Spec(A) Spec(A 0 ) is nite, generically étale. Remark This theorem is obvious in mixed characteristic (hence "generically of characteristic zero"). However, in the algebraization process, it is also used (see below). 19/ 36

20 Equicharacteristic upper bound: dim p (K) dim(a) + dim p (k) Algebraization 1/2 X = Spec(A) X 0 = Spec(A 0 ) is nite, generically étale. X 0 A d k (o). Let R X 0 ramication locus. WMA: point (t 1,..., t d 1 ) / R so (Weierstraÿ) R V (r), r monic polynomial in k[[x 1,..., x d 1 ]][x d ]. 20/ 36

21 Reminder on Weierstraÿ theorem Theorem (Weierstraÿ, circa 1880) 1. Let κ be a local complete ring (e.g. a eld) with maximal ideal m, and f κ[[t 1,..., t n]] with f (u κ[[t n]] ) tn N mod. (m, t 1,..., t n 1 ). Then f = unit P, where P κ[[t 1,..., t n 1 ]][t n]. 2. For each element f κ[[t 1,..., t n]] non zero mod m, there exists a κ-linear automorphism α, dened by α(t i ) = t i + t c i n for i = 1,..., n 1 (and suitable c i 's), and α(t n) = t n such that α(f ) is as in (1). 21/ 36

22 Equicharacteristic upper bound: dim p (K) dim(a) + dim p (k) Algebraization 2/2 X = Spec(A) X 0 = Spec(A 0 ) is nite, generically étale. X 0 A d k (o). Let R X 0 ramication locus. WMA: point (x 1,..., x d 1 ) / R so (Weierstraÿ) R V (r), r monic polynomial in k[[x 1,..., x d 1 ]][x d ]. In particular: V (r) comes from f X 0 := Spec(k[[x 1,..., x d 1 ]]{x d }). The r-adic completion of f X 0 is X 0. The ("algebraized") pair ( f X 0, V (r)) is henselian, we can use R. Elkik's theorem to descend X X 0 to e X f X 0. 22/ 36

23 Reminder on Renée Elkik's theorem Denition A pair (X = Spec(A), Y = V (I )) is henselian if for every polynomial f A[T ], every simple root of f in A/I lifts to a root in A. Theorem (Renée Elkik, 1973) Let (X = Spec(A), Y = V (I )) be an henselian pair with A noetherian. Let b X be the completion of X along Y and b Y be the corresponding closed subscheme. Assume for simplicity that the complement U of Y in X is connected. Then b U := b X b Y is also connected and the map π 1 (U) π 1 ( b U) is an isomorphism. Renée Elkik Solutions d'équations à coecients dans un anneau hensélien. Annales scientiques de l'école normale supérieure, / 36

24 Equicharacteristic upper bound: proof 1/3 Set d = dim(a), r = dim k Ω 1 k, n = r + d. ` dimp K = dim K Ω 1 K + {0, 1}? d + ` dim p(k) = r + {0, 1}. dim K Ω 1 easy K = dim(a) + dim k Ω 1 k (= d + r) 1?! 0. H r+1 p (k) = 0 H n+1 p (K) = 0 (applied to K /K). Reminder If dim κ Ω 1 κ = r, dim p(κ) = r, i κ /κ nite, Hp r+1 (κ ) = 0 where H r+1 p (κ) is the cokernel of the map: Ω r κ Ω r κ/dω r 1 κ : a db 1 dbr (a a p ) db 1 dbr. b 1 b r b 1 b r Assume H r+1 p (k) = 0 (and similarly for nite extension), and take an element in Ω n K. Want to show it belongs to the image of (modulo exact forms). 24/ 36

25 Equicharacteristic upper bound: proof 2/3 As above: d = dim(a), r = dim k Ω 1 k, n = r + d, assume Hr+1 p (k) = 0, and consider ω Ω f n K, where ω Ωn A /torsion and f m A {0}. Let A 0 = k[[x 1,..., x d ]] as in O. Gabber's theorem and we have by algebraization a cartesian diagram X = Spec(A) e X = Spec( e A) nite, gen. étale X 0 = Spec(A 0 ) f X0 = Spec( f A 0 ) where f A 0 = k[[x 1,..., x d 1 ]]{x d }. We may assume f A 0 (by taking norms). 25/ 36

26 Equicharacteristic upper bound: proof 3/3 Up to a change of the rst coordinates (Weierstraÿ), we may assume f k[[x 1,..., x d 1 ]][x d ] and is monic. (In particular, it belongs to fa 0 = k[[x 1,..., x d 1 ]]{x d }.) It follows, that A is the f -adic completion of e A. Hence, ω f = ` ω f Ω ne A /tors. + `? m A Ω n A /tors. Surjectivity of? Second term: easy. Cf. a = (a + a p + a p2 + ). First term: Frac e A of transcendence degree one over Frac k[[x 1,..., x d 1 ]] use K. Kat o's theorem and induction. 26/ 36

27 Mixed-characteristic upper bound: cd p (K) dim(a) + dim p (k) Easier: X = Spec(A) X 0 = Spec(A 0 ) (A 0 = C[[x 1,..., x d 1 ]], C Cohen (discrete valuation) ring, nite and generically étale (obvious). Diculty: we don't want the ramication locus to be, for example, V (p) X 0 ( no hope to apply Weierstraÿ, to algebraize). Solution: use H. Epp's theorem to make X generically étale over the "special ber" V (p). (This is part of Ofer Gabber's method to prove niteness of cohomology.) Algebraization of cohomology class more subtle than the algebraization of the "denominator" f above (uses formal/henselian comparison theorem). 27/ 36

28 Reminder on Helmut Epp's theorem Theorem (Helmut Epp, 1973) Let T S a dominant morphism of complete traits, of residue characteristic p > 0. Assume the residue eld κ S is perfect and that the maximal perfect subeld of κ T is algebraic over κ S. Then there exists a nite extension of traits S S such that has reduced special ber over S. Helmut Epp Eliminating wild ramication. Inventiones mathematicæ, T := (T S S ) ν red Franz-Viktor Kuhlmann A correction to Epp's paper Elimination of wild ramication Inventiones mathematicæ, / 36

29 Mixed-characteristic upper bound: cd p (K) dim(a) + dim p (k) Sketch 1/2 Let k 0 the maximal perfect subeld of the residue eld k of A (A is complete, normal). Apply H. Epp's result over W (k 0 ), at the generic points of V (p), to get reduced special ber. Lemma Let X be a noetherian normal scheme. Every generically reduced Cartier divisor is reduced. Get by O. Gabber's theorem X p Spec(k[[x 1,..., x d 1 ]]) nite, generically étale. Lift to X X 0 := Spec(C(k)[[x 1,..., x d 1 ]]. (C(k) = Cohen ring.) By construction, ramication locus (downstairs) doesn't contain V (p). Weierstraÿ contained in V (f ), f C(k)[[x 1,..., x d 2 ]][x d 1 ] (up to change of coordinates). 29/ 36

30 Upper bound: cd p (K) dim(a) + dim p (k) = d + r Sketch 2/2 We can algebraize X X 0 (R. Elkik) into ex f X 0 = Spec(C(k)[[x 1,..., x d 2 ]]{x d 1 }). We want to show that H d+r+i (K, Z/p) = 0, for all i > 0. Choose c. Extend c to an algebraizable locus of X? (I.e. does it come from e K?) Apply K. Kat o's theorem and equal characteristic case in the codimension one point (in the special ber), to make the "pole" locus small in V (p). Lemma Let B be a discrete valuation ring, B h its henselization, and K (resp. K h ) the respective fraction elds. If the image of an element c H j (Spec(K)ét, Z/N) is zero in H j (Spec(K h )ét, Z/N), then c belongs to the image of the restriction morphism H j (Spec(B)ét, Z/N) H j (Spec(K)ét, Z/N). 30/ 36 Weierstraÿ / R. Elkik + Fujiwara-Gabber (+ K. Kat o and induction) to conclude.

31 Reminder on Kazuhiro Fujiwara and Ofer Gabber's theorem Theorem (Kazuhiro Fujiwara and O. Gabber) For a noetherian henselian pair (X = Spec(A), Y = V (I )) and a torsion étale sheaf F on U := X Y, the canonical morphism is an isomorphism. H a (Uét, F ) H a ( b Uét, b F ) Observe that we don't make hypothesis on the torsion. Kazuhiro Fujiwara. Theory of tubular neighborhood in étale topology. Duke mathematical journal, / 36

32 32/ 36 Esta terminado.

33 Proof of O. Gabber's theorem 1/4 Theorem (O. Gabber, conf. L. Illusie, lemma 8.1) Let A be an integral local complete noetherian ring of dimension d with residue eld k. There exists a subring A 0 of A, isomorphic to k[[t 1,..., t d ]] such that Spec(A) Spec(A 0 ) is nite, generically étale. Let κ be a eld of representative of k in A (to be changed later) lifting B i (i I ) in A of a p-basis b i of k (hence non unique if k isn't perfect). For all nite subset e I, let κ e := κ p (B i, i / e) ltered decreasing family of conite sub-κ p -extension of κ, such that eκ e = κ p. 33/ 36

34 Proof of O. Gabber's theorem 2/4 κ e := κ p (B i, i / e), B i lifting p-basis, e nite; eκ e = κ p. Let t 1,..., t d A a system of parameters K e := Frac κ e[[t p 1,..., tp d ]] Kκ := Frac κ[[t 1,..., t d ]] K. Observe: \ K e = Kκ p e big e rk K Ω 1 K/K e = rk Kκ Ω 1 K κ/k e. rk K Ω 1 = rk K/K e A Ω 1 A/κ e [[t p 1,...,tp ]] (gen. rank i.e. rk K (K A )), d rk Kκ Ω 1 K κ/k e = d + rk κ Ω 1 κ/κ e Hence: rk A Ω 1 A/κ e [[t p 1,...,tp ]] = dim(a) + rkκ Ω1 κ/κ e = d + #e for some nite set e. d (Reminder: Ω 1 generated by the db κ/κ e i, i e.) By changing the lifting B i of b i, i e (i.e. B i B i + (m i m A )), we can make the db i linearly independent in Ω 1 A/κ e [[t p 1,...,tp ]] A K. d (Use: d`m A ) generically generates Ω 1 A.) Using the corresponding elds of representatives (still denoted by κ), we achieve: 34/ 36 rk A Ω 1 A/κ[[t p 1,...,tp ]] = dim(a). d

35 Proof of O. Gabber's theorem 3/4 rk A Ω 1 A/R κ = dim(a). (R κ := κ[[t 1,..., t d ]]) Let f 1,..., f d A such that the df i form a basis of Ω 1 A/R κ K. WLOG: f i m A. Take t i := t p i (1 + f i ). Observe: dt i = t p i df i. Then, A is nite, generically étale over the subring A 0 = κ[[t 1,..., t d ]]. 35/ 36

36 Proof of O. Gabber's theorem 4/4 This completes the proof of: Theorem (O. Gabber) Let A be a local complete noetherian integral ring of dimension d with residue eld κ. There exists a subring A 0 of A, isomorphic to κ[[t 1,..., t d ]] such that Spec(A) Spec(A 0 ) is nite, generically étale. 36/ 36