Then, for any open subsetUofX, we have (1.1.1) cd`(U)≤2d−1

XVIIIA. Cohomological dimension : First results Luc Illusie

version du 2016-11-14 à 13h36TU (19c1b56)

Table des matières

1. Bound in the strictly local case and applications. . . 1 2. Proof of the main result. . . 2 Références. . . 4

In this exposé we establish Gabber’s bound on cohomological dimension stated in the introduction (in the comments on the proof of the finiteness theorem).

1. Bound in the strictly local case and applications

Theorem 1.1. — LetXbe a strictly local, noetherian scheme of dimensiond > 0, and let`be a prime number invertible onX. Then, for any open subsetUofX, we have

(1.1.1) cd`(U)≤2d−1.

Recall that, for a schemeS, cd`(S)(`-cohomological dimensionofS) denotes the infimum of the integersn such that for all`-torsion abelian sheavesFonS, and alli > n, Hⁱ(S, F) =0.

Corollary 1.2. — LetX=SpecAbe as in1.1, and assumeAis a domain, with fraction fieldK. Then

(1.2.1) cd`(K)≤2d−1.

Indeed, it suffices to show that ifFis a finitely generatedF`-module overη = SpecK, then H<sup>i</sup>(η, F) = 0 fori > 2d−1. Butηis a filtering projective limit of affine open subsetsUα ofX,Fis induced from a locally constant constructibleF`-sheafF_α0 onU_α0, and Hⁱ(η, F) = lim−→Hⁱ(U_α, F_α), whereF_α = F_α0|U_α forα ≥ α₀ ([SGA 4VII5.7]).

Remark 1.3. — (a) The proof shows that, givenXas in1.1, withXintegral, then, if (1.1.1) holds for any affine open subsetU, (1.2.1) holds, too.

(b) IfXis an integral noetherian scheme of dimensiond, with generic point SpecK, and`is a prime number invertible onX, then cd`(K)≥d([SGA 4X2.5]). Gabber can prove that under the assumptions of1.2one has cd`(K) =d(seeXVIIIB).

Corollary 1.4. — LetYbe a noetherian scheme of finite dimension,f:X→Ya morphism of finite type, and`a prime number invertible onY. Then

cd`(Rf_?)<∞,

i.e. there exists an integerNsuch that for any`-torsion abelian sheafFonX,R^qf?F=0forq > N.

Proof of1.4. We may assumeYaffine. CoveringXby finitely many open affine subsetsUi(0≤i≤n), and using the alternating ˇCech spectral sequence

E^pq₁ =M

R^q(f|Ui₀···ip)?(F|Ui₀···ip)⇒R^p+qf?F,

whereUi₀···ip=Ui₀∩· · ·∩Ui_p, we may assumefseparated. Repeating the procedure, we may assumeXaffine.

Choose an immersionX → PⁿY, and replacePⁿY by the scheme-theoretic closure ofX. We get a commutative diagram

X

f

j //X

g

Y

withjopen andgprojective of relative dimension≤n. By the proper base change theorem we have cd`(Rg?)≤ 2n. By the Leray spectral sequence R^pg?R^qj?F⇒R^p+qf?Fit thus suffices to prove1.4forj, in other words, we may assume thatfis an open immersion. Letdbe the dimension ofY. Letybe a geometric point ofY, and let

1

2

U=Y(y)×Y Xbe the corresponding open subset of the strictly local schemeY(y) (of dimension≤d, that we may assume to be> 0). Then

Rⁱf_?(F)_y=Hⁱ(U, F)

(where we still denote byFits inverse image onU). The conclusion follows from1.1.

Remarks 1.5. — (a) Under the assumptions of 1.1, if Xis quasi-excellent and Uis affine, then by Gabber’s affine Lefschetz theorem (exp.XV,1.2.4) we have cd`(U)≤d. More generally, see exp.XVII,3.2.1for a proof of1.1forXquasi-excellent.

(b) Gabber can show that, under the assumptions of1.1, one has cd`(U)≥difUis not empty and does not contain the closed point and that for eachnsuch thatd≤n≤2d−1, there exists a pair(X, U)as in1.1, with Uaffine, such that cd`(U) =n(by (a), forn > d,Xis not quasi-excellent). These results are proved inXVIIIB.

2. Proof of the main result

Lemma 2.1. — LetXbe as in1.1, and letxbe the closed point ofX. Then (1.1.1) holds forU=X−{x}.

Proof. — It suffices to show that for any constructibleF`-sheafFonU, Hⁱ(U, F) =0fori ≥2d. LetXbbe the completion ofXat{x}and setUb :=Xb×XU=Xb−{x}. LetbFbe the inverse image ofFonUb. By Gabber’s formal base change theorem ([Fujiwara, 1995, 6.6.4]), the natural map

Hⁱ(U, F)→Hⁱ(U,b bF)

is an isomorphism for alli. Therefore we may assumeXcomplete, and in particular, excellent. Let(f1,· · · , fd) be a system of parameters ofX, and letU_i=X_fi, so thatU=S

1≤i≤dU_i. Consider the (alternate) ˇCech spectral sequence

E^pq₁ =M

H^q(Ui₀···ip, F)⇒H^p+q(U, F),

withU_i0···ip=U_i0∩ · · · ∩U_ipas above. By definition,E^pq₁ =0forp≥d. On the other hand, asXis excellent, by Gabber’s affine Lefschetz theorem (exp.XV,1.2.4), we haveE^pq₁ =0forq≥d+1. ThereforeE^pq₁ =0for p+q≥2d, hence Hⁱ(U, F) =0fori≥2d.

Lemma 2.2. — LetXbe a noetherian scheme of finite dimension,Y a closed subset,`a prime number invertible onX.

Then, for any`-torsion sheafFonX,

HⁱY(X, F) =0 for

i >sup_x∈Y(cd`(k(x)) +2dimOX,x).

In particular,

cd`(X)≤sup<sub>x∈X</sub>(cd`(k(x)) +2dimOX,x).

Proof. — Forp≥0, letΦ^pbe the set of closed subsets ofYof codimension≥pinX. We haveΦ^p=∅forp >

dim(X). Consider the (biregular) coniveau spectral sequence of the filtration(Φ^p)(cf. [Grothendieck, 1968, 10.1]),

(2.2.1) E^pq₁ =H^p+q_Φp/Φ^p+1(X, F)⇒H^p+q_Y (X, F).

We have

E^pq₁ = M

x∈Y^(p)

H^p+q_{x} (X_x, F|X_x),

where Y^(p) denotes the set of points ofY of codimension p in X, and X_x = SpecOX,x. For x ∈ Y^(p) (i.e.

dimOX,x=p), letxbe a geometric point abovex. Consider the diagram {x}

i_x //X_(x)

j U

oo

{x} ^ix //Xx oo ^j U ,

whereU=X_x−{x},U=X_(x)−{x}. We have

RΓ_{x}(X_x, F|X_x) =RΓ({x},Ri^!_x(F|X_x)).

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The stalk of Ri^!x(F|Xx)atxis

Ri^!x(F|Xx)x=Ri^!x(F|X(x)), as(Rj?(F|U))x=Rj_?(F|U)x. We thus have a spectral sequence

(2.2.2) E^rs₂ =H^r(k(x),R^si^!_x(F|X_(x)))⇒H^r+s_{x} (Xx, F|Xx), It suffices to show that, in the initial term of (2.2.1),

(2.2.3) H^p+q_{x} (Xx, F|Xx) =0

forp+q > cd`(k(x)) +2p. Ifp = 0, then Ri<sup>!</sup><sub>x</sub>(F|X<sub>(x)</sub>) = F<sub>x</sub>, and, in (2.2.2),E<sup>rs</sup><sub>2</sub> = 0 fors > 0,E<sup>r0</sup><sub>2</sub> = 0 for r >cd`(k(x)), so (2.2.3) is true in this case. Assumep > 0. We have

(2.2.4) R^si^!_xF=H^s−1(U, F|U)

for s ≥ 2, where, as above,U = X(x) −{x}. By 2.1, H^s−1(U, F|U) = 0 for s− 1 ≥ 2p, hence, by (2.2.4), R^si^!_x(F|X_(x)) =0andE^rs₂ =0fors≥2p+1. Ifr+s≥cd`(k(x)) +2p+1ands≤2p, thenr >cd`(k(x)), hence E^rs₂ =0as well. Therefore, by (2.2.2), (2.2.3) holds, which finishes the proof.

Proof of1.1. We prove1.1by induction ond. Forn≥0consider the assertion

Gn :For every strictly local, noetherian schemeXof dimensionn, all open subsetsUofXand any prime number` invertible onX, we havecd`(U)≤sup(0, 2n−1).

Letd > 0. AssumeGnholds forn < d, and let us proveGd. LetXbe as in1.1. If(Xi)_1≤i≤rare the reduced irreducible components ofXand Ui = U×XXi, we have cd`(U) ≤ sup(cd`(Ui)), hence we may assumeX integral. Letxbe the closed point ofX, andU=X−{x}the punctured spectrum. Letj:V→Ube a nonempty open subset ofU, andFbe a constructibleF`-sheaf onV. AsF = j<sup>?</sup>j!F, by2.1it suffices to show that, for any constructibleF`-sheafLonU, the restriction map

(∗) Hⁱ(U, L)→Hⁱ(V, j^?L)

is an isomorphism fori≥2d. LetY=U−V. Consider the exact sequence Hⁱ_Y(U, L)→Hⁱ(U, L)→Hⁱ(V, j^?L)→Hⁱ⁺¹_Y (U, L).

By 2.2, we have Hⁱ_Y(U, L) = 0 fori > sup_y∈Y(cd`(k(y)) +2dimOX,y). For y ∈ Y, denote byZthe closed, integral subscheme of Xdefined by the closure of{y}in X. AsXis integral andV nonempty, Zis a strictly local scheme of dimensionn < d, with generic pointy. By1.3(a) andG<sub>n</sub> (inductive assumption), we have cd`(k(y))≤2n−1. We have2n−1+2dimOX,y≤2d−1. Hence, fori≥2d, Hⁱ_Y(U, L) =Hⁱ⁺¹_Y (U, L) =0, and (∗) is an isomorphism, which finishes the proof.

Remark 2.3. — Gabber has an alternate proof of1.1, based on the theory of Zariski-Riemann spaces. By2.2, it suffices to show1.2. Here is a sketch, pasted from an e-mail of Gabber to Illusie of 2007, Aug. 15 :

"ForY →Xproper birational with special fiberY₀, consideri:Y₀→Yandj:η→Y,ηthe generic point. We have by proper base change a spectral sequence

H^p(Y0, i^?R^qj?F)→H^p+q(η, F)

forFan`-torsion Galois module. We take the direct limit and get a spectral sequence involving cohomologies on the étale topos ofZRS₀ defined as the limit of étale topoi ofY₀or viewingZRS₀as a locally ringed topos and applying a universal construction in the book of M. Hakim. The limit of the R^qj_?F isR^q(η → ZRS)_?F.

By a classical result of Abhyankar, also proved in Appendix 2 of the book of Zariski-Samuel Vol. II, ifRis a noetherian local domain of dimensiondandVa valuation ring ofFrac(R)dominatingR, the sum of the rational rank and the residue transcendence degree is at mostd. For a strictly henselian valuation ringVwith residue characteristic exponentpand value groupΓ, the absolute Galois group ofFrac(V)is an extension of the tame part (product for`prime not equal topofHom(Γ,Z`(1)))by ap-group, so the `-cohomological dimension is the dimension of Γ tensored with the prime fieldF`, which is at most the dimension of Γ tensored with the rationals. IfAis an`-torsion sheaf on the étale topos ofZRS0, letδ(A)be the sup of transcendence degrees of points where the stalk is non-zero. I claim thatHⁿ(ZRS0, A)vanishes forn > 2δ(A). One reduces it to the finite type case (passage to the limit[SGA 4VI8.7.4]) using that theδof the direct image ofAtoY0is at most δ(A). InY0the transcendence degrees over the closed point ofXare at mostd−1by the dimension inequality.

Summing up, for the limit spectral sequence theq-th direct image sheaf restricted to the special fiber hasδat mostmin(d−1, d−q), giving vanishing for certainE^p,q₂ and the result."

4

Références

[Fujiwara, 1995] Fujiwara, K. (1995). Theory of tubular neighborhood in étale topology.Duke Math. J., 80(1), 15–57.↑2 [Grothendieck, 1968] Grothendieck, A. (1968). Le groupe de Brauer III : exemples et compléments. In J. Giraud, A. Gro-

thendieck, S. L. Kleiman, M. Raynaud, & J. Tate (éds),Dix exposés sur la cohomologie des schémas, volume 3 desAdvanced studies in Pure Mathematics(pp. 88–188). Masson et Cie, North-Holland.↑2

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