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Schemes of dimension 2: obstructions in non Abelian cohomology
Bénaouda Djamai
To cite this version:
Bénaouda Djamai. Schemes of dimension 2: obstructions in non Abelian cohomology. Pure Mathe- matical Sciences, 2017, 6, pp.39 - 45. �10.12988/pms.2017.711�. �hal-01768672�
Obstructions in
Non Abelian Cohomology
Bénaouda DJAMAI UFR de Mathématiques
Université de Lille 1 Sciences et Technologies
F-59665 Villeneuve d’Ascq Cedex France Abstract
Let f : X → Y be a proper and a flat morphism with fibers of dimension1 andX regular of dimension 2. Suppose that the geometric fibers off are connected, and the generic fiber is smooth. In this paper we consider a reductive group G onX and use results of Artin-Tate to find the obstruction for aG-gerb onX to come from anf∗G-gerb onY. Subject Classification: 14F20 Keywords:Non Abelian Cohomology, Gerbs, Obstruction
1 Introduction
Let X be regular scheme of dimension 2 and Y a smooth irreducible curve defined over a perfect field k or the spectrum of the ring of integers of a number field. Suppose that the generic fiber is smooth and that the geometric fibers are connected
We will consider etale topology on X and Y and suppose f∗(OX) ' OY. This is a condition of normalization, because if it is not met, we can consider Spec (f∗(OX)) instead ofY.
This condition implies, in particular, f∗(Gm,X)'Gm,Y.
LetAbe an abelianX-group. From the Leray spectral sequence associated to f, Hp(Y, Rqf∗(A)) ⇒ Hp+q(X, A), we obtain the following exact sequence in lower dimensions:
0 //H1(Y, f∗(A)) //H1(X, A) //H0(Y, R1f∗(A))
//H2(Y, f∗(A)) //H2(X, A)tr //H1(Y, R1f∗(A))
//H3(Y, f∗(A))
(1)
2 Bénaouda DJAMAI
whereH2(X, A)tr :=Ker{H2(X, A)−→H0(Y, R2f∗A)}
In case A =Gm,X and under our assumptions, we have R2f∗Gm,X = 0 by [12, Artin’s Theorem(3.2)], and therefore H2(X, Gm,X)tr = H2(X, Gm,X). So (1) becomes:
0 //P ic(Y) //P ic(X) //P ic(X/Y)
//Br(Y) //Br(X) //H1(Y, P)
//H3(Y, Gm,Y) //H3(X, Gm,X)
(2)
whereP =R1f∗(GmX) etP ic(X/Y) =H0(Y, P), which finally we write 0 //S //Br(Y) //Br(X) //H1(Y, P) //T //0
whereS =Coker(P ic(X)−→P ic(X/Y)andT =Ker{H3(Y, Gm)−→H3(X, Gm)}
(for example, iff admits a section, S =T = 0).
We will now study the analogous of this exact sequence when A is not abelian. More precisely, we will consider a reductive group G and reduce its 2-cohomology to a maximal torusT.
Let L be a lien locally, for the etale topology, represented by a reductive groupG. We know by [3, Prop 3.2 p 75] thatL is represented by a quasi-split reductiveX-groupGL, and therefore, on some extensionGLadmits a maximal torusTL isomorphic to a product of d copies of the multiplicative groupGm.
We can suppose then, by taking a finite etale extension, that G = Ge is a split X-group scheme and thatL is represented by G.e
For the next, we setH2(·,G) =e H2(·, lien(G))e andH2(X,G)e tr the subset of H2(X,G)e composed by the classes of gerbs G bound by lien(G), such thate there exists a refinement R of Y such that, for every U ∈ Ob(R), the fiber of G in f−1(U)is not empty (cf. [6], Chapitre V,3.1.9.3).
LetZ(G)e be the center of Ge and consider next diagram:
H0(Y, R1f∗Z(G))e //
H2(Y, f∗Z(G))e
//H2(X, Z(eG))tr
i(2)∗
//H1(Y, R1f∗Z(eG))
H0(Y, R1f∗Te) //
H2(Y, f∗Te)
O //H2(X,Te)tr
O //H1(Y, R1f∗Te)
O
H0(Y, R1f∗G)e H2(Y, f∗G)e //H2(X,G)e tr //?
(3)
where // is the Giraud’s relation defined in [6], and the question mark is becauseH1(k, R1π∗GX) does not exist, since R1π∗GX has no more a group structure.
By [4, Cor.(3.4) P 74], for every geometric point of y ∈Y, all the classes of the fiber(Ret2f∗G)y are trivial. Hence:
Proposition 1.1. H2(X,G)e tr =H2(X,G).e
2 Main results
We can now complete the last line of diagram (3):
H2(Y, f∗G)e //H2(X,G)e tr =H2(X,G)e
by defining an obstruction set to pull back a class fromH2(X,G)e toH2(Y, f∗G).e Consider diagram:
H2(Y, f∗Z(G))e u //H2(X, Z(G))e tr
λ //H1(Y, R1f∗Z(G))e
µ
H2(X,Te)tr =Br(X)d
//H1(Y, R1f∗eT) =H1(Y, P)d
H2(Y, f∗eG) //H2(X, eG)tr //?
(4) Let q ∈ H2(X,G)e tr. Because H2(X,G)e tr is stable under the simply tran-
sitive action of H2(X, Z(G))e tr ([6]), we have q = α·ε where ε = [T ors G]e ∈ H2(X,G)e and α∈H2(X, Z(G)) =e H2(X, Z(G))e tr.
Theorem 2.1. The map
H2(X,G)e −→Im(λ)⊂H1(Y, R1f∗Z(G))e q λ(α)
defines the appropriate obstruction, i.e, the sequence
H2(Y,G)e −→H2(X,G)e −→Im(λ)−→0 is exact.
It should be noted in passing that when Im(λ) = 0, we find proposi- tion(3.1.10), chap.V of [6], and more precisely Corollary(3.1.10.3).
In particular:
4 Bénaouda DJAMAI
Corollary 2.2. Under assumption of corollary [4, Cor.(3.4) P 74], if the fibers of f :X −→Y are rational, then every class q∈ H2(X,G)e comes from a class q0 ∈H2(Y, f∗G)e
Proof. For every point y∈Y, by setting f−1(y) =Fy, we have : (R1f∗ZG)e y = lim←−−
U3y
H1(f−1(U), Z(G)) =e H1(Fy, Z(G))e 'Hom
π1(Fy), f∗Z(G)e
whereU runs over etale neighborhoods of y.
But Hom
π1(Fy), f∗Z(G)e
=Hom
π1(P1), f∗Z(G)e
= 0.
Corollary 2.2 extends results of corollaries(3.13) and (3.14 of [4], where the residue fieldk where supposed algebraically closed or finite.
Now let µ : H1(Y, R1f∗Z(G))e −→ H1(Y, R1f∗(Te)). We can consider also as set of obstruction the imageIm(µ◦λ)⊂H1(Y, R1f∗(Te)).
Proposition 2.3. The sequence
H2(Y,G)e −→H2(X,G)e −→Im(µ◦λ)−→0
is exact. Obstruction defined by Im(µ◦λ) is the same as the one defined by Im(λ).
However, we will see in theorem(2.5) later that it is more flexible then that defined byIm(λ).
Proof. If q = α· ε ∈ H2(X,G)e is pulled back to q0 = β ·ε0 ∈ H2(Y, f∗G)e with ε0 = [T ors(Y, f∗G)]e and β ∈ H2(Y, f∗Z(G)), thene λ(α) =λ(u(β)) = 0 ∈ H1(Y, R1f∗Z(G)), hencee µ◦λ(α) = 0.
Conversely, suppose that µ◦λ(α) = 0.
We have a canonical isomorphism of sheaves of sets between R1f∗(Te) and the sheaf of maximal sub-gerbs of the direct image of the gerbT ors(X,Te)by f [6, Lemme 3.1.5].
Moreover, if [G] ∈ H2(X,Te) (resp. H2(X,Te)tr) is the class of an X-gerb G, the sheaf Ger(G)of maximal sub-gerbs of the direct image f∗(G) ofG par f is a pseudo-torsor (resp. a torsor) underR1f∗(Te) [6, Ex. 3.1.9.2].
letGTe be the gerb representing the image of α in H2(X,Te)tr =H2(X,Te).
Since (µ◦λ)(α) = 0, the sheaf Ger(G
Te) of maximal sub-gerbs of the direct image f∗(GTe) admits a section. If G represents the class q, there exists an X-morphism of gerbs:
GTe −→ G induced by the inclusion:
T ,e →Ge
The gerb morphism G
Te −→ G induces in turn anY-morphism of sheaves Ger(G
Te)−→Ger(G) whereGer(G) is the sheaf of maximal sub-gerbs of f∗G
So every Y-section ofGer(G
Te)induces anY-section of Ger(G), i.e,q can be pulled back to a class in H2(Y,G).e
Under assumptions of [4, Cor.(3.4) P 74]), suppose now that f admits a section. We have then the following exact sequence:
0 //Br(Y) //Br(X) //H1(Y, P) //H3(Y, Gm) //H3(X, Gm,X)
{{
which, compared with exact sequence [12, Theorem(3.1)]:
0→Br(Y)→Br(X)→X1(K, J)→0 provides the isomorphism: X1(K, J)'H1(Y, P)
Here, K is the function filed of Y, J is the jacobian variety of the generic fiber andX1 is the Tate-Shafarevich group.
From diagram (4), we get :
Im(µ◦λ)⊂H1(Y, R1f∗(Te)'H1(Y, Pd)'X1(K, J)d Theorem 2.4. In the exact sequence:
H2(Y, f∗G)→H2(X, G)→Im(µ◦λ)→0
the sub-group Im(µ ◦ λ) ⊂ X1(K, J)d is a sub-group of obstruction to a class q = α ·ε ∈ H2(X, G) to come from a class in H2(Y, f∗G). The map H2(X,G)e −→Im(µ◦λ)can be interpreted as the application which associates to a class G the class of the sheaf of maximal sub-gerbs Ger(GTe) of the direct image of G
Te by f.
One can summarize this situation in the following diagram : G ∈H2(X,G)e
++
//[Ger(G
Te)]∈Im(µ◦λ)⊂H1(Y, R1f∗Te))
Ger(G)∈?
Let’s recall that the Artin-Tate conjecture states thatX1(K, J)is finite. Our obstructions takes then values in a finite group. Therefore, if the order Z(G)e and the order of X1(K, J) are coprime, this obstruction vanishes. Hence:
Theorem 2.5. Under Artin-Tate conjecture, if |Z(G)|e and the order of X1(K, J) are coprime, where K denotes the functions field of Y, then all the classes inH2(X,G)e come from H2(Y, f∗G).e
Example 2.6. If Ge = SLn, we have: Te = (Gm)n−1 and Z(G) =e µn. So there is an infinity of integers for which theorem 2.5 is true.
6 Bénaouda DJAMAI
References
[1] M. Demazure & A. Grothendieck. Propriétés générales des schémas en groupes, volume 151 of Lecture Notes in Mathematics. Springer Verlag, 1970.
[2] Jean-Claude Douai. Cohomologie galoisienne des groupes semi-simples definis sur les corps globaux. C.r Acad. Sci, A 281:1077–1080, 1975.
[3] Jean Claude Douai. 2-cohomolgie galoisienne des groupes semi-simples.
Th‘ese de doctorat, Université de Lille1, 1976.
[4] Jean Claude Douai. Suites exactes déduites de la suite spectrale de leray en cohomoligie non abélienne. Journal of Algebra, 79:68–77, 1981.
[5] Jean Giraud. Méthode de la descente. Mémoires de la S.M.F, 2, 1964.
[6] Jean Giraud. Cohomologie non abelienne, volume 179 of Grundlheren Math. Wiss. Springer-Verlag, 1971.
[6] A. Grothendieck.Revetements étales et groupe fondamental,Lecture Notes in Mathematics. 224 Springer Verlag, 1971.
[7] A. Grothendieck. Groupe de Brauer II. Dix exposés sur la cohomologie des schémas, 1968.
[7] A. Grothendieck. Groupe de Brauer III. Dix exposés sur la cohomologie des schémas, 1968.
[8] A. Grothendieck and J.-L. Verdier. Théorie des topos et cohomologie étale des schémas, volume 269 of Lecture Notes in Mathematics. Springer Verlag, 1972.
[9] J.S Milne.Comparison of the Brauer group with the Tate-Safarevic group.
J. Fac. Sci. Univ. Tokyo (Shintani Memorial Volume) IA 28 (1982), 735- 743. .
[10] Jean Pierre Serre.Cohomologie Galoisienne. Lecture Notes in Mathemat- ics. Spinger, 1997.
[11] S.S. Shatz. Profinite groups, Arithmetic and Geometry. Annals of Math- ematics Studies, Princeton University Press, 1972.
[12] J. Tate. On the conjecture of Birch and Swinerton-Dyer and a geometric analog. Séminaire N. Bourbaki, exp. no306, p. 415-440., 1964-1966.
