The Bloch-Ogus spectral sequence

LetXbe an algebraic variety (in particular, it is irreducible and we can speak of its function field). IfX is defined overC, we can consider two topologies onX(C), namely the Euclidean (or analytic) topology and the Zariski topology. We will denote X_an, resp. X_Zar, the topological spaceX(C) equipped with the Euclidean topology, resp. the Zariski topology.

As Zariski open sets are open for the Euclidean topology, the identity ofX(C) is a continuous map

f :Xan→XZar.

Given any abelian groupA, the Bloch-Ogus spectral sequence is the Leray spectral sequence off, abutting to the cohomologyH_Bⁱ(X, A) :=Hⁱ(Xan, A). It starts with

E₂^p,q(A) =H^p(XZar,H^q(A)),

whereH^q(A) is the sheaf onX_Zar associated with the presheafU 7→H_B^q(U, A). The Betti cohomology groupsH_Bⁿ(X, A) =Hⁿ(X_an, A) thus have a filtration, (which is in fact whenX is smooth the coniveau filtration,) namely the Leray filtration for whichGr_L^pH_B^p+q(X_an, A) = E_∞^p,q, the latter group being a subquotient ofE^p,q₂ .

A fundamental result of Bloch-Ogus [11] is the Gersten-Quillen resolution for the sheaves H^q(A). It is constructed as follows: For any varietyY, we denote byHⁱ(C(Y), A) the direct limit over all dense Zariski open setsU ⊂Y of the groupsH_Bⁱ(U, A):

Hⁱ(C(Y), A) = lim

∅̸=U⊂Y,open→

H_Bⁱ (U, A). (1.6)

Let nowZ be a normal irreducible closed algebraic subset ofX, and letZ^′be an irreducible reduced divisor ofZ. At the generic point ofZ^′, bothZ^′ andZ are smooth. There is thus a residue map∂:Hⁱ(C(Z), A)→Hⁱ⁻¹(C(Z^′), A). It is defined as the limit over all pairs of dense Zariski open setsV ⊂Zreg, U ⊂Z_reg^′ such thatU ⊂V ∩Z_reg^′ , of the residue maps

ResZ,Z^′ :Hⁱ((V \V ∩Z^′)an, A)→Hⁱ⁻¹(Uan, A).

If nowZ^′ ⊂Z is a divisor, withZ not necessarily normal along Z^′, we can introduce the normalization n : Ze → Z with restriction n^′ : Z^′′ → Z^′, where Z^′′ = n⁻¹(Z^′), and then define∂:Hⁱ(C(Z), A)→Hⁱ⁻¹(C(Z^′), A) as the composite

Hⁱ(C(Z), A)∼=Hⁱ(C(Z), A)e →^∂ Hⁱ⁻¹(C(Z^′′), A)ⁿ

′∗

→Hⁱ⁻¹(C(Z^′), A). (1.7) In (1.7), the pushforward morphism

n^′_∗:Hⁱ⁻¹(C(Z^′′), A)→Hⁱ⁻¹(C(Z^′), A)

is defined by restricting to pairs of Zariski open sets U ⊂ Z_reg^′′ , V ⊂ Z_reg^′ such that n^′ restricts to a proper (in fact, finite) morphism U → V. More precisely, as Z^′′ is not necessarily irreducible, we should in the above definition write Z^′′ = ∪jZ_j^′′ as a union of irreducible components, and take the sum over j of the morphisms (1.7) defined for each Z_j^′′.

For each subvariety j:Z ,→X, we consider the group Hⁱ(C(Z), A) as a constant sheaf supported on Z and we get the corresponding sheaf j_∗Hⁱ(C(Z), A) on XZar. Finally, we observe that we have a natural sheaf morphism

Hⁱ(A)→Hⁱ(C(X), A)

where we recall that the second object is a constant sheaf onX_Zar. This sheaf morphism is simply induced by the natural mapsHⁱ(U_an, A)→Hⁱ(C(X), A) for any Zariski open set U ⊂X, given by (1.6). The residue maps have the following property: LetD₁, D₂⊂Y be two smooth divisors in a smooth variety, letZ be a smooth reduced irreducible component ofD1∩D2 and letα∈H_Bⁱ(U, A), whereU :Y \(D1∪D2). Then

Res_Z(Res_D1(α)) =−Res_Z(Res_D2(α)), (1.8) where on the leftZ is seen as a divisor inD1, and on the right it is seen as a divisor ofD2. Considering the case whereY ⊂X is the regular locus of any subvariety of codimensionk ofX,D, D^′⊂Y are of codimensionk+ 1, andZ ⊂D∩D^′ ⊂Y is of codimensionk+ 2 in X, we conclude from (1.8) that for anyi, the two sheaf maps

∂:⊕codimY=kHⁱ(C(Y), A)→ ⊕codimD=k+1Hⁱ⁻¹(C(D), A)

and

∂:⊕codimD=k+1Hⁱ⁻¹(C(D), A)→ ⊕codimZ=k+2Hⁱ⁻²(C(Z), A) satisfy∂◦∂= 0.

Theorem 1.16. (Bloch-Ogus, [11]) LetX be smooth. The complex 0→ Hⁱ(A)→Hⁱ(C(X), A)→ ⊕Dirred

codimD=1

Hⁱ⁻¹(C(D), A)→. . .→ ⊕Zirred

codimZ=i

H⁰(C(Z), A)→0(1.9) is an acyclic resolution ofHⁱ(A).

It is clear that this resolution is acyclic. Indeed, all the sheaves appearing in the resolution are acyclic, being constant sheaves for the Zariski topology on algebraic subvarieties ofX. Note that the codimensionisubvarietiesZofX appearing above are all irreducible, so that H⁰(C(Z), A) =Aand the global sections of the last sheaf appearing in this resolution is the groupZⁱ(X)⊗Aof codimensionicycles with coefficients inA.

Theorem 1.16 says first that the sheaf map Hⁱ(A)→Hⁱ(C(X), A) is injective, which is by no means obvious. The meaning of this assertion is that if a classα∈H_Bⁱ(U, A) vanishes on a dense Zariski open setV ⊂U, thenU can be covered by Zariski open setsVisuch that α_|V_i = 0. This is a moving lemma for the support of cohomology.

We now come back to the Bloch-Ogus spectral sequence and describe the consequences of this theorem, following [11].

Theorem 1.17. (i) For any two integersp > q, one has E₂^p,q(A) =H^p(X_Zar,H^q(A)) = 0.

(ii) For p≤q, one has

H^p(X_Zar,H^q(A)) = Ker (∂:⊕codimZ=pH^q−p(C(Z), A)→ ⊕codimZ=p+1H^q−p−1(C(Z), A)) Im (∂:⊕codimZ=p−1H^q−p+1(C(Z), A)→ ⊕codimZ=pH^q−p(C(Z), A))(1.10). (iii) The groupH^p(X,H^p(Z))is isomorphic to the groupZ^p(X)/algof codimensionpcycles ofX modulo algebraic equivalence.

Proof. (i) Indeed, Theorem 1.16 says thatH^q(A) has an acyclic resolution of lengthq.

(ii) As (1.9) is an acyclic resolution ofH^q(A), the complex of global sections of (1.9) has degreepcohomology equal toH^p(XZar,H^q(A)).This is exactly the contents of (1.10).

(iii) We use (ii), which gives in this case

H^p(X_Zar,H^p(Z)) = ⊕codimZ=pH⁰(C(Z),Z)

Im (∂:⊕codimZ=p−1H¹(C(Z),Z)→ ⊕codimZ=pH⁰(C(Z),Z)). We already mentioned that the numerator is the groupZ^p(X). The proof is concluded by recalling the following two facts :

(1) A cycle Z of codimensionponX is algebraically equivalent to 0 if it belongs to the group generated by divisors homologous to 0 in the (desingularization of a) subvarieties of codimensionp−1 ofX.

(2) A divisorD in a smooth complex manifold is cohomologous to 0 if and only if there exists a degree 1 integral Betti cohomology classαonX\ |D| such that Resα=D. Here we denote by|D|the support ofD.

The vanishing (i) in Theorem 1.17 is very important. Let us give some applications taken from [11]. We will give further applications in Section 1.2.3. First of all, by the vanishing (i), we conclude that there is no nonzero Leray differentialdr, r≥2 starting from E₂^p,p(Z) = H^p(XZar,H^p(Z)). It follows that E_∞^p,p(Z) is a quotient of the group H^p(XZar,H^p(Z)).

Futhermore, by the same vanishing (i) above, the Bloch-Ogus filtration onH_B^2p(X,Z) has L^p+1= 0, and thus L^pH_B^2p(X,Z) =Gr_L^pH_B^2p(X,Z) =E_∞^p,p(Z). We conclude that there is a natural composite map

H^p(XZar,H^p(Z))→E_∞^p,p(Z),→H_B^2p(X,Z). (1.11)

It is proved in [11] that, via the identification given by Theorem 1.17, (iii), this map is the cycle class map in Betti cohomology. Note that by definition, the kernel of the cycle class mapZ^p(X)/alg→H_B^2p(X,Z) is the Griffiths group Griff^p(X). We finally have the following result for codimension 2 cycles which describes the kernel of the cycle class map.:

Theorem 1.18. [11] LetX be a smooth variety overC. There is a natural exact sequence H_B³(X,Z)→H⁰(XZar,H³(Z))→H²(XZar,H²(Z))→H_B⁴(X,Z).

Proof. The maps are the natural ones. The first map is given by restriction to Zariski open sets. The second map is the differentiald₂ of the Bloch-Ogus spectral sequence and the last map is the one appearing in (1.11) and just identified with the cycle class map. The proof of the exactness follows from inspection of the Bloch-Ogus spectral sequence. The kernel of the map H²(X_Zar,H²(Z)) = E_∞^2,2 → H_B⁴(X,Z) must be in the image of some d_r and obviously onlyr = 2 is possible. This shows exactness in the third term. Finally, by the vanishing of Theorem 1.17,(i), the only nonzerodrstarting from H⁰(XZar,H³(Z) is d2. It follows that Kerd2=E_∞^0,3, and this is a quotient ofH_B³(X,Z). This shows exactness in the second term.

1.2.2 Unramified cohomology

The following definition was first introduced in [16] in the setting of ´etale cohomology.

Definition 1.19. Let X be an algebraic variety overCand letAbe an abelian group. Then H_nrⁱ (X, A) =H⁰(XZar,Hⁱ(A)).

This definition can be made in fact over other fields, with Betti cohomology replaced by

´etale cohomology. IfAis finite, andX is overC, ´etale and Betti cohomology compare natu-rally. The advantage of Betti cohomology is that we can consider integral coefficients, while

´etale cohomology needs coefficients likeZℓwhich are projective limits ofZ/lⁿZ. However a big advantage of ´etale cohomology is that it fits naturally with Galois cohomology. In fact, we have a natural isomorphism

lim→

U⊂X,open

H_etⁱ(U, A) =H_Galⁱ (C(X), A), (1.12) whereAis finite, and the direct limit is over the dense Zariski open sets ofX. The term on the right is the cohomology of the Galois group of the fieldC(X) with coefficients inA,The term on the left is the analogue of what we defined to beHⁱ(C(X), A) in the Betti context.

IfAis finite, then

H_etⁱ (U, A)∼=H_Bⁱ(U, A) henceHⁱ(C(X), A) =H_Galⁱ (C(X), A).

One consequence of Theorem 1.16 is the following formula for unramified cohomology:

this is actually cohomology without residues.

Proposition 1.20. Assuming X smooth over C, one has

H_nrⁱ (X, A) = Ker (Hⁱ(C(X), A)→ ⊕^∂ codimZ=1Hⁱ⁻¹(C(Z), A). (1.13) In particular, the restriction mapH_nrⁱ (X, A)→H_nrⁱ (U, A)is injective for any Zariski dense open setU of X.

Proof. Looking at Definition 1.19, this is a particular case of formula (1.10).

We now get the following important consequence:

Theorem 1.21. Unramified cohomology groupsH_nrⁱ (X, A)are birational invariants of smooth projective varieties.

We should make precise here that we consider complex varieties over C if we want to work with Betti cohomology and any coefficients, and that for more general fields, we use

´etale cohomology and have to restrict coefficients as mentioned below.

Proof of Theorem 1.21. This is an immediate application of Proposition 1.20 and 1.3, be-cause formula (1.13) shows that the natural restriction mapH_nrⁱ (X, A)→H_nrⁱ (U, A) is injec-tive whenU is a dense Zariski open set ofX, and that it is an isomorphism if codim (X\U ⊂ X)≥2. One uses of course the obvious contravariant functoriality of unramified cohomol-ogy.

We refer to Section 2.3.2 for the proof that unramified cohomology is in fact a stable birational invariant. The following example shows that unramified cohomology generalizes Artin-Mumford invariant to higher degree.

Proposition 1.22. Let X be a smooth projective complex variety. Then

H_nr² (X,Q/Z) = TorsH²(Xan,OX^∗an), (1.14) where O^∗Xan is the sheaf of invertible holomorphic functions on Xan, is the Brauer group of X. In particular, if X is rationally connected, H_nr² (X,Q/Z) ∼= TorsH_B³(X,Z) is the Artin-Mumford group ofX.

Proof. Let us show the following precise version of (1.14):

H_nr² (X,Z/nZ) =n−Tors (H²(X_an,O^∗X_an)). (1.15) Consider the exact sequence

0→Z/nZ→ O^∗Xan → O^∗Xan →1,

where the second map is x 7→ xⁿ and Z/nZ is identified with the group of n-th roots of unity. The associated long exact sequence shows that

n−TorsH²(Xan,OX^∗an)∼=H²(Xan,Z/nZ)/Imcln, where

cln :H¹(Xan,O^∗X_an) =H¹(X,OX^∗) = CH¹(X)→H²(Xan,Z/nZ)

is the cycle class modulon. We consider the Bloch-Ogus exact sequence for the sheafZ/nZ onX_an. TheE₂^p,q-terms in degree 2 are, by Theorem 1.17, (i)

E₂^0,2=H⁰(X_Zar,H²(Z/nZ)) =H_nr² (X,Z/nZ), E₂^1,1=H¹(X_Zar,H¹(Z/nZ)).

The last term maps toH²(Xan,Z/nZ) as all the higherdr vanish on it, again by Theorem 1.17, (i). Nodr forr≥2 starts from or arrives toE₂^0,2, by Theorem 1.17, (i) again. Hence E₂^0,2 =E_∞^0,2 is the quotient ofH²(Xan,Z/nZ) by the image ofE₂^1,1. One then proves that this image is Imcln.