REFINED UNRAMIFIED HOMOLOGY OF SCHEMES STEFAN SCHREIEDER Abstract.

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STEFAN SCHREIEDER

Abstract. We introduce refined unramified cohomology of algebraic schemes and show that it interpolates between Borel–Moore homology, algebraic cycles, and classical unramified cohomology. We prove comparison theorems that identify certain cycle groups with refined unramified cohomology groups.

This generalizes previous results of Bloch–Ogus, Colliot-Th´el`ene–Voisin, Kahn, Voisin, and Ma that concerned cycles of low (co-)dimensions on smooth projective varieties.

As an application, we prove that for complex algebraic schemes, Bloch’s coniveau filtration on torsion cycles modulo algebraic equivalence has the property that the graded quotients are determined by higher Abel–Jacobi invariants. This is an analogue of Green’s conjecture [Gre98,Voi99] for torsion cycles modulo algebraic equivalence. It follows for instance that a homologically trivial torsion cycle with trivial transcendental Abel–Jacobi invariant on a smooth complex projective variety has coniveau 1 in the sense of Bloch.

1. Introduction

Unramified cohomology of a smooth variety is classically defined as the subgroup of the cohomology of the generic point given by all classes that have trivial residues at all codimension one points, see [CTO89, CT95]. This definition has roots in the work of Artin–Mumford [AM72] and has ever since had many applications to rationality problems, see e.g. [AM72,CTO89,HPT18, Sch19a,Sch19b].

By the Gersten conjecture for étale cohomology, proven by Bloch–Ogus [BO74], q-th unramified cohomology identifies to the E^0,q₂ -term of the Leray spectral sequence E₂^p,q ⇒H^p+q(X) associated toXét→XZar. Moreover, over algebraically closed fields, E₂^p,p = (CH^p(X)/ ∼alg)⊗H⁰({pt}), which furnished a connection between unramified cohomology and algebraic cycles, showing for instance that the second Griffiths group of a smooth complex variety can be computed by unramified cohomology, see [BO74]. Using in addition the Bloch–Kato conjecture, proven by Voevodsky, this approach allowed Colliot-Thélène–Voisin [CTV12] to compute the failure of the integral Hodge conjecture for codimension 2 cycles in terms of unramified cohomology. A similar result holds for the failure of the integral Tate conjecture for codimension 2 cycles by Kahn [Kah12]. Still relying on the Ger- sten and Bloch–Kato conjectures, Voisin [Voi12] and Ma [Ma17] found a relation to torsion classes in the Griffiths group of codimension 3 cycles.

The known relations between unramified cohomology and algebraic cycles concern only cycles of low (co-)dimensions. In fact, for large p, the E₂^p,p-term is far away from allE₂^0,q-terms in the coniveau spectral sequence, and so it seems hard

Date: December 10, 2021.

2010Mathematics Subject Classification. primary 14C25; secondary 14F20, 14C30.

Key words and phrases. Algebraic Cycles, Integral Hodge Conjecture, Unramified Cohomology.

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to relate those terms. Also, unramified cohomology of a smooth projective variety is a birational invariant, while cycle groups are in general not.

This paper arose from the observation that the relations between unramified cohomology and algebraic cycles from [BO74, CTV12, Kah12, Voi12, Ma17] have more elementary proofs, not relying on the Gersten conjecture [BO74], nor on the Bloch–Kato conjecture proven by Voevodsky. Our proofs naturally lead to the notion of refined unramified cohomology, which generalizes unramified cohomology and gives a new perspective on it. With this notion at hand, our arguments work for cycles of arbitrary codimensions and even on singular schemes. Our approach is robust and works over arbitrary fields. This leads to an interesting dichotomy: we find that refined unramified cohomology ocomputes over algebraically closed fields Chow groups modulo algebraic equivalence, while over finitely generated fields it computes (`-adic) Chow groups on the nose.

As an application, we prove over algebraically closed fields versions of Green’s conjecture [Gre98, Voi99] for torsion cycles modulo algebraic equivalence: there is a finite filtration on the torsion subgroup of the Chow group of algebraic cycles modulo algebraic equivalence such that the graded pieces are determined by higher Abel–Jacobi invariants, see Theorems1.2and8.2.

In [Sch20b], we use the theory developed in this paper to produce the first example of a smooth complex projective variety whose Griffiths group contains infinite 2-torsion, showing in particular that the torsion subgroup of Griffiths groups is in general not finitely generated.

1.1. Refined unramified (co-)homology. LetV be a suitable category of Noe- therian schemes (cf. Definition3.1below), and letA ⊂ModR be a full subcategory of R-modules for some ring R. We fix a twisted Borel–Moore homology theory H_∗^BM(−, A(n)) onV with coefficientsA∈ A that satisfy a few natural properties as outlined in Section3 below. Concrete examples are:

• Borel–Moore homology of the underlying analytic space of algebraic schemes overC;

• `-adic pro-´etale Borel–Moore homology [BS15] on arbitrary algebraic schemes.

We define twisted Borel–Moore cohomology byHⁱ(X, A(n)) :=H_2d^BM

X−i(X, A(dX− n)), wheredX = dim(X). These groups admit restriction maps with respect to open immersionsU ,→X with dim(U) = dim(X). IfX is smooth and equi-dimensional, Poincar´e duality identifiesHⁱ(X, A(n)) in all our examples with ordinary cohomology groups.

For anyX ∈ V, we consider the increasing filtration

F₀X ⊂F₁X⊂ · · · ⊂F_dim_XX =X, where F_jX :={x∈X |codim(x)≤j}, and codim(x) := dimX−dim({x}). EachF_jX may be seen as a pro-object in the category of schemes, but it is in general not a scheme. Nonetheless, we may define As in [BO74], one defines

Hⁱ(F_jX, A(n)) := lim//

F_jX⊂U⊂X

Hⁱ(U, A(n)),

where U ⊂ X runs through all Zariski open subsets containing FjX. Note that the cohomology ofFjX is a natural generalization of the cohomology of the generic point ofX. We then define thej-th refined unramified cohomology groups ofX by

H_j,nrⁱ (X, A(n)) := im(Hⁱ(F_j+1X, A(n))→Hⁱ(F_jX, A(n))).

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Note thatH_j,nrⁱ (X, A(n)) =Hⁱ(X, A(n)) forj ≥dimX.

If X is a smooth variety, then by Lemma 4.5 below we recover traditional unramified cohomology forj= 0:

H_0,nrⁱ (X, A(n)) =H_nrⁱ (X, A(n)).

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1.2. Cycles over the complex numbers. Let nowX be an algebraic scheme of dimensiond_X over C. We denote the underlying analytic space by X_an. For any abelian groupA, let

Hⁱ(X, A(n)) :=H_2d^BM_X_−i(Xan, A), (1.2)

where H_∗^BM denotes Borel–Moore homology. By [Ful98, §19.1], there is a cycle class map

clⁱ_X : CHⁱ(X) //H²ⁱ(X,Z(i)),

where CHⁱ(X) := CHd_X−i(X). We let Griffⁱ(X) := ker(clⁱ_X)/ ∼alg and put Zⁱ(X) := coker(clⁱ_X).

IfX is smooth and equi-dimensional, thenH_2d^BM

X−i(Xan, A)'H_singⁱ (Xan, A) and the above map agrees with the usual cycle class map in singular cohomology.

Theorem 1.1. Let X be a separated scheme of finite type over C and compute refined unramified cohomology via Borel–Moore homology as in (1.2).

(1) There are canonical isomorphisms

Zⁱ(X)_tors'H_i−2,nr²ⁱ⁻¹ (X,Q/Z(i))

H_i−2,nr²ⁱ⁻¹ (X,Q(i)) , Griffⁱ(X)' H_i−2,nr²ⁱ⁻¹ (X,Z(i)) H²ⁱ⁻¹(X,Z(i)) . (2) There is a transcendental Abel–Jacobi map

λⁱ_tr: Griffⁱ(X)tors // H²ⁱ⁻¹(X,Q/Z(i)) Nⁱ⁻¹H²ⁱ⁻¹(X,Q(i)),

where Nⁱ⁻¹H²ⁱ⁻¹(X,Q(i)) = ker(H²ⁱ⁻¹(X,Q(i))→H²ⁱ⁻¹(F_i−2X,Q(i))).

IfX is a smooth projective variety, this agrees with Griffiths’ transcendental Abel–Jacobi map[Gri69]on torsion cycles. Its kernelTⁱ(X) := ker(λⁱ_tr)is canonically isomorphic to

Tⁱ(X)'H_i−3,nr²ⁱ⁻² (X,Q/Z(i))/GⁱH_i−3,nr²ⁱ⁻² (X,Q/Z(i)).

In the above theorem,GⁱH_i−3,nr²ⁱ⁻² (X,Q/Z(i))⊂H_i−3,nr²ⁱ⁻² (X,Q/Z(i)) denotes the subspace generated by classes that admit a liftα∈ H²ⁱ⁻²(F_i−2X,Q/Z(i)) whose imageδ(α)∈H²ⁱ⁻¹(F_i−2X,Z(i)) via the Bockstein map lifts toH²ⁱ⁻¹(X,Z(i)), cf.

Definition6.16and Lemma6.17below.

The above theorem contains the aforementioned results from [BO74, CTV12, Voi12,Ma17] as the special case wherei= 2 in item (1) andi= 3 in (2), and where X is a smooth projective variety. As alluded to above, our proofs of the more general statements above are simpler. In particular, item (1) which generalizes [CTV12] only uses Hilbert 90, but not the Bloch–Kato conjecture in any degree, cf. Remark3.8 below. Moreover, item (2) only uses the Bloch–Kato conjecture in degree 2, proven by Merkurjev–Suslin, while [Voi12,Ma17] needed Bloch–Katon in higher degree.

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The fact that Theorem1.1 treats cycles of arbitrary codimension leads to new results on the integral Hodge conjecture for uniruled varieties that we discuss in Section9.1below.

1.3. Higher Abel–Jacobi mappings and Green’s conjecture for torsion cycles. The Chow group CHⁱ(X) of a smooth complex projective variety X is classically studied through the cycle class map clⁱ_X. The kernel ker(clⁱ_X) is further studied via Griffiths’ Abel–Jacobi mappings AJⁱ_X [Gri69]. While clⁱ_X and AJⁱ_X suffice to describe codimension one cycles, work of Mumford [Mum69] on zero- cycles on surfaces withh^2,0 >0 shows that this fails (heavily) already fori = 2.

Moreover, Nori [No93] showed that even up to algebraic equivalence, Chow groups are in general not fully described by clⁱ_X and AJⁱ_X; not even up to torsion.

To overcome this problem, Green conjectured that the rational Chow groups CHⁱ(X)_Q of smooth complex projective varieties should carry a finite decreasing filtrationF^∗CHⁱ(X)_QwithF⁰= CHⁱ(X)_Q,F¹= ker(clⁱ_X)_Q,F²= ker(AJⁱ_X)_Q, and Fⁱ⁺¹= 0, such that there are higher Abel–Jacobi mappingsψⁱ_jonF^jCHⁱ(X)_Qwith

ker(ψ_jⁱ) =F^j+1CHⁱ(X)_Q. (1.3)

It is expected that the filtration F^∗ should agree with the conjectural Bloch–

Beilinson filtration [Jan88, Conjecture 11.1]. Green proposed explicit candidates forψ_jⁱin [Gre98], but Voisin showed [Voi99] that Green’s maps do not have the envi- sioned property (1.3). It remained open whether a refinement of Green’s invariants satisfies Green’s conjecture.

In analogy with Grothendieck’s coniveau filtration on cohomology [Gro68], Bloch [Blo85] introduced an interesting coniveau filtration on Chow groups. His filtration yields a natural coniveau filtrationN^∗ on Griffiths groups:

N^jGriffⁱ(X) := im

lim//Griff^i−j(Z) // Griffⁱ(X) ,

whereZ⊂X runs through the reduced closed subschemes with dim(X)−dim(Z) = j. Since Griff¹(Z) = 0 by Theorem1.1, Nⁱ⁻¹Griffⁱ(X) = 0 and so we obtain a filtration of the form

0 =Nⁱ⁻¹⊂Nⁱ⁻²⊂ · · · ⊂N¹⊂N⁰= Griffⁱ(X).

This is two steps shorter than the (conjectural) filtration F^∗ on CHⁱ(X)_Q above:

we loose one step because Griffⁱ(X)⊂CHⁱ(X)/ ∼_alg is already the kernel of the cycle class map and another step by quotiening out algebraic equivalence.

For any separated scheme of finite type overC, Theorem1.1yields a map λⁱ_tr: Griffⁱ(X)tors //Jⁱ_tr(X)tors:= H²ⁱ⁻¹(X,Q/Z(i))

N¹H²ⁱ⁻¹(X,Q(i)), (1.4)

becauseNⁱ⁻¹H²ⁱ⁻¹(X,Q(i))⊂N¹H²ⁱ⁻¹(X,Q(i)). If X is smooth projective, this map factorizes through Griffiths’ transcendental Abel–Jacobi map [Gri69].

We aim to construct higher versions of the above Abel–Jacobi mapping. To this end, note that the natural pushforward map induces an isomorphism

ι_∗: lim//

Z⊂X

Griff^i−j(Z)_tors ^∼ // N^jGriffⁱ(X)_tors, (1.5)

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whereZ⊂X runs through the reduced closed subschemes with dim(X)−dim(Z) = j, see Lemma6.27. Applying the map (1.4) to the subschemesZ ⊂X, we thus get in the limit a higher transcendental Abel–Jacobi map on torsion cycles

λⁱ_j,tr:N^jGriffⁱ(X)tors //Jⁱ_j,tr(X)tors:= lim//

Z⊂X

J^i−j_tr (Z)tors, (1.6)

whereZ⊂X runs through the reduced closed subschemes with dim(X)−dim(Z) = j. (At this step it is crucial that Theorem1.1works for singular schemes, because the subschemesZ ⊂X will in the limit never be smooth.)

As mentioned above, item (1) in Theorem1.1 does not rely on the Bloch–Kato conjecture and item (2) relies (for cycles of arbitrary codimension) only on the Bloch–Kato conjecture in degree 2, i.e. Merkurjev–Suslin’s theorem. It is thus natural to wonder if one could get stronger results if one added the Bloch–Kato conjecture in arbitrary degree. This question was posed to us by Claire Voisin.

We propose the following answer, which together with the versions over arbitrary algebraically closed fields (see Theorem 8.2below) is the main result of this paper and the only place where the full strength of Voevodsky’s work [Voe11] is used.

Theorem 1.2. Let X be a separated scheme of finite type overC. Then for any i andj,

N^j+1Griffⁱ(X)_tors= ker

λⁱ_j,tr:N^jGriffⁱ(X)_tors //Jⁱ_j,tr(X)_tors .

The above theorem establishes Green’s conjecture for torsion cycles modulo algebraic equivalence, cf. [Gre98,Voi99]; a slightly different version whereλⁱ_tris replaced byλⁱ_tr is contained in Theorem8.5below.

Note thatJⁱ_0,tr(X)tors=Jⁱ_tr(X)tors and soN¹Griffⁱ(X)tors= ker(λⁱ_tr).As a direct consequence, we thus obtain the following, which may be seen as generalization of the injectivity of the transcendental Abel–Jacobi map on codimension 2 cycles, proven by Merkurjev–Suslin [MS83,§18].

Corollary 1.3. LetX be a smooth complex projective variety and letz∈Griffⁱ(X) be a torsion cycle with trivial transcendental Abel–Jacobi invariant. Then z has coniveau 1 in the sense of Bloch, i.e. there is a codimension one subschemeZ ⊂X such that z is the pushforward of a homologically trivial torsion cycle onZ.

Examples of torsion-cycles with trivial transcendental Abel–Jacobi invariants have been constructed in [Tot97, SV05,Sch20b]. The geometric consequence that follows form the above corollary is non-trivial in each of these examples.

In [Sch20b], we show that the 2-torsion subgroup of Griffⁱ(X) may be infinite fori≥3. Since for anyn≥1, then-torsion subgroup of J^i−j_tr (Z)tors from (1.4) is finite for any closed subscheme Z ⊂ X, we see that in (1.5) and (1.6) the direct limit is necessary and we can in general not find a single subscheme Z ⊂X that would suffice our purposes.

We will prove in this paper that im λⁱ_j,tr

= 0 wheneverj <2i−1−dimX. Theorem1.2has therefore the following consequence.

Corollary 1.4. Let X be a separated scheme of finite type overC. Then

Griffⁱ(X)_tors=N¹Griffⁱ(X)_tors=· · ·=N^jGriffⁱ(X)_tors for allj≤2i−1−dimX.

For instance, the above corollary says that any torsion curve class in the Griffiths group ofX is the pushforward of a torsion class on some threefold contained inX.

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1.4. Cycles over arbitrary fields. Let k be a field and let πX : X → Speck be a separatedk-scheme of finite type and dimensiond. LetXpro´etdenote Bhatt–

Scholze’s pro-étale site of X, see Section 5.1 below. The `-adic generalization of (1.2) is given byHⁱ(X, µ^⊗n_`r ) :=Hî−2d(Xproét, π_X^! µ^⊗n−d_`r ),

Hⁱ(X,Z`(n)) :=H^i−2d(Xpro´et, π^!_XZb`(n−d)), (1.7)

Hⁱ(X,Q`/Z`(n)) := colimrHⁱ(X, µ^⊗n_`r ) andHⁱ(X,Q`(n)) :=Hⁱ(X,Z`(n))⊗_Z_`Q`. If X is smooth and equi-dimensional, these groups coincide with the continuous

´

etale cohomology groups of Jannsen [Jan88].

We will see that there is an`-adic cycle class map clⁱ_X : CHⁱ(X)⊗_ZZ` //H²ⁱ(X,Z`(i)) (1.8)

that coincides with Jannsen’s cycle class map in continuous ´etale cohomology (see [Jan88, Lemma 6.14]) ifX is a smooth variety over k. The kernel of clⁱ_X modulo algebraic equivalence is denote by Griffⁱ(X)_Z_`. Our main result is as follows.

Theorem 1.5. Let k be a field and let ` be a prime invertible in k. Let X be a separated scheme of finite type overkand compute (refined unramified) cohomology via (1.7). Then the cycle class map (1.8) has the following properties:

(1) There are canonical isomorphismscoker(clⁱ_X)'H_i−1,nr²ⁱ (X,Z`(i))and Tors(coker(clⁱ_X))'H_i−2,nr²ⁱ⁻¹ (X,Q`/Z`(i))/H_i−2,nr²ⁱ⁻¹ (X,Q`(i)).

(2) Ifkis a purely inseparable extension of a finitely generated field, then ker(clⁱ_X)'H_i−2,nr²ⁱ⁻¹ (X,Z`(i))/H²ⁱ⁻¹(X,Z`(i))

and there is an Abel–Jacobi map

λⁱ_tr : Tors(ker(clⁱ_X)) //H²ⁱ⁻¹(X,Q`/Z`(i))/Nⁱ⁻¹H²ⁱ⁻¹(X,Q`(i)), with kernelker(λⁱ_tr)'H_i−3,nr²ⁱ⁻² (X,Q`/Z`(i))/GⁱH_i−3,nr²ⁱ⁻² (X,Q`/Z`(i)).

(3) Ifkis algebraically closed, then

Griffⁱ(X)_Z_` 'H_i−2,nr²ⁱ⁻¹ (X,Z`(i))/H²ⁱ⁻¹(X,Z`(i)) and there is a transcendental Abel–Jacobi map on torsion-classes λⁱ_tr: Tors(Griffⁱ(X)_Z_`) //H²ⁱ⁻¹(X,Q`/Z`(i))/Nⁱ⁻¹H²ⁱ⁻¹(X,Q`(i)).

If X is a smooth projective variety, this map is induced by Bloch’s Abel–

Jacobi map on torsion cycles[Blo10]. Its kernel is isomorphic to ker(λⁱ_tr)'H_i−3,nr²ⁱ⁻² (X,Q`/Z`(i))/GⁱH_i−3,nr²ⁱ⁻² (X,Q`/Z`(i)).

The filtrations N^∗ and G^∗ in the above theorem are defined similarly as in Theorem1.1.

The proof of the above result uses Bhatt–Scholze’s pro-étale theory [BS15] and the computation of ker(λⁱ_tr) uses as before Merkurjev–Suslin’s theorem. A key ingredient for item (2) is Néron’s Mordell–Weil theorem [Nér52].

The second isomorphism in item (1) generalizes a result of Kahn [Kah12] who proved it fori= 2 andX smooth projective. Item (3) shows that over algebraically closed fields, refined unramified cohomology computes`-adic Chow groups modulo algebraic equivalence, which is in complete analogy with Theorem1.1. In contrast, items (1) and (2) show that over finitely generated fields, refined unramified cohomology computes`-adic Chow groups on the nose. This furnishes a useful tool to

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study Chow groups via cohomological methods, because any identity in the Chow group holds over some finitely generated field.

Remark 1.6. Beilinson conjectured that the cycle class map in (1.8) has torsion kernel ifX is smooth projective andkis finitely generated, cf.[Jan90,§11.6].

This implies the Bloch–Beilinson conjecture, see[Jan94, Lemma 2.7]. By Theorem 1.5, Beilinson’s conjecture translates to H_i−2,nr²ⁱ⁻¹ (X,Q`(i))/H²ⁱ⁻¹(X,Q`(i)) = 0.

In the special case i = 2, the conjecture is thus equivalent to the surjectivity of H³(X,Q`(2))→H_nr³ (X,Q`(2))for smooth projective varieties over finitely generated fields. The latter would imply the Bloch–Beilinson conjecture for surfaces and hence Bloch’spg-conjecture [Voi03, Conjecture 11.2]for smooth complex projective surfaces.

1.5. Relation to higher unramified cohomology. We explain in this section the connection of refined unramified cohomology to classical higher unramified cohomology of Bloch–Ogus, given as the higher Zariski cohomology groups of the sheaves Hⁱ studied in [BO74]. We explain our result over the complex numbers, but it holds more generally whenever the Bloch–Ogus resolution from [BO74] holds true, see Proposition10.1below.

Proposition 1.7. LetX be a smooth complex variety. Then for anyi andn, and any abelian groupA, there is a canonical long exact sequence

. . . H_j−1,nr^i+j−1(X, A(n)) //H_j−2,nr^i+j−1(X, A(n)) //H^j(X_Zar,Hⁱ_X(A(n)))

//H^i+j_j,nr(X, A(n)) //H^i+j_j−1,nr(X, A(n)). . . , whereHⁱ_X(A(n))denotes the Zariski sheaf onX, associated toU //H_singⁱ (Uan, A(n)).

The above result shows that up to some extension data, refined unramified cohomology determines all Zariski cohomology groups H_Zar^j (X,Hⁱ_X(A(n))). As a special case, we get:

Corollary 1.8. LetX be a smooth complex variety of dimensiond. For anyi≥0, there is a canonical isomorphismH_i,nr^d+i(X, A(n))'Hⁱ(X,H^d_X(A(n))).

The group that appears in the above corollary is the only higher unramified cohomology group for which geometric interpretations in terms of algebraic cycles were known before, see [CTV12, Ma17]. Those known geometric interpretations are implied by Theorem1.1.

1.6. Homology or cohomology? Recall from Section1.1 that the theory of refined unramified cohomology relies on a twisted Borel–Moore homology theory H_∗^BM(−, A(n)) with corresponding Borel–Moore cohomology theoryHⁱ(X, A(n)) :=

H_2d^BM

X−i(X, A(dX−n)). If X is smooth and equi-dimensional, then Hⁱ(X, A(n)) agrees with ordinary cohomology, but in general it is different. Since everything is based on Borel–Moore homology, we think about the theory developed in this paper as refined unramified homology, hence the title of the paper.

The reason we wrote this paper nonetheless cohomologically is twofold: On the one hand, this yields formulas that are closer to those in [CTV12, Kah12, Voi12, Ma17], which motivate this paper. On the other hand, and more importantly, the application of the theory in [Sch20b,Sch21] concern smooth projective varieties and use the identification of Borel–Moore cohomology with ordinary cohomology. This allows one to make use of cup products, which will be crucial (and which requires

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a cohomological formulation). Writing this paper homologically would thus make it significantly harder to read those applications.

After all it is a matter of formal manipulations to rewrite this paper homologically, but note that it will not be enough to just useHⁱ(X, A(n)) =H_2d^BM

X−i(X, A(d_X− n)), one should also change the indices in the filtration F_∗X to make the indices in the resulting formulas in Theorems 1.1 and 1.5 appealing. Unfortunately, the translation between the homological and the cohomological notation is tedious, so that we restrict ourselves to only one version here.

2. Notation

A field is said to be finitely generated, if it is finitely generated over its prime field.

For a Noetherian schemeX, we write X_(i) for the set of all points x∈X with dim({x}) = i. We then define X⁽ⁱ⁾ := X_(d_X_−i), where dX = dimX. That is, x∈X lies inX⁽ⁱ⁾ if and only if dimX−dim({x}) =i. Note that this is slightly non-standard, as it does not imply that the codimension ofxdefined locally inX is i, but it has the advantage that the Chow group CHⁱ(X) := CHd_X−i(X) (see [Ful98]) is the quotient ofL

x∈X⁽ⁱ⁾[x]Zby rational equivalence, where [x]Zdenotes the freeZ-module with generator [x].

An algebraic scheme is a separated scheme of finite type over a field. A variety is an integral algebraic scheme.

WheneverGandH are abelian groups (orR-modules for some ring R) so that there is a canonical mapH →G(and there is no reason to confuse this map with a different map), we writeG/H as a short hand for coker(H →G).

For an abelian groupG, we denote byG[`^r] the subgroup of`^r-torsion elements, and by G[`^∞] := S

rG[`^r] the subgroup of elements that are `^r-torsion for some r≥1. We further write Tors(G) orG_tors for the torsion subgroup ofG.

3. `-adic Borel–Moore cohomology theories

Definition 3.1. LetV be a category of Noetherian schemes such that the morphisms are given by open immersionsU ,→X of schemes withdim(U) = dim(X). We call V constructible, if for anyX ∈ V, the following holds:

• ifY ,→X is an open or closed immersion, then Y ∈ V;

• ifX ∈ V is reduced, then the normalization ofX is also inV.

Definition 3.2. Let V be a constructible category of Noetherian schemes as in Definition3.1. LetRbe a ring and letA ⊂ModRbe a full subcategory ofR-modules withR∈ A. A twisted Borel–Moore cohomology theory onV with coefficients inA is a family of contravariant functors

V // ModR, X //Hⁱ(X, A(n)) with i, n∈ZandA∈ A (3.1)

that are covariant in Aand such that the following holds:

P1 ForX, Y ∈ V and any proper morphismf :X →Y of schemes of relative codimension c= dimY −dimX, there are functorial pushforward maps

f_∗:H^i−2c(X, A(n−c)) //Hⁱ(Y, A(n)), compatible with pullbacks along morphisms inV.

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P2 For any pair (X, Z)of schemes in V with a closed immersion Z ,→X of codimensionc= dim(X)−dim(Z)and with complement U withdim(X) = dim(U), there is a Gysin exact sequence

. . . //Hⁱ(X, A(n)) //Hⁱ(U, A(n)) ^∂ // H^i+1−2c(Z, A(n−c)) ^ι^∗ // Hⁱ⁺¹(X, A(n)) //. . . where∂is called residue map andι_∗is induced by proper pushforward from

(P1). The Gysin sequence is functorial with respect to pullbacks along open immersionsV ,→X withdimV = dim(V∩X) = dimX anddim(V∩Z) = dimZ. Iff :X⁰ →X is proper withZ⁰ =f⁻¹(Z)anddimX⁰ = dim(X⁰\ Z⁰), then the proper pushforward along f induces an exact ladder between the Gysin sequence of(X⁰, Z⁰)and(X, Z), respectively.

P3 For any X∈ V andx∈X, the groups Hⁱ(x, A(n)) := lim//

∅6=Vx⊂{x}

Hⁱ(Vx, A(n)), (3.2)

whereV_xruns through all open dense subsets of{x} ⊂X, satisfyHⁱ(x, A(n)) = 0 fori < 0. Moreover, there are isomorphisms H⁰(x, A(0))'A that are functorial inA, and forA=Rthere is distinguished class[x]∈H⁰(x, R(0)) (called fundamental class) such thatH⁰(x, R(0)) = [x]Ris generated by[x].

Remark 3.3. We warn the reader that even if Specκ(x)∈ V, the cohomology of a point x∈X in (3.2) may not agree withHⁱ(Specκ(x), A(n)).

Remark 3.4. The main example of a Borel–Moore cohomology theory is given by Hⁱ(X, A(n)) :=H_2d^BM_X_−i(X, A(dX−n)) with dX = dim(X),

where H_∗^BM denotes a suitable Borel–Moore homology theory. In all applications we have in mind, Poincar´e duality ensures that Borel–Moore cohomology as defined above identifies to ordinary cohomology groups ifX is equi-dimensional and smooth over a field.

One could use the above formula to rewrite the whole theory homologically, which would be more accurate at many (but not all) places; see Section 1.6for an expla- nation why we nonetheless prefer to write this paper cohomologically.

In what follows, we will usually writeµ^⊗n_`r :=Z/`^r(n).

Definition 3.5. Let V be a constructible category of Noetherian schemes as in Defition 3.1. Let ` be a prime and letR=Z`. A twisted Borel–Moore cohomology theory on V with coefficients inA ⊂Mod_Z_` as in Definition3.2is called `-adic, if Z`,Q`, Q`/Z`, and Z/`^r for allr≥1 are contained in A, such that the following holds:

P4 Functoriality in the coefficients induces isomorphisms of functors lim//

r

Hⁱ(−, µ^⊗n_`r ) ^∼ // Hⁱ(−,Q`/Z`(n)) and Hⁱ(−,Z`(n))⊗_Z_`Q` ∼ // Hⁱ(−,Q`(n)).

P5 For any X∈ V, there is a long exact Bockstein sequence

. . . //Hⁱ(X,Z`(n)) ^×`^r// Hⁱ(X,Z`(n)) //Hⁱ(X, µ^⊗n_`r ) ^δ //Hⁱ⁺¹(X,Z`(n)) ^×`^r// . . . where Hⁱ(X,Z`(n))→ Hⁱ(X, µ^⊗n_`r ) is given by functoriality in the coefficients and whereδ is called the Bockstein map. This sequence is functorial with respect to proper pushforwards and pullbacks along morphisms inV.

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P6 For any X ∈ V and x ∈ X, there is a map : κ(x)^∗ → H¹(x,Z`(1)) such that Hilbert 90 holds in the sense that the induced map : κ(x)^∗ → H¹(x, µ^⊗1_`r ) is surjective. Moreover, for X ∈ V integral with generic point η, there is a unit u ∈ Z` such that for any regular point x ∈ X⁽¹⁾, the natural composition

κ(η)^∗ // H¹(η,Z`(1)) ^∂ // H⁰(x,Z`(0)) = [x]Z`,

where∂ is induced by (P2) and the last equality comes from (P3), mapsf to[x](u·ν_x(f)), whereν_x denotes the valuation onκ(η)induced byx.

P7 For any X∈ V andx∈X, there is a map(κ(x)^∗)^⊗2→H²(x,Z`(2))such that the induced map (κ(x)^∗)^⊗2 →H²(x, µ^⊗2_`r ) is surjective, i.e. a version of Merkurjev–Suslin’s theorem holds.

LetX ∈ V be integral and let U ⊂X be a big open subset, i.e. dim(X\U)<

dimX −1. Then H²(U,Z`(1))'H²(X,Z`(1)) by (P2) and (P3) (see Corollary 4.7below). Taking the direct limit over allU and usingH⁰(x,Z`(0)) = [x]Z` from (P3), we find that the proper pushforwards from (P1) induce a cycle class map

ι_∗: M

x∈X⁽¹⁾

[x]Z` //H²(X,Z`(1)).

(3.3)

The following two options are of particular interest to us:

P8.1 IfX is integral and regular, the kernel of (3.3) is given byZ`-linear combi- nations of algebraically trivial divisors.

P8.2 IfX is integral and regular, the kernel of (3.3) is given byZ`-linear combi- nations of principal divisors.

Definition 3.6. LetV be a constructible category of Noetherian schemes, see Def- inition 3.1. An `-adic twisted Borel–Moore cohomology theory H^∗(−, A(n))on V as in Definition 3.5is adapted to algebraic equivalence, if (P8.1) holds, and it is adapted to rational equivalence, if (P8.2) holds.

In addition to `-adic theories, we will also need the following integral variant.

To this end, we perform in each of the statements (P4)–(P7), (P8.1), and (P8.2) the formal replacement of symbols:

Z` Z, Q` Q, `^r r,

and denote the corresponding statements by (P4’)–(P7’), (P8.1’), and (P8.2’), respectively.

Definition 3.7. Let V be a constructible category of Noetherian schemes (see De- fition 3.1) and let R = Z. A twisted Borel–Moore cohomology theory on V with coefficients in A ⊂Mod_Z as in Definition3.2is called integral, if Z, Q,Q/Z, and Z/rfor allr≥1are contained inA, such that items (P4’)–(P7’) hold. The theory is adapted to algebraic (resp. rational) equivalence, if item (P8.1’) (resp. (P8.2’)) holds true.

Remark 3.8. Item (P7) correpsonds to the Bloch–Kato conjecture in degree 2, proven by Merkurjev–Suslin. This property will only be used to ensure thatH³(x,Z`(2)) is torsion free, see Lemma 4.10 below. That result in turn will, as indicated in the introduction, only be used to compute the kernel of the transcendental Abel–Jacobi mapping. For instance, the result will not be used in Theorem6.5, which describes the failure of the integral Hodge or Tate conjecture for cycles of arbitrary degree.

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4. Definition of refined unramified cohomology and simple consequences

In this section, we fix a constructible category V of Noetherian schemes as in Definition 3.1. We further fix a ring R and a twisted Borel–Moore cohomology theoryH^∗(−, A(n)) onV with coefficients in a full subcategory A ⊂Mod_R, as in Definition3.2.

For X ∈ V we write F_jX := {x ∈ X | codim(x) ≤ j}, where codim(x) :=

dim(X)−dim({x}). We then define

Hⁱ(FjX, A(n)) := lim//

FjX⊂U⊂X

Hⁱ(U, A(n)),

where the direct limit runs through all open subschemes U ⊂ X with F_jX ⊂U. Since the direct limit functor is exact, many of the properties ofH^∗(X, A(n)) remain true forFjX in place ofX.

Form≥j, classes onFmX may be restricted toFjX, giving rise to a decreasing filtrationF^∗ onHⁱ(F_jX, A(n)), which for m≥j is given by:

F^mHⁱ(F_jX, A(n)) := im Hⁱ(F_mX, A(n)) //Hⁱ(F_jX, A(n)) . (4.1)

The key definition is as follows.

Definition 4.1. The j-th refined unramified cohomology of X ∈ V with respect to a twisted Borel–Moore cohomology theoryH^∗(−, A(n))on V with coefficients in a full subcategory A ⊂M od_R, is given by

H_j,nrⁱ (X, A(n)) :=F^j+1Hⁱ(FjX, A(n)) = im Hⁱ(Fj+1X, A(n)) //Hⁱ(FjX, A(n)) . Following Grothendieck, the coniveau filtration onHⁱ(X, A(n)) is defined by

N^jHⁱ(X, A(n)) := ker(Hⁱ(X, A(n))→Hⁱ(Fj−1X, A(n))).

There is a similar coniveau filtration on refined unramified cohomology, defined as follows.

Definition 4.2. Let X ∈ V and fix a twisted Borel–Moore cohomology theory H^∗(−, A(n))on V with coefficients in a full subcategoryA ⊂M odR. The coniveau filtrationN^∗onHⁱ(FjX, A(n))andH_j,nrⁱ (X, A(n))is the decreasing filtration which forh≤j+ 1 is defined by

N^hHⁱ(FjX, A(n)) := ker Hⁱ(FjX, A(n))→Hⁱ(F_h−1X, A(n))

, and N^hH_j,nrⁱ (X, A(n)) := ker H_j,nrⁱ (X, A(n))→Hⁱ(F_h−1X, A(n))

. 4.1. Consequence of the Gysin sequence.

Lemma 4.3. Let X tY ∈ V with dimX = dimY and let A ∈ A. Then the canonical map given by pullback is an isomorphism:

Hⁱ(XtY, A(n)) ^' // Hⁱ(X, A(n))⊕Hⁱ(Y, A(n)).

Proof. LetiX (resp.iY) denote the inclusions of X (resp.Y) into XtY. By the Gysin sequence (P2), we have an exact sequence

Hⁱ(X, A(n)) ⁱ^X∗// Hⁱ(XtY, A(n)) ⁱ

∗

Y // Hⁱ(Y, A(n)).

Functoriality of this sequence with respect to proper pushforward and pullbacks along morphisms inV shows that i_X∗ andi^∗_Y admit splittings. Hence, the above sequence is part of a short exact sequence that splits, which proves the lemma.

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Lemma 4.4. LetX ∈ V andA∈ A. Then for anyn∈Zandm, j≥0, the Gysin sequence in (P2) induces a long exact sequence

. . . //Hⁱ(Fj+mX, A(n)) //Hⁱ(F_j−1X, A(n)) ^∂ // lim//

Z⊂X codim(Z)=j

H^i+1−2j(FmZ, A(n−j)) ^ι^∗ // . . . ,

where the direct limit runs through all closed reduced subschemes Z ⊂X of codimension codim(Z) = dimX−dimZ =j.

Proof. This follows immediately from (P2) by taking direct limits twice. We explain the details for convenience of the reader. Let Z ⊂ X be closed with dimZ = dimX −j and let U = X \Z. Let further W ⊂ Z be closed of dimension dimW = dimZ−m−1 = dimX−j−m−1. By (P2), we get an exact sequence

. . . //Hⁱ(X\W, A(n)) //Hⁱ(U, A(n)) ^∂ // H^i+1−2j(Z\W, A(n−j)) ^ι^∗ // . . . . Note thatZ is not necessarily equi-dimensional and so the inverse limit overU = X\Z where Z runs through all subschemes Z ⊂X of codimension codim(Z) = dimX−dimZ =jagrees withF_j−1X. Taking the direct limit overZ of the above sequence, we thus find

. . . //Hⁱ(X\W, A(n)) //Hⁱ(Fj−1X, A(n)) ^∂ // lim//

Z⊂X codim(Z)=j

H^i+1−2j(Z\W, A(n−j)) ^ι^∗ // . . . .

Taking now direct limits over W, and using that direct limits commute with each other, we arrive at an exact sequence

. . . //Hⁱ(Fj+mX, A(n)) //Hⁱ(F_j−1X, A(n)) ^∂ // lim//

Z⊂X codim(Z)=j

H^i+1−2j(FmZ, A(n−j)) ^ι^∗ // . . . ,

as claimed. This proves the lemma.

Lemma 4.5. Let X ∈ V andA∈ A. Then for anyj, n∈Z, (P2) induces a long exact sequence

. . . //Hⁱ(FjX, A(n)) //Hⁱ(Fj−1X, A(n)) ^∂ // M

x∈X^(j)

H^i+1−2j(x, A(n−j)) ^ι^∗ // Hⁱ⁺¹(FjX, A(n)) //. . . , whereι∗(resp.∂) is induced by the pushforward (resp. residue) map from the Gysin

exact sequence (P2).

Proof. Using additivity from Lemma4.3, this identifies to the special case m= 0

in Lemma4.4.

Corollary 4.6. Let X ∈ V. Then for any n ∈ Z and j, m ≥ 0, the following canonical sequence is exact

lim//

Z⊂X codim(Z)=j

H_m,nr^i−2j(Z, A(n−j)) ^ι^∗ // H_j+m,nrⁱ (X, A(n)) //H_j−1,nrⁱ (X, A(n)),

where the direct limit runs through all closed reduced subschemes Z ⊂X of codimension codim(Z) = dimX−dimZ =j.

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Proof. The composition of the two arrows in the corollary is zero by Lemma 4.4.

Conversely, assume thatα∈H_j+m,nrⁱ (X, A(n)) maps to zero inH_j−1,nrⁱ (X, A(n)).

By Lemma 4.4, α=ι_∗ξ for someξ ∈H^i−2j(F_mZ, A(n−j)) and some Z ⊂X of codimensionj. Sinceαis unramified, Lemma4.5shows that

ι_∗(∂ξ) =∂(ι_∗ξ) = 0∈ M

x∈X^(j+m+1)

H^{i−2j−2m−1}(x, A(n−j−m−1)), (4.2)

where the first equality uses that the Gysin sequence is functorial with respect to proper pushforwards (see (P2)), so that ι∗ and∂ commute. But this implies that the class

∂ξ∈ M

x∈Z^(m+1)

H^{i−2j−2m−1}(x, A(n−j−m−1))

vanishes, as the above right hand side is a subgroup of the right hand side of (4.2), andι_∗identifies to the inclusion. Hence, Lemma4.5impliesξ∈H_m,nr^i−2j(Z, A(n−j)), as we want. This concludes the proof of the corollary.

Corollary 4.7. Let X ∈ V and A∈ A. ThenHⁱ(F_jX, A(n))'Hⁱ(X, A(n))for allj ≥ di/2e.

Proof. SinceHⁱ(x, A(n)) vanishes for i <0 by (P3), Lemma4.5implies Hⁱ(FjX, A(n))'Hⁱ(F_j−1X, A(n))

for all j with j > di/2e. This proves the corollary by induction on j, because Hⁱ(F_jX, A(n)) =Hⁱ(X, A(n)) forj ≥dim(X).

Corollary 4.8. LetX ∈ V andA∈ A. Assume that there is a non-negative integer c, such that for any X ∈ V and x ∈X_(j), Hⁱ(x, A(n)) = 0 for i > j+c. Then Hⁱ(F_jX, A(n)) = 0for alli >dimX+j+c.

Proof. Our assumption implies by Lemma4.5thatHⁱ(FjX, A(n))'Hⁱ(Fj−1X, A(n)) for all j with i > j+ dimX+c. Hence, Hⁱ(FjX, A(n))'Hⁱ(F0X, A(n)) for all j with i > j+ dimX+c. But Hⁱ(F0X, A(n)) = 0 for alli > dimX by Lemma 4.3 and our assumption, because F0X is the union of the generic points of the maximal-dimensional components ofX. This proves the corollary.

Lemma 4.9. Let X ∈ V andA∈ A. Letw∈X^(p−1) with closureW ⊂X and let τ : ˜W →W be the normalization with generic point ηW˜ ∈W˜. Then the following diagram commutes for all integers iandn

Hⁱ(w, A(n)) =Hⁱ(ηW˜, A(n)) ^∂ //

_

L

w∈f˜ W⁽¹⁾Hⁱ⁻¹( ˜w, A(n−1))

τ∗

L

x∈X^(p−1)Hⁱ(x, A(n)) ^∂◦ι^∗ //L

x∈X^(p)Hⁱ⁻¹(x, A(n−1)),

where the vertical arrow on the left is the natural inclusion, the vertical arrow on the right is induced by the proper pushforward maps from (P1), the upper horizontal arrow is induced by the residue map in (P2) and the lower horizontal arrow is given by

M

x∈X^(p−1)

Hⁱ(x, A(n)) ^ι^∗ // H^i+2p−2(Fp−1X, A(n+p)) ^∂ // M

x∈X^(p)

Hⁱ⁻¹(x, A(n−1)), whereι_∗ resp.∂ is the pushforward resp. residue map induced by (P2).

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Proof. Note thatW ∈ V and henceWf∈ V, cf. Definition3.1. The lemma is thus a direct consequence of the functoriality of the Gysin sequence (P2) with respect

to proper pushforwards (P1), as required in (P2).

4.2. Torsion-freeness of the cohomology of points. In this section we fix a prime`and assume that the twisted Borel–Moore cohomology theoryH^∗(−, A(n)) on V is `-adic, see Definition 3.5. It is an observation of Bloch (which partly motivated the Bloch–Kato conjecture, see [Blo10, end of Lecture 5]) that properties (P5)–(P7) have the following important consequence.

Lemma 4.10. LetV be a constructible category of Noetherian schemes as in Defi- nition3.1. Fix a prime`and assume thatH^∗(−, A(n))is an`-adic twisted Borel–

Moore cohomology theory on V as in Definition 3.5. Then for any X ∈ V and x∈X,Hⁱ(x,Z`(i−1))is torsion-free for 1≤i≤3.

Proof. Taking direct limits of abelian groups is exact, so that property (P5) implies that

Hⁱ(x,Z`(i−1))[`^r]'coker(Hⁱ⁻¹(x,Z`(i−1)) //Hⁱ⁻¹(x, µ^⊗i−1_`r )).

This vanishes fori= 1, as in this case we have by (P5) an exact sequence H⁰(x,Z`(0)) ^×`^r// H⁰(x,Z`(0)) //H⁰(x, µ^⊗0_`r )

which by (P3) identifies toZ`

×`^r

→ Z`→Z/`and so the last arrow is surjective.

By (P6), there is a surjection : κ(x)^∗ ////H¹(x, µ^⊗1_`r ) which factors through H¹(x,Z`(1)) and so the above cokernel also vanishes fori= 2. By (P7), the same argument yields vanishing of the above cokernel for i = 3. This concludes the

proof.

Remark 4.11. We emphasize that (P7) (i.e. Merkurjev–Suslin’s theorem) is only needed for torsion-freeness of H³(x,Z`(2)).

Remark 4.12. The above proof shows more generally thatHⁱ⁺¹(x,Z`(i))is torsion- free if there are surjections(κ(x)^∗)^⊗i ////Hⁱ(x, µ^⊗i_`r)that factor throughHⁱ(x,Z`(i)).

In particular, Hⁱ⁺¹(x,Z`(i))is torsion-free if a version of the Bloch–Kato conjecture holds in degree i in the sense that there is map K_i^M(κ(x)) → Hⁱ(x,Z`(i)) which induces isomorphisms K_i^M(κ(x))/`^r ' Hⁱ(x, µ^⊗i_`r). It folows from Voevod- sky’s proof of the Bloch–Kato conjecture [Voe11] that the theories that we discuss in Proposition 5.6and5.9below have this property.

5. Examples

In this section we discuss some examples of cohomology functors that satisfy the properties from Section3.

5.1. `-adic Borel–Moore pro-´etale cohomology.

5.1.1. Continuous étale cohomology of Jannsen. LetX be a scheme over a field k and let Ab(Xét)^Nbe the abelian category of inverse systems of abelian étale sheaves on the small étale siteX_´_et ofX. This category has enough injectives (see [Jan88]) and we may consider the functor

lim◦Γ : Ab(X´et)^N // Ab, (Fr) //lim

←−r

Γ(X, Fr).

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Jannsen then defines the continuous ´etale cohomology groups H_contⁱ (X,(F_r)) := Rⁱ(lim◦Γ)((F_r)).

These groups are closely related to the corresponding ´etale cohomology groups via the following canonical short exact sequence (see [Jan88,§1.6]):

0 //R¹lim Hⁱ⁻¹(Xét, Fr) //H_contⁱ (Xét,(Fr)) //limHⁱ(Xét, Fr) //0, (5.1)

where lim denotes the inverse limit functor overr.

By [Jan88, (3.27)], we have the following Kummer exact sequence in Ab(X´et)^N: 0 //(µ^⊗n_`r )r //(Gm,×`)r //(Gm,id)r //0,

(5.2)

where ` is a prime invertible in k. Taking cohomology, the boundary map of the corresponding long exact sequence yields maps

:H⁰(X,Gm) //H_cont¹ (X,Z`(1)) and c1: Pic(X) //H_cont² (X,Z`(1)), (5.3)

whereH_contⁱ (X,Z`(n)) :=H_contⁱ (X,(µ^⊗n_`r )r).

5.1.2. Pro-étale cohomology of Bhatt and Scholze. For a scheme X we denote by X_pro´_et the pro-étale site of X formed by weakly étale maps of schemes U → X (withU of not too big cardinality), see [BS15, Definition 4.1.1 and Remark 4.1.2].

Since every étale map is weakly étale, there is a natural map of associated topoi ν : Shv(Xproét) //Shv(Xét).

(5.4)

The pullback ν^∗ : D⁺(Xét) → D⁺(Xproét) on bounded below derived categories is fully faithful and the adjunction id → Rν∗ν^∗ is an isomorphism, see [BS15, Proposition 5.2.6]. For a sheaf F ∈ Ab(Xproét) of abelian groups on Xproét, one defines

Hⁱ(Xpro´et, F) := RⁱΓ(Xpro´et, F),

where RⁱΓ denotes the i-th right derived functor of the global section functor F //Γ(X, F).

If the transition maps in the inverse system (F_r)∈Ab(X_´_et)^Nare surjective, then there is a canonical isomorphism

Hⁱ(Xpro´et,limν^∗Fr)'H_contⁱ (X´et,(Fr)), (5.5)

see [BS15,§5.6].

5.1.3. Constructible complexes in the pro-´etale topology. We present in this section some parts of the six functor formalism on constructible complexes of Bhatt and Scholze in the special case of algebraic schemes, i.e. separated schemes of finite type over a field, which suffices for our purposes. In Remark 5.3 below we add some comments on the more general setting from [BS15].

LetX be an algebraic scheme over a fieldkand recallν from (5.4). For a prime

`invertible ink, let

Zb`(n) := limν^∗µ^⊗n_`r ∈Ab(Xpro´et) (5.6)

and write Zb` := bZ`(0). Note that Zb` is a sheaf of rings on Xproét and Zb`(n) are Zb^`-modules, which are in fact locally free (e.g. they are free on the pro-étale cov- ering X_k →X). We may then consider the derived category D(Xproét,bZ`) of the abelian category Mod(Xproét,Zb`) of sheaves of bZ`-modules on Xproét. A complex K ∈D(Xproét,Zb`) is constructible, if it is complete, i.e.K →^' R lim(K⊗^L

Zb_` Z/`^r),

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andK⊗^L

Zb_`Z/`^r'ν^∗Krfor a constructible complexKr∈D(Xét,Z/`^r), see [BS15, Definition 6.5.1]. The full subcategory spanned by constructible complexes is de- noted byDcons(Xproét,Zb`)⊂D(Xproét,Zb`). Constructible complexes are bounded, see [BS15, Lemma 6.5.3].

For a morphismf :X →Y of algebraic schemes, Rf_∗ respects constructibility and is right adjoint tof_comp^∗ :Dcons(Yproét,Zb`)→Dcons(Xproét,Zb`), which is given by pullback followed by (derived) completion, see [BS15, Lemma 6.7.2]. There is also a functor Rf_! : D_cons(X_pro´_et,Zb`) → D_cons(Y_pro´_et,bZ`) (see [BS15, Definition 6.7.6]) with a right adjoint f^! : Dcons(Yproét,Zb`) →Dcons(Xproét,bZ`), see [BS15, Lemma 6.7.19]. Iff is proper, Rf!= Rf∗ (by definition).

To explain the construction off^! in [BS15], note that the pullback ν^∗:Dcons(X´et,Z/`^r) ^' // Dcons(Xpro´et,Z/`^r) (5.7)

is an equivalence (see paragraph after [BS15, Definition 6.5.1]). Using this, we will freely identify complexes on the two sides with each other. For instance, we will freely identify µ^⊗n_`r on Yét with its pullbackν^∗µ^⊗n_`r to Yproét. Let now K ∈ Dcons(Yproét,Zb`) with truncationKr=K⊗^L_Z

`Z/`^rand letf_r^! :Dcons(Yét,Z/`^r)→ Dcons(Xét,Z/`^r) be the exceptional pullback on the étale site, induced by f, cf.

[SGA4.3, ´Expose XVIII]. Since any constructible complex of sheaves ofZ/`^r-modules onX_pro´_etis also a constructible complex ofZb`-modules onX_pro´_et, we may by (5.7) identify f_r^!Kr with an object inDcons(Xpro´et,Zb`). By [BS15, Lemma 6.7.18], the natural reduction mapsK_r→K_mform≤rmake (f_r^!K_r) into a projective system and so, following Bhatt–Scholze (see [BS15, Lemma 6.7.19]), one may define

f^!K:= R limf_r^!Kr∈Dcons(Xpro´et,Zb`).

(5.8)

The above construction implies that many properties known from the ´etale site carry over to the pro-´etale site.

Lemma 5.1. Letf :X →Y be a morphism between algebraic schemes over a field kand let`be a prime invertible ink. Then the following holds inD_cons(X_pro´_et,Zb`):

(1) Iff is weakly ´etale or a closed immersion, thenf_comp^∗ 'f^∗; (2) Iff is ´etale, thenf^! 'f^∗'f_comp^∗ ;

(3) Ifg:Y →Z is another morphism, then there is a natural isomorphism of functorsf^!g^! ^' // (g◦f)^!;

(4) Iff is smooth of pure relative dimension d, then there is a canonical isomorphism of functorsf_comp^∗ (d)[2d] ^' // f^!, where f_comp^∗ (n) :=f_comp^∗ (− ⊗

bZ`

Zb`(n)).

(5) Let f be smooth of pure relative dimension d. Then for any ´etale map j:U →X, the diagram

(f◦j)^∗_comp(d)[2d] ^' //(f ◦j)^!

j_comp^∗ f_comp^∗ (d)[2d] ^' //

OO

j_comp^∗ f^!'j^!f^!

OO

commutes, where the horizontal maps are induced by the canonical isomorphisms from item (4) and the vertical arrows are induced by the canonical maps given by functoriality off_comp^∗ andf^!.