The equivalence of several conjectures on independence of $\ell$

We consider several conjectures on the independence of $\ell$ of the \'etale cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main result is that independence of $\ell$ of the Betti numbers $h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for cycles on smooth projective varieties. We give several other equivalent statements. As a surprising consequence, we prove that independence of $\ell$ of Betti numbers for smooth quasi-projective varieties implies the same result for arbitrary separated finite type $k$-schemes.


Introduction
Let k be a field, and let X be a k-variety. For every prime number invertible in k, there is an associated étale cohomology group H * c (Xk, Q ) defined using the geometry of -power degree covers of X. The main question we want to consider is the following. If k = C and X is smooth, this easily follows from the functorial comparison isomorphisms [SGA4 III , exp. XI, th. 4.4(iii)] H * et (X, Q ) H * sing (X(C), Q) ⊗ Q Q . The result for arbitrary X over C can be deduced from this using hypercoverings, cf. [Del74b, 6.2.8]. The Lefschetz principle proves the result for any field k of characteristic 0, since étale cohomology is insensitive to extensions of algebraically closed fields [SGA4 III , exp. XVI, cor. 1.6].
On the other hand, if k is finite and X is smooth and proper, then the Weil conjectures [Del74a], [Del80] imply that h ié t (Xk, Q ) can be read off from the zeta function of X, and thus does not depend on . The question for arbitrary k-varieties X is a well-known open problem [Kat94,p. 28,(2a)], [Ill06,3.5(c)].
The homological standard conjecture [Gro69, §4, Remarks (3)] is known to imply the result in the following two cases: (i) X is proper; (ii) X is the complement of a simple normal crossings divisor D in a smooth projective varietyX.
Indeed, (ii) is explained in [Kat94,, and (i) is an application of de Jong's alterations [dJo96]. Even assuming the homological standard conjecture, the result for an arbitrary variety X does not seem to appear in the literature (although it may have been known to experts). One cannot simply combine the arguments of (i) and (ii); see Remark 6.7.
We improve these conditional results in three ways: • we replace the homological standard conjecture by a weaker assumption; • we prove independence of of h i c (Xk, Q ) for every separated finite type k-scheme X; • we prove a converse as well.
Theorem 1. Let k be an algebraically closed field. If k =F p , then the following are equivalent: (1) For every smooth projective k-scheme X, the kernel of the cycle class map cl : CH * Q (X) → H * (X, Q ) is independent of ; (2) For all smooth projective k-schemes X and Y , any α ∈ CH * Q (X × Y ), and any i, the rank of α * : H i (X, Q ) → H * (Y , Q ) 1 is independent of ; (3a) For every separated finite type k-scheme X and any i, the dimension of H i c (X, Q ) is independent of ; (3b) For every smooth quasi-projective k-scheme X and any i, the dimension of H i c (X, Q ) is independent of . Moreover, if these hold when k =F p for some prime p (resp. for every prime p), then they hold over any algebraically closed field of characteristic p (resp. any algebraically closed field).
This result is given in Theorem 4.6 and Remark 4.8 below. This gives many new angles to the independence of question. The implication (3b) ⇒ (3a) is particularly surprising; the proof goes through (1) and (2).
We also have an extension to crystalline cohomology. In fact, we work with an arbitrary Weil cohomology theory (see Definition 1.1) satisfying some additional axioms (see Axiom 3.6), at the expense of restricting to k =F p . (Developing 'Weil cohomology theories with specialisation' would take us too far afield.) However, our methods do not say anything about independence of of the dimensions h i (X, Q ) of the (usual) cohomology groups H i (X, Q ), except in the proper (resp. smooth) case where it coincides with (resp. is dual to) compactly supported cohomology.
The idea of (1) ⇒ (2) is that the rank of a linear map f : V → W is the largest r ∈ Z ≥0 such that r f 0. Although the functors H * (−, Q ) : Mot k → gVec do not preserve wedge products (see Remark 1.10), algebraicity of the Künneth projectors [KM74] decomposes a cycle α ∈ CH * Q (X × Y ) as α even ⊕ α odd . Then r α even (resp. S r α odd ) acts on cohomology as r α even, * (resp. r α odd, * ), so the rank of the map α * : H * (X, Q ) → H * (Y , Q ) is determined by the vanishing or nonvanishing of cl( r α even ) and cl(S r α odd ) for various r.
To prove (2) ⇒ (3a), we use a variant of the classical hypercovering argument [SGA4 II , exp. V bis ], [Del74b,§5,6]: if X • → X is a proper hypercovering, then there is a hypercohomology spectral sequence (1) E p,q 1 = H q (X p , Q ) ⇒ H p+q (X, Q ). If each X p is smooth projective, then (1) degenerates on the E 2 page for weight reasons, so h i (X, Q ) is determined by the ranks of the maps on the E 1 page.
However, a proper hypercovering by smooth projective schemes can only exist if X is proper. In general, again using de Jong's alterations [dJo96], one can construct a proper hypercovering X • → X where each X i is the complement of a simple normal crossings divisor Z i in a smooth projective k-schemeX i . There is a different spectral sequence [Kat94, computing the compactly supported cohomology of X i in terms ofX i and the components of Z i ; its dual then computes the cohomology of X i . However, if we then compute (1), the purity argument no longer applies.
Instead, we choose a compactification X →X first, with closed complement V , and we produce a morphism of simplicial schemes v • : V • →X • , where V • (resp.X • ) is a hypercovering of V (resp.X). Then the simplicial mapping cone of v • computes the compactly supported cohomology of X, by comparing the long exact sequence for the mapping cone with that for the triple (X, X, V ). This allows us to apply the purity theorem as in the argument above for X proper, which finishes the proof of (2) ⇒ (3a).
Finally, for the implication (3b) ⇒ (1) we prove that any cycle α ∈ CH d (X) can be written as a difference [Z 1 ] − [Z 2 ] with the Z i irreducible; see Corollary 7.5. Letting U be the complement of Z 1 ∪ Z 2 , we relate 1 We do not write H i (Y , Q ) because α * does not always take H i to H i ; see Definition 1.7. the vanishing of cl(α) to the dimension of H 2d+1 c (U , Q ). There are only two possible cases depending on whether cl(Z 1 ) and cl(Z 2 ) are linearly independent or linearly dependent; in the latter case the linear relation is determined by intersection numbers.

Outline of the paper
In Section 1 we give a brief review of Weil cohomology theories (Definition 1.1) and pure motives (Definition 1.5). Section 2 contains a review of simplicial schemes and mapping cones, which play a role 2 in (2) ⇒ (3a). In Section 3 we state the additional axioms on our Weil cohomology theory for the arguments to work; see Axiom 3.6.
The main theorem will be stated in Section 4 (see Theorem 4.6). We then proceed to prove the implications of Theorem 4.6 as outlined in the introduction above, in the following cyclic order: The implication (3a) ⇒ (3b) is trivial; each of the others will take up one section (Section 5, Section 6, and Section 7 respectively).

Notation and conventions
If k is a field, then a k-variety will mean a finite type, separated, geometrically integral k-scheme. A pair (X, H) is called a projective k-scheme if X is a projective k-scheme and H a very ample divisor on X.
In the main theorems, the base field k will be assumed algebraically closed, because standard references on Weil cohomology theories have this running assumption, and establishing the general framework would take us too far astray.
The category of smooth projective k-varieties will be denoted by SmPrVar k , and the category of smooth projective k-schemes will be denoted by SmPr k . The latter can be obtained from the former as the category of formal finite coproducts (if k is algebraically closed), cf. Example 3.5. The category of Chow motives is denoted by Mot k ; its definition will be recalled in Definition 1.5. Morphisms in this category are typically denoted by α : X Y .
If K is a field, then Vec K denotes the category of K-vector spaces, gVec K the category of Z-graded K-vector spaces, and gAlg K the category of Z-graded (unital, associative) K-algebras. The objects of gAlg K we encounter will always be graded-commutative and vanish in negative degrees.
We write Ab(C ) for the category of abelian group objects in a category C with finite products. All topoi 3 will be Grothendieck topoi, i.e. the topos of sheaves (of sets) on a small category with a Grothendieck topology (or pretopology). We write Topos for the (strict) 2-category of topoi, whose objects are topoi, whose 1-morphisms are (geometric) morphisms of topoi, and whose 2-morphisms are natural transformations between the inverse image functors (equivalently, between the direct image functors).
We write Shv for the (strict) 2-category whose objects are pairs (X, F ) of a topos X with an abelian object F in X, whose 1-morphisms (X, F ) → (Y , G ) are pairs (f , φ) of a 1-morphism f : X → Y and a morphism φ : f * G → F of abelian objects, and whose 2-morphisms (f , φ) → (g, ψ) are given by natural transformations η : f * ⇒ g * such that ψ•η G = id F •φ. We think of it as a "fibred 2-category" Shv → Topos, whose fibre above the topos X is Ab(X) op (with only identity 2-morphisms). In a similar way, we define a category Comp of pairs (X, K) of a topos X with a complex K of abelian objects on X.
2 Weizhe Zheng provided an easier argument without mapping cones; see Remark 6.8. We kept the original argument as it proves more, and the techniques might be useful elsewhere.
3 We work with topoi instead of sites because they have better formal properties, but all arguments could also be carried out using Grothendieck (pre)topologies. We ignore set-theoretic issues; they can for instance be dealt with using universes.
The equivalence of several conjectures on independence of 5 The equivalence of several conjectures on independence of 5

Pure motives and Weil cohomology theories
This is a review of the theory of pure motives, cf. e.g. Kleiman [Kle72], Jannsen [Jan92], or Scholl's excellent survey [Sch94]. We also give a brief review of Weil cohomology theories; see [Kle94,§3] for more details. Following standard references, we will assume that k is algebraically closed. Our setup is slightly more general than [Kle94,§3], in that we allow smooth projective k-schemes with multiple components. Definition 1.1. Let k be an algebraically closed field, and let K be a field of characteristic 0. A Weil cohomology theory is a functor H : SmPr op k → gAlg K satisfying the following axioms. (W1) Each H i (X) is finite-dimensional and vanishes for i < 0 and i > 2 dim X; (W2) There is a trace map tr X : H 2 dim X (X) → K that is an isomorphism if X is irreducible, and takes 1 to 1 if X = Spec k. If all components of X have the same dimension d, then the natural pairing (W4) There are cycle class maps cl :  [GM87]. See also [Ill94] for an expository account and additional references.
Let ∼ be an adequate equivalence relation (cf. [Sam60]) finer than homological equivalence for every Weil cohomology theory, e.g. ∼ is rational, algebraic, or smash-nilpotent equivalence. For a variety X and i ∈ Z ≥0 , we will write CH i (X) for algebraic cycles of codimension i modulo ∼, and CH i Q (X) for CH i (X) ⊗ Q. We will omit further mention of ∼ unless it plays a role in the arguments. Definition 1.4. Let X and Y be smooth projective k-schemes, and assume that X has connected components X 1 , . . . , X m of dimensions d 1 , . . . , d m respectively. Then the group of correspondences of degree r (modulo ∼) from X to Y is An element α ∈ Corr r (X, Y ) is a correspondence from X to Y , and is denoted α : X Y . There is a natural composition of correspondences: If the degree r in the superscript is omitted, it will be assumed 0.
Definition 1.5. The category of Chow motives (modulo ∼) is the category whose objects are triples (X, p, m), where X is a smooth projective k-scheme, p ∈ Corr(X, X) a projector (i.e. p 2 = p), and m ∈ Z an integer. Morphisms (X, p, m) (Y , q, n) from (X, p, m) to (Y , q, n) are given by We denote the category of Chow motives by Mot k . The motive (Spec k, id, −1) is called the Lefschetz motive, and is denoted by L. We write L n for (Spec k, id, −n), which can also be defined as L ⊗n using the tensor product of Remark 1.9. Remark 1.6. There is a functor SmPr op k → Mot k associating to every smooth projective k-scheme X the motive (X, id, 0), and to every map f : X → Y the (class of the) graph Γ f ∈ Corr(Y , X). Definition 1.7. If H is a Weil cohomology theory with coefficient field K, then Poincaré duality gives an isomorphism Together with the Künneth formula this gives isomorphisms If α ∈ Corr r (X, Y ), then under these isomorphisms cl(α) induces a pushforward In particular, a projector p ∈ Corr(X, X) induces a projector on H * , and we extend H to a functor Mot k → gVec K by setting where for a graded vector space Given a morphism α : (X, p, m) (Y , q, n), we define Hα to be the graded map given by the pushforward α * : pH i+2m (X) → qH i+2n (Y ) as in (1.2).
Some Weil cohomology theories have further structure (e.g. a Hodge structure or a Galois action), and these structures are typically preserved by pushforward along cycles (if everything is given the correct 'Tate twist'). We will consider this additional structure understood, and we will not use separate notation for the corresponding enriched functor.
Example 1.8. The cohomology H i (L) of the Lefschetz motive is 0 if i 2 and one-dimensional if i = 2. It's often thought of as the compactly supported cohomology of A 1 (or the reduced cohomology of P 1 ), and is equipped with the corresponding Galois action or Hodge structure.
Remark 1.9. The category Mot k has a tensor product given by Thus, we also get symmetric and alternating products S n and n by considering the projectors 1 n! σ σ and 1 n! σ (−1) σ σ respectively on X n . Remark 1.10. If H is a Weil cohomology theory with coefficient field K, then the functor H : Mot k → gVec K is a tensor functor if we equip gVec K with the tensor product as in super vector spaces: on objects, it is given by the usual graded tensor product, but the swap is given by To see that this makes H into a tensor functor, note that the Künneth isomorphism is given by the map Indeed, because of the sign in τ V ,W , the symmetriser and antisymmetriser get swapped in odd degree.
Remark 1.12. The category Mot k has binary biproducts; for example One can construct (X, p, m) ⊕ (Y , q, n) when m n as well, by replacing X by X × P n−m or Y by Y × P m−n , using that L d = (pt, id, −d) is a summand of (P d , id, 0) for d ≥ 0 (see e.g. [Sch94, 1.13]), and using the tensor product of Remark 1.9 to reduce to the case m = n.
However, the category Mot k is not in general abelian, and in fact for our choices of ∼ this is either false or open; see for example [Sch94, Corollary 3.5] for the case where ∼ is rational equivalence and k is not isomorphic toF p . Lemma 1.13. Let X and Y be smooth projective schemes over a field k, and let α ∈ Corr(X, Y ). Then the map Proof. This is clear from the definition of α * , cf. Definition 1.7. Lemma 1.14. Let X be an n-dimensional smooth projective k-scheme, and let α ∈ Corr(X, X). Then where ∆ X ⊆ X × X is the diagonal. In particular, it is a rational number that does not depend on H.
Proof. The first statement is well-known, and the second follows.
, and is independent of the Weil cohomology theory H.
Proof. One easily checks that the coefficients of the characteristic polynomial Hence, if p i ∈ Corr(X, X) denotes the i-th Künneth projector, then applying Lemma 1.14 to n−j (p i • α) gives the result.

Simplicial topoi and mapping cones
We will use the following (possibly non-standard) terminology: Definition 2.1. Let D be a small category, and let C be a category. Then a D-object in C is a functor D op → C . If D is the category ∆ + of finite (totally) ordered sets with monotone maps (resp. the subcategory ∆ of nonempty objects), then a D-object is an augmented simplicial object (resp. simplicial object) of C .
If X • is an (augmented) simplicial object, then X n denotes the value of X • on the set [n] = {0, . . . , n}. Giving an augmented simplicial object (X n ) n≥−1 is equivalent to giving a simplicial object X • = (X n ) n≥0 together with a map X • → X −1 to the constant simplicial object with value X −1 .
Remark 2.2. If C is a 2-category (e.g. Topos or Shv; see Notation), then the correct definition of a D-object is a pseudofunctor X • : D op → C . By the 'Grothendieck construction', this should correspond to some sort of fibred object. For example, a D-topos corresponds to a functor F : X → D that is a fibration and cofibration, such that each fibre X i is a topos, and for any morphism By [SGA4 II , exp. V bis , déf. 1.2.8 and prop. 1.2.12], this gives rise to a total topos Γ(X). Abelian objects in Γ(X) correspond to enrichments of the pseudofunctor X • : D op → Topos through D op → Shv, and similarly for complexes.
Remark 2.4. If D = ∆, then following the notation of simplicial topological spaces we will write Ab(X • ) (resp. Ch(X • )) for Ab(Γ(X)) (resp. Ch(Γ(X))) and write H i (X • , K) for H i (Γ(X), K). For a bounded below complex 4 K on X • , there is a spectral sequence whose E 1 page is the alternating face complex on the cosimplicial abelian groups H q (X * , K| X * ); see for example [SGA4 II , exp. V bis , cor. 2.3.7 and 2.3.9]. For computation, it's useful to have a refined version of this statement: Lemma 2.5. Let X • be a simplicial topos, and let K be a bounded below complex on X • with H i (X n , K j | X n ) = 0 for all j ∈ Z, n ∈ N, and i > 0. Then the complex .3], the constant simplicial topos is the terminal simplicial topos, so we get a functor θ : Again since Set is terminal, the terminal morphism Γ : Γ(X) → Set factors as On the level of derived categories of abelian objects, we get for all n ∈ N, all j ∈ Z, and all i > 0. Thus, all K j are acyclic for RΓ(θ) * , so The result follows since Rε * is computed by the totalisation of the alternating face complex [SGA4 II , exp. V bis , cor. 2.3.6].
Definition 2.6. Let C be a category with finite coproducts and a terminal object * , and let f • : The mapping cone satisfies 4 Note that a complex K on X • is bounded below iff the complexes on the components X i are bounded below uniformly in i.
A slightly different construction is given in [Del74b, 6.3.1], but the two are homotopy equivalent as simplicial objects in C . Similarly, the mapping cylinder cyl • (f ) is the pushout of Example 2.7. Let f • : Y • → X • be a morphism of simplicial topoi, let K and L be abelian objects (resp. complexes) on X • and Y • respectively, and let φ : Remark 2.8. The terminal object in Topos is Set [SGA4 I , exp. IV, §4.3], whose abelian objects are just abelian groups. Moreover, 2-coproducts in Topos are given by the product of categories [SGA4 I , exp. IV, exercice 8.7(bc)]. The terminal object in Shv or Comp is the pair (Set, 0). Thus, in the description of (2.1), the complex C n (φ) on C n (f ) equals 0 on * = Set, equals L| Y n on each of the components Y n , and equals K| X n on the component X n .
Lemma 2.9. Let X • and Y • be simplicial topoi, and let f • : Y • → X • be a morphism of simplicial topoi. Let K and L be bounded below complexes on X • and Y • respectively, and let φ : f * • K → L be a morphism. Then there is a canonical distinguished triangle In the setting of simplicial topological spaces, this is [Del74b, 6.3.3].
Proof. View (φ : f * • K → L) as a bounded below complex K on the (←)-topos X = (Γ(Y ) → Γ(X)). Let K → I be an injective resolution. This means that I = (ψ : f * • I → J) for bounded below complexes of injectives I and J on X • and Y • respectively, together with a commutative square Since e X,! and e Y ,! are exact by Remark 2.3, we conclude that I = e * X I and J = e * Y I are still injective [Stacks, Tag 015Z]. Since e * X and e * Y are exact, i : K → I and j : L → J are still resolutions. Similarly, K| X n → I| X n and L| Y n → J| Y n are injective resolutions for any n ∈ N. Finally, functoriality of the mapping cone and mapping cylinder gives resolutions where all terms C n (ψ) and cyl n (ψ) are injective (but we do not know whether C • (ψ) and cyl • (ψ) themselves are injective).
From the descriptions of (2.1), (2.2), and Remark 2.8, we see that the maps Y The homotopy equivalence (cyl • , cyl • (ψ)) (X • , I) induces a chain homotopy equivalence be a commutative diagram of simplicial objects in Comp where K, K , L, and L are bounded below. We may view this as a bounded below complex K on the Proceeding as above gives a natural morphism from the sequence (2. 3) for f • to the same sequence for f • (which again does not depend on any choices).

All-or-nothing motives and additional axioms
As in Section 1 we will assume k is an algebraically closed field. We want to apply the previous section in the case where X • is a simplicial scheme, viewed as a simplicial topos with the pro-étale topology [BS15] (resp. the crystalline topology [Ber74]), and K and L are the constant sheaves Q (resp. the structure sheaf O X/W (k) ). The recipe of Definition 2.6 and Remark 2.8 tells us to consider the sheaf on C • (f ) given by 0 on the components * of C • (f ) and by Q (resp. O −/W (k) ) on all other components.
Thus, we need to equip the mapping cone of a morphism of simplicial schemes with a mild motivic structure, interpreting the sheaf 0 on * as a variant of the zero motive. This motivates the following ad hoc notion.
Definition 3.1. An all-or-nothing motive (X, p) is a smooth projective k-scheme together with a locally constant function p : X → {0, 1}. The set of morphisms f : (X, p) → (Y , q) of all-or-nothing motives (X, p), (Y , q) whose underlying schemes X and Y are connected is given by with composition given by f • 0 = 0. (The contravariance is consistent with Remark 1.6.) In general, if X = X i and Y = Y j with the X i and Y j irreducible, we set The category of all-or-nothing motives is denoted Mot aon k . Remark 3.2. If X i are the components of X, then we think of (X, p) as the pure motive (X, p, 0) by identifying p with the projector in Corr(X, X) given by where 0 • f = 0 whenever this makes sense. If F : C → D is a functor to a category D with a terminal object * , then there is a unique extension of F to a functor F : Definition 3.4. Let C be a category. Define the category Coprod(C ) of formal finite coproducts in C whose objects are diagrams X : I → C from a finite discrete category I, and morphisms from X : I → C to Y : J → C are given by Mor(X i , Y j ).
If F : C → D is a functor to a category D with finite coproducts, then there is a unique extension (up to isomorphism) of F to a functor F : Coprod(C ) → D taking X : I → C to the coproduct i F(X i ). If X : I → C is an object of Coprod(C ), then X is the coproduct of the one-object diagrams X i , so we may write X = i X i . An initial object of Coprod(C ) is given by the empty diagram X : ∅ → C , and if * is a terminal object of C , then it is also terminal in Coprod(C ).
Example 3.5. We have an equivalence SmPr Coprod(SmPrVar k ), and Mot aon k can be defined as (Coprod(SmPrVar k { * })) op . In particular, any functor F : SmPrVar k → D to a category D with finite coproducts and a terminal object * extends uniquely to a functor F : (Mot aon k ) op → D such that F(X, 0) = * for X irreducible and F(X, p) = i F(X i , p| X i ) if the X i are the components of X. This gives an alternative method to extend a Weil cohomology H to a functor H : Mot aon k → gVec K , cf. Remark 3.2. Now we are ready to state the additional axioms on our Weil cohomology theory. (A2) For any X ∈ C , we have H i c (X) = 0 for i > 2 dim X. If X is smooth projective of dimension n and Z ⊆ X is a reduced closed subscheme whose irreducible components Z 1 , . . . , Z r have dimension d, then H 2d (Z) K r and H 2d (X) → H 2d (Z) K r is Poincaré dual to the map (A3) For a cosimplicial all-or-nothing motive (X • , π • ), there is a graded K-vector space H * (X • , π • ) that is computed by a spectral sequence 1 is the alternating face complex on the cosimplicial K-vector space H q (X • , π • ). If all π p are equal to the constant function 1, then we write H * (X • ) for H * (X • , π • ).
(A4) (Cohomological descent for proper hypercoverings) If X • → X is a proper hypercovering of a proper k-scheme X such that all X n are smooth projective k-schemes, then the pullback map is an isomorphism.
is a morphism of cosimplicial all-or-nothing motives, then there is a long exact sequence extends uniquely to a pseudofunctor F : (Mot aon k ) op → Shv that maps * to the pair (Set, 0) and preserves finite coproducts (see Remark 2.8). This corresponds to a fibred and cofibred category E → Mot aon k whose fibre (X, p) pro-ét over a connected all-or-nothing motive (X, p) is X pro-ét if p = 1 and Set if p = 0, along with a sheaf F on Γ(E) whose restriction to (X, p) is Q if p = 1 and 0 if p = 0.
The sheaf F on Γ(E) pulls back to Γ((X • , π • ) pro-ét ), and the spectral sequence of Remark 2.4 then reads For any all-or-nothing motive (X, π), we have H q ((X, π) pro-ét , F ) = H q (X, π), where the right hand side is defined by Remark 3.2. This gives the required spectral sequence of (A3 Finally, in the situation of (A6), the functoriality statement in Lemma 2.9 immediately reduces us to the case of the trivial hypercoverings X • → X and Z • → Z given by the constant simplicial schemes X n = X and Z n = Z. (This does not preserve the hypothesis that each X n and Z n is smooth projective, but étale cohomology is defined and has all the desired properties also for simplicial objects in Sch sep, f.t./k .) In fact, since i * is exact, the same goes for Γ(i) * : Ab(Z • ) → Ab(X • ), so we may replace the morphism using the functoriality assertion of Lemma 2.9. An injective resolution This gives a commutative diagram with exact rows of complexes on X pro-ét , whose vertical maps are quasi-isomorphisms, where C * and cyl * denote the dual constructions to Definition 2.6, applied to the rows of (3.3) viewed as morphisms of cosimplicial objects in Ch(X pro-ét ). The terms in the bottom row of (3.4) are injective, so the long exact sequence of (A1) is computed by

This agrees with the sequence (2.3) for the constant resolution (3.3), giving the commutative diagram (3.1).
This construction is functorial for commutative diagrams (3.2) since pullback along Y → X preserves the short exact sequence 0 → j ! Q → Q → i * Q → 0 and by the functoriality statement in Lemma 2.9. Remark 3.9. The above axioms are exactly what we need, but possibly not the most natural choice. It's likely that our axioms (or an alternative set of sufficient axioms) can be deduced from Cisinski-Déglise's axioms for a mixed Weil cohomology theory [CD12]. Somewhat surprisingly, cohomological descent for proper hypercoverings (A4) is indeed always satisfied [CD19, Corollary 17.2.6].

Independence of Weil cohomology theory
As in Section 1, we assume that k is an algebraically closed field, and we fix an adequate equivalence relation ∼ that is finer than homological equivalence for any Weil cohomology theory. From now on, we will fix Weil cohomology theories H and H (see Definition 1.1) with coefficient fields K and K respectively. We will always assume that H and H satisfy the additional properties (A1-6) of Axiom 3.6. Rk r (X, Y ): for any α ∈ Corr r (X, Y ) and any i, the ranks of α * on H i and H i agree.
Kün(X): for each i, there exists a cycle p ∈ Corr(X, X) inducing the i th Künneth projector on both H * and H * .
The reliance of these properties on the chosen Weil cohomology theories H and H will be implicit, and we will make no further mention of it.

Remark 4.3.
For Kün(X), note that such a cycle p need not be a projector in Corr(X, X). We only know that the cycle class map sends it to a projector in both End(H * (X)) and End(H * (X)).

Remark 4.4.
If k =F p and X is smooth proper, then we know Dim c (X) (for any Weil cohomology theories H, H), because the dimension can be read off from the zeta function. On the other hand, Rk c (X, Y ) is still unknown even when X and Y are smooth and projective. We also know Kün(X) for X smooth projective overF p , by [KM74]. Hence by Corollary 1.15, the characteristic polynomial of α ∈ Corr(X, X) is independent of the Weil cohomology theory. In particular, if α = p is a projector, this implies that dim H i (X, p, 0) is independent of H.  (1) For all smooth projective k-schemes X, we have Cl(X).
(2a) For all smooth projective k-schemes X and Y , we have Rk * (X, Y ); (2b) For all smooth projective k-schemes X and Y , we have Rk(X, Y ); (2c) For all smooth projective k-schemes X, we have Rk(X, X); (3a) For all separated finite type k-schemes X and Y , we have Dim c (X) and Rk c (X, Y ); (3b) For all smooth, quasi-projective k-schemes X, we have Dim c (X).
The outline of the rest of the article is as follows. In each of the following sections, we will prove one of the implications, often in a more refined version. We will prove the implications in the following cyclic order: The implications (1) ⇔ (2a), (2b) ⇒ (3a), and (3b) ⇒ (1) will be the contents of the following three sections (Section 5, Section 6, Section 7) respectively.
Remark 4.7. The proof of (1) ⇒ (2a) relies on Kün(X), which currently is known only when k =F p . Implication (2b) ⇒ (3a) uses the Weil conjectures and a hypercovering argument, so also does not generalise to other fields (but see Remark 4.8). The implication (3b) ⇒ (1) holds over an arbitrary algebraically closed field.
Remark 4.8. Suppose H and H are given by étale cohomology. If S is an irreducible scheme,s andη are a geometric point and a geometric generic point, is a prime invertible on S, and X → S is a smooth proper morphism of schemes, then the smooth and proper base change theorems give isomorphisms  (1) and (2abc) forF p for one prime p (resp. every prime p) imply the same result for any algebraically closed field k of characteristic p (resp. any algebraically closed field k). Moreover, for étale cohomology the argument in (2b) ⇒ (3a) can be refined to deduce (3a) over an arbitrary algebraically closed field k from (2b) overF p , again using spreading out and the specialisation isomorphism (4.1). The argument for (3b) ⇒ (1) works over any algebraically closed field, so we see that the case k =F p is the essential one.
Remark 4.9. If X is smooth projective over k and ∼ is rational equivalence, we get a ring isomorphism . Although Cl(X) is known for divisors, it seems that this cannot be used to deduce the statement in general.

Cycle classes and ranks
In this section, k is an arbitrary algebraically closed field. However, we will soon assume that Kün holds, which is currently only known for k =F p [KM74].
Theorem 5.1. Let X and Y be smooth projective k-schemes.
Proof. We have Corr r (Spec k, X) = CH r Q (X), and a cycle α ∈ CH r Q (X) maps to zero under the cycle class map cl : CH r Q (X) → H 2r (X) if and only if α * : H * (Spec k) → H * (X) is zero (and similarly for H * ); see Lemma 1.13. Now (1) follows from the assumption that H * (Spec k) is 1-dimensional, so the only possibilities for the rank of α * are 0 and 1, corresponding to α * = 0 and α * 0 respectively. For (2), let i and r be given, and let p ∈ Corr(X, X) (resp. q ∈ Corr(Y , Y )) be an element acting on H * and H * as the i th (resp. i + 2r th ) Künneth projector. For α ∈ Corr r (X, Y ), we get an induced element Moreover, the map (qαp) * : H i (X) → H i+2r (Y ) agrees with the map induced by α (and the same holds for H i (X) → H i+2r (Y )). Denote this map by α i (on both H i and H i ). First assume i is even, and consider the induced maps j (qαp) : j X j Y for various j. By Remark 1.11, we have a decomposition Corollary 5.2. Let k be an algebraically closed field such that Kün(X) holds for all smooth projective k-schemes X (e.g. k =F p [KM74]). Then the following are equivalent: (1) Cl(X) holds for all smooth projective k-schemes X; (2) Rk * (X, Y ) holds for all smooth projective k-schemes X and Y .

Ranks and dimensions
In this section, k will denote an arbitrary field. The main result of this section (Theorem 6.6) assumes that k =F p , because its proof relies on the Weil conjectures. The idea is to use alterations to produce smooth hypercoverings that compute the cohomology of arbitrary separated finite type k-schemes. Proof. Let X →X be a Nagata compactification [Nag62]. ReplacingX by the closure of X inX, we may assume that X is dense inX. LetȲ be a relative Nagata compactification of Y →X [Nag63]. Again, we may assume that Y is dense inȲ . ThenȲ is proper overX, hence proper over k sinceX is. This proves the first statement. The second statement follows because the scheme theoretic image of the morphism of proper X-schemes Y →f −1 (X) is closed. Since Y is also dense inf −1 (X) (in fact, inȲ ), this forces equality. (2) each D 1 (i) is quasi-projective over k and regular.
Proof. We will construct D 1 → D 0 as a functor D : (J × [1]) op → Sch k with the desired properties. For i ∈ J and J ⊆ J , write J <i = {j ∈ J | j < i} and J ≤i = {j ∈ J | j ≤ i}, and set If D is defined on (J × [1]) op and i ∈ J is arbitrary, we write Since this is a finite limit, it exists in Sch k and is separated and of finite type. We always have L ∅ = D 0 , and we have L J ≤i (i) = D 1 (i) and write L(i) := L J <i (i) when these are defined.
By induction on the size of the finite poset J <i , we construct D 1 (i) as a cone over the restriction of D to C J <i ,i = (J ≤i × [1]) \ (i, 1), such that moreover the natural map D 1 (i) → L(i) is proper and surjective. If i is minimal (i.e. J <i = ∅), take D 1 (i) D 0 (i) an alteration [dJo96, Theorem 4.1]. Since L(i) = L ∅ (i) = D 0 (i), the map D 1 (i) → L(i) is proper and surjective by construction. Now take i ∈ J arbitrary, and assume D 1 (j) has been defined and D 1 (j) → L(j) is proper and surjective for all j with |J <j | < |J <i |; in particular for all j ∈ J <i . Then L(i) is defined, and we take D 1 (i) L(i) any alteration [dJo96]. The required functoriality D 1 (i) → D 1 (j) for j < i comes from the fact that L(i) is a cone over the restriction of D to C J <i ,i .
If J ⊆ J <i is downward closed and j ∈ J is a maximal element, then J \ {j} is downward closed, and we have in particular L J (i) → L J\{j} (i) is proper and surjective. By induction on |J|, we conclude that L(i) → D 0 (i) and hence D 1 (i) → D 0 (i) is proper and surjective. Finally, D 1 (i) is a regular, quasi-projective k-scheme by construction.
We note that the proof above uses nothing special about schemes, and it generalises without difficulty to the categorical setting of [SGA4 II , exp. V bis , 5.1.4].
Proof. All ranks in question only depend on the stupid truncations σ ≥i σ ≤i+1 of the complexes. For d i A , d i B , and d i C this is clear, and for δ i this follows because of the factorisation noting that precomposing by surjections and postcomposing by injections does not alter ranks. Now the snake lemma gives a long exact sequence Additivity of dimension in short exact sequences gives Subtracting (6.1) from (6.2) gives the result.
Proof. Apply Lemma 6.4 to the short exact sequence noting that the boundary homomorphism of this sequence is H i (f ).
Theorem 6.6. Assume k =F p . If Rk(X, Y ) holds for all smooth projective k-schemes X and Y , then (1) Dim c (X) holds for every separated finite type k-scheme X; (2) Rk c (X, Y ) holds for all separated finite type k-schemes X and Y .
Proof. Let X and Y be separated k-schemes of finite type, and let f : Y → X be a proper morphism. Choose a commutative diagram as in Lemma 6.1 whereX andȲ are compatible compactifications of X and Y respectively such that X (resp. Y ) is dense in X (resp.Ȳ ) with complement V (resp. W ), and g denotesf | W . Applying Corollary 6.3 to the right hand square of (6.3), we get a commutative diagram (6.4) of augmented simplicial schemes such that each W i is smooth projective for i ≥ 0 and W • is a proper hypercover of W −1 = W , and similarly for V ,X, andȲ . By Axiom 3.6 (A1), (A5), and (A6), we get a commutative square of long exact sequences (6.5) whose vertical maps are isomorphisms. Thus H i c (X) is computed by H i (C • (v)), which by (A3) is computed by the hypercohomology spectral sequence (6.6) E p,q ). IfX p and V p are defined over some finite field k 0 for all p ≤ i + 1, then the computation of H i (C • (v)) only involves maps between cohomology groups of smooth projective varieties defined over k 0 . Moreover, the action of the |k 0 |-power geometric Frobenius (as an algebraic cycle) on H q (C p (v)) is pure of weight q [KM74, Corollary 1(2)] (see also Corollary 1.15).
Since Ext i K [x] (M, N ) = 0 for all i and for all K[x]-modules M and N that are pure of different weights, the spectral sequence (6.6) degenerates on the E 2 page and the filtration on E ∞ canonically splits: where E p,q 2 is given by .
. By assumption and by Remark 4.2, each of these numbers is independent of the Weil cohomology theory H. Then the same holds for dim E p,q 2 and therefore also for dim H i c (X), which proves (1) for X. By (A3), the natural map f • : C • (v) → C • (w) of cosimplicial all-or-nothing motives induces a morphism of spectral sequences f •, * : E p,q * (X • ) → E p,q * (Y • ). On the E 1 pages, this gives commutative diagrams degenerate. So even knowing Rk(Y , Z) for smooth quasi-projective k-schemes Y and Z does not imply Dim c (X) (or its variant Dim(X) for cohomology H i ) through this method.
The above argument is a way around this problem. The author is not aware of a place in the literature where this argument is carried out, but variants of it might have been known to experts.
Remark 6.8. To prove Theorem 6.6 (1) and (2) only in the case where X and Y are proper, one can carry out the proof above without talking about (simplicial) mapping cones. Indeed, applying the proof above to the morphism f • : Y • → X • (of simplicial schemes) instead of the morphism f • : C • (v) → C • (w) (of cosimplicial all-or-nothing motives) shows that the dimensions of H i (X) and H i (Y ) as well as the rank of Applying this to the closed immersion v : V →X as in (6.3), we see that so finally dim H i c (X) does not depend on H. This recovers Theorem 6.6 (1) without ever using simplicial mapping cones. (The mapping cone of complexes is still implicitly used through Corollary 6.5.) One can also obtain Theorem 6.6 (2) in this way (at least for étale or crystalline cohomology), by looking at the different weight parts as in (6.7) and applying Corollary 6.5 in the case where the complexes themselves are mapping cones.
This requires a different kind of axiom comparable to (A6), using the mapping cone of complexes instead. However, mapping cones in the derived category are only unique up to non-unique isomorphism, and to get a commutative diagram like (6.5) one needs to choose mapping cones functorially. We use the ones from simplicial schemes, but there are other possibilities. For example, Cisinki-Déglise's mixed Weil cohomologies [CD12] work in the dg setting, which may well be enough to make this part of the argument work.

Dimensions and cycle class maps
In this section, the ground field k is allowed to be arbitrary again. We start with a Bertini irreducibility theorem. We do some extra work in Lemma 7.2 to avoid extending the base field, using the Bertini irreducibility theorem of Charles-Poonen [CP16] as well as the classical one [Jou83, Theorem 6.3(4)].
The main application of these Bertini theorems is Corollary 7.5, which we use to prove the implication (3b) ⇒ (1) of Theorem 4.6. The idea is that if Z ⊆ X is an effective codimension m cycle on a smooth projective variety (X, H), then for m general sections H 1 , . . . , H m ∈ |H| containing Z, the intersection H 1 ∩ . . . ∩ H m contains only one new component Z , which is smooth away from Z (in particular reduced). Therefore, which realises [Z] as a difference of two irreducible cycles. We do something similar for an arbitrary (not necessarily effective) cycle α ∈ CH m (X).
We suggest the reader skip ahead to Corollary 7.5 on a first reading. If X is geometrically normal, then X i ∩ X j = ∅ for i j. In this case, the final statement of (Irr) is automatic: we clearly have Z i ⊆ X i ∩ Zk, and all other components Z j of Zk are disjoint from X i ∩ Zk.
Lemma 7.2. Let (X, H) be a projective k-scheme all of whose components have dimension n ≥ 2, let Z ⊆ X be a geometrically reduced closed subscheme of pure dimension 0, n, and let Y 1 , . . . , Y s ⊆ X be integral subschemes of dimension < n that are not contained in Z. Assume that X is smooth away from Z and at the generic points of Z. Then for d 0, there exists an element D ∈ |dH| containing Z such that • D is smooth away from Z; • D is smooth at the generic points of Z; • the divisor D \ Z of X \ Z satisfies (Irr); • D does not contain any of the Y j .
Proof. Write X sing and Z sing for the singular (non-smooth) loci of X → Spec k and Z → Spec k respectively. By assumption, X \ X sing contains U = X \ Z, as well as the generic points of Z. Since Z is geometrically reduced, Z sing has dense open complement. Thus, the open subset W = X \ (X sing ∪ Z sing ) is smooth of dimension n, contains U , and W ∩ Z is smooth and dense. Let Y be the union of the zero-dimensional Y j , and note that Y ∩ Z = ∅.
If k is infinite, choose d 0 such that I Z (dH) is globally generated, and let V be the image of H 0 (X, I Z (dH)) in H 0 (X, O X (dH)), i.e. |V | is the linear system in |dH| of sections vanishing on Z. Since I Z (dH) is globally generated, the base locus of |V | is exactly Z, so |V | defines a morphism Note that φ is a composition of a locally closed immersion and a coordinate projection, hence unramified. Thus [Jou83, Corollary I.6.11(2,3)] shows that the locus of D ∈ |V | = P(V )(k) such that D ∩ U is smooth and satisfies (Irr) is dense open.
Since the Y j are not contained in Z, the locus of D ∈ |V | not containing Y j is dense open. Similarly, the locus of s ∈ H 0 (X, I Z (dH)) whose restriction to I Z /I 2 Z (dH) is nonzero is dense open. Thus we can find a k-point of P(V ) satisfying all desired properties, since a dense open subset of P(V ) has a k-point if k is infinite.
If k is finite, we apply [Wut17, Theorem 2.1], where Wutz's X, Y , Z, k, , m are our W , Y , Z, − 1, , n respectively. This shows that the set has positive density µ > 0 as d → ∞. In particular, such D are smooth at the generic points of Z, since Z has (pure) dimension . Applying Bertini's irreducibility theorem [CP16, Theorem 1.2] to X \ Z shows that the set of D such that D \ Z satisfies (Irr) in X \ Z has density 1.
Finally, if Y j is positive-dimensional, then the ideal sheaf I Z∩Y j ⊆ O Y j is nonzero because Y j Z. The map I Z → I Z∩Y j is surjective by the second and third isomorphisms theorems. Hence, for d 0, the map is surjective. The dimension of the right hand side is (eventually) a polynomial of degree dim Y j > 0 in d, so codim ker(φ d ) → ∞ as d → ∞.
Hence, the functions that vanish on Y j have density 0 as d → ∞. Therefore, the intersection of the three sets has positive density µ.
where Z ⊆ X is (geometrically) reduced, satisfies (Irr), does not contain any of the Z i and Y j , and Z \ Z i is smooth.
Proof. Let Z = Z i . We apply Lemma 7.2 inductively on the codimension m of the Z i to find sections D i of |d i H| containing Z such that D i is smooth away from Z and at the generic points of Z and does not contain any of the Y j , and D i \ Z satisfies (Irr). Thus the multiplicity of D i at the generic points of Z is 1, so for an effective cycle α none of whose components is contained in Z .
If Z is the closure of D i \ Z , then restricting to X \ Z , we find that α = [Z], where all coefficients are 1 since D i \ Z is smooth. Then Z is geometrically reduced by [Stacks,Tag 020I]. Finally, Z satisfies (Irr) because D i \ Z does, and it does not contain any of the Y j since D i \ Z does not contain any Y j \ Z by the choice of the D i .
Theorem 7.4. Let k be a perfect field, let (X, H) be a smooth projective k-scheme of equidimension n, let α ∈ CH m (X) be a pure dimensional cycle, and let e 0 ∈ Z. If m 0, n, then there exists a (geometrically) reduced subscheme Z ⊆ X satisfying (Irr) and e ≥ e 0 such that where Z i , Z i ⊆ X are pairwise distinct integral subschemes of codimension m, and n i , n i ∈ Z ≥0 (to start with, we may take e = 0). We will apply Lemma 7.3 a few times.
First, by induction on z = (n i − 1) + (n i − 1), we will reduce to the case where z = 0. Indeed, if z > 0, then one of the n i or n i is ≥ 2. Say n 1 ≥ 2; the case n i ≥ 2 is similar. Applying Lemma 7.3 to Z 1 with {Y j } := {Z i | i > 1} ∪ {Z i }, we can write [Z 1 ] = dH m − [Z ], where Z does not contain any of the Z i and Z i , and Z is generically smooth (in particular reduced). Adjoining the irreducible components of Z to the Z i and changing e to e − d, we have reduced z by one, because the new components coming from Z all have coefficient 1. After finitely many steps, we get z = 0, so all n i and n i are equal to 1. Choosing the d i large enough so that e + d ≥ e 0 gives the result.