REFINED HEIGHT PAIRING

A BSTRACT . For a d -dimensional regular proper variety X over the function ﬁeld of a smooth variety B over a ﬁeld k and for i ≥ 0 , we deﬁne a subgroup CH i ( X ) (0) of CH i ( X ) and construct a “reﬁned height pair-ing”

If B is a smooth projective curve and we compose (1) with the degree map, we get a ‫/1[ޚ‬ p]-valued pairing (with values in p −s ‫ޚ‬ for some integer s ≥ 0), which relates to the one constructed by Beilinson [1987, §1].Beilinson [1987, p. 5] asked what happens when trdeg(K /k) > 1: (1) gives one answer to this question.
The quotient CH i (X )/ CH i (X ) [0] is finitely generated.When we vary (X, i), CH i (X ) [0] defines an adequate equivalence relation for smooth projective K-varieties, which a priori depends on the choice of B. Its saturation CH i (X ) (0) lies between the subgroups CH i alg (X ) and CH i num (X ) of algebraically and numerically trivial cycles, hence equals CH i num (X ) when i = 1, d.We conjecture that this holds for all i, and prove it in further special cases (Theorem 5.5(ii)).One can show that it would follow in general from the Tate conjecture, or the Hodge conjecture in characteristic 0, for cycles of codimension < i, although we don't include a proof here.More generally, one might hope that Lemma 1.1 below induces pairings in Ab ⊗ ‫ޑ‬ where F * CH * (X ) is the conjectural Bloch-Beilinson-Murre filtration [Jannsen 1994], the case n = 0 (resp.1) being the intersection pairing (resp.( 1)).
Works following Néron's seminal paper [1965] have much relied on l-adic cohomology to analyse or define height pairings (because of the cohomological definition of Hasse-Weil L-functions): for δ = 1, this is the case in [Schneider 1982] (i = 1, X an abelian variety), [Bloch 1984] and [Beȋlinson 1987].This is also the case in the work of Damian Rössler and Tamás Szamuely [2022], which is the direct inspiration of this one: they construct a pairing where l is a prime number invertible in k and CH i l (X ) denotes cycles homologically equivalent to 0 with respect to l-adic cohomology.By contrast, our approach here is completely cycle-theoretic and very close in spirit to Moret-Bailly's geometric height [1985, chapitre III, définition 3.2]; it relies on Fulton's marvellous theory of Gysin maps [1984, Chapters 6 and 8].This gives a different flavour to the definitions because numerical and homological equivalence have rather opposite functoriality under specialisation, as described in detail by Grothendieck in [SGA 6 1971, 7.9 and 7.13].See Remark 2.7.
Comparing various definitions of height pairings is a highly nontrivial issue, which is solved only in a few cases: for example, as far as I know those defined by Bloch [1984] and Beilinson [1987] have still not been checked to agree.Schneider [1982] compares an l-adic height pairing [loc. cit., p. 298] with the Néron-Tate height by comparing each to an intermediate Yoneda pairing [loc. cit., p. 502] where A 0 is the connected component of the identity of the Néron model A of the abelian variety A (= X here).
In Proposition 2.11, I show that (1) and (2) are compatible (at least in characteristic 0) on a common subgroup CH * B,l (X ) of CH * l (X ) and CH * (X ) [0] via the cycle class map Pic(B) → H 2 ét (B k , ‫ޑ‬ l (1)): this is what Rössler and Szamuely [2022, Proposition 6.1] had checked in the special case where X/K has a smooth model, by using a variant of Proposition 2.8 here.In Theorem 5.10, I show that (1) is the opposite of Silverman's refined height pairing [1994, Theorem III.9.5(b)] in the classical case of an elliptic curve X over the function field of a smooth projective curve B over an algebraically closed field k.
Another case where a compatibility should not be hard to show is that of [Moret-Bailly 1985].
Note that (1) is finer than (2) inasmuch as it takes homologically trivial cycles on B into account.This extra structure is presumably arithmetically significant; it is studied in Section 6E in the case d = 1, B projective.
It may seem disturbing that (1) is essentially integral, while the classical height pairing is usually rational: this may be "explained" by (3) which is integral but takes values on the subgroup of finite index A 0 (B) ⊆ A(K ).In this spirit, I show in Remarks 5.13(a) that in the elliptic curve case mentioned above, CH 1 (X ) [0] contains N 0 (B) as a subgroup of finite index, where N 0 is the identity component of the Néron model of X .
The raison d'être of [Beȋlinson 1987;Bloch 1984] was to refine the conjectures of Tate [1965] on the orders of poles of zeta functions at integers by describing special values at these integers, when K is a global field.Thus one might like to extend (1) to the case where B is regular and flat over ‫.ޚ‬I consider this as beyond the scope of this article for two reasons: • The present method fails in this case even if one is given a regular projective model f : X → B of X , because Fulton's techniques do not define an intersection product on X , except when δ = 1 and f is smooth [1984, p. 397].One does get an intersection product with ‫ޑ‬ coefficients, by using either K-theory as in [Gillet and Soulé 1987, 8.3], or alterations and deformation to the normal cone as in Andreas Weber's thesis [2015, Corollary 4.2.3 and Theorem 4.3.3]; it is possible that the present approach may be adapted by using one of these products.
• However, the main point in characteristic 0 is to involve archimedean places to get a complete height pairing whose determinant has a chance to describe the special values as mentioned above: this is what was done successfully in [Bloch 1984;Beȋlinson 1987] when δ = 1.In higher dimensions, one probably would have to use something like Arakelov intersection theory (see [Rössler and Szamuely 2022, Conjecture 7.1] for a conjectural statement).
I leave these issues to the interested readers.Rather, I hope to show here that height pairings in the style of (1) also raise interesting geometric questions.These are discussed in Section 6, which is closely related to [Kahn 2014, Question 7.6].
Contents.Up to Section 4F, we assume k perfect; this assumption is removed in the said subsection.In Definition 2.2, we introduce subgroups CH i (X ) 0 of admissible cycles in the Chow groups of a kmodel f : X → B of f ′ : X → Spec K , with X smooth; when B is projective, CH i (X ) 0 contains numerically trivial cycles (Proposition 2.5) and in general it contains locally homologically trivial cycles in the sense of Beilinson [1987, 1.2] (Proposition 2.6).From the intersection pairing on X , pushed forward to CH 1 (B), we then get, thanks to Proposition 2.8, a height pairing ⟨ , ⟩ f defined on the groups CH i (X ) 0 ).This is a pairing of genuine abelian groups.We prove in Propositions 3.6 and 3.8 that the CH i (X ) 0 f and ⟨ , ⟩ f are independent of f and compatible with the action of correspondences, and in Proposition 3.9 that they behave well with respect to base change.The group CH i (X )/ CH i (X ) 0 is finitely generated (Proposition 3.11).
If we are in characteristic 0, the construction is finished since X always admits a smooth model by resolution of singularities (Proposition 4.1).In characteristic p > 0, there turns out to be quite a bit of work to get a pairing in general after suitably inverting p, by using Gabber's refinement of de Jong's theorem: the general height pairing (4-1) is defined in Theorem 4.14; as said above, it makes sense in the category Ab ⊗ ‫/1[ޚ‬ p].Functoriality and base change extend to this pairing (Theorem 4.14).
In Section 5, we investigate Conjecture 5.1: CH i (X ) [0] is of finite index in CH i num (X ), the group of cycles numerically equivalent to 0 (the inclusion is always true by Lemma 4.3(d)); we prove it for i = 1, d in Theorem 5.6(b) (see Theorem 5.5(ii) for other cases).In Section 5C, we also relate (1) to the classical Néron-Tate height pairing in the case where X is an elliptic curve and B is a smooth projective curve.
In Section 6, we study the height pairing (2-9) in the basic case i = 1.If B is projective, it leads to a coarser pairing (6-2) between the Lang-Néron groups LN(Pic 0 X , K /k) and LN(Alb X , K /k) with values in N 1 (B), codimension 1 cycles modulo numerical equivalence (Theorem 6.2).When δ = 1, a version of this pairing involving an ample divisor is negative definite (Theorem 6.6): one should compare this with a result of Shioda [1999] when d = 1.See also Theorem 6.6 for a conjectural statement when δ > 1.We finally get an intriguing homomorphism from LN(Pic 0 X , K /k) to homomorphisms between certain abelian varieties in (6-6).
Notation and conventions.We try and follow Fulton's notation [1984] as much as possible.In particular, given a morphism of k-schemes f : X → Y , we write γ f for the associated graph morphism X → X × k Y and δ X for γ 1 X ; if f admits refined Gysin morphisms as in [loc.cit., Chapters 6 and 8], we write them f ! and sometimes use the notation f * for ordinary Gysin morphisms.
We usually abbreviate the notation × k (fibre product over k) to ×, and re-establish it when it may be confused with other fibre products.
We shall encounter k-schemes essentially of finite type, being of finite type over some localisation of B. We shall sometimes commit the abuse of treating them as if they were of finite type: for example, call them smooth even if they really are essentially smooth, and take (refined) Gysin morphisms associated to morphisms between them even if these morphisms are not of finite type.This is easily justified by the fact that Chow groups commute with inverse limits of open immersions [Bloch 2010, Lemma IA.1].
1.An elementary reduction 1A.Intersection on regular K-schemes.Let K be a field.If char K = 0, every regular K-scheme X , separated of finite type, is smooth, so the intersection theory of [Fulton 1984, Chapter 8] applies.Here we point out that this is also true in characteristic p > 0: it will be needed in and after Section 4B.
We may assume K to be finitely generated over its (perfect) subfield k = ‫ކ‬ p , and X (regular) to be irreducible of dimension d.We may find a smooth connected separated k-scheme B of finite type with generic point η = Spec K , and a dominant morphism f : X → B with X k-smooth, of generic fibre X .We have the intersection pairing of [Fulton 1984, §8.1] which commutes with base change by [Fulton 1984, Proposition 6.6(c) and 8.3(a)].Then (1-1) induces an intersection product on X by passing to the limit.If f 1 : X 1 → B 1 is another choice, then B and B 1 share a common open subset with isomorphic fibres, so this intersection product is independent of the choice of (B, f ).Suppose moreover X and f proper.Composing (1-1) with f * , we get a pairing For the same reason, numerical equivalence makes sense on X via (1-2), and does not depend on any choice.
1B.The set-up.Let now k be any perfect field; we place ourselves in the situation (B, X , f ) of Section 1A with f proper, and let f ′ : X → η be the generic fibre of f .In particular, the observations of Section 1A apply to X .
For a subscheme Z of B, write X Z = f −1 (Z ), ι : X Z → X for the corresponding immersion and f Z : X Z → Z for the projection induced by f .We extend these notations to pull-backs by a morphism Z → B when there is no ambiguity in the context.
We shall use the case r = 1 of this lemma in the rest of this paper.
Remarks 1.2.(a) Let Z be the locus of nonsmoothness of f .If f ′ is smooth, f (Z) is a proper closed subset of B, hence contains only finitely many points of B (1) , the set of codimension 1 points of B.
(b) If δ = 1, any proper surjective morphism ϕ from an irreducible k-variety V to B is flat [Hartshorne 1977, Chapter II, Proposition 9.7]; in general, this is true after base-changing to the local scheme at any point b ∈ B (1) .If F ⊂ V is the (closed) locus of nonflatness of ϕ, the closed subset ϕ(F) is therefore of codimension ≥ 2 in B. This shows that one may reduce to ϕ flat by removing a closed subset of codimension ≥ 2 from B. This technique may be applied to f if necessary; a variant will be used in the proof of Proposition 3.11.
Let CH i num (X ) denote the subgroup of CH i (X ) formed of cycles numerically equivalent to 0; write j for the inclusion X → X .

The refined height pairing
We keep the set-up of Section 1B.
The definitions of f !agree when f is of several of these forms at the same time, e.g., [loc.cit., Proposition 8.1.2].The assignment f → f ! is functorial in certain cases, many of which are summarised in [loc. cit.,Example 17.4.6].
Since it is difficult to find a unified statement of all these compatibilities in [Fulton 1984], we shall strive to give precise references for all those we use; the above reminder should only be viewed as a guide to the reader.
We shall very often use the following situation, that we record as a lemma.
−−−→ T be a Cartesian square of k-schemes, where g is proper and f is an l.c.i.morphism.Then: (c) If f and g are two composable l.c.i.morphisms, then (g Proof.This follows from [Fulton 1984, Theorem 6.6(c)]. 1); write Z = {b}.Recall the cap-product [Fulton 1984, p. 131] where ι is the closed immersion X Z → X .Take l = δ + i − 1. Composing with ( f Z ) * , we get a pairing (2-1) We record two useful formulas: which follows from Lemma 2.1 applied to the Cartesian diagram where ι ′ is the closed immersion Z → B.
Definition 2.2.With the above notation, we set 1) , and We call the cycles in CH i (X ) 0 admissible.
Remarks 2.3.(a) One should be careful that CH i (X ) 0 does not contain Ker j * in general.For example, let B = A 1 = Spec k[t] and let X be the hypersurface in B × ‫ސ‬ 2 with (partly) homogeneous equation t X 2 0 = X 1 X 2 .Then the pull-back of the curve (t = X 1 = 0), viewed as a codimension 1 cycle on X , to the curve (t = X 2 = 0), is the point (0, (1 : 0 : 0)) which is not numerically equivalent to 0. On the other hand, if f is smooth above Spec O B,b for a b ∈ B (1) , then any element of Ker j * vanishes when restricted to X b thanks to [Fulton 1984, §20.3].So this caveat only involves finitely many exceptional b.
b , thanks to Lemma 1.3.We shall not use these facts in the present paper, so the rather long proof is omitted (see [Kahn 2023]).
(c) Let b ∈ B (1) .Suppose that all the irreducible components X λ b of X b are of dimension d and smooth over k(b).Then it is easy to see that α ≡ b 0 if and only if κ !λ α ∈ CH i num (X λ b ) for all λ, where κ λ : X λ b → X is the inclusion.Our initial approach to the refined height pairing was based on such models; they are not necessary anymore.

We obviously have
Lemma 2.4.The quotient CH i (X )/ CH i (X ) 0 is torsion-free.□ 2C.Comparison with numerical and homological equivalence. 1), and Z = {b} as above.Let β ∈ CH δ+i−1 (X Z ).We have this time is injective since Z , as an irreducible divisor on a smooth projective variety, is not numerically equivalent to 0 (compare [Debarre 2001, Chapter I, Theorem 1.21]).Therefore ⟨α, β⟩ b = 0, as requested.□ Let now l be a prime number invertible in k.We have a composition where the first map is the (geometric) cycle class map.Write CH i l (X ) (resp.CH i (X ) 0 B,l for the kernel of the first map (resp. of their composition): the latter group is introduced by analogy to [Beȋlinson 1987, 1.2], which is the special case δ = 1, k algebraically closed.We obviously have CH i l (X ) ⊆ CH i (X ) 0 B,l .The following is parallel to Proposition 2.5, without assuming B projective.It will be used in Proposition 2.11 and in Remarks 5.4(a) and 3.12.
this means that j * α is l-adically homologically equivalent to 0, hence also numerically equivalent to 0. This part of the proof works in all characteristics.
We now give the sequel of the proof in characteristic 0: to oversimplify, it follows by functoriality from the fact that the cycle class map is injective in codimension 0 (sic).(So this argument is geometrically cheaper than the one for Proposition 2.5.) We may assume k finitely generated and choose an embedding of k in ‫.ރ‬By Artin's comparison theorem, where H B denotes Betti (or analytic) cohomology.Let b ∈ B (1) , and let Z , ι, β be as in Definition 2.2.
To show that ⟨α, β⟩ b = 0 in CH 0 (Z ) − → ∼ CH 0 (Z ‫ރ‬ ), we may assume k = ‫ރ‬ and drop all Tate twists.In [Fulton 1984, Chapter 19], a cycle class map cl is defined for Chow groups of complex, possibly singular, varieties, with values in their Borel-Moore homology and we have the formula cl(α since cl commutes with push-forwards, by definition and [Fulton 1984, Lemma 19.1.2]. It now suffices to show that the right hand side of (2-6) vanishes since CH δ−1 (Z ) → H 2δ−2 (Z ) is injective, as one sees by reducing to Z smooth by removing from it a proper closed subset.For this, it suffices to show that the pairing (2-7) given by (x, y) We switch by Poincaré duality from the Borel-Moore homology of X Z (resp.Z ) to the cohomology of the smooth variety X (resp.B) with supports in X Z (resp. in Z ).Then (2-7) becomes the composition where ∩ is the usual cap-product.The (global) trace map f * factors as a composition where Tr f is the local trace map in étale cohomology for the proper morphism f .Thus, (2-8) factor through the map In positive characteristic, the leap of faith is that (2-5) and (2-6) hold for the cycle class maps defined in l-adic Borel-Moore homology [Laumon 1976, §6].The commutation with push-forwards causes no problem, and (2-5) indeed appears in [Laumon 1976, Theorem (7.2)], except that the extraordinary cap-product • ι (defined in [Verdier 1976, 2.1.1]using intersection multiplicities) should be shown to agree with Fulton's.(This is suggested in the notes and references of [Fulton 1984, Chapter 19]; see also [loc. cit., p. 382].) 1his being accepted, the same argument goes through.□ Remark 2.7.As a referee pointed out, there is an important conceptual difference between CH i (X ) 0 B,l and CH i (X ) 0 : by the smooth and proper base change, we have the equality for any open subset U ⊆ B over which f is smooth.Thus, the condition α ∈ CH i (X ) 0 B,l for α ∈ CH i (X ) only has to be checked at the generic fibre and at the "bad fibres" of f .This contrasts with the case of CH i (X ) 0 , see Remarks 2.3(a).See also Remarks 5.4 further down.
2D. Global height pairing.The following proposition is the key point of this paper.
for some proper closed subset Z ⊂ B, where ι : X Z → X is the inclusion.We may assume that β ′ is the class of an irreducible cycle, hence take Z irreducible.If codim B Z > 1, the result follows from Lemma 1.1.If Z = {b} for b ∈ B (1) , the conclusion follows from (2-3).□ The proof of the following lemma is in the same spirit, so we include it here.It will be used in the proof of Proposition 3.9(ii).
Lemma 2.9.Let b 1 , . . ., b n be a finite set of points on B (1) and let Z = {b 1 , . . ., b n }.Then one has ( f Z ) * (α • ι β) = 0 for any α ∈ CH i (X ) 0 and any β ∈ CH δ+i−1 (X Z ), where ι is the closed immersion Proof.We may assume that β is the class of an irreducible cycle β ′ ; then β ′ is supported on X Z r for some r , where Z r = {b r }.Let κ : X Z r → X Z be the corresponding closed immersion, and let ι r = ικ: by applying again Lemma 2.1 to the obvious Cartesian square involving κ, we get the identity We shall see in the next section (Propositions 3.6 and 3.8) that neither CH i (X ) 0 f nor ⟨ , ⟩ f depends in the choice of f .2E.Comparison with the pairing of Rössler-Szamuely.
On the other hand, the height pairing of [Rössler and Szamuely 2022] is defined on and the height pairing of Rössler and Szamuely is defined by (2-14) on these subgroups.Let where X 1 is the generic fibre of X 1 . If where δ ! is the refined Gysin morphism from [Fulton 1984, §6.2] associated to the (regular immersion) diagonal 2 Except that f is assumed projective in [Corti and Hanamura 2000]; proper is sufficient to apply its formalism.
and the notation for the projections is self-evident.
As usual, one can generalise this to "graded correspondences" and reduce these graded correspondences to ordinary ones if one wishes, by using the projective bundle formula [Fulton 1984, Theorem 3.3(b)].
Since is also a regular immersion of the same codimension as δ (namely, dim X 2 ), we may apply Lemma 2.1(b) which gives If the f i are smooth, we also have a "classical" composition of correspondences à la Deninger-Murre [1991]: The pro-open immersion j defines a functor to the category of Chow correspondences over K from the full subcategory of CHC(B) consisting of those f : X → B whose generic fibre is smooth.
Proof.(a) We use (3-3).We have the Cartesian square in which all morphisms are l.c.i.morphisms, hence by [Fulton 1984, Proposition 6.6(b)], and finally by definition of the intersection product on smooth varieties [Fulton 1984, p. 131].As a special case of (3-1), take X 3 = B: we get pairings compatible via j * with the usual action of correspondences over K , by Lemma 3.1(c).For clarity, we repeat (3-1) in this special case: where γ p 2 is the graph of p 2 := p 1,2 2 .We also write ψ * for ( t ψ) * .As an even more special case, when X 1 = B: writing β rather than ψ, we recover the pairing (1-2) Proof.For clarity, write δ i for the diagonal map X i → X i × k X i .As in the proof of Proposition 3.6, let p i be the projection X 1 × B X 2 → X i .Developing, the identity to be proven is We now observe that since X 2 is smooth, γ p 2 is also a regular embedding in (3-4), hence δ ! 2 = γ * p 2 (nonrefined Gysin map) by Lemma 2.1(b) (see also (3-3)); similarly, δ ! 1 = γ * p 1 .The expression γ * p i (x × y) is also written x • p i y in [Fulton 1984, Definition 8.1.1](cf.proof of Proposition 2.8).The formula to be proven therefore becomes There is a much more conceptual proof by interpreting both sides as compositions of correspondences: we then have by the associativity of •.
3B. Independence from the model and functoriality.
Lemma 3.5.Let b ∈ B (1) and Z = {b} as usual.For ψ Proof.(a) Let us first draw the diagram of Cartesian squares underlying the coming computation: It already explains the use of δ ! 1 in the definition of ψ ! .Now where the third equality follows as usual from Lemma 2.1(a).
(b) First where λ is the regular embedding X 1 × B X 2 → X 1 × X 2 , so that γ λ is the composition of the bottom row in the diagram of Cartesian squares Here (a) follows from Lemma 2.1(b) applied to (3-7), (b) from Lemma 2.1(a), (c) from Lemma 2.1(b) again (applied twice), and (d) from Lemma 2.1(c). Next where (a) follows from Lemma 2.1(b) applied to (3-4) and (b) follows from Lemma 2.1(a) applied to the Cartesian square , we are left to prove the equality For this we draw the diagram of Cartesian squares, similar to (3-8): Here the composition of the bottom row is γ λ , up to permuting X 1 and X 2 .By Lemma 2.1(b), ( t γ ι 1 ) ! and γ !p 1 both compute the refined Gysin map corresponding to the arrow (κ, p 1,Z ), and also ) ! ; we conclude by applying Lemma 2.1(c) to the bottom row once again.□ Proposition 3.6.Let f 1 : X 1 → B, f 2 : X 2 → B be two proper morphisms with generic fibres X 1 , X 2 of dimensions d 1 , d 2 , where X 1 and X 2 are smooth; let r ∈ ‫ޚ‬ and let γ ∈ CH d 2 +r (X 1 × K X 2 ) be a Chow correspondence of degree r .Then for any i ≥ 0. In particular, f does not depend on f .Proof.First, (i) (resp.(ii)) follows from (3-9) by considering t γ (resp.by taking X 1 = X 2 = X , γ = X ).To prove (3-9), we may assume that γ is the class of an integral cycle ⊂ X 1 × K X 2 .
Let j i : X i → X i be the corresponding immersions, and ψ be the closure of in and it suffices to show that ψ * α ∈ CH i+r (X 1 ) 0 for any α ∈ CH i (X 2 ) 0 .Formula (3-10) shows that j * 1 (ψ * α) ∈ CH i+r num (X 1 ); the other condition follows from Lemma 3.5(b).□ Remark 3.7.If B is projective, Lemma 3.5(a) is sufficient for the proof of Proposition 3.6 by using (2-3), as in the proof of Proposition 2.5.
Proposition 3.8.The pairing (2-9) does not depend on the choice of f (we drop f from its notation from now on).Moreover, in the situation of Proposition 3.6 with r = 0, we have the identity Proof.As in the proof of Proposition 3.6, the first claim follows from the second by taking X 1 = X 2 = X , γ = X .For the second claim, we take γ and ψ as in the proof of Proposition 3.6.Then (3-11) follows from Lemma 3.3 applied to lifts α and β of α and β in CH i (X 2 ) 0 and CH d 1 −i+1 (X 1 ) 0 , respectively.□ 3C.Base change.
Proposition 3.9.Consider a commutative diagram where f 1 , f 2 satisfy the hypotheses of Section 1, ḡ is finite surjective and g proper; we assume that the diagram of generic fibres, X 1 is Cartesian (in particular, g is generically finite).Then, for all i ≥ 0, one has: (iv) One has the identities for any i ≥ 0 and any 1 and Z = {b}.Let β ∈ CH δ+i−1 (X 1,Z ), f 1,Z : X 1,Z → Z be the restriction of f 1 and ι 1 : X 1,Z → X 1 be the closed immersion: we need to prove that ( f 1,Z ) * (g * α • ι 1 β) = 0. Let T = ḡ(Z ) and h : Z → T be the (finite surjective) projection: it suffices to show that h * ( f 1,Z ) * (g * α • ι 1 β) = 0 ∈ CH 0 (T ).This follows from the computation 0 where h : X 1,Z → X 2,T is the restriction of g and ι 2 is the inclusion X 2,T → X 2 , in which (ii) The inclusion j * 2 Z = {b} and ι 2 : X 2,Z → X 2 , f 2,Z : X 2,Z → Z be the inclusion and the projection.Let α ∈ CH i (X 1 ) 0 and β ∈ CH δ+i−1 (X 2,Z ): we need to prove that By Lemma 2.9, we have where h and h are the restrictions of g and ḡ, gives the identity of push-forwards Therefore, it suffices to prove the identity (projection formula) For this, consider the commutative diagram of Cartesian squares Applying Lemma 2.1 to the two bottom squares yields first We are now left to show the identity where the right hand side stems from the top part of the diagram (with vertical arrows pointing upwards).
, both by Lemma 2.1.Here, t γ denotes the transpose of a graph (graph composed with the switch of factors).Finally, This follows from (i) and (ii) by the projection formula g * g * = deg(g) (generic degree), and Lemma 2.4.
Proof.It suffices to show the first claim.We proceed in several steps.
(1) Suppose Therefore the claim for X implies the claim for X ′ , and conversely in the latter case.
(3) In general, let B be a compactification of B and X f − → B a projective morphism extending f (in the sense that X = X × B B).
Again by [de Jong 1996, Theorem 4.1], alter X ′ into a smooth projective k-variety X 1 .We are now in the situation of (2).
• By Remarks 1.2(b), the alteration B 1 → B becomes flat, hence finite, after removing from B a closed subset F of codimension ≥ 2. Let B ′ = B − F and X ′ , B ′ 1 , X ′ 1 be the corresponding base changes of X , B 1 and X 1 .By (2), the claim is true for X 1 ; therefore it is also true for X ′ 1 by (1).By Proposition 3.9 (i), (ii), the projection X ′ 1 → X ′ induces maps between CH i (X ′ )/ CH i (X ′ ) 0 and CH i (X ′ 1 )/ CH i (X ′ 1 ) 0 , whose composition is multiplication by [K 1 : K ].Since CH i (X ′ )/ CH i (X ′ ) 0 is torsion-free by Lemma 2.4, it is finitely generated, and so is CH i (X )/ CH i (X ) 0 by reapplying (1).□ Remark 3.12.Proposition 2.6 gives a more direct proof of Proposition 3.11 in characteristic 0, by the comparison theorem between Betti and l-adic cohomology.

3E.
A vanishing result.Let l be a prime number invertible in k.For any smooth k-variety V , there are cycle class maps with values in Jannsen's continuous étale cohomology which are compatible with pull-backs, push-forwards and products [Jannsen 1988, (3.25) and (6.14)]. 3emma 3.13.Suppose k finitely generated.Then the composition of cl 1 with the projection where the bottom map is part of the Milnor exact sequence of [Jannsen 1988, (3.16)] and CH 1 (V ) ∧ is the l-adic completion of CH 1 (V ).The Kummer exact sequences imply the injectivity of (cl 1 ) ∧ .Since k is finitely generated, CH 1 (V ) is a finitely generated abelian group, which implies that α has finite kernel of order prime to l. Hence the same holds for cl 1 .On the other hand, the choice of a 0-cycle of nonzero degree on V (e.g., a closed point), plus transfer, provide a map ρ : is multiplication by some integer m > 0. Since CH 1 (k) = 0, the naturality of the cycle class map implies that ρ • cl 1 = 0. Hence the lemma.□ The following proposition will be used in the proof of Proposition 6.8.
Proof.We may assume k to be the perfect closure of a finitely generated field.We use the spectral sequences of [Jannsen 1988, Theorem (3.3)] They are compatible with the action of correspondences, in particular with products and push-forwards.Thus, if F • H cont is the filtration on H cont induced by the spectral sequence, we have . We conclude by Lemma 3.13.□ Question 3.15.When B is projective, can one prove Proposition 3.14 with CH l replaced by CH num , without assuming the standard conjectures?
3F. Local height pairing.In this context, there is not much to say.Let f be as in Section 1.Let [Beȋlinson 1987, Lemma 2.0.1]).One may then extend by bilinearity and get an expression of ⟨ , ⟩ as the class of a divisor.We leave it to the interested reader to refine Lemma 3.3 to this local height pairing in the style of [Bloch 1984, (A.2)].

Extension to the general case
Let X be regular, connected and proper over K .In the previous section, we defined subgroups CH i (X ) 0 ⊂ CH i (X ) and pairings (1) assuming the existence of a k-smooth model X of X , proper over B. Proof.Start from an integral proper model f : X → B of X/K .Let U ⊆ X be the regular locus of X /k: it is open [EGA IV 2 1965, corollaire 6.12.6] and since X is regular, we have X ⊂ U .By hypothesis, we may find X 1 regular over k and a projective morphism π : X 1 → X such that π |π −1 (U ) : π −1 (U ) → U is an isomorphism.Then the immersion X → X lifts to X → X 1 , and X 1 is the desired smooth model of X (since k is assumed to be perfect).□ 4B.Positive characteristic.Here we cannot directly use de Jong's theorem [1996] to replace Hironaka resolution, because there is no control in this theorem on the centre of the alteration.Instead we must proceed more carefully.
Definition 4.2.Let X be an integral proper K-scheme.
(a) X is good (relatively to B) if it admits a k-regular proper model by sending a pair (ϕ, N ) to ψ; = N −1 ϕ; its image is contained in the subgroup formed of those homomorphisms ψ : A → B ⊗ R such that ψ(A) ⊆ N −1 B for some N ̸ = 0, with B = B/torsion.If B is torsion-free, ρ is injective with the above image.
(b) In any category, the commutativity of a diagram (i.e., the equality of two arrows) is equivalent to the commutativity of a family of diagrams of sets, thanks to Yoneda's lemma.In the category of modules over a ring R, one can test such commutativity on elements, because the R-module R is a generator.
In the sequel, we shall extend identities such as (3-11), (3-13) and (3-14) to Ab ⊗ R. However this category is not Grothendieck (note that abelian groups with finite exponent are not closed under infinite direct sums), so reasoning with "elements" is abusive.Writing out the above identities as commutative diagrams in Ab is straightforward, but cumbersome.(For example,(3)(4)(5)(6)(7)(8)(9)(10)(11) We shall therefore sometimes make the abuse of talking of such identities in Ab ⊗ R when we mean the corresponding commutative diagrams.
In Theorem 4.14, we shall use a local-to-global result for these localisations (Corollary 4.8 below).
Theorem 4.6.Let H be a module over an integral domain R with quotient field Q. Suppose given, for each maximal ideal m ⊂ R, an element f m ∈ H m , all of which become equal in Q ⊗ R H . Then there exists at most one element f ∈ H which becomes equal to f m in H m for every m; f exists provided (i) H is torsion free, or (ii) R is Noetherian and S = Supp(M tors ) is a finite set of maximal ideals.
(Counterexample without Hypothesis (ii): R = ‫,ޚ‬ H = m ‫/ޚ‬m, f m = 1 m .)Proof.Uniqueness.Let f, f ′ verifying the condition.Then f and f ′ become equal in H m for all m.This means that, for every m, there exists Existence.We may write f m = r −1 m fm with fm ∈ H and r m ∈ R − m; again, the r m generate the unit ideal of R.
In case (ii), write T = H tors for notational simplicity.Considering H/T , we find Claim 4.7.The monomorphism T → m∈S T m is surjective.
Proof.For each m ∈ S, let T m = Ker(T → m ′ ̸ =m T m ′ ): we must show that T = T m .Let t ∈ T ; by assumption, the radical of Ann(t) (the annihilator of t) is of the form m∈S ′ m for a subset S ′ of S. By [Bourbaki 1985, IV.2.5, proposition 9] Coming back to the proof of case (ii), the claim yields an element t ∈ T such that t m = 1 m ⊗ f 0 − f m for all m ∈ S; then f = f 0 − t yields the desired element.□ Corollary 4.8.Let A, B ∈ Ab and R be a subring of ‫.ޑ‬ Suppose given, for each prime number l not invertible in R, a morphism f l : A → B in Ab ⊗ ‫ޚ‬ (l) , all of which become equal in Ab ⊗ ‫.ޑ‬ Then there exists at most one morphism f : A → B in Ab ⊗ R which becomes equal to f l in Ab ⊗ ‫ޚ‬ (l) for every l; f exists provided B is l-torsion free for almost all l not invertible in R.
Proof.Apply Theorem 4.6 to H = Hom(A, B)⊗ R, noting that the hypothesis on B implies the hypothesis on H . □ 4D.p-covers.
Definition 4.9.Let X be an integral proper K-scheme.A p-cover of X is a finite family (π l : X l → X ), indexed by prime numbers l ̸ = p and such that (i) for each l, π l is an admissible alteration of generic degree d l prime to l; (ii) gcd l (d l ) is a power of p.
(b) Given two p-covers (π l ), (π ′ l ), there exists a third p-cover (π ′′ l ) such that, for each l, π ′′ l factors through π l and π ′ l .(c) Given a p-cover (π l ) and an admissible morphism f 1 : X 1 → X , there exists a p-cover (π 1,l ) of X 1 such that the composition X 1,l → X 1 → X factors through X l for each l.
Proof.(a) We use Gabber's refinement of de Jong's alteration theorem [Illusie and Temkin 2014, Theorem 2.1]: given a model X of X and a prime number l ̸ = p, we may find an alteration X l → X with X l regular (hence smooth over k) and of generic degree d l prime to l; the induced alteration π l : X l → X is then admissible of generic degree d l .Considering the other prime divisors of d l different from p, we may find a finite number of l and π l such that the gcd of the d l is a power of p. Definition 4.11.We set Proposition 4.12.(a) If X is regular, CH i (X )/ CH i (X ) 0 is an extension of a finitely generated abelian group by a torsion group of p-power exponent, and CH i (X )/ CH i (X ) [0] is finitely generated with primeto-p torsion.
(b) Let (π l ) be a p-cover of X , and let α ∈ CH i (X ).Then α ∈ CH i (X ) [0] if and only if π * l α ∈ CH i (X l ) [0]  for each l.
(c) Propositions 3.6 and 3.9 (i), (ii), (iii) extend to all regular X after replacing CH i (X ) 0 by CH i (X ) [0] .Proof.(a) Given a p-cover (π l ), since (π l ) * π * l is multiplication by d l for each l, Ker(CH i (X )/ CH i (X ) 0 → l CH i (X l )/ CH i (X l ) 0 ) is killed by a power of p, say p s , and the first claim follows from Proposition 3.11.The second follows by definition of CH i (X ) [0] .(b) The condition is necessary by definition; the converse follows from Proposition 4.10 (c), as in (a).
Proof.(a) Suppose first that k is the perfect closure of a field k 0 finitely generated over ‫ކ‬ p , and that B = B 0 ⊗ k 0 k for some smooth k 0 -variety B 0 .Then CH 1 (B 0 ) is a finitely generated abelian group [Kahn 2006], and CH 1 (B 0 ) ⊗ ‫/1[ޚ‬ p] does not change under purely inseparable extensions; in particular, CH 1 (B) ⊗ ‫/1[ޚ‬ p] has finite torsion and a fortiori verifies the hypothesis of Corollary 4.8.The result then follows from this theorem and Lemma 4.13.
In general, the situation is defined over such a subfield of k, so reduces to the first case.
(b) Let X 1 , X 2 be (proper) regular, and let γ ∈ CH dim X 2 (X 1 × K X 2 ).We need to prove the analogue of (3-11), where ⟨ , ⟩ i is the height pairing of X i .By the uniqueness statement of Corollary 4.8, it suffices to prove this identity after localising at l for all l ̸ = p.Let π i : X i,l → X i (i = 1, 2) be two admissible alterations of generic degrees d i prime to l, and let By Lemma 4.13, we have, with obvious notation, where (a) used (3-11) for γ l .The identity of Proposition 3.9(iv) is extended in similar fashion.□ We shall use the following fact in the proof of Theorem 6.2: Example 4.15.Suppose that X is an abelian variety.For a ∈ X (K ), write τ a for the translation by a.It yields a self-correspondence of degree 0 still denoted by τ a , and we have the obvious formula t τ a = τ −a .This yields the identity (see Remarks 4.5(b)) ⟨τ * a α, β⟩ = ⟨α, τ * −a β⟩ for (α, β) ∈ CH i (X ) [0]  × CH d+1−i (X ) [0] .
Remark 4.16.The functoriality of Proposition 4.12(c) means that the subgroups CH i (X ) [0] , for varying X and i, define an adequate equivalence relation on algebraic cycles with integral coefficients on smooth projective K-varieties.This adequate relation a priori depends on the choice of B, but see Conjecture 5.1 and Remarks 5.4 below.
4F. Extension to imperfect fields.Let X, K , B be as in the introduction, but relax the assumption that k is perfect; specifically, we assume k imperfect of characteristic p. Write k p (resp.K p , B p , X p for the perfect closure of k (resp.for We define CH i (X ) [0] as the inverse image of CH i (X p ) [0] under the pull-back morphism CH i (X ) → CH i (X p ).We claim that the pairing (4-1) for X p induces a similar pairing for X , with the same properties.
Since the homomorphism λ : CH 1 (B) → CH i (B p ) has p-primary torsion kernel and cokernel, this is trivial if we accept to replace CH 1 (B) by CH 1 (B) ⊗ ‫/1[ޚ‬ p] (note that Ker λ and Coker λ do not have finite exponent, so λ is not an isomorphism in Ab[1/ p]).We can avoid this, however, by observing that all constructions involved in constructing (4-1) for X p and proving its properties are defined over some finite subextension of k p /k.
• or X is "of abelian type" (i.e., its homological motive is isomorphic to a direct summand of the motive of an abelian variety).
Theorem 5.6.(a) One has Of course, (b) follows from (a) (using Matsusaka's theorem [1957] in the case i = 1).To prove (a), we first reduce to the case where X has a smooth model X as in Section 4: this is automatic if char k = 0 by Proposition 4.1, and if char k > 0 we first reduce to k perfect as in Section 4F, then we can use Proposition 4.10(a) and a transfer argument.
We now give ourselves a model f : X → B of X with X smooth.The proof is in two steps.
Step 1. Assume d = 1 and two sections c0 , c1 of f are given.Let c 0 , c 1 be their generic fibres and α Lemma 5.7.There exists an integer N > 0 such that N α ∈ CH 1 (X ) 0 .
Proof X ).We now need to find N > 0 and ξ ∈ Ker j * such that N α + ξ ∈ CH 1 (X ) 0 .We shall look for ξ in the form where ι b : X Z b → X is the inclusion (with Z b = {b} as usual) and each ξ b is a linear combination of classes of irreducible δ-dimensional components X λ Z b of X Z b (almost all ξ b will be 0).For this, I claim that the method of [Silverman 1994, III.8] extends to this case: The first thing to check is that the hypothesis of [loc.cit., Proposition III.8.3] is verified, namely that 1) .For simplicity, write Z and ι instead of Z b and ι b .Up to removing a proper closed subset from Z , we may assume it smooth.In the Cartesian square of the diagram and ι ′ is the inclusion Z → B, the top horizontal map g i is a regular embedding of codimension δ + 1 as the composite of the two regular embeddings Here we use that the embedding d i is regular [EGA IV 4 1967, proposition 19.1.1] where where the X λ Z are the irreducible components of X Z of dimension δ: this follows from [Fulton 1984, Example 1.8.1] by induction on the number of components.The second thing to observe is that the statement and proof of [Silverman 1994, Proposition III.8.2] apply verbatim, namely that the quadratic form α → ⟨ι * α, α⟩ b on CH δ (X Z ) is negative, with kernel generated by [X Z ].Indeed, this is a local computation so we can consider the fibre of X over Spec O B,b and simply apply the said proposition.(The fact that f * O X = O B , which is used in its proof, follows from the fact that X is geometrically connected since it has rational points, and that B is normal.) We can now find N and ξ just as in [Silverman 1994, Proposition III.8.3].□ Step 2. The general case.Let α ∈ CH i alg (X ).By [Achter et al. 2019, Lemma 3.8], there exist an integer s ≥ 0, a smooth projective K-curve C, two rational points c 0 , c 1 ∈ C(K ) and an element y ∈ CH i (C × X ) such that p s α = (c * 0 − c * 1 )y (recall that p is the exponential characteristic of k).(projection formula), and the intersection numbers of (P) with the components of X b are ≥ 0, this implies that P meets exactly one component of X b , with multiplicity 1.By definition, X (K ) 0 is the set of P such that (P) meets the same component of X b as (0) for all b ∈ B (1) .Equivalently, deg ((P) − (0)) • [X λ b ] = 0 for all b and all such components.By (2-3), this degree is none else than ⟨(P) − (0), X λ b ⟩ b , so we get that P ∈ X (K ) 0 ⇒ (P) − (0) ∈ CH 1 (X ) 0 ⇒ P − 0 ∈ CH 1 (X ) 0 .
(b) What we use here is that P = τ P • 0 (5-3) for all P ∈ X (K ), where τ P is the translation by P [Silverman 1994, Proposition III.9.1].This already implies that X (K ) 0 is a subgroup of X (K ).

The pairing in codimension 1
In this section, we assume X projective and geometrically irreducible.Recall that δ = trdeg(K /k) = dim B. We shall study the height pairing (4-1) for i = 1, in Ab ⊗ ‫;ޑ‬ note that CH i (X ) (0)  = CH i num (X ) for i = 1, d by Theorem 5.6.6A.A general result.We write T (X ) ⊂ CH d num (X ) = CH d (X ) 0 for the Albanese kernel.For an abelian K-variety A, write Tr K /k A for its K /k-trace and LN(A, K /k) = A(K )/(Tr K /k A)(k) for its Lang-Néron group: it is finitely generated by the Lang-Néron theorem [1959].We shall need the following classical fact: Lemma 6.1.The Albanese map a X : CH d (X ) 0 → Alb X (K ) has a cokernel of finite exponent.
(b) The pairing (2-1) makes sense for any b ∈ B (replacing CH δ+i−1 (Z ) by CH δ+i−r (Z ) if b ∈ B (r ) ), and defines an equivalence relation α ≡ b 0 if ⟨α, β⟩ b = 0 for any β (b) The statement means that ϕ defines a functor ϕ * : CHC(B) → CHC(C), given by fibre product.It is defined on objects by the smoothness of ϕ, and on morphisms because smooth morphisms are flat.To check that it respects composition involves chasing in the Cartesian cube obtained by pulling back the square of B-schemes (3-2) along the morphism C × B C → B, and then further pulling back along the diagonal δ ′ : C → C × B C; this latter operation is unnecessary if C is an open subset of B. The first step involves [Fulton 1984, Proposition 6.6] as in the proof of (a), to take care of the flat l.c.i morphisms C × B (X i × B X j ) → X i × B X j ; the second step uses the fact that δ ′ is a regular immersion.(c) This follows from (a), (b) and [Bloch 2010, Lemma IA.1], since U × B X is smooth over U for a suitable open subset U of B for X as in the statement.□ Remark 3.2.The associativity of the composition • is not proven in [Corti and Hanamura 2000].It will not be used here and is left to the reader.See nevertheless Remark 3.4.
(b) and (c) These are proven similarly to (a).□ 4E.The refined height pairing (characteristic p).