Integrality of the Chern character in small codimension

We prove an integrality property of the Chern character with values in Chow groups. As a consequence we obtain, for a prime number p, a construction of the p-1 first homological Steenrod operations on Chow groups modulo p and p-primary torsion, over an arbitrary field. We provide applications to the study of correspondences between algebraic varieties.


Introduction
The Grothendieck Riemann-Roch theorem of [BFM75] can be expressed as the existence of the homological Chern character, a collection of group morphisms ch i : K ′ 0 (X) → CH i (X) Q , which are compatible with projective push-forwards. Here X is a (possibly singular) algebraic variety, K ′ 0 (X) the Grothendieck group of coherent sheaves, and CH i (X) Q the Chow group of X of algebraic cycles of dimension i, with Qcoefficients. Compatibility with pull-backs is expressed by a Riemann-Roch type formula involving the Todd class.
Given a prime number p, one can formulate the following statement describing a p-integrality property of the homological Chern character.
Conjecture. Let X be a variety, and n a positive integer. Then p [n/(p−1)] · ch dim X−n [O X ] ∈ im(CH dim X−n (X) Z (p) → CH dim X−n (X) Q ).
The topological analog of this conjecture was stated and proved by Adams in [Ada61, Theorem 1]. A more elementary proof of this topological result was later given by Atiyah ([Ati66, Theorem 7.1]), under the quite restrictive assumption that X is a CW-complex whose homology has no torsion. Although our approach is more akin to Atiyah's than Adams's (where in particular Steenrod operations were used), no condition of such nature will appear.
In [Haub,Theorem 4.4], we proved the conjecture for p = 2 and n = 1. In the present article, we prove it for n < p(p − 1), and any p (section 3). In particular we recover the previously obtained result, using an argument of different nature.
We discuss in Appendix A how this result can be extended to the more general setting of a scheme of finite type over a regular base.
Let us mention that the conjecture is an immediate consequence of the formula expressing compatibility of the homological Chern character with pull-backs when X is smooth, or more generally a local complete intersection variety. This remark can be used to prove the conjecture in the presence of some form of resolution of singularities. Using a result of Gabber on existence of regular alterations, we prove the conjecture for all n, when the characteristic of the base field is not p.
The n-th Steenrod operation on Chow groups modulo p is an endomorphism of the Chow group with Z/p-coefficients which lowers dimension of cycles by n(p − 1). The Steenrod operations constitute an efficient tool for studying rationality properties of projective homogeneous varieties.
The classical constructions of the operations modulo p ( [Boi08], [Bro03], [EKM08], [Lev07], [Voe03]) do not work over a field of characteristic p. In [Haub], we constructed a weak form of the operation for p = 2 and n = 1, over any field. We later obtained the full version of this operation in [Hau10]. In the present paper, we construct a weak form of the n-th Steenrod operation modulo p, for p an arbitrary prime and n = 1, · · · , p − 1, over any field.
More precisely, assuming that the conjecture above holds true for n ≤ m(p −1), and given any (possibly singular) variety X over an arbitrary field, we construct for each integer i such that 0 < i ≤ m, a morphism of graded abelian groups T i : CH • (X) ⊗ Z/p → CH •−i(p−1) (X) ⊗ Z/p.
Here CH d is the quotient of the Chow group CH d by its torsion subgroup. These morphisms are compatible with proper push-forwards, external products, and extension of the base field. We also provide a Riemann-Roch type formula expressing compatibility with pull-backs along local complete intersection morphisms.
Thus we construct operations T 1 , · · · , T p−1 unconditionally, and operations T i for all i > 0 when the characteristic of the base field is not p. The operation T i corresponds -up to a sign -to the i-th Steenrod operation modulo p when i ≤ p. More generally the total operation i T i corresponds to the inverse of the total Steenrod operation j S j . We explain how to recover the operation induced by S j on reduced Chow groups (the reduced Steenrod operation) from the operations T i .
In the last part of this paper, we prove the degree formula corresponding to the operations T i . Since we only proved the conjecture in general for small values of n, we need to work with varieties of dimension ≤ p(p − 1) (note that the dimension p(p − 1) is allowed). We first give a bound on the p-adic valuation of the index of a projective variety in terms of its dimension and of the Euler characteristic of its structure sheaf. Then we deduce from the degree formula that the Severi-Brauer

Notations
Varieties. We fix a base field k. A variety is a finite type, separated, quasiprojective scheme over k. A morphism of varieties is a morphism of schemes over k. We will denote by Var/k this category. We also denote by Proj/k the category with the same objects, but with morphisms only those which are projective. The function field of an integral variety X shall be denoted by k(X).
Integral elements. Given a rational number α, we write [α] for the greatest integer ≤ α.
Let A be an abelian group. For any commutative ring R we write A R for A⊗ Z R, and, if S is another commutative ring containing R, we write Note that if A is a ring, then A R⊂S is a subring of A S .
Let p be a prime number. When R = Z (p) (the subring of Q consisting of fractions with denominator prime to p), S = Q, and x ∈ A Q − {0}, we denote by v p (x) the greatest integer r such that If A is commutative ring, M an A-module, a ∈ A Q , and m ∈ M Q , we have Grothendieck groups of schemes. We denote by K 0 (−) the presheaf of rings on Var/k which associates to a variety X the Grothendieck ring of locally-free coherent O X -modules. We denote by K ′ 0 (−) the functor Proj/k → Ab which associates to a variety X the Grothendieck group of coherent O X -modules. Here, and in the rest of the paper, the notation Ab stands for the category of abelian groups.
Local complete intersection morphisms. (See [Ful98, Appendix B.7.6]) These are the morphisms of varieties f : Y → X which can factored as p • i, with i a regular closed embedding, and p a smooth morphism. Such a morphism admits a virtual tangent bundle where T p is the tangent bundle of p, and N i the normal bundle of i. This element does not depend on the choice of the factorization. There are pull-backs f * : CH(X) → CH(Y ) and f * : We call a variety X a local complete intersection variety if its structural morphism x : X → Spec(k) is a local complete intersection morphism. We denote by T X ∈ K 0 (X) the virtual tangent bundle T x of x.
Topological filtration. Let X be a variety, and d an integer. We denote by K ′ 0 (X) (d) the subgroup of K ′ 0 (X) generated by those O X -modules whose support has dimension ≤ d. It coincides with the subgroup generated by elements of type i * [O Z ], where i : Z ֒→ X is a closed embedding, and dim Z ≤ d.
When f : Y → X is a projective morphism, we have . When f : Y → X is a local complete intersection morphism of relative dimension n, we have ([Gil05, Theorem 83 and Lemma 84]) . The associated graded group is denoted by There is a natural transformation of functors Proj/k → Ab (2) ϕ − : CH(−) → gr K ′ 0 (−) which respects the gradings ([Ful98, Example 15.5]). It is moreover compatible with pull-backs along local complete intersection morphisms, external products, and extension of the base field (see [Hauc,Section 3] Chern character. The homological Chern character is a natural transformation of functors Proj → Ab ch : K ′ 0 (−) → CH(−) Q . This is the map τ of [Ful98, Theorem 18.3]. Its component of dimension i will be Adams operations. (See [Sou85]) Let l ∈ Z − {0}. The l-th homological Adams operation is a natural transformation of functors Proj → Ab Part 1.

Integrality of the Chern character
We fix an prime number p. We say that p-integrality holds in codimension ≤ c if the following statement is true. Statement 1.1. Let X be a variety and d ≥ 0 an integer. Then for all integers n such that 0 ≤ n ≤ c, and all elements x ∈ K ′ 0 (X) (d) , we have v p (ch d−n (x)) ≥ − n p − 1 .
We prove in section 3 that p-integrality holds in codimension ≤ p(p − 1) − 1. Moreover, we prove in section 4 that p-integrality holds in codimension ≤ m for all m when the characteristic of the base field is not p. We expect that this assumption is superfluous.

Adams operations and Chern character
We fix an element l ∈ Z − {0}. The sole purpose of this section is to prove, in Proposition 2.4, the relation for n ≥ 0, This is certainly very classical, at least so is the cohomological counterpart of this statement. Due to the lack of suitable reference, we give some details in this section. We shall need to introduce some notations, but except in Appendix A, the only thing used in the sequel will be the relation (3).
The cohomological (i.e. usual) l-th Adams operation is a morphism of presheaves of rings on Var/k sending the class of a line bundle to its l-th tensor power.
When X is connected, the ring K 0 (X) has an augmentation rank : K 0 (X) → Z sending the class of a vector bundle to its rank.
Bott's class. Consider the Laurent polynomial in the variable x There is a unique morphism of presheaves of abelian groups on Var/k such that for all line bundles L, Note that this does not follow [Sou85], where θ l (L) = t l [L].
K-theory with supports. When X is a closed subvariety of a variety W , the notation K X 0 (W ) stands for the Grothendieck group of complexes of locally free O W -modules, which are acyclic off X, modulo the classes of acyclic complexes. It has a natural structure of a K 0 (W )-module. We denote by CH(X → W ) the bivariant Chow group (see [Ful98,Definition 17.1]). There are maps Assume that W is smooth, and that x ∈ K X 0 (W ). Then the homological Chern character is given by the formula ([Ful98, Theorem 18.3, (3)]), We now prove a few lemmas that will be used to obtain (3).
Proof. Assume that L is a line bundle. Then Thus the claimed formula holds when u is the class of a line bundle. Both sides of this formula define morphisms of presheaves of abelian groups on Var/k Therefore we can conclude with the splitting principle.
Lemma 2.2. Let X be a variety. We have in CH(X) Q , for all u ∈ K 0 (X), Proof. For a line bundle L, we have proving the formula for u the class of a line bundle. The lemma follows as above from the splitting principle.
Proposition 2.4. Let n ≥ 0 be an integer. We have ch n •ψ l = l −n · ch n .
Proof. It will be enough to prove that the two sides of the formula have the same effect on any element of K ′ 0 (X), for X a connected variety. Choose a closed embedding X ֒→ W into a smooth connected variety W of dimension d. Then, The first equality is (4) and (5) . The second follows from [Ful98, Proposition 18.1, (c)]. For the third one, we use Lemma 2.1. The fourth one uses Lemma 2.2 and Lemma 2.3. The last one is again (4).

Integrality in arbitrary characteristic
Proof. When l = p is a prime number, this is [Hauc, Lemma 6.1]. Actually we obtained this result using the formula which is true when p is an arbitrary element of Z − {0}. Therefore we can obtain the proposition using the proof of [Hauc, Lemma 6.1].
Alternatively, one can assume that X is integral and d = dim X. Then we have an exact sequence Proof. We must prove Statement 1.1 with c = p(p − 1) − 1. For all d, the case n = 0 immediately follows from [Ful98, Theorem 18.3, (5)]. In particular the theorem is proved when d = 0. We can therefore proceed by induction on d, and assume that n > 0.
For all i ≥ 1, define an integer Since 0 < n < p(p − 1), we can use Lemma 3.3 below to choose a integer l, prime to p, such that , and m ≥ 0. Then applying l d+m · ch d−n to the relation above, and using Proposition 2.4, we obtain in Because of (7), we get v p (ch d−n (x)) = v p (ch d−n (α)) + a p (n).
Lemma 3.3. Let p be a prime number. Then there exists an integer l, prime to p, with the following properties: Proof. Take for l an integer whose class modulo p 2 generates the cyclic group (Z/p 2 ) × . Then one checks easily that the class of l modulo p generates the cyclic group (Z/p) × .
Remark 3.4. In contrast with [Haub] where the normalization procedure was used, we do not use geometric arguments here. We only use basic relations between Adams operations, Chern character, and filtration on K-theory. Using [AA66, (1.5)] in place of Proposition 3.1, and [Ada62, Theorem 5.1, (vi)] in place of Proposition 2.4, one could proceed as above to replace the assumption "X is without torsion" by "n < p(p − 1)" in [Ati66, Theorem 7.1], thus giving an elementary proof of the p-primary part of [Ada61, Theorem 1], for small n.
Note also that the approach of [Haub] is limited by nature to schemes whose normalization morphism is of finite type, while Theorem 3.2 is valid for more general schemes (see Appendix A).

Consequences of Gabber's Theorem
In this section, we use the theorem of Gabber asserting that, over a field of characteristic different from p, any integral variety X admits a regular alteration of degree prime to p. This means that there exists a projective morphism Z → X, generically finite of degree prime to p, with Z regular. Some details can be found in [Gab] and [Ill].
If d ≥ 0 is an integer, the d-th Todd number is Proposition 4.1. Let X be a local complete intersection variety. Then, for all n Proof. Let x : X → Spec(k) be the structural morphism of X. Then by [Ful98,Theorem 18.3, (4) and (5)], Then the proposition follows from Lemma 6.3 below.
Theorem 4.2. Let p be a prime different from the characteristic of the base field. Then p-integrality holds in codimension ≤ c, for any c.
Proof. We must prove Statement 1.1 with c arbitrary large. We proceed by induction on d, the case d = 0 being obvious. It is enough to consider the case X integral of dimension d, and x = [O X ]. Using Gabber's theorem, we choose f : Z → X a projective morphism, with Z a regular connected variety of dimension d, and such that [k(Z) : k(X)] = λ is prime to p. Then there is an element We conclude using Proposition 4.1 for Z, and the induction hypothesis on δ.
The combination of Theorem 3.2 and Theorem 4.2 implies: Corollary 4.3. Let X be a variety over a field of characteristic p, and d be an integer. Then for all integers n such that 0 ≤ n < p(p − 1), and all elements Part 2. Operations on Chow groups

Construction
We proceed as in [Haub] to derive operations from the integrality property of the Chern character stated above. This follows [Ati66, Section 7].
Let A be an abelian group. Then the natural map is an isomorphism. This group can also be described as A modulo p and p-primary torsion. By this we mean the quotient of A by p · A + T , where T is the subgroup of elements a ∈ A such that p n · a = 0 for some integer n > 0.
Using Statement 1.1 for d = q − 1 and n = i(p − 1) − 1, we see that this element belongs to p · CH q−i(p−1) (X) Z (p) ⊂Q . This gives a natural transformation of functors Proj → Ab Composing on the left with the map (2), and on the right with the inverse of the map (9) with A = CH q−i(p−1) (−), we obtain a natural transformation of functors Proj → Ab It will be convenient to agree that is the quotient map modulo the image modulo p of p-primary torsion.

The inverse Todd genus
We consider the power series in the variable x There is a unique morphism of presheaves of abelian groups on Var/k such that for L a line bundle, Individual components are denoted Using the terminology of [Pan04], this class will be the inverse Todd genus of the operation T = i T i constructed in the previous section. In this section we describe a relation between the characteristic class r (p) and the Todd class.
Remark 6.1. The inverse Todd genus of the usual Steenrod operation i S i is defined by the power series W (p) (x) = 1 + x p−1 . One easily checks that x · R (p) (x) is the left inverse of x · W (p) (x) for the law given by composition of power series with coefficients in Z/p.
We begin by stating an easy argument that we will use repeatedly below.
Lemma 6.2. Let A be a commutative ring, M an A-module, a u ∈ A Q and m u ∈ M Q for u = 0, · · · , i(p − 1) satisfying Then we have an equality in M Z (p) ⊂Q ⊗ Z/p, Proof. We compute, for all u, using (1), This integer is always ≥ −i, hence the elements delimited by parenthesis in the formula of the statement belong to the submodule M Z (p) ⊂Q ⊂ M Q . Moreover when u is not divisible by p − 1, this integer is > −i. This gives the requested formula modulo p.
Lemma 6.3. We have, for all integers n, and all x ∈ K 0 (X) v p (Td n (x)) ≥ − n p − 1 .
Proof. Let X be a variety. Then the set of elements 1 + y with We have a morphism of presheaves of abelian groups on Var/k When L is a line bundle, we have by definition of the Todd class, Then ν vanishes on the classes of line bundles, because of Lemma 6.5 below. The splitting principle allows us to conclude.
Proposition 6.4. Let j ≥ 0 be an integer, X a variety, and x ∈ K 0 (X). Then the elements p j · Td j(p−1) (−x) and r By Lemma 6.3, we can consider the association It is clearly compatible with arbitrary pull-backs. It sends sums to products because the total Todd class has the same property, and because of Lemma 6.2 and Lemma 6.3.
Let L be a line bundle. Then by definition of the Todd class, we have Using Lemma 6.5 below, we see that the j-th coefficient is an element of Z (p) . Moreover we see that it belongs to pZ (p) if and only if j(p − 1) + 1 is not a power of p. Otherwise, when say j(p − 1) + 1 = p i , we see using Lemma 6.6 below, that the j-th coefficient is (−1) i modulo pZ (p) . In view of (10), the characteristic class ρ coincides with r (p) on the classes of line bundles, and the claim follows from the splitting principle. Lemma 6.6. Let i ≥ 0 be an integer, and p a prime number. We have Proof. We proceed by induction on i. For i = 0 this is immediate. Let i ≥ 1 and consider the set U i = {1, 2, · · · , p i − 1, p i }. The product of the all the integers sitting between two consecutive multiples of p is congruent to −1 modulo p, by Wilson's Theorem. Thus the product of all elements of U i which are non-divisible by p is congruent modulo p to (−1) p i−1 = −1. The product of the p i−1 elements of U i which are divisible by p is p p i−1 · (p i−1 )!. Hence we have By the induction hypothesis, this is (−1) i , as requested.

Properties
Let p be a prime number. In this section, we assume that integrality holds in codimension ≤ m(p−1), and describe some functorial properties of the operations T 0 , · · · , T m constructed in section 5.
Proposition 7.1. Let f : Y → X be a local complete intersection morphism. Let i an integer such that 0 ≤ i ≤ m. Then we have Proof. We have, by [Ful98,Theorem 18.3, (4)], In view of Lemma 6.3, Lemma 6.2, and Theorem 3.2, we have, in the group Since the map (2) is compatible with f * , Proposition 6.4 allows us to conclude. Applying Proposition 7.1 to the structural morphism of a local complete intersection variety, we obtain the formula computing the effect of the operation T i on regular cycles.
Corollary 7.2. Let X be a local complete intersection variety (e.g. a regular variety). Let i an integer such that 0 ≤ i ≤ m. Then Proposition 7.3. Let X and Y be two varieties over the same field. Then for x ∈ CH d (X), and y ∈ CH e (Y ), and 0 ≤ i ≤ m, we have

By Lemma 6.2, this gives in CH
We can conclude using the fact that ϕ is compatible with external products.
Assume that X is a local complete intersection variety. Define, for i ≤ m, the i-th cohomological operation T i by the formula Proposition 7.4. Let X and Y be local complete intersection varieties over a common field, and i an integer such that 0 < i ≤ m.
(i) For x ∈ CH(X), y ∈ CH(Y ), we have (ii) When X is smooth, x, y ∈ CH(X), we have (iii) The operation T 0 is the identity.
(v) Assume that X is smooth, and let x ∈ CH 1 (X). Then Proof. Statement (i) follows from Proposition 7.3, statement (vi) from Proposition 7.1, and (vii) from the fact that the operations T i are compatible with projective pushforwards. Then (ii) follows from (i) and (vi), statement (iv) from (vi) and the obvious computation for X = Spec(k). Statement (iii) is a consequence of the definition of T 0 . We now prove (v). We can assume that x = [Z] for some integral closed subvariety Z of X. Since X is locally factorial, Z is a locally principal divisor. In particular the closed embedding f : Z ֒→ X is regular, and Z is a local complete intersection variety. Then, using (iv) and (vii), where T f = −f * [O X (Z)], we obtain for i > 0, This is zero when i > 1 by (iv). When i = 1, this is, by (iii), Finally, we mention Proposition 7.5. Let L/k be a field extension. Then, for all varieties X, all x ∈ CH(X), and i such that 0 ≤ i ≤ m, we have Proof. This follows from [Haub, (2) and Theorem 3.1, (d)].

Relation with Steenrod operations
In this section we compare the operations T i with the Steenrod operations constructed by other methods. We therefore make the assumption that the characteristic of the base field is different from p.
Consider the i-th homological Steenrod operation of, say, [Bro03] S i : CH • (−) ⊗ Z/p → CH •−i(p−1) (−) ⊗ Z/p, and write S = i S i . Since S 0 = id, we have S = id +f , where for every variety X the induced endomorphism f X of CH(X) ⊗ Z/p is nilpotent. We write T ′X for the left inverse of S X ; this defines a natural transformation of functors Var/k → Ab with components We have T ′ 0 = id and, for i > 0, one can express T ′ i inductively as Proposition 8.1. We have for i ≤ p T ′ i = (−1) i · S i . Proposition 8.2. Assume that X is a smooth variety. Then Proof of Proposition 8.1 and Proposition 8.2. Let Z be a variety. It is proven in [Mer03, Lemma 5.2 and Proposition 3.1] that the homological Steenrod operations S • induce an action of the Steenrod Z/p-algebra S on CH(Z)⊗Z/p. The operation S j acts as the element P j ∈ S (in case p = 2 we write P j for Sq 2j ) . Using (11), we see that the action of T ′ j on CH(Z) ⊗ Z/p coincides with the action of the conjugate c(P j ) of P j , in the sense of [Mil58,§7]. Let b i for i ∈ N − {0} be indeterminates. If R = (r 1 , r 2 , · · · ) is a sequence of non-negative integers, we write b R for the product b 1 r 1 b 2 r 2 · · · . By [Mil58, Corollary 6], the morphism We obtain Proposition 8.1 by noticing that for i ≤ p the element P i is the only basis element of type P R of the same dimension as c(P i ).
Moreover by [Mer03, Proposition 5.3, (2)], we obtain where d is the total Chern class R c R · b R evaluated at b i = (−1) i . Inspection of the value of d on line bundles reveals that d = r (p) , establishing Proposition 8.2.
Proposition 8.3. Assume that the base field is of characteristic different from p. Let i be any integer ≥ 0. Then we have a commutative diagram Proof. By Lemma 8.4 below and Proposition 7.5, we can assume that the base field is perfect. Since both operations are compatible with proper push-forwards, it will be enough to compare their values in CH d−i(p−1) (X) Z⊂Q ⊗ Z/p on the class [X], when X is an integral variety of dimension d. Using Gabber's theorem, we find a projective morphism f : Z → X, with Z smooth of dimension d, and f dominant with [k(Z) : k(X)] = λ an integer prime to p. Let ν ∈ (Z/p) × be the inverse modulo p of λ. Then, using Corollary 7.2, A similar computation, and Proposition 8.2, show that T ′ i [X] has the same value, as requested.
Lemma 8.4. Let X be a variety over a field k of characteristic = p. Let F/k be a perfect closure. Then scalars extension induces an injective morphism CH(X) Z⊂Q ⊗ Z/p → CH(X F ) Z⊂Q ⊗ Z/p.
Proof. Let L/k be a finite field extension. A transfer argument shows that CH(X) Q → CH(X L ) Q is injective, hence so is CH(X) Z⊂Q → CH(X L ) Z⊂Q , and therefore so is . When the degree of the extension L/k is prime to p, the transfer induces a splitting of this monomorphism, so that By [Sri96,Theorem 5.20 and Proposition 5.26], the group CH(X F ) is the direct limit of the groups CH(X L ) with L/k finite subextension of F/k. Such an extension L/k is of degree prime to p because it is a purely inseparable extension of a field of characteristic = p. Using the fact that the functor A → A Z⊂Q ⊗ Z/p commutes with any exact functor commuting with tensor products, we see that the natural map is an isomorphism. The lemma follows. K(X) = ker(CH(X) → gr K ′ 0 (X)). Then K(X) consists of torsion elements. Let K p (X) be its image in CH(X) ⊗Z/p. One can ask the following question: "what is the set E of operations Q on Chow groups modulo p such that µ • Q vanishes on K p (X), for all X?" By construction, we have T i (K p (X)) = 0, hence a consequence of Proposition 8.3 is that T ′ i ∈ E, for all i. On the other hand when p = 2 and j > 1, then the cohomological Steenrod square S 2 j −1 does not belong to E (see [Hau10,Remark 8.4]).
This can also be used to obtain informations on gr K ′ 0 (X). For example, a direct consequence of Proposition 8.3 is that gr 2 K ′ 0 (D) is contains non-zero 3-torsion, where D is the variety of [Sem08] (over a field of characteristic zero).
Remark 8.6. One can use the operations T i to recover some of the results of [Hauc]. Assume that X is a variety and L/k an extension of the base field such that CH(X L ) is torsion-free. Then T i is an endomorphism of CH(X L ) ⊗ Z/p. We set S 0 = id, and define inductively, for n > 0 and x ∈ CH(X L ) ⊗ Z/p, Let Ch(X) = im(CH(X) ⊗ Z/p → CH(X L ) ⊗ Z/p) be the reduced Chow group modulo p. Since the operations T i commute with extension of the base field, they send the subgroup Ch(X) ⊂ CH(X L ) ⊗ Z/p to itself. Therefore the association x → S n (x) induces an operation of this subgroup, the n-th reduced Steenrod operation.

Part 3. Degree formula and applications
We now describe some consequences of the results obtained above. In order to make unconditional statements, we restrict ourselves to applications of the fact that p-integrality holds in codimension ≤ p(p − 1) − 1 (Theorem 3.2).
We fix prime number p. When X is a projective variety, we let n X be the positive generator of the image of the degree map CH(X) → Z. We also set n X (p) = p vp(n X ) .
Let x : X → Spec(k) be the structural morphism of X. We denote by χ(O X ) the integer corresponding to x * [O X ] under the identification K ′ 0 (Spec(k)) = Z. In other words

Index and Euler characteristic
We begin by giving a restriction on the possible values of the index n X of a projective variety X, in terms of two integers which do not change under field extensions: the dimension of X and the Euler characteristic χ(O X ).
Proposition 9.1. Let X be a projective variety of dimension < p(p − 1). Then Proof. We have, by Theorem 3.2, which means that there is an integral zero-cycle x ∈ CH 0 (X), and an integer λ prime to p such that . Taking degrees, we obtain n X |λ · p [dim X/(p−1)] · χ(O X ), whence the result.

Correspondences modulo p
Let X and Y be varieties, with Y projective and X integral. A correspondence α : X Y is an element α ∈ CH(X × Y ). The multiplicity of α is the image of α under the composite 10.1. Strong p-incompressibility. Let p be a prime number. Following [Kar10, p.150], we call an integral projective variety X strongly p-incompressible if the existence of a correspondence X Y of multiplicity prime to p, with Y an integral projective variety of dimension ≤ dim X and such that v p (n Y ) ≥ v p (n X ), implies that dim Y = dim X, and that there exists a correspondence Y X of multiplicity prime to p.
Proposition 10.1. Let p be a prime number, and i an integer such that 0 < i ≤ p. Let X be an integral projective variety with v p (n X ) ≥ i. Assume that χ(O X ) is prime to p. Then dim X ≥ i(p − 1). In case of equality, X is strongly pincompressible.
Proof. Assume that dim X < i(p − 1). Then in particular dim X < p(p − 1), and Proposition 9.1 gives a contradiction. Hence dim X ≥ i(p − 1). Now we assume that dim X = i(p − 1). We proceed as in the proof of [Mer03, Theorem 7.2] -but as we include the case i = p, we cannot use the relation p · t p (X) = 0.
Let Y be an integral projective variety of dimension ≤ dim X, with a correspondence X Y of multiplicity prime to p, and such that v p (n Y ) ≥ v p (n X ). We can assume that there is an integral variety Z with dim Z = dim X, and two projective morphisms f : Z → X and g : Z → Y ֒→ Y ′ , such that deg f is prime to p. Here Y ′ is the product of Y with a projective space of appropriate dimension: in particular dim Y ′ = dim X and n Y ′ = n Y . Since by Theorem 9.4 we have deg f · t p (X) = t p (Z) mod n X (p), we see that t p (Z) is non-zero modulo n X (p), hence modulo n Y (p). Now by the same theorem we have hence deg g is non-zero and g is dominant, so that Y = Y ′ and dim Y = dim X. Then Z gives a correspondence Y X of multiplicity prime to p. For further reference, let us mention that we also obtain that t p (Y ) is non-zero modulo n Y (p).
Let X be the Severi-Brauer variety of a central division algebra of degree p n , with p a prime number. Then X is strongly p-incompressible when the characteristic of the base field is not p ([Mer03, Example 7.2, Theorem 7.2]). According to [Kar10, Example 2.3], it is unknown whether X is strongly p-incompressible when the characteristic is p. We can treat the case n = 1 with Proposition 10.1: Corollary 10.2. The Severi-Brauer variety of a central division algebra of prime degree p is strongly p-incompressible.
10.2. Index reduction formula. We mentioned in [Haua, §4.1] that the index reduction formula of [Zai10, Corollary (B)] can be generalized to base fields of arbitrary characteristic. The results obtained here allow us to additionally remove all assumptions of regularity: Then there is no correspondence X Y of multiplicity prime to p.
Proof. The first part of Proposition 10.1 implies that dim X = i(p − 1), hence dim Y ≤ dim X. If such a correspondence existed, the last sentence of the proof of Proposition 10.1 would imply that χ(O Y ) is prime to p, a contradiction.
10.3. Rost's correspondences. The operations T i can be used to extend the case n = 1 in [Ros06,Theorem 9.1] to arbitrary fields.
Theorem 10.4. Let X be a smooth projective variety of dimension p−1. Assume that X has a special correspondence in the sense of [Ros06, Definition 5.1], with b = 1. Assume moreover that X has no zero-cycle of degree prime to p. Then deg(r (p) (−T X ) · [X]) = 0 mod p 2 .
By Remark 9.3, this means that the integer χ(O X ) is prime to p.
Proof. We only sketch how [Ros06, Proof of Theorem 9.1] can be adapted. This proof only involve cycles of dimension ≤ 2p − 3, in particular the only Steenrod operation used is the first one. Moreover since the degree map factors through the Chow group modulo torsion, there is no loss in using the operation with values in this group. All the properties needed are contained in Proposition 7.4. Appendix A. Schemes over a regular base Let S be a regular connected scheme of finite Krull dimension. Then, as explained in [Ful98, §20.1], most results of [Ful98] for varieties over fields extend to arbitrary schemes of finite type over S. There are only two difficulties : -the lack of external product in general.
-one should use relative dimension dim S instead of the usual dimension.
In particular the homological Chern character can be constructed in this generality. The purpose of this appendix is to explain how Theorem 3.2 (and thus the construction of the operations T i , i = 0, · · · , p−1) can be extended to this setting.
Let V S be the category of schemes quasi-projective over S. Homological Adams operations are constructed in [Sou85] for objects of V S . The results of section 2 extend without difficulty to the category V S .
However, both proofs of Proposition 3.1 used the fact that X contains a nonempty open regular subscheme. We now give a proof that works for arbitrary objects of V S , using the notations introduced in the beginning of section 2.
Proposition A.1. Let l ∈ Z − {0}, X ∈ V S , and x ∈ K ′ 0 (X) (d) . Then ψ l (x) − l −d · x ∈ K ′ 0 (X) (d−1) ⊗ Z[1/l]. Proof. As before, one can assume that X is integral and d = dim S X, and we have the exact sequence − → K ′ 0 (Spec(k(X))) ⊗ Z[1/l] → 0. Choose a closed embedding X ֒→ W , with W connected and smooth over S. Let x be the generic point of X, and write W x for Spec(O W,x ). Then the map u * in the sequence above may be identified with the map Since W x is a local scheme, the rank map K 0 (W x ) → Z is an isomorphism. The rank of θ l (−T W/S )) is l − rank T W/S ([Hau10, (2), p.6]). Since W is flat over S, we have dim S W = tr. deg.(k(W )/k(S)) = rank T W/S . Thus θ l (−T W/S )) acts on K {x} 0 (W x ) by multiplication by l − dim S W . By [GS87, A4') p.263, see also Proof of Proposition 5.3], ψ l {x} is multiplication by l codim(X,W ) . This gives, using [Ful98, Lemma 20.1, (2)], u * • ψ l = l codim(X,W )−dim S W · u * = l − dim S X · u * .
We can now conclude using the exact sequence above.
As a consequence, Theorem 3.2 immediately extends to the category V S . Finally let X a scheme of finite type over S (possibly non quasi-projective). It follows from [Ful98, Lemma 18.3, (1)] that there is always an envelope X ′ → X over S, such that X ′ ∈ V S . Moreover we see, using [Ful98, Lemma 20.1, (3)], that K ′ 0 (X ′ ) (d) → K ′ 0 (X) (d) is surjective, for any d. Using this remark, Theorem 3.2 is easily extended to the category of schemes of finite type over S.