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Isotropic motives

In this article we introduce the local versions of the Voevodsky category of motives with Z/p-coefficients over a field k, parameterized by finitely-generated extensions of k. We introduce the, so-called, flexible fields, passage to which is conservative on motives. We demonstrate that, over flexible fields, the constructed local motivic categories are much simpler than the global one and more reminiscent of a topological counterpart. This provides handy"local"invariants from which one can read motivic information. We compute the local motivic cohomology of a point, for p=2, and study the local Chow motivic category. We introduce local Chow groups and conjecture that, over flexible fields, these should coincide with Chow groups modulo numerical equivalence with Z/p-coefficients, which implies that local Chow motives coincide with numerical Chow motives. We prove this Conjecture in various cases.


Introduction
The category of algebraic varieties is rich and marvelous, but not additive. In a sense that one can't add morphisms between varieties. The program to "linearize" the algebro-geometric world was first introduced by A.Grothendieck in the 1960-ies, who proposed the category of Chow motives. It is a close relative of the category of correspondences, where objects are smooth projective varieties and morphisms are algebraic cycles on the product modulo rational equivalence. The result is a tensor additive category, as we can add (and subtract) algebraic cycles and multiply them externally. Moreover, one doesn't have to limit oneself to only rational equivalence of cycles. Instead, it is possible to consider algebraic, numerical, or homological equivalence, and actually, the theory of Chow groups here can be substituted by any oriented cohomology theory (in the sense of [11,Def. 3.1.1] or [9,Def. 1.1.2]). Chow motives of varieties split into smaller pieces which permit to express in a precise form some of the similarities observed in the behavior of different varieties. In particular, the Tate-motives appear, responsible for the cellular structure. The above Grothendieck's category has innumerate remarkable applications, but it deals with smooth projective varieties only. At the same time, in topology motives were known already for a long time in full generality in the form of singular complexes of topological spaces. This problem was solved by V.Voevodsky in [19] who constructed the triangulated category of motives DM(k) over a field k (around the same time, alternative constructions were proposed by M.Hanamura and M. Levine). This category of Voevodsky contains the Grothendieck's category of Chow motives as a full additive subcategory closed under direct summands, and, in particular, permits to study these "pure" objects by triangulated methods. Voevodsky supplied his category with many flexible tools and his motives found numerous bright applications, most notably, to the proof of Milnor's and Bloch-Kato Conjectures.
In this article, we study Voevodsky category DM(k; F p ) with finite coefficients over a field of characteristic zero. We introduce the local versions DM(E/k; F p ) of this category, corresponding to all finitely generated field extensions E/k together with the natural localization functors ϕ E : DM(k; F p ) → DM(E/k; F p ). In the case of a trivial extension we get the isotropic motivic categories DM(E/E; F p ) and localization functors can be specialized further to isotropic functors ψ E : DM(k; F p ) → DM(E/E; F p ). Such isotropic versions (in the appropriate situation) appear to be much simpler than the original global category, and permit to obtain "local" invariants of motives, residing in a rather simple world. the latter given by Q i (r {i} ) = 1 and Q i (r {j} ) = 0, for i = j. Thus, the answer is the same for all flexible fields, and all these cohomology elements are "rigid", as we can get 1 from any such non-zero element by applying appropriate combination of Milnor's operations. The answer is also remarkable in the sense that Milnor's operations are encoded into the structure of isotropic motivic category (in the form of their "inverses" r {i} 's). The computation is done with the help of Voevodsky technique used in the proof of Milnor's conjecture, and our answer explains to some extent why Milnor's operations played such an important role in Voevodsky proof -see Theorem 3.5. Finally, the answer is drastically different from the "global" one and the localization functor H * , * ′ M (k, F 2 ) → H * , * ′ M (k/k; F 2 ) is zero outside the bi-degree (0) [0]. We are restricted to the prime p = 2, since in our calculations the crucial role is played by [15,Cor. 3], and there is no analogue of this statement for p > 2.
It appears that isotropic Chow motives are closely related to the numerical equivalence of cycles with F p -coefficients. We conjecture that, in the case of a flexible field, isotropic Chow groups Ch * k/k (describing Homs between such motives) coincide with the Chow groups (with F p -coefficients) modulo numerical equivalence Ch * N um -see Conjecture 4.7. This would imply that the category of isotropic Chow motives is equivalent to the category of Chow motives modulo numerical equivalence (with finite coefficients). We prove this Conjecture for divisors, for varieties of dimension 5, and for cycles of dimension 2 -Theorem 4.11. In particular, this implies that isotropic Chow groups are finite-dimensional in these situations. This also shows that the projection Ch * ։ Ch * k/k factors through the 3-rd theory of higher type Ch * (3) (where Ch * (0) = Ch * is the original theory of rational type and Ch * (1) = Ch * alg is the algebraic version) -see Definition 4.3 and Proposition 4.25. The proof of Theorem 4.11 constitutes the bulk of this article. It is by induction on the dimension of a variety X. Using the moving technique introduced in Section 6, we show that after an appropriate blow-up, any class u numerically equivalent to zero may be represented by a cycle supported on a smooth connected divisor Z and numerically trivial already on Z. An important step here is to represent u by the class of a smooth connected subvariety and numerically annihilate certain characteristic classes of it -cf. Corollaries 6.11, 6.12. Interestingly, the latter is achieved by a combination of appropriate blow-ups and Steenrod operations, depending on a prime involved.
The paper is organized as follows. After briefly discussing flexible fields in Section 1.2, in Section 2 we introduce the local motivic category with F p -coefficients as well as its Chow-motivic version. In Section 3 we study the isotropic motivic cohomology of a point, while Section 4 is devoted to the study of isotropic Chow groups and the respective Chow-motivic category. In Section 5 we expand the definition of local motivic category beyond prime coefficients. Finally, in Section 6 we prove various geometric results used in Section 4.
Acknowledgements: I'm very grateful to J.-L.Colliot-Thélène for useful discussions, to C.Voisin for helpful comments and drawing my attention to [13,Theorem 5], and to J.I.Kylling, P.A.Østvaer, O.Röndigs and S.Yagunov for fruitful interaction during my visit to the University of Oslo which led to the inclusion of Section 5. Also, I'm very grateful to T.Bachmann for the valuable remarks on the previous version of the paper.

Notations and conventions
Everywhere below k will denote a field of characteristic zero. Sm k -the category of smooth quasi-projective varieties over k. Ch * -the Chow groups CH * /p with finite coefficients, where p is some prime (in Section 6, p will be replaced by an arbitrary natural number n). DM(k; F p ) will denote the triangulated category of Voevodsky motives over a field - [19], [3] and DM gm (k; F p ) will denote the full triangulated subcategory of geometric motives in it. L -the Lazard ring, that is, the coefficient ring of the universal formal group law.

Flexible fields
Traditionally, algebraic geometry was considered over algebraically closed fields. Over such fields, every algebraic variety (of finite type) has a rational point, which simplifies many things. In the case of a general field the standard approach is to consider the passage to its algebraic closure. Note, however, that (torsion) information is lost under such a passage. One of the aims of the current paper is to convince the reader that there are other directions one can pursue. Namely, I propose to move instead in the direction of the, so-called, flexible fields. Such fields have the advantage that one doesn't have to distinguish between the ground field and finitely-generated purely transcendental extensions of it. This helps with many algebro-geometric constructions. Definition 1.1 Let us call a field k flexible if it is a purely transcendental extension of countable infinite degree of some other field: k = k 0 (t 1 , t 2 , . . .) = k 0 (P ∞ ).
Note, that any finitely generated extension of a flexible field k is itself flexible. Indeed, such an extension is defined by finitely many generators and relations, which can "spoil" only finitely many of the original transcendental generators. Thus, all the points of the large Nisnevich site of Spec(k) are flexible. On the other hand, we have: The natural restriction functor DM(k 0 ; F p ) → DM(k; F p ) is conservative. So, we can substitute a field by a flexible one without loosing any motivic information. △ The main property of flexible fields we will need is the following obvious observation.

Proposition 1.3
Let k be a flexible field, X -variety of finite type over k, and E/k be a finitely-generated purely transcendental field extension. Then there exists a commutative diagram with horizontal maps isomorphisms (over some subfield k 0 ).
Proof: Let k = k 0 (t 1 , t 2 , . . .). Then X is defined over some finitely-generated purely transcendental extension F of k 0 such that k/F is purely transcendental. That is, there is a variety X of finite type over F , such that X k = X. Since extensions k/F and E/F are isomorphic, we get what we need.

Motivic category of a field extension
Everywhere below DM(k; F p ) will denote the triangulated motivic category of Voevodsky over Spec(k) with F p -coefficients (see [19], [3]). We will construct the local versions of this category, corresponding to all finitely-generated field extensions E/k, or, in other words, to all points of the big Nisnevich site over Spec(k). The local motivic categories will be obtained as full localizing subcategories of a global one by application of certain projectors. These projectors will be produced usingČech simplicial schemes.
Let P be a smooth variety over k. TheČech simplicial schemeČech(P ) has graded components (Čech(P )) n = P ×(n+1) with faces and degeneracy maps given by partial projections and partial diagonals. This object is an analogue of the contractible space EG in topology, and it will be contractible in Morel-Voevodsky homotopic motivic category as long as P has a rational point, while in general, it "measures" how far we are from acquiring such a point. In particular, it is a form of a point, since it certainly contracts over algebraic closure. Let us denote the motive ofČech(P ) as X P . The natural projectioň Cech(P ) → Spec(k) provides the morphism X P → T to a trivial Tate-motive, which is an isomorphism if and only if P has a zero cycle of degree 1 (modulo p, in our case) (a "weak form" of a rational point) - [14,Thm 2.3.4]. This gives an exact triangle where X P and X P are mutually orthogonal idempotents: Thus, the functors of tenzor product with these objects: are projectors. This defines the semi-orthogonal decomposition of the category DM(k; F p ) as an extension of X P ⊗ DM(k; F p ) by X P ⊗ DM(k; F p ), as there are no Hom's from the latter subcategory to the former one (by [14,Thm 2.3.2], which is, basically, [20,Lem. 4.9]). For different varieties, these projectors naturally commute and we have canonical (co-associative, respectively, associative) identifications X P ⊗ X Q = ← X P ×Q and X P ⊗ X Q = → X P Q (note, that endomorphism rings of X V and X V are either F p , or zero -[14, Thms 2.3.2, 2.3.3], and such an endomorphism is fixed by the map X V → T , respectively, T → X V ). Thus, tenzor product of any (finite) number of such objects can be always expressed as X R ⊗ X S , for some R and S. Each X P corresponds to a sub-sheaf χ P of the constant sheaf T = F p on the big Nisnevich site over Spec(k) defined as follows. For a smooth connected quasi-projective variety X, χ P (X) = F p , if P has a zero-cycle of degree 1 over k(X), and it is zero, otherwise. Equivalently, χ P (X) = F p exactly when X P ⊗M (X) = 0 ([14, Thms 2.3.6, 2.3.3]). Respectively, X P corresponds to the quotient sheaf χ P = T /χ P . We can introduce an order on the set of X Q 's as follows: Definition 2.1 We say that X Q X P if any of the following equivalent conditions is satisfied: (1) The natural map X Q ∼ = ← X Q ⊗ X P is an isomorphism; (2) The natural map X Q ⊗ X P ∼ = ← X P is an isomorphism; (3) The map X Q → T factors through X P → T ; (4) P has a zero-cycle of degree 1 over the generic point of every component of Q; Here (1) ⇔ (2) is automatic from the definition; (2) ⇒ (5) follows from the description of χ P above; (5) ⇒ (4) follows from the fact that χ Q (Q l ) = F p , for any connected component Q l of Q; (4) ⇒ (1) is [14,Thm 2.3.6]; (1) ⇒ (3) is straightforward; and, finally, Note, that the relation X Q X P is obviously transitive. In the case of connected varieties, this relation may be formulated in terms of the respective field extensions (generic points). Let E/k and F/k be two finitely-generated extensions of a field of characteristic zero. Let P/k and Q/k be smooth projective varieties whose function fields are identified with E and F . Definition 2.2 We say that F/k E/k, if there exists a correspondence of degree 1 (with F p -coefficients) Q P . We call extensions "equivalent" F/k ∼ E/k, if F/k E/k and E/k F/k.
By the composition of correspondences, the property F/k E/k is transitive. It is also equivalent to the condition X Q X P above. If k/l is a field extension, then F/k E/k implies that F/l E/l.
Let P be some smooth variety (of finite type) over k. Let Q be the disjoint union of all connected varieties Q/k, such that X Q X P (so, it is a smooth variety, but with infinitely many components), and let X Q be the motive of the respectiveČech simplicial scheme, which is still an idempotent in DM(k; F p ), and X Q be the complementary idempotent. Define We can view Υ P as a colimit of projectors X Q ⊗ X P , where Q runs over all smooth projective varieties of finite type with X Q X P .
Note, that if P is not connected up to equivalence, that is, if P can't be substituted by a connected variety with the same X , then Υ P = 0. Indeed, let P 1 be a "minimal" component, that is, X P 1 X P i implies that X P 1 = X P i . Suppose, there exists another component P 2 with X P 2 X P 1 . LetP 1 be the union of all the components equivalent to P 1 . Then for Q 1 = P \P 1 and Q 2 =P 1 we have: Now we can define the local motivic category corresponding to a finitely-generated extension E/k (cf. [18,Sect. 4]). Let P/k be a smooth connected variety with k(P ) = E. Definition 2.3 Define the "motivic category of the extension E/k" as the full localizing subcategory DM(E/k; F p ) = Υ P ⊗ DM(k; F p ) of DM(k; F p ), and the "local geometric category" DM gm (E/k; F p ) as the full thick triangulated subcategory of DM(E/k; F p ) generated by (local) motives of smooth projective varieties.
This definition doesn't depend on the choice of a smooth model P , since X P depends on k(P ) only. Moreover, it depends only on the ∼-equivalence class of an extension E/k.
In the case of a trivial extension, we obtain (cf. [18,Sect. 4]): The "isotropic motivic category" is the full localizing subcategory DM(k/k; F p ) of DM(k; F p ) given by Υ Spec(k) ⊗ DM(k; F p ), while the geometric version DM gm (k/k; F p ) is the full thick triangulated subcategory of it generated by (isotropic) motives of smooth projective varieties. Now, we can read the information about motive by looking at local versions of it. Namely, we get a collection {ϕ E | E/k − f.g. extension} of localization functors parameterized by all points of the big Nisnevich site over Spec(k). These can be further specialized to isotropic realizations The following result shows that there are no unexpected objects in the isotropic geometric category.
Proposition 2.5 The category DM gm (k/k; F p ) is the idempotent completion of the full subcategory Υ Spec(k) ⊗ DM gm (k; F p ) of DM(k; F p ).
Proof: We need to prove that Υ Spec(k) ⊗ DM gm (k; F p ) is closed under cones. For this, it is sufficient to show that, for any objects U, V of DM gm (k; F p ) and a map f : Composing the map f with U → X Q ⊗U we obtain a map g : U → X Q ⊗V with the property , for some smooth varieties Y and r > n. Since U and V are geometric, for sufficiently large n, there are no Hom's from U to ( X Q ) >n ⊗ V . Hence, the map g can be lifted to a map f ′ : U → ( X Q ) n ⊗ V , which, in turn, can be lifted to a geometric map f : U → ( X Q ) n ⊗ V , for some anisotropic variety Q of finite type over k. Since X Q ⊗ ( X Q ) n = X Q , for any n 0, we obtain that We can describe Hom's from geometric isotropic motives as follows. For an object X of DM(k; F p ) and some idempotent ξ, we will denote by the same letter the image of X in ξ ⊗ DM(k; F p ).
Proposition 2.6 Let U ∈ Ob(DM gm (k; F p )) and V ∈ Ob(DM(k; F p )). Then where the colimit is taken over all the functors X S ⊗ : In other words, Q runs over all anisotropic varieties over k.
, and the latter can be identified with Hom DM(k;Fp) (U, X Q ⊗ V ), where Q is the disjoint union of all anisotropic varieties over k. But since U is geometric, any map U → X Q ⊗ V factors through U → X Q ⊗ V , for some anisotropic Q of finite type, and the map U → X Q ⊗ V vanishes when extended to a map to X Q ⊗ V if and only if there exists an anisotropic Q ′ with X Q X Q ′ , such that the composition U → X Q ⊗ V → X Q ′ ⊗ V is zero. Thus, our Hom-group can be identified with the where the colimit is taken over a directed system (as Geometric motives vanishing in the local category can be also detected by projectors corresponding to varieties of finite type. Let P be smooth connected variety with E = k(P ). Proposition 2.7 An object U of DM gm (k; F p ) vanishes in DM(E/k; F p ) if and only if there is a variety Q of finite type over k, with X Q X P and X Q ⊗ X P ⊗ U = 0.
Proof: If Υ P ⊗ U = 0, then ( X Q ⊗ U ) E = 0, where, as above, Q is the disjoint union of all smooth connected varieties Q over k, with X Q X P . That means that the projection (X Q ⊗ U → U ) E has a section (from the right). But since U is geometric, such a section will factor through some section of (X Q ⊗ U → U ) E for some variety Q of finite type over k with X Q X P . Hence, ( X Q ⊗ U ) E = 0 (as U E is a direct summand of X Q ⊗ U E , and so, X Q ⊗ U E is a direct summand of it as well, but the latter object is stable under X Q ⊗ while the former one is killed by it). But, according to [14,Thm. 2.3.5], the functor Since X Q ⊗ M (Q) = 0, the projection to the isotropic motivic category DM(k/k; F p ) kills the motives of all anisotropic varieties over k. Hence, the name of this category.

Remark 2.8
The isotropic motivic category DM(k/k; F p ) is the Verdier localization of DM(k; F p ) modulo the localizing subcategory A generated by motives of anisotropic varieties 1 . Indeed, an object U of DM(k; F p ) vanishes in DM(k/k; F p ) if and only if U = U ⊗ X Q , where Q is the disjoint union of all connected anisotropic varieties Q/k. Hence, U belongs to A, since this subcategory is a tensor ideal. By the universal property of the Verdier localization, We have functoriality for the "denominator" of the extension E/k. Suppose, we have a tower of fields L ⊂ F ⊂ E, and P/L, Q/L are smooth projective varieties with L(P ) = E and X Q X P . Then Q remains anisotropic over L(P ), and so, over F (P ) (since F ⊂ L(P )). Hence, X Q | F X P | F . Thus, we get a natural functor The following result shows that, in the case of a flexible ground field, we can pass from (geometric) local realizations {ϕ E | E/k − f.g. extension} to isotropic realizations {ψ E | E/k − f.g. extension} without loosing any information.
Proposition 2.9 Let E/k be a finitely-generated extension of a flexible field. Then the functor is conservative on the image of ϕ E .
Proof: Let us start with purely transcendental extensions. Lemma 2.9.1 Let E/L/k be a tower of finitely generated extensions where L/k is a purely transcendental extension of a flexible field. Then the functor is conservative on the image of ϕ E .
Proof: Let L = k(A n ), and E = k(R) for some smooth variety R/k. Let U ∈ Ob(X R ⊗ DM gm (k; F p )) be an object vanishing in DM gm (E/L; F p ). Then, according to the Proposition 2.7, there exists a variety Q/L of finite type such that X Q X R L and U L ⊗ X Q = 0 in DM(L; F p ). The condition X Q X R L means that we have an L-correspondence α : Q R L of degree one, and there is no such correspondence in the opposite direction. Since k = k 0 (P ∞ ) is flexible, varieties R and Q are, actually, defined over F and M = F (A n ), respectively, where extensions k/F/k 0 are purely transcendental and F/k 0 is moreover finitely generated. By the same reason, we can assume that the geometric object U is defined over F , while the correspondence α is defined over M . So, there exist varieties R/F , Q/M , an object U of X R ⊗ DM gm (F ; F p ) and a degree one M -correspondence α : Q R M such that R| k = R, Q| L = Q, U | k = U and α| L = α. Note, that we still have: X Q X R M (since α is defined over M and by functoriality), and U M ⊗ X Q = 0 (since the restriction DM(M ; F p ) → DM(L; F p ) is conservative). But the extension M/F can be embedded into k/F making k/M purely transcendental. Let Q ′ be a variety over k obtained from Q using this embedding. Then X Q ′ X R (since k/M is purely transcendental) and U ⊗ X Q ′ = 0 in DM(k; F p ). Hence, U = 0 in DM(E/k; F p ).
Using Lemma 2.9.1 our problem is reduced to the case of a finite extension. In this situation, the statement is true for an arbitrary field. Lemma 2.9.2 Let E/L be a finite extension of fields. Then the functor is conservative.
Proof: Let E = L(P ) for some smooth connected 0-dimensional variety P . Let U ∈ Ob(X P ⊗ DM(L; F p )) be an object vanishing in DM(E/E; F p ). Then, for the disjoint union Q of all anisotropic varieties over E, we have: U E ⊗ X Q = 0 in DM(E; F p ). Consider a smooth L-variety Q given by the composition Q → Spec(E) → Spec(L). We have a natural map Q → Q E . This means that X Q X Q E , and so, Clearly, X Q X P . Suppose, these are equal. Then there exists a commutative diagram where F is an extension of E of degree prime to p. But since [E : L] is finite, the composition Spec(F ) → Q → Spec(E) has the same (prime to p) degree. This contradicts to the fact that Q is anisotropic. Hence, X Q X P , and so, X Q X Q , where Q is a disjoint union of all L-varieties Q with X Q X P . Thus, This finishes the proof of the Proposition 2.9.
Another type of functoriality we have is the following one. Let k(A)/k be a purely transcendental extension of k. Then we have a natural functor DM(E/k; F p ) → DM(E(A)/k(A); F p ).
One just needs to observe that the inequality X Q X P is preserved under the passage from k to k(A).
It is natural to ask, in which situations will our localization functors be conservative? Since the passage from k 0 to k = k 0 (t 1 , t 2 , . . .) is conservative, and any finitely generated extension E of k has the form E = E 0 (t N , . . .), for some finitely-generated extension E 0 of k 0 , and by Proposition 2.9, the non-triviality of {ϕ E | E/k−f.g. extension} on X 0 | k implies the non-triviality of {ψ E | E/k−f.g. extension} on X 0 | k , which in turn, implies the non-triviality of {ψ E 0 | E 0 /k 0 − f.g. extension} on X 0 , implying the non-triviality of {ϕ E 0 | E 0 /k 0 − f.g. extension} on X 0 . Thus, for a given geometric object X 0 /k 0 , where (a) means that X 0 is in the kernel of the family {ϕ E 0 | E 0 /k 0 − f.g. extension}, etc..

Remark 2.11
Restricting the functors ψ E to the tenzor triangulated subcategory DQM gm generated by motives of smooth projective quadrics, and specializing it further, one gets the functors of T.Bachmann Φ E : DQM gm → K b (T ate(F 2 )) to the category of bi-graded F 2 -vector spaces -see [1]. This can be deduced from the fact that the functor ψ E maps the subcategory DQM gm to the subcategory of geometric Tatemotives in DM(E/E; F p ) (by [18,Prop. 4.9]). The functors Φ E , constructed originally by completely different methods, were shown by T.Bachmann to be conservative [1,Thm. 31]. In particular, the collection {ψ E | E/k − f.g. extension} is conservative on DQM gm . Our approach also permits to see conservativity on this and other similar subcategories. Namely, it follows from [18,Prop. 4.9] that, for any object A of DQM gm , there exists a finite filtration by idempotents on the unit object T , such that associated graded idempotents map A to geometric Tate-motives. And this collection of associated graded idempotents (having a form X Q ⊗ X P , for some smooth varieties P and Q, with P -connected) acts conservatively (as the unit object is an extension of them). It remains to observe that, for geometric Tate-motives, the triviality of X Q ⊗ X P ⊗ A is equivalent to the triviality of Υ P ⊗ A ∈ Ob(DM(E/E; F p )), for E = k(P ). △

Remark 2.12
If the ground field k 0 is algebraically closed, then there exists only one ∼-equivalence class of finitely-generated extensions of k 0 (the trivial one). Thus, there is only one "local" point and only one localization functor ϕ : DM(k 0 ; F p ) −→ DM(k 0 /k 0 ; F p ) which is an equivalence of categories (as there are no anisotropic varieties over k 0 ). Thus, in this case, the family {ϕ E 0 | E 0 /k 0 − f.g. extension} is conservative, but it does not provide any interesting information. △ The collection {ϕ E | E/k − f.g. extension} is not conservative, in general.
Indeed, for an arbitrary p, such a group is generated by [ Thus, M (C) = 0 in Chow N um (E; F 2 ), for any extension E/k. Hence, by Theorem 4.11 (1), it is zero in Chow(E/E; F 2 ) which is a subcategory of DM(E/E; F 2 ). So, all isotropic realizations ψ E ( M (C)) are trivial. At the same time, M (C) is non-trivial, since the (complex) topological realisation of it is non-trivial (has non-zero H 1 T op ). Alternatively, one can see that the restriction to the algebraic closure M (C)| k is non-trivial. Note, that the choice of a prime was essential here.
(2) Refining the previous example, we can show that even the combination is not conservative on DM gm (k; F 2 ). In the above situation, consider some non-trivial quadratic extension F = k √ a and P = Spec(F ) π → Spec(k). Let α = {a} ∈ K M 1 (k)/2 and M α be the "completely" reduced Rost-motive -see the proof of Theorem 3.5. This motive fits into the exact triangle M α → X α [1] → X α → M α [1], where X α = X P . Let us show that U = M (C) ⊗ X α = 0. Indeed, such a triviality is equivalent to the fact that the projection X P × M (C) → M (C) has a section. And since M (C) is a pure motive (= Chow motive), any such section is liftable to a section of P × M (C) → M (C). This would mean that the projector ρ defining M (C) is in the image of the natural map π * : Ch 1 (C × C × P ) → Ch 1 (C × C). Note however, that Ch 1 (C × C| k ) = [∆] · F 2 ⊕ [pt × C] · F 2 ⊕ [C × pt] · F 2 (since k-points of the Jacobian form a 2-divisible group), and so the map res k •π * : Ch 1 (C × C × P ) → Ch 1 (C × C| k ) is zero (since the action of the Galois group on Ch 1 (C × C| k ) is trivial, which implies that res k •π * = 2 · res k/F ). On the other hand, ρ| k = 0, since it is non-zero even in the topological realization. Hence, ρ is not in the image of res k •π * and M (C) ⊗ X α = 0. Notice, that ψ E (U ) = 0, since ψ E ( M (C)) = 0, while res k (U ) = 0, since res k ( X α ) = 0. Thus, we have produced a non-trivial example on which the needed combination of functors vanishes, but so far, not a geometric one.
. In particular, V is geometric and all the above functors vanish on it. It remains to show that V = 0. Note that, since U = 0, the homology Hom DM(E;F 2 ) (T ( * )[ * ′ ], U ) considered for all finitely generated extensions E/k, is non-trivial. At the same time, this homology is zero for * ′ < * (below the main diagonal). This implies that V = 0. Indeed, if it would be zero, then the homology of U would be [1]-periodic, which is not the case. △

Local Chow motivic category
Let X be a scheme of finite type over k. We can define its' isotropic Chow groups as where M c (X) is the motive with compact support of X -see [19]. For smooth varieties, we have from duality: ). The theory Ch k/k has natural pull-backs and push-forward maps coming from the respective maps between motives of varieties, which satisfy all the axioms of [9, Def. 1.1.2] (since these follow from the properties of motives). Finally, we have the excision axiom (EXCI), claiming that for a scheme X with the closed subscheme Z and open complement U , there is an exact sequence: This follows from the Gysin exact triangle [19, (4.1.5)]: [1] and the fact that the map Ch * ։ Ch * k/k is surjective, which follows from Proposition 2.16 below. Thus, Ch * k/k is an oriented cohomology theory (with excision) on Sm k in the sense of [17, Def. 2.1].
Definition 2.14 Let Q be a scheme (of finite type) over k and n ∈ N. We say that Q is "n-anisotropic", if the degrees of all closed points of Q are divisible by n.
Schemes which don't have this property will be called not n-anisotropic, while we will reserve the term isotropic for a scheme having a zero-cycle of degree 1 (mod n). Below almost everywhere we will be dealing with n = p-prime, and so isotropic will be the complement to anisotropic. Unless specified, the term anisotropic will mean p-anisotropic, for some prime p.
Definition 2.15 Let X be a scheme over k, and x ∈ Ch r (X). We call x "anisotropic", if there exists a proper morphism f : Y → X from a p-anisotropic scheme Y and a class y ∈ Ch r (Y ) such that x = f * (y).
For fields of characteristic zero and X projective, x is anisotropic if and only if it is a push-forward of the generic cycle from some smooth projective anisotropic variety over k.
Isotropic Chow groups can be alternatively described as follows.
Proof: By Proposition 2.6, Hom DM(k/k;Fp) (T (r)[2r], M c (X)) is the colimit of the groups where Q runs over all anisotropic varieties over k. Recall, that we have an exact triangle By Thus, Hom DM(k/k;Fp) (T (r)[2r], M c (X)) = Ch r (X)/I, where I is the subgroup generated by the images of (π Q ) * , for all anisotropic varieties Q/k. In other words, we mod-out all anisotropic classes.
The isotropic motivic category DM gm (k/k; F p ) has a pure part.
Definition 2.17 Define the "isotropic Chow motivic category" Chow(k/k; F p ) as the full additive subcategory of DM gm (k/k; F p ) -the image of Chow(k; F p ) under the natural projection Thus, the objects of Chow(k/k, F p ) can be identified with direct summands of motives of smooth projective varieties over k, while the morphisms are described as follows.
Proposition 2.18 Let X and Y be smooth projective k-varieties. Then Proof: If B is an object of DM gm (k; F p ) with the dual B ∨ , and A, C are objects of DM(k; F p ), then we have a functorial identification Hence, for the projector ρ Q = X Q ⊗ we also have a functorial identification We can describe Chow motives disappearing in the isotropic category.

Remark 2.19
An object U of Chow(k, F p ) vanishes in Chow(k/k, F p ) if and only if it is a direct summand in the motive of a (smooth projective) anisotropic variety 2 . Indeed, a direct summand U of M (P ) vanishes in Chow(k/k, F p ) if and only if the identity map id U : U → U does. By Propositions 2.16 and 2.18, this means that the map ∆ U : T → U ⊗ U ∨ factors through (the motive of) a smooth projective anisotropic variety Q. Consequently, U is a direct summand of M (Q) ⊗ U , which, in turn, is a direct summand of M (Q × P ), and the latter variety is still anisotropic. △ From Propositions 2.16 and 2.18 we obtain: Corollary 2.20 All "local" morphisms between (isotropic) Chow motives are defined "globally".
We will have a closer look at the category Chow(k/k, F p ) in Section 4.

Local motivic cohomology of a point
In this Section we will compute the motivic cohomology of a point in the isotropic motivic category, for p = 2. This will be achieved by substituting all anisotropic k-varieties in the colimit of Proposition 2.6 by norm-varieties for non-zero pure symbols from K M * (k)/2 (anisotropic Pfister quadrics, in our case). This makes the problem amenable to calculation due to Voevodsky technique. Moreover, the resulting answer, drastically different from the "global" one, in turn, sheds some light on this technique.
The starting point is the following statement, which is a slight modification of the result of J.-L.Colliot-Thélène and M.Levine [4,Theorem 3]. We provide a somewhat different proof. Statement 3.1 Let B be an anisotropic (mod n) projective variety. Then, over some finitely generated purely transcendental extension, it can be embedded into an anisotropic hypersurface of degree n.
Proof: Embed B into a projective space. Passing to a Veronese embedding, we can assume that all the relations in the projective coordinate ring of B are generated by quadratic ones, or in other words, that B is defined by quadrics. Then it will be also defined by hypersurfaces of degree n (in our P m ), for any n 2. Let P r = Proj |D| be the projective system of hypersurfaces of degree n containing B. I claim that the generic element of this linear system is an anisotropic hypersurface. Consider Y ⊂ (P m \B) × P r defined by Y = {(x, H)|x ∈ H}. Then Y is a projective bundle Proj (P m \B) (V ) over (P m \B), where V is a co-dimension one subbundle in the trivial bundle |D|. Let Y η be the generic fiber of the projection Y → P r . This is exactly (Q η \B), where Q η is the generic hypersurface of degree n passing through B. Note, that the degree (mod n) is well-defined on the zero-cycles on Y η , since B is anisotropic. By the projective bundle theorem, CH * (Y ) is a free module over CH * (P m \B) with the basis 1, ρ, . . . , ρ r−1 , where ρ = c 1 (O(1)). On the other hand, we have a surjective ring homomorphism CH * (Y ) ։ CH * (Y η ) which is zero on ρ (as this class is supported on a hypersurface in P r ). Thus, we obtain the surjective map CH * (P m \B) ։ CH * (Y η ) which sends the class c ∈ CH * (P m \B) to the restriction of π * (c) to Y η , where π is our projective bundle fibration. In particular, c ∈ CH 1 (P m \B) is mapped to a zero cycle on Y η whose degree is equal to the intersection number of c and any hypersurface from our linear system (which, again, makes sense, since B is anisotropic). Hence, it is a zero cycle of degree 0 (mod n). Thus, the degrees of all zero-cycles on Y η are divisible by n, and so, the same is true about Q η = Y η ∪ B.
Corollary 3.2 Let k be a flexible field, U ∈ Ob(DM gm (k; F p )) and V ∈ Ob(DM(k; F p )). Then where the colimit is taken over all the functors ⊗ X Q , where Q runs over all anisotropic hypersurfaces of degree p over k. This system is directed.
Proof: Let B be any anisotropic variety over k. By the Statement 3.1, there exists a purely transcendental field extension E/k and anisotropic hypersurface Q over E such that X B | E X Q . Let k = k 0 (P ∞ ). Then there exists a diagram of purely transcendental extensions of fields with the extensions of the bottom row finitely generated, such that the variety B is defined over l, while the variety Q and the correspondence B Q (of degree 1) are defined over L. But we can embed L into k over l so that k/L will be purely transcendental. Thus, we obtain an anisotropic hypersurface Q ′ over k together with a correspondence B Q ′ of degree 1. Hence, the set of anisotropic hypersurfaces of degree p form a final subsystem in the system of all anisotropic varieties over a flexible field. And final subsystem of a directed system is directed where the colimit is taken over all the functors ⊗ X Qα , where α runs over all non-zero pure symbols from K M * (k)/2 and Q α is the respective Pfister quadric. This is a directed system.
Proof: By [15, Cor. 3] (see also [8]), every anisotropic quadric Q (over any field k) can be embedded into an anisotropic Pfister quadric Q α over an appropriate purely transcendental extension of finite transcendence degree. If now k is flexible, then arguing as in the proof of Corollary 3.2, we can embed Q into some anisotropic Pfister quadric Q α ′ over k. Thus, the set of anisotropic Pfister quadrics form a final subsystem in the system of all anisotropic varieties over a flexible field, which, again, must be directed.
From the fact that the system in Corollary 3.3 is directed, as a by-product, we obtain the following result (which, of course, is a simple consequence of Statement 3.1 and [15,Cor. 3], and can be even seen from the latter result alone): Proposition 3.4 Let k be a flexible field, and {α l } l∈L be a finite collection of non-zero pure symbols from K M * (k)/2. Then there exists a non-zero pure symbol α ∈ K M * (k)/2 divisible by every α l .
Using Corollary 3.3, we can compute the cohomology of a point in isotropic motivic category for p = 2. For a non-zero pure symbol α ∈ K M r (k)/2, let us denote as DM(α/k; F 2 ) the full triangulated subcategory X α ⊗ DM(k; F 2 ), where X α = X Qα and Q α is the respective Pfister quadric. Hom's between Tate-objects in this category can be computed as follows.
In particular, it has a natural ring structure. The multiplicative structure on Q −1 (n)[r {n+1} ] ⊗ F 2 R α is provided by r I = i∈I r {i} and the identity:

For a motivic category D with Tate-objects, let us denote as End
Theorem 3.5 Let α ∈ K M m (k)/2 be a non-zero pure symbol. Then . From this point, all the Hom's will be in the category DM(k; F 2 ), unless specified otherwise, so I will omit it from notations. Let M α be the respective Rost motive ( [12]). We have natural maps whose composition is zero. Cutting out the respective Tate-motives from M α and tensoring the result by X α and X α , respectively, we obtain: Here we are using the fact that M α ⊗ X α = 0 and that M α ⊗X α = 0, which are equivalent to the exactness of the left triangle - [22,Thm 4.4]. Let us denote the above half of the octahedron as ♦. Note, that since there are no Hom's from X α to X α ( * )[ * ′ ], we can naturally identify groups Hom( For each 0 i < m − 1, let β ∈ K M i+1 (k)/2 be any pure symbol dividing α. We obtain a similar map Below we will see that this map doesn't depend on the choice of the divisor β.
For any smooth projective R, we have the natural (homological) action of the Steenrod algebra on Hom(T, X R ( * )[ * ′ ]) and the natural (cohomological) action of it on Hom( X R , T ( * )[ * ′ ]), which commute with the maps X R → X S (for X R X S ). In particular, we have the action of the Milnor's operations Q i . If R is a ν i -variety, then by the arguments of V.Voevodsky [22,Cor 3.8], the differential Q i is exact on Hom( X R , T ( * )[ * ′ ]). By the same arguments, it is exact on Hom(T, X R ( * )[ * ′ ]). In particular, in our case, . It has the natural right action of A = End( X α ) as well as the left action of A = End(X α ). In particular, there is a right action by η and a left one by µ. I claim that these are mutually inverse. Indeed, to see that for any f : , which is (after rotating by 90 • ): a + 1 by the Beilinson-Lichtenbaum "conjecture" (note, that X α disappears in the etale topology). Considering Hom's from is exactly the > 0 diagonal part of it. Thus, N combines the homology and cohomology groups of X α .
Since there are no Hom's from T to X α (a)[a + 1], the map Hom(T, T (a)[a]) ։ Hom(T, X α (a)[a]) is surjective, and so, the 0-th diagonal of A (or, which is the same, the 0-th diagonal of N ) as a K M * (k)/2module is generated by 1 -the unit of this ring. Let R α be this 0-th diagonal. From µ − η-periodicity, the diagonal number (−2 m−1 ) (where η resides), as a K M * (k)/2-module, is generated by η. Since the differential Q m−1 is exact on A, we obtain that 1 is covered by the image of Q m−1 (since A is concentrated in non-positive diagonals). But the only non-zero element of the needed bi-degree is η. Thus, Q m−1 (η) = 1. Applying the same arguments to the symbol β (considered above), we obtain that Q i (η β ) = 1, and so, Q i (r {i} ) = 1.
For any I ⊂ (m − 2), denote r I := i∈I r {i} and In particular, Q i (η) = 0, for any 0 i m − 2 (since this element is below r I i ), and since η generates D −2 m−1 , all the differentials Q i , for i m − 2 are trivial on this diagonal. Applying the same arguments to the divisor β of degree (i + 1), we obtain that Q l (η β ) = 0, and hence, Q l (r {i} ) = 0, for l < i (the fact that Q l (r {i} ) = 0, for l > i, is obvious, since A is concentrated in non-positive diagonals). Combining it with the (external) co-multiplication identity for Milnor's operations: where 2 I = i∈I 2 i , we obtain that Q I (r I ) = 1, for any I ⊂ (m − 2). Let us show that D −2 I is a free module over D 0 = R α generated by r I . Let s be the smallest element of I (which we assume to be m−1, if I is empty). Decreasing induction on s. The (base) s = m−1 follows from µ − η-periodicity. The (step): let 2 J = 2 I + 2 s . Since I\s and J consist of elements larger than s, by inductive assumption, D −(2 I +2 s ) = r J · D 0 and D −(2 I −2 s ) = r I\s · D 0 (where we denote r {m−1} := η).
In particular, Q s is trivial on D −(2 I +2 s ) , and from the exact sequence D −(2 I +2 s ) From µ − η-periodicity we obtain that A is a free module over R α generated by η k · r I , for k 0 [20,Lem 6.4]. The generator γ is identified with some element of degree m, which must coincide with the symbol α (since α vanishes over k(Q α ) and there exists exactly one non-zero element of the respective degree in D 1 ). Hence, as a K M * (k)/2-module, R α can be identified with the principal ideal of K M * (k)/2 generated by α. So, as a ring, R α = (K M * (k)/2)/(Ker(·α)). This gives the description of A (as well as N ) as a module over K M * (k)/2 and over Steenrod algebra. Finally, the equation r 2 {i} = r {i+1} · ρ follows from the co-multiplication identity for Milnor's operations (1).
As a by-product, we obtain the description of motivic cohomology of X α (known already from the original version of [10] and [23,Thm 5.8]) but now enhanced with the structure of a module over motivic homology of X α .
It is a free R α -module with the basis r −l {m−1} · r I , for l > 0 and I ⊂ (m − 2). Now we can compute the "local" motivic cohomology H * , Theorem 3.7 Let k be a flexible field. Then Proof: As we know, our colimit (from Corollary 3.3) is taken over a directed system. Let α ∈ K M m (k)/2 and β = α · {b} ∈ K M m+1 (k)/2 be non-zero pure symbols. Consider the restriction Then res = X β ⊗ is a ring homomorphism respecting Steenrod algebra action, and, in the notations of Theorem 3.5, for I ⊂ {0, . . . , m − 2}, we have: res(r I ) = r I and res(r {m−1} · r I ) = r I∪{m−1} . Indeed, res sends r ∅ to r ∅ , while r I and r I (respectively, r {m−1} · r I and r I∪{m−1} ) are the only elements (in the source and the target) which are mapped to r ∅ via Q I (respectively, Q I∪{m−1} ). Also, res : R α → R β is the natural projection, corresponding to the map α · K M * (k)/2 ·{b} −→ β · K M * (k)/2. Combining this with Corollary 3.3 we obtain that Then, from {a, −a} = 0, both {a} and {−a} belong to N . This implies that {−1} ∈ N . Since, over a flexible field, an element a as above always exists, we obtain that N contains K M 1 (k)/2, and so, coincides with the augmentation ideal What is remarkable here, the Milnor's operations are intertwined into the very fabric of the local motivic category. Also, all the non-zero elements of H * , * ′ M (k/k; F 2 ) are "rigid", in the sense that the identity map of the unit object of the local category can be obtained from any such element using Milnor's operations, and so these classes disappear only together with the category itself.
We can now compare "local" and "global" motivic cohomology of a point: . [j] O O Note, that in contrast to "global" motivic cohomology of a point residing in the I-st quadrant, the "local" version resides in the III-rd one. In particular, the global-to-local map Our ring generators r {i} are related by the action of the Steenrod algebra. Namely, since (modulo [21], and ρ disappears locally, anyway, we obtain from bi-degree considerations that Sq 2 i+1 (r {i+1} ) = r {i} . In particular, Sq 1 Sq 2 . . .
Remark 3.8 Despite some similarities between the (complex) topological realization functor and isotropic functors, there is a difference in the way they handle τ . Namely, τ T op = 1, while ψ E (τ ) = 0 (in the case of a flexible field). △ The only obstacle which prevents us from performing the same calculations for odd primes is the lack of the analogue of [15,Cor. 3] in this situation. In particular, it would be sufficient to have a positive answer to the following: Question 3.9 Let Q be an anisotropic hypersurface of degree p over k. Is it true that, over some finitelygenerated purely transcendental extension E/k, the kernel contains a non-zero pure symbol?

Isotropic category of Chow motives
In this Section we will study in details local Chow motives. As we will see, over a flexible field, these resemble in many respects their topological counterparts, and are closely related to the numerical equivalence of cycles with finite coefficients. In particular, the Hom's between such local pure motives are expected to be not larger than Hom's between their topological realizations, and so finite-dimensional. We will prove it in various situations. Let us start by introducing some "gradual" approach to the numerical equivalence of cycles, which will permit to measure our progress towards the goal.

Theories of higher types and numerical equivalence
Let A be a commutative ring and L ϕ −→ A be some formal group law with A-coefficients. Denote as A * (0) := Ω * ⊗ L A the respective free theory in the sense of Levine-Morel [9, Rem. 2.4.14]. By [17,Prop. 4.7] this is a theory of rational type. We call this type 0. We are going to introduce the theory A * (n) of type n as some quotient of A * (0) . This is based on the following construction (cf. That is, we have a collection of A * -correspondences ρ λ : Q λ Spec(k). Construct the new theory A * Γ as follows: In other words, we mod out all the elements of the form β * (α * (a λ ) · u), where u is an arbitrary element of A * (Q λ × X) and α and β are natural projections: One can check that the resulting theory A * Γ will be an oriented cohomology theory in the sense of [17, be a smooth projective variety and a ∈ A * (Q). We say that a Num ∼ 0, if π * (a · b) = 0, for any b ∈ A * (Q). Now we can introduce the theories of higher types. Definition 4.3 Consider the collection Γ n = {Q λ , a λ } λ∈Λ , where Q λ runs through all smooth projective k-varieties of dimension 2n − 1 and a λ ∈ A * (Q λ ) runs over all elements So, the images of correspondences from Γ n are covered by those from Γ n+1 . Hence, we get a chain of surjections with the colimit A * N um . Here A * N um is obtained from A * by moding-out all classes Num ∼ 0 on all varieties.

Remark 4.4
For n = 1, the theory A * (1) is, by definition, the algebraic version A * alg . In particular, The meaning of the theory A * N um is described by the following universal property. Proof: By [17,Prop. 4.8], the canonical morphism of theories G : A * (0) ։ A * is surjective (this morphism is induced by the canonical morphism Ω * → A * of [9, Thm 1.2.6]). And by the same universality of algebraic cobordism such morphism is unique. In particular, there is a unique morphism of theories Thus, A * N um plays the role opposite to that of A * (0) , and can be denoted as A * (∞) , while any generically constant theory A * with the formal group law ϕ is canonically squeezed between A * (0) and A * (∞) : and the latter provides an alternative way of describing such theories A * .

Numerical equivalence modulo p and isotropy
Everywhere below X is a smooth projective variety over a field k of characteristic zero.
In the case of a theory Ch * = CH * /p, we will denote the numerical equivalence ∼ to stress that we consider finite coefficients. Thus, Ch * N um (X) = Ch * (X)/N , where N is the ideal of elements ∼ 0 if, for any y ∈ Ch(X), the deg(x · y) = 0 ∈ F p . By the very definition, we have a non-degenerate pairing Ch N um (X) × Ch N um (X) −→ F p defined by (x, y) → deg(x · y). Moreover, since deg(x · y) may be defined on the level of the (complex) topological realization, the kernel of the topological realization functor is contained in N , and so, Ch N um (X) is a sub-quotient of the topological cohomology H T op (X; F p ) of X. In particular, Ch N um (X) is finite-dimensional F p -vector space. Note, however, that this sub-quotient depends on the ground field k, since for different fields, the images of the topological realization functor will be different.
The theory Ch * N um inherits the action of the reduced power operations from Ch * , as follows from the Proposition 4.6 below. Recall, that we have reduced power operations P i : Ch * → Ch * +i(p−1) (see [2] and [21]) commuting with pull-back morphisms, and the respective homological operations P i : Ch * → Ch * −i(p−1) commuting with push-forwards. These are connected as follows:  Proof: Induction on i. Since P 0 = id, we have the (base) i = 0. (step) We need to show that deg(P i (u) · v) = 0, for any class v of complementary dimension. By Cartan's formula, . By the inductive assumption, we have that deg(P j (u) · P i−j (v)) = 0, for any j < i. And the degree of the first summand can be rewritten as The second summand is zero by the inductive assumption, while the degree of P i (u · v) is the same as that of P i (π * (u · v)), where π : X → Spec(k) is the natural projection. The latter degree is equal to zero for any i > 0. The induction step is proven.
If x is an anisotropic cycle, then x for some projective map f : Z → X from anisotropic variety Z, and some x ′ ∈ Ch * (Z). But then deg(x·y) = deg(f * (x ′ )·y) = deg(x ′ ·f * (y)) = 0, since all zero cycles on Z have zero degree (modulo p). Thus we get the surjective map: . I conjecture that, over flexible fields, these two cohomology theories, in reality, coincide.  Since the degree pairing is defined over the algebraic closure and even in the topological realization, we obtain that the numerical Chow groups over E are sub-quotients of the respective groups over E, which, in turn, are sub-quotients of topological cohomology: Thus, our Conjecture implies, that any object of Chow(k; F p ) which vanishes over k, or even in the topological realization, should vanish in every isotropic category Chow(E/E; F p ).
In contrast, "non-pure" motives behave differently. For example, the idempotent, corresponding to the projector π E : DM(k; F p ) → DM(E/k; F p ) is mapped via ψ E to the unit object of the category DM(E/E; F p ). On the other hand, its' restriction to k is trivial, since X Q | k = 0, for any non-empty Q (note that E = E, since k is flexible, so the respective directed system contains non-empty Q's). Consequently, all the subcategories DM(E/k; F p ), for all finitely-generated E/k, are killed by the restriction to the algebraic closure functor DM(k; F p ) → DM(k; F p ). There are geometric examples as well: the motive M α of Section 3 vanishes over k, but is non-zero in the isotropic motivic category. △ Remark 4.10 Although, the numerical Chow motivic category Chow N um (k; F p ) is a sub-quotient of a topological category (that is, of the category of graded F p -vector spaces), it is more interesting. In particular, it is not generated by a single object (Tate-motive). This is reflected by the absence of the Kunneth formula. Namely, for smooth projective X and Y , the product map is not an isomorphism (⇔ not surjective), in general. △ Our aim is to prove the following result.
Item (2) (1) Is it true that any U ∈ Ob(Chow(k; F p )) which vanishes in the (complex) topological realization is zero?
(2) Is it true that any f ∈ End Chow(k;Fp) (V ) which vanishes in the (complex) topological realization is nilpotent?
As we saw, by Conjecture 4.7, for a flexible field, the triviality of the topological realization should imply the triviality of all local realizations.
Remark 4.14 Note, that the stronger variant of Question 4.13(2) fails. Namely, C.Soule and C.Voisin produced an example of a class c ∈ Ch 3 (X), for some smooth projective X, such that c vanishes in H 6 T op (X; F p ), but c is not smash-nilpotent (that is, c ×r = 0 ∈ Ch 3r (X ×r ), for any r) -[13, Theorem 5].
If we are interested, instead, only in the triviality of local realizations, it is sufficient to take the image x of any torsion class x from CH 1 (X), whose topological realization in H 2 T op (X; F p ) is non-trivial. Then, since the degree pairing is defined integrally, x| E Num(p) ∼ 0, for any E/k. Thus all local realizations of x are trivial by Theorem 4.11 (2). At the same time, its topological realization is non-trivial, and so, not smash-nilpotent. Hence, x is not smash-nilpotent either. △ A weaker variant of Question 4.13 with "topological realization" replaced by the "restriction to the algebraic closure" is a safer bet. In this form, the question (2) becomes the Rost Nilpotence Conjecture.
We start the proof of Theorem 4.11 with the case of divisors (item (2)), which is, actually, the base for the whole technique. ∼ 0. Then u = 0 ∈ Ch 1 k/k (X). Proof: Adding to an effective divisor representing u a p-multiple of a very ample divisor, we may assume that u is represented by a very ample divisor D. Let (P n ) ∨ = Proj |D| be the projective linear system of D. This defines the embedding of X into P n . From Statement 6.3, the embedding ι : D η → X of the generic representative of our linear system into X induces a surjective map ι * : Ch 1 (X) ։ Ch 0 (D η ). But since D Num(p) ∼ 0, the degree of the zero-cycle ι * (v) is zero, for any v ∈ Ch 1 (X). Hence, D η is anisotropic. Thus, over k((P n ) ∨ ), our class u is represented by the class of an anisotropic divisor D η . Since k is flexible, it is represented by an anisotropic class already over k (by Proposition 1.3).
Since the theory Ch * k/k has the structure of push-forwards and pull-backs, we obtain: Corollary 4.16 Let k be a flexible field. The projection Ch * ։ Ch * k/k passes through Ch * alg = Ch * (1) . Proof: A class u ∈ Ch * (X) is algebraically equivalent to zero, if it can be presented as f * (y · g * (v)), where X f ←− X × C g −→ C are natural projections, C is a smooth projective curve, y ∈ Ch * (X × C) and v ∈ Ch 0 (C) is a zero cycle of degree zero. Since v Num(p) ∼ 0, Proposition 4.15 implies that v = 0 ∈ Ch * k/k (C), and so, u = 0 ∈ Ch * k/k (X).
Our general strategy of proving that the class u

Num(p)
∼ 0 is anisotropic will be to find an appropriate blow-up π : X → X, so that π * u may be represented by a cycle supported on a smooth connected divisor Z ⊂ X, which is Num(p) ∼ 0 already on Z. Then we use induction on the dimension of X and the fact that u = π * π * u. In order to achieve this, we will need first to present u by the class of a smooth connected subvariety S, and make the appropriate characteristic classes of it Proposition 4.18 Let k be a flexible field and X be a smooth projective variety over k of dimension 3. Let u ∈ Ch 1 (X) be Num(p) ∼ 0. Then u = 0 ∈ Ch k/k;1 (X).
Proof: We will show that there is a blow-up π : X → X such that π * (u) is represented by the class of a smooth anisotropic curve on X. Since π * π * (u) = u, this will show that the class u is anisotropic.   2 We may assume that u is represented by the class of a smooth (possibly, disconnected) curve S which is contained in some smooth (possibly, disconnected) surface E on X, and the curve S satisfies also: deg(c 1 (N S⊂X ) · [S]) = 0.
Proof: By Lemma 4.18.1, we can assume that u = [S], where S ⊂ X is a smooth curve, and moreover, deg(c 1 (N S⊂X ) · [S]) = 0. Consider X = Bl S (X) with the projection π : X → X. Then π * (u) is supported on E = P S (N S⊂X ), which is a smooth surface. More precisely, it is represented by ρ+ε * (c 1 (N S⊂X )), where ε : E → S is the natural projection and ρ = c 1 (O(1)) = −[E]. By adding a p-multiple of a very ample divisor, we can assume (by Statement 6.4) that π * (u) is represented by a very ample divisor on every component of E, and so, is represented by a smooth curve S ′ on E. Moreover, Since k is flexible, we may assume that the divisor E ′′ of X is defined already over k. Let π : X → X be the embedded desingularization of E ′′ . Let E be the proper pre-image of E ′′ and Y be the proper pre-image of S. Then π * (u) is equal to [ S] plus some classes supported on the special divisors of our blowup. But these special divisors are anisotropic (since the singularities were). Hence, modulo anisotropic classes, π * (u) is equal to the class   ∼ 0. Then u = 0 ∈ Ch k/k;1 (X).
(base) The case n = 1 is trivial. The cases n = 2 and n = 3 follow from Propositions 4.15 and 4.18, respectively.
Lemma 4.20. 1 We may assume that u is represented by a class supported on some smooth (possibly, disconnected) divisor Z on X.
Proof: By Corollary 6.2, we may assume that u is represented by the class of a smooth curve S on X. Consider the blow-up π : X = Bl S (X) −→ X. Then π * (u) is supported on the special divisor E = P S (N S⊂X ) which is smooth.
Lemma 4.20. 2 We may assume that u is represented by a class supported on some smooth connected divisor Z on X such that the restriction Ch 1 k/k (X) ։ Ch 1 k/k (Z) is surjective. Proof: By Lemma 4.20.1, we may assume that u is represented by the class of a curve S contained in a smooth divisor Z. By Statement 6.5 and flexibility of k, S is contained in some irreducible divisor Z ′ , smooth outside an anisotropic subset, and such that the restriction Ch 1 k/k (X) ։ Ch 1 k/k (Z ′ \S) is surjective. Since dim(Z ′ ) − 1 > 1 = dim(S), we also get the surjection Ch 1 k/k (X) ։ Ch 1 k/k (Z ′ ). Let π : X → X be the embedded resolution of singularities of Z ′ . Since the singularities of Z ′ were anisotropic, so will be the special divisors of X. Let Z ′ be the proper pre-image of Z ′ and S be the proper pre-image of S. Then Ch * k/k (Z ′ ) = Ch * k/k ( Z ′ ), and so, the map f * : Ch 1 k/k (X) ։ Ch 1 k/k ( Z ′ ), induced by the natural projection f : Z ′ → X, is surjective. The image of π * (u) in Ch * k/k ( X) = Ch * k/k (X) is represented by the class of S supported on Z ′ .
Since the restriction, j * : Ch 1 k/k (X) ։ Ch 1 k/k (Z) is surjective, u = j * (u ′ ), for some class u ′ ∈ Ch k/k;1 (Z), and u Num(p) ∼ 0 on X, we get that u ′ Num(p) ∼ 0 on Z. As Z is smooth connected projective variety of dimension n − 1, by the inductive assumption, u ′ = 0 ∈ Ch k/k;1 (Z). Then the class u is equal to 0 ∈ Ch k/k;1 (X) as well. Proposition 4.20 is proven. Proof: Our strategy will be to find an appropriate blow-up π : X → X such that π * (u) is supported on some smooth connected divisor Z and is Num(p) ∼ 0 on it. By Corollary 6.2, we may assume that u is represented by the class of a union of smooth complete intersections of very ample divisors with components meeting transversally. In such a situation (of transversal smooth components), let us denote as c where all the divisor classes involved are very ample, and so, components can be made transversal. Then , while t i,j · t k,l = 0, for {i, j} = {k, l}. Now, for all pairs (i, j) of distinct numbers let us choose signs ε (i,j) ∈ ±1 with the condition that ε (i,j) + ε (j,i) = 0. Let us substitute classes u i by u ′ i := u i + j =i ε (i,j) t i,j . Note, that the sum i u ′ i is still equal to the i u i = π * (u). On the other hand, Since these classes u ′ i are divisors on (connected) E i , we can make them very ample (by adding a pmultiple of some very ample divisor) and so, can move them around and make them smooth connected and transversal to any given subvariety. Let S ′ i be the generic representative of the respective linear system | u ′ i | on E i . It is a smooth connected surface on E i representing u ′ i . We now have a divisor with strict normal crossings E = ∪ i E i and a surface S ′ = ∪ i S ′ i on it, with S ′ i smooth connected, transversal to each other and to the other components of E.
Since deg( u ′ i · u ′ j ) = 0, for i = j, by the arguments of the proof of Proposition 4.15, all the intersections S ′ i ∩ S ′ j are anisotropic. Since k is flexible, we may assume that S ′ i is defined already over k (by Proposition 1.3). Finally, denoting γ i = π * (c 1 (N S i ⊂X )), since ρ i ρ j γ i , ρ i ρ 3 j , ρ i ρ j ρ k are zero, for distinct i, j, k, we have

If [S]
. After all, we managed to present our cycle by the class of the union of smooth surfaces transversal to each other and all intersections anisotropic. It remains to use Statement 6.6.  c 1 (N S⊂X )) + ρ). We have: c 1 (N S⊂ X ) = 2ρ + π * S (c 1 (N S⊂X )), while c 1 (N F ⊂ X ) = ε * (c 1 (N S⊂X )). Hence, (taking into account that π(E) is zero-dimensional), On the other hand, By Lemma 4.21.2, there exists a blow-up µ : X → X, such that µ * π * ([S]) ∈ Ch 2 k/k (X) is represented by a smooth connected surface S contained in a smooth connected divisor Z with ∼ 0 on X, and R is connected, we have that ∼ 0 on X, for m > 0.
Substituting F by the generic representative of the (very ample) linear system |F | on E, by the proof of Proposition 4.15, we may assume that the intersection S ∩ F is anisotropic. Then, by Statement 6.6, there exists a blow-up µ : X → X such that µ * π * ([S]) is represented by the class of a smooth connected surface S contained in a smooth connected divisor Z, and such that c m ∼ 0, for m 0. Now Proposition 4.21 follows from Corollary 6.12 and flexibility of k.

Proposition 4.22
Let k be a flexible field and X be a smooth projective k-variety of dimension 5. If u ∈ Ch 2 (X) is ∼ 0, then u = 0 ∈ Ch k/k;2 (X).
Proof: The strategy, as usual, is to find an appropriate blow-up, so that the pull-back of u is supported on some smooth connected divisor Z and is ∼ 0 already on Z. −→ S l is the proper transform of S l and V = P 2 Q is a subvariety of the Q-fiber of the P 3 -bundle E → R. Here S l is a complete intersection (x 1 + ρ)(x 2 + ρ)x 3 , while V is a complete intersection −ρ 2 · x 3 . We can move [Q] along R and make V disjoint from S l (it is automatically disjoint from other components). On S l , x i · ρ = 0 (since π S ( S ∩ E) is zero-dimensional) and ρ 2 = −π * S [Q], while on V = P 2 Q , x i = 0 and ρ 2 is the class of a section Q → P 2 Q . Hence, the Since p = 2, by choosing d appropriately, we can always make the total degree of c 2 (N S⊂X ) · [S] to be zero (in F p ), while keeping all the components complete intersections. The Chern roots of N S l ⊂ X are x i + j =i ρ {i,j} , i = 1, 2, 3. Moreover, S l is still a complete intersection: , and on S l we have identities: given by ρ · π −1 {i,j} (Q {i,j} ) and contained in the {i,j} x m (m as above). We can move the class [Q {i,j} ] along R {i,j} , and so can make P 2 Q {i,j} disjoint from S l (and it is automatically disjoint from the other components of S). The Chern roots of N V ⊂ X are −ρ, 0, ρ, and so, Let us apply the above construction to every component S l of S, and denote the respective objects by the subscript l. Now, after applying µ * π * , the degree of c 2 (N S⊂X ) · [S] is concentrated in the Fcomponents, where [F l ] is given by −α l · α l · ρ l . Then [F ] = l (−α l · α l · ρ l ). But since ρ l · α k = 0, and α l · α k = 0, for l = k, this can be rewritten as −α · α · ρ, where α = l α l and ρ = l ρ l . We can substitute −α, α and ρ by very ample divisors and represent . Being a complete intersection on E, this class may be represented by a smooth surface S on E. Note also, that ρ(ρ 2 + c 1 ρ) · [E] = 0 and that the Chern roots of N S i ⊂ X are −ρ, ρ, ρ + c 1 , and so, the deg(c 2 (N S i ⊂ X ) · [ S i ]) is still zero.
By Statement 6.5 and flexibility of k, there is an irreducible divisor Z, containing S, smooth outside an anisotropic closed subscheme of S, and such that the restriction Ch * ( X) ։ Ch * (Z\ S) is surjective. Let µ : X → X be the embedded desingularization of Z and S (note, that we may assume that no component of S belongs to the singular locus of Z, since this locus is anisotropic). Let Z and S be the proper pre-image of Z and S, respectively. Then, Z is smooth connected and, modulo anisotropic , we obtain that the group Ch 2 k/k (Z) is generated by the image of j * : Ch 2 k/k (X) → Ch 2 k/k (Z) and the classes [S i ] of all the connected components of S. Since S k/k (X) (since S coincides with S, Z coincides with Z and X coincides with X modulo anisotropic subvarieties) and ρ · [ S i ] = 0 ∈ Ch * ( X).
Hence  (step) Let dim(X) > 5. By Corollary 6.2, we may assume that u is represented by a class of a smooth (possibly, disconnected) surface S on X. Let π : X → X be the blow-up at S, with the special divisor E. Then π * (u) has support on a smooth divisor E, and is represented there by a class of some surface S ′ (possibly, non-smooth). By Statement 6.5 and flexibility of k, there is an irreducible divisor Z, containing S ′ , smooth outside an anisotropic closed subscheme of S ′ , and such that the restriction Ch * ( X) ։ Ch * (Z\S ′ ) is surjective.
Let µ : X → X be the embedded resolution of singularities of Z, with Z and S -the proper transforms of Z and S ′ , respectively. Then Z is a smooth connected divisor on X and µ * ([S ′ ]) is represented by [S] ∈ Ch * k/k (X) supported on Z (since the remaining ingredients are anisotropic). Since Z, respectively, S, coincides with Z, respectively, S ′ , modulo anisotropic subvarieties, the restriction ∼ 0 on Z as well. By inductive assumption, [S] is represented by the class of an anisotropic surface on Z, and so, on X. Induction step and Proposition 4.23 are proven. ∼ 0 on X.
Proof: We need to treat separately the case p = 2 and that of odd primes. Then On the other hand, (c 3 where P l is the reduced power operation (modulo 2). By Proposition 4.6, since [S ′ ]  , where S is the proper transform of S and F = P 3 is a divisor on E given by ρ. We can make S and F transversal. If [S] = xy, then S is a complete intersection (x + ρ)(y + ρ), and F is a complete intersection −ρ · ρ. On S we have: ρ · x = ρ · y = 0, while ρ 3 is the minus class of a point. Then Similarly, Proof: It is sufficient to treat the case of a single complete intersection [S] = xy. Then Substituting S by S exp and blowing up the zero-cycles 5 2 c 3 1 (N S⊂X ) and 1 2 c 1 c 2 (N S⊂X ) on it (note, that p = 2), we obtain [S ′′ ] = π * ([S]) such that all the components of S ′′ are still smooth complete intersections transversal to each other, and by Sublemma 4.24.1.1, Applying this procedure twice, we obtain c 3 1 = 0 and c 1 c 2 = 0. Lemma 4.24.1 is proven. Let S ′ = S ∪ F ∪ G. Here ρ {i,j} satisfies: ρ 4 {i,j} + ρ 3 {i,j} (x i + y i + x j + y j ) + x i y i x j y j = 0, where [S k ] = x k y k . Of course, one can get rid of the G-term by considering This will work for odd primes. But for p = 2, this term really makes a difference.

again, since [S]
Num(p) ∼ 0 on X. Using the same properties, we obtain: ∼ 0 and c 2 1 (N F ⊂ X ) · [F ] = 0. Proof: Using the fact that the centers of the blow-up π were 1-dimensional, we obtain: We may assume that our class is represented by the class of S ′ = S ∪ F ∪ G, where S, F and G are smooth (possibly, disconnected) and transversal to each other (note, that we don't have triple intersections of S i 's), and [G] = (−ρ) · ρ, for some divisor ρ. Moreover, both S and F have trivial ∼ 0. As a next step, we will combine S ∪ F into a single component. Proof: Adding to S i a p-multiple of a very ample divisor on Y i , we may assume that S i is given by a very ample divisor on Y i (this doesn't change the characteristic classes (mod p)). Let S ′ i be the generic representative of the linear system |S i |. Then S ′ i is connected and, by Statement 6.3, the restriction , all the characteristic classes are preserved. Finally, since k is flexible, we may assume that S ′ is defined over k.  ∼ 0 onX.
(1) Let π : X → X be the blow-up in the components of S, with the components E i π i −→ S i of the special divisor, and . Consider the classes [E 1 ](ρ 1 −ρ 2 +π * 1 c 1 (N 1 )) and [E 2 ](ρ 1 +ρ 2 +π * 2 c 1 (N 2 )). We may assume that these classes are represented by the classes of smooth connected subvarieties S ′ 1 and S ′ 2 , contained in E 1 and E 2 , respectively, and transversal to each other and to E {1,2} . Let At the same time, since ρπ * S c 1

Num(p)
∼ 0 on S, as we saw above, and [S]c 3 1

Num(p)
∼ 0 on R. Using the same arguments, ∼ 0, and The case (p = 2, 3) is done. By blowing-up S, we may assume that S ⊂ Y , where Y is smooth connected divisor on X. Note, that the new c 3 1 , c 1 c 2 and c 2 1 characteristic classes are the pull-backs of the old ones, and so, are still numerically trivial. By Statement 6.5 and flexibility of k, there exists an irreducible and smooth outside an anisotropic subscheme (of S) divisor Z containing S, such that the restriction Ch 1 (X) ։ Ch 1 (Z\S) is surjective. Let π : X → X be an embedded desingularization of Z and S, with proper pre-images Z and S, which are smooth connected subvarieties. Since singularities were anisotropic, the maps π * : Ch * k/k (X) ∼ 0 already on Z. Since this class is a complete intersection on Z, the intersection of generic representatives of the respective (very ample) linear systems is anisotropic by Statement 6.3. The respective subvariety S is smooth connected, and c 3 1 and c 1 c 2 characteristic classes are preserved. Since k is flexible, we may assume that our varieties are defined over k. Lemma 4.24.3 is proven. Proof: By Lemma 4.24.3, we may assume that S is smooth connected variety with the numerically trivial c 3 1 , c 1 c 2 and c 2 1 characteristic classes. It remains to make c 1 numerically trivial. We need to treat separately the case p = 2 and that of odd primes. (p = 2) Let R be the generic representative of the (very ample) linear system | 1 2 c 1 (N S⊂X )| on S. It is a smooth connected surface and, by Statement 6.3, we have the surjection Ch 1 (S) ։ Ch 1 (R). Since k is flexible, we may assume that it is defined over k. Let π : X → X be the blow-up at R, with the special connected divisor E and ρ ∼ 0 on S and Ch 1 (S) ։ Ch 1 (R). On the other hand, on R = S ∩ E, and moreover, we may assume the intersection S ∩ F to be anisotropic. By Statement 6.6, there exists a blow-up µ : X → X, such that µ * ([ S ∪ F ]) is represented by the class of a smooth connected subvariety S and such that the characteristic classes of S in Ch * k/k are µ * of the respective classes of S ∪ F . In particular, the c 1 -class of it is numerically trivial.

Thick local categories
In this section we extend the definition of local motivic category to arbitrary finite coefficients Z/n and introduce the thick versions of it which have better conservativity properties.
Definition 5.1 Let n ∈ N. Let P and Q be smooth varieties of finite type over k. We say that X Q n > X P , if P is n-isotropic over every generic point of Q, while Q is n-anisotropic over some generic point of P .
Let E/k be some finitely generated extension and P be smooth connected variety with k(P ) = E. Let Q n be the disjoint union of all smooth connected varieties Q of finite type with X Q n > X P and Υ n P := X Q n ⊗ X P .
Generalizing Definition 2.3, we can define the local motivic category with Z/n-coefficients. If we are interested in a p-localized situation, we can define the thick local motivic categories.

Definition 5.3
The p-local motivic category of thickness r and F -coefficients In particular, the local category with Z/p r -coefficients DM(E/k; Z/p r ) is the p-local motivic category of thickness r and Z/p r -coefficients DM({E/k} r ; Z/p r ).
Since X Q n > X P implies that X Similarly, for the thick local Chow groups, we have: CH {k/k} r (X; F ) = CH(X; F )/(p r -anisotropic classes).
Since every n-anisotropic class is numerically equivalent to zero modulo n, we obtain the surjection: Question 5.5 Let k be flexible. Is it true that CH k/k (X; Z/p r ) = CH N um (X; Z/p r )?
Using the arguments of Proposition 4.15 and Corollary 4.17, we see that the answer is positive for divisors and for zero-cycles. In particular, the projection CH(X; Z/p r ) ։ CH k/k (X; Z/p r ) factors through CH alg (X; Z/p r ). Also, it is easy to see that it is true for cycles of dimension 1, provided p > 2, and so, the mentioned projection factors through CH (2) (X; Z/p r ), in this case.
With the increase of r (and fixed s), the family of functors for all f.g. extensions E/k, becomes more and more conservative. But the target categories are getting more complicated. At the same time, the categories DM(E/k; Z/p r ) (that is r = s) should be simpler, and the natural strategy to improve conservativity is to pass to Z (p) -coefficients. The local motivic category with Z (p) -coefficients DM(E/k; Z (p) ) should be the local category of ∞ thickness DM({E/k} ∞ ; Z (p) ). This should be defined as a limit lim r DM({E/k} r ; Z (p) ) of categories of finite thickness and will be considered in a separate paper.
6 Auxiliary results

Up to blow-up, Chow ring of a variety is generated by divisors
The following result is crucial for most of our constructions. Theorem 6.1 Let X be a smooth projective variety and y ∈ CH r (X). Then there exists a blow-up π : X → X such that π * (y) is a Z-polynomial in divisor classes.
Proof: Induction on the dim(X). Below we will denote this induction as Ind 1 .
(Ind 1 step) Can assume that r > 0. Then y has support on some divisor. By blowing X up we can assume that this divisor has strict normal crossings, and by the following Lemma, can assume that y has support on a smooth divisor D. Lemma 6.1.1 If y = i y i , and the statement is true for each y i , then it is true for y.
Proof: Suppose, for each i, there exists such blow-up π i : X i → X that π * i (y i ) is a polynomial in divisorial classes. Then, by the results of Hironaka [7], there exists a blow-up π : X → X, which covers π i , for each i. Then, clearly, π * (y) is a polynomial in divisorial classes.
Let now j D : D → X be a smooth divisor and y is supported on D. Let us say that the pair (D, X) has a special structure with the base B if D is a consecutive projective bundle over B where the canonical line bundles O(1) of all these fibrations are the restrictions of some line bundles from X. Every pair (D, X) possesses a "trivial" special structure with the base B = D. We will prove our statement by the induction on the dim(B). We will denote this induction as Ind 2 below. (Ind 2 base) dim(B) = 0. Then D is a disjoint union of consecutive projective bundles whose all canonical bundles O(1) are restrictions of some line bundles from X. Then CH * (D) is generated by j * D (c 1 (L)), for some line bundles L on X. Since y = (j D ) * (y), for some y ∈ CH * (D), it is a polynomial in divisorial classes.
(Ind 2 step) We have: y = (j D ) * (y), where y = l ε * (u l ) · r l with u l ∈ CH * (B), ε : D → B is the natural projection, and r l is a monomial in ρ t 's, where ρ t = c 1 (O(1) t ) is the restriction of some divisorial class from X. By Lemma 6.1.1, we can assume that y = ε * (u), where u ∈ CH * (B).
Since dim(B) < dim(X), by the inductive assumption of Ind 1 , there exists a blow-up τ : B → B, such that τ * (u) is a polynomial in divisorial classes. We have a cartesian diagram of blow-ups: Let ϕ : X → X be a blow-up of X in the same centers as τ D , and j D : where v m are supported on the components E m of the special divisor E = ∪ m E m of the blow-up ϕ -see [5, Thm 6.7] (or [16,Prop. 5.27]). Let X m be a variety obtained after m blow-ups with the special divisor j Em : E m → X m of the m-th blow-up and the projection ϕ m : X → X m whose restriction to E m is the blow-up α m : E m → E m . Then (again by [5,Thm 6.7]), the image of has support on ∪ n>m E n , while the map is surjective (since ∪ n m E n \ ∪ n>m E n is an open subvariety of E m ). Hence, the map covers the image of (j E ) * . The pair (E m , X m ) has a special structure with the base of dimension smaller than B (namely, the center of the m-th blow-up of τ ), where c 1 (O(1)) of the external projective fibration is the restriction of [−E m ] from X m , and all the other (internal) canonical bundles are induced by the special structure on (D, X), and so, are restrictions of some divisorial classes from X. By the inductive assumption of Ind 2 , any element in (j Em ) * CH * (E m ) is a polynomial in divisorial classes over some blow-up X m → X m . By Lemma 6.1.1, we obtain that m v m is a polynomial in divisorial classes over some blow-up of X. Hence, it remains to deal with (j D ) * τ * D (ε * (u)). We know that τ * D (ε * (u)) = ε * τ * (u) is a polynomial in divisorial classes by construction. Let us denote this element again as y and the divisor (on which it is supported) as D. By Lemma 6.1.1, we can assume that y is a monomial x 1 · . . . · x r−1 in very ample divisorial classes on D. We will use induction on r. Below it will be denoted as the Ind 3 . (Ind 3 step) Consider the chain of co-dimension 1 regular embeddings D ] represents the restriction of x 1 to Y k . Construct the chain of blow-ups X ←− . . . in co-dimension-two centers Z k → X k−1 inductively as follows. The special divisor E k of π k is a consecutive projective line fibration E k ϕ k → Y k and Z k+1 = ϕ −1 k (Y k+1 ). In particular, E k ε k → Z k is a projective line fibration and ϕ k = ε 1 • . . . • ε k . Let i k = i 1 • . . . • i k and y k = ϕ * k (i k ) * (y) be the class supported on E k . Then we have a cartesian diagram with E k = E k and y k+1 = ε * k+1 α * y k . By (the Chow group version of) [16,Prop. 5.27], we have: Here, by the inductive assumption of Ind 3 , the first summand is expressible as a polynomial in divisorial classes over some blow-up of X k+1 , so (using Lemma 6.1.1) the question about y k supported on E k → X k is reduced to the question about y k+1 supported on E k+1 → X k+1 . Thus, it is sufficient to show that our statement is true for y k supported on E k → X k , for at least, one k. But for k = dim(X) − r, the class (i k ) * (y) is zero by dimensional reasons. Hence, the (Ind 3 step) is proven. This implies (Ind 2 step) and (Ind 1 step). The Theorem is proven.
Corollary 6.2 Let X be a smooth projective variety and y ∈ CH r (X). Then there exists a blow-up π : X → X, such that π * (y) is represented by a linear combination of classes of smooth complete intersections of very ample divisors which are transversal to each other. In particular, for r > dim(X)/2, it is represented by the difference of classes of two smooth disjoint subvarieties.

General position results
In this section we present various general position arguments which permit to replace cycles by the classes of connected subvarieties and, in some cases, reduce the anisotropy of a class [S] to the numerical triviality of some characteristic classes of S.
The following simple and well-known "Chow group shadow" of the Lefshetz theorem is one of our key ingredients. Statement 6.3 Let X be a scheme with a map X f → P n and ι : D η → X be the generic hyperplane section of X (over k((P n ) ∨ )). Then the pull back ι * : CH * (X) ։ CH * (D η ) is surjective. If X is a smooth variety, so is D η .
Proof: Consider Y ⊂ X × (P n ) ∨ given by Y = {(x, H)|f (x) ∈ H}. Then Y is a projective bundle: Y = Proj X (V ), where V = W/O(−1) (with (P n ) ∨ = P(W )). Let Y η be the generic fiber of the projection Y → (P n ) ∨ . This is the "generic" hyperplane section D η . By the projective bundle theorem and localization, we have surjections CH * (X)[ρ] ։ CH * (Y ) ։ CH * (Y η ), where ρ = c 1 (O(1)). Since ρ is the pull-back of the class of a hyperplane on (P n ) ∨ , it is zero in CH * (Y η ). Thus, we get the surjection CH * (X) ։ CH * (Y η ). Finally, if X is smooth, so is Y and Y η . Statement 6.4 Let X be an irreducible quasi-projective variety and n ∈ N. Let E be an effective Cartier divisor on X, and S ⊂ E be a closed subscheme. Then there exists a very ample divisor D such that the linear system |E + n · D| S consists of very ample divisors and S is the base set of it.
Proof: There are very ample divisors Statement 6.5 Let n ∈ N, X be a smooth projective connected variety of dimension d and E = ∪ i E i be a divisor on it with strict normal crossings outside an n-anisotropic closed subscheme, with, possibly, reducible components E i , and let S i ⊂ E i be some closed subschemes such that, for any subset I of the set of indices, dim(S I ) < d − #(I), where S I = ∩ i∈I S i . Then, over some purely transcendental f.g. extension of k, there is a divisor Z = ∪ i Z i , where [Z i ] = [E i ] ∈ CH 1 (X; Z/n), S i ⊂ Z i , for any I, the variety Z I = ∩ i∈I Z i is irreducible, and Z has strict normal crossings outside an n-anisotropic closed subscheme of S. Moreover, the restriction CH * (X) ։ CH * (Z i \S i ) is surjective.
Proof: By the Statement 6.4, for any i, there exists a very ample divisor D i , such that the linear system Φ i = |E i + n · D i | S i consists of very ample divisors and has the base set S i . Let Z i be the generic element of this linear system (defined over k(P(Φ i ))). Clearly, [Z i ] = [E i ] ∈ CH 1 (X; Z/n) and S i ⊂ Z i . Let us show that Z I = ∩ i∈I Z i is irreducible. We will prove by induction on #(I) that, for any i ∈ I, the restriction CH 0 (Z I\i ) ։ CH 0 (Z I ) is surjective. The (base) I = ∅ is empty. (step) The linear system Φ i defines an embedding of X\S i into a projective space, and for any subscheme T ⊂ X defined over some field L and Z i defined over L(P(Φ i )) as above, we have: dim(T ∩ (Z i \S i )) dim(T ) − 1. By the Statement 6.3, we have a surjection CH 0 (Z I\i \Z I\i ∩ S i ) ։ CH 0 (Z I \Z I\i ∩ S i ). We know that all the components of Z I have dimensions d − #(I). On the other hand, the scheme Z I\i ∩ S i is the union ∪ i∈J⊂I Y J , where Y J = (∩ j∈J S j ) ∩ (∩ j∈I\J (Z j \S j )). Since dim(Y J ) dim(S J ) − #(I\J), we obtain that dim(Z I\i ∩ S i ) max(dim(S J ) − #(I\J)|i ∈ J ⊂ I) < d − #(I). Hence, CH 0 (Z I ) = CH 0 (Z I \Z I\i ∩ S i ), and we get the surjection CH 0 (Z I\i ) ։ CH 0 (Z I ). Induction step is proven.
Thus, we have the surjection CH 0 (X) ։ CH 0 (Z I ), and since X is irreducible, so is Z I . Our system Φ i contains E i + |n · D i |. The generic representative of |n · D i | is n-anisotropic (by Statement 6.3 and the arguments from the proof of Proposition 4.15). Thus, the generic representative G i of E i + |n · D i | has only n-anisotropic singularities and, modulo n-anisotropic subvarieties, the divisor G = ∪ i G i has strict normal crossings. Consequently, the generic representative Z i of our system |E i + n · D i | Y will have only n-anisotropic singularities too, and the divisor Z = ∪ i Z i will have strict normal crossings modulo n-anisotropic subvarieties. Indeed, we have a divisor W on X × P , where P = i P(Φ i ) parameterizes (combinations of) elements of our linear systems. The fiber over the generic point of P is Z, while the fiber over some special point is G. Let R be a d.v.r. with the fraction field K and residue field κ, and W R be some divisor on X × Spec(R), with fibers W K and W κ over the generic and closed point of Spec(R), respectively. Having a closed point T of W K , consider the closure of it in W . We get a proper morphism f : T → Spec(R) of relative dimension zero, whose fiber over Spec(κ) consists of points t l with multiplicities e l -these are the specializations of T . The specialization of a singular point is singular, and specialization of a point where components are not transversal has the same property. is. This shows that Z should be a divisor with strict normal crossings at every point which is not n-anisotropic. Since our linear system Φ i is very ample on X\S i , the divisor Z has strict normal crossings outside S. By the same reason and by Statement 6.3, we have the surjection CH * (X) ։ CH * (Z i \S i ).
This permits to represent some cycles of co-dimension 2 by single components.
Everywhere below we will denote: Ch * k/k (X) := CH * (X, Z/n)/(n − anisotropic classes), where n is some natural number (which should be clear from context). Statement 6.6 Let S = ∪ i S i be the union of smooth connected transversal subvarieties of codimension 2 on a smooth projective variety and n ∈ N. Suppose, that all the intersections S i ∩ S j are n-anisotropic.
Then over some f.g. purely transcendental extension of k there exists a blow-up µ : X → X, such that µ * ([S]) ∈ Ch * k/k (X) is represented by the class of a smooth connected variety S whose characteristic classes in Ch * k/k are µ * of the characteristic classes of S. Moreover, if S = ∪ i S i and T = ∪ j T j are two subvarieties as above which are transversal to each other, then there exists a blow-up µ : X → X with the properties as above for both S and T and such that S and T are smooth connected, transversal to each other, and S ∩ T is connected.
Proof: Let π : X → X be the blow-up in all the components S i of S, with E i π i −→ S i the components of the special divisor and ρ i = −[E i ]. Then, by [5,Prop. 6.7], π * ([S]) = [F ], where F = ∪ i F i , F i is supported on E i and [F i ] = [E i ](ρ i + π * i (c 1 (N S i ⊂X ))). Note, that E = ∪ i E i is a divisor with strict normal crossings, with all the intersections E i ∩ E j n-anisotropic. Note, that c 1 (N F i ⊂ X ) = π * i (c 1 (N S i ⊂X )) and c 2 (N F i ⊂ X ) = π * i (c 2 (N S i ⊂X )). Hence, the same is true about all other characteristic classes. Since E is smooth outside an n-anisotropic subscheme, by Statement 6.5 (where we consider E as a single component), over some f.g. purely transcendental extension of k, there is an irreducible divisor Z, containing F , smooth outside an n-anisotropic closed subscheme of F . Let ε : X → X be an embedded desingularization of Z. Let Z and F be the proper pre-images of Z and F . Then ε * : Ch * k/k ( X) = → Ch * k/k (X) is an isomorphism and ε * ([F ]) = [F ] ∈ Ch 2 k/k (X) is supported on the smooth connected divisor Z. By adding an n-multiple of a very ample divisor, we can substitute [F ] by a very ample divisor on Z. Let S be the generic representative of the linear system |F | on Z. Then S is smooth and connected. Since, modulo n-anisotropic subvarieties, X coincides with X and F with F , the characteristic classes of S in Ch * k/k (X) are ε * of the respective classes of F . For the pair of subvarieties S = i S i and T = j T j , consider the blow-up π : X → X at all the components of S and T . Then the special divisor E = E S ∪ E T has strict normal crossings, where E S and E T are smooth outside n-anisotropic subschemes. Applying Statement 6.5 to F S ∪ F T contained in E S ∪ E T (where we consider E S and E T as single components), we obtain a divisor Z = Z S ∪ Z T containing S ∪T , with strict normal crossings outside an n-anisotropic subscheme, with irreducible Z S , Z T and Z S ∩ Z T . Resolving n-anisotropic singularities and non-transversalities of Z, as above, we obtain the needed smooth connected transversal subvarieties S and T having the needed characteristic numbers. Since Z S ∩ Z T is irreducible, the intersection S ∩ T is connected. Statement 6.7 Let n ∈ N, X be projective irreducible variety, smooth outside an n-anisotropic closed subscheme, and S ⊂ Z ⊂ X be embeddings of co-dimension 1 of irreducible subvarieties, smooth outside n-anisotropic closed subschemes and such that S is not n-anisotropic. Then, over some f.g. purely transcendental extension of k, there exists Z ′ , such that S ⊂ Z ′ ⊂ X has the same properties, [Z ′ ] = [Z] in Ch 1 k/k (X) and the restriction Ch * k/k (X) ։ Ch * k/k (Z ′ \S) is surjective. Proof: This is a particular case of Statement 6.5, aside from the fact that we permit X to have anisotropic singularities. But the same proof works. Outside some closed n-anisotropic subscheme T of X, Z is a Cartier divisor and, by the Statement 6.4, there exists a very ample divisor D on X\T , such that the linear system Φ = |Z + n · D| S\S∩T on X\T consists of very ample divisors and has the base set S\T . This linear system defines an embedding of X\(S ∪ T ) into a projective space. Let Z ′ be the closure in X of the generic element of this linear system (defined over k(P(Φ))). Since S is not n-anisotropic, S\S ∩ T is non-empty, and so, Z ′ contains S. Clearly, [Z ′ ] = [Z] ∈ Ch 1 k/k (X) = Ch 1 k/k (X\T ). By Statement 6.3, we have a surjection Ch * k/k (X) ։ Ch * k/k (Z ′ \S), and Z ′ is irreducible. The same arguments as in the proof of Statement 6.5 show that Z ′ has only n-anisotropic singularities.  where X i are irreducible varieties smooth outside some closed proper n-anisotropic subschemes, the maps ((j i ) * , (g i+1 ) * ) : Ch * k/k (X i−1 ) ⊕ Ch * −1 k/k (X i+1 ) ։ Ch * k/k (X i ) are surjective, and [X i+1 ] = [X i+1 ] ∈ Ch 1 (X i ). Let π 0 : X ′ 0 → X 0 be the embedded desingularization of X r ⊂ X r−1 ⊂ X r−2 ⊂ . . . ⊂ X 1 ⊂ X 0 , and ε i : X ′ i → X i be the proper pre-images (with ε 0 = π 0 ). Since special divisors are n-anisotropic, we have isomorphisms ε * i : Ch * k/k (X i ) = → Ch * k/k ( X i ). Noting that X i is not n-anisotropic, by blowing X i at n-anisotropic centers, we may resolve the indeterminacies of the maps X i k/k (X m+1 where α l i = (j l i ) * , β l i = (g l i ) * (π l−1 i ) * and β-maps shift the codimension by (+1). The maps (α l i , β l i+1 ) : Ch * k/k (X l i−1 ) ⊕ Ch * k/k (X l−1 i+1 ) ։ Ch * k/k (X l i ) are surjective. Either at some stage we will get an n-anisotropic X m r , in which case we are done, or we can iterate the process q = dim(X) + 1 times. SetX i = X q i , etc. Then Ch * k/k (X i ) is generated by the elements of the form ω(x), where ω is a composition of α's and β's and x ∈ Ch * k/k (X 0 ). Here we are using the fact that the number of β's in such a composition can't be more than dim(X i ) (as each β increases the codimension by 1), and so, the chain has to start with X 0 . We also have maps γ l i = (j l i ) * and δ l i = (π l−1 i ) * (g l i ) * fitting commutative diagrams (recall, that (π m i ) * and (π m i ) * are isomorphisms) Using these relations, one can reduce ω(x) to the form θ(x), where θ is a combination of α q i 's and γ q j 's. The restriction of such an element to Ch * k/k (X q r ) is f * 0 (x) times a monomial in c 1 (N X q i ⊂X q i−1 ) = c 1 (N i )'s, where each factor c 1 (N i ) corresponds to a loop α q i γ q i in θ. Thus, the image f * j : Ch * k/k (X j ) → Ch * k/k (X r ) as a Ch * k/k (X 0 )-module is generated by monomials in c 1 (N i )'s. Since such a monomial corresponds to a closed path from X q j to itself, these will be exactly j-good monomials. Also, we need to observe that c 1 (N 1 ) = c 1 (NX 1 ⊂X 0 ) is the restriction of a class fromX 0 . Finally, from the same relations, we get that (f 0 ) * ĉ n = γ q 1 . . . γ q rĉ n = γ 0 1 . . . γ 0 r c n = c n · [X r ]. Hence, the image of (f 0 ) * (·ĉ m )(f j ) * : Ch * k/k (X j ) → Ch * +r k/k (X 0 ) as a Ch * k/k (X 0 )-module is generated by elements c n · [X r ], where n = m + l and l runs over all j-good vectors.