Rost Projectors and Steenrod Operations

Let X be an anisotropic projective quadric possessing a Rost projector ‰. We compute the 0-dimensional component of the total Steenrod operation on the modulo 2 Chow group of the Rost motive given by the projector‰. The computation allows to determine the whole Chow group of the Rost motive and the Chow group of every excellent quadric (the results announced by Rost). On the other hand, the computation is being applied to give a simpler proof of Vishik's theorem stating that the integer dimX + 1 is a power of 2.

M. Rost noticed that certain smooth projective anisotropic quadric hypersurfaces are decomposable in the category of Chow motives into direct sum of some motives.The smallest (in some sense) direct summands are called the Rost motives.For example, the motive of a Pfister quadric is a direct sum of Rost motives and their Tate twists.The Rost projectors split off the Rost motives as direct summands of quadrics.In the present paper we study Rost projectors by means of modulo 2 Steenrod operations on the Chow groups of quadrics.The Steenrod operations in motivic cohomology were defined by V. Voevodsky.We use results of P. Brosnan who found in [1] an elementary construction of the Steenrod operations on the Chow groups.
As a consequence of our computations we give description of the Chow groups of a Rost motive (Corollary 8.2).This result (which has been announced by M. Rost in [11]) allows to compute all the Chow groups of every excellent quadric (see Remark 8.4).
We also give an elementary proof of a theorem of A. Vishik [3, th. 6.1] stating that if an anisotropic quadric X possesses a Rost projector, then dim X + 1 is a power of 2 (Theorem 5.1). is odd if and only if we don't carry over units while adding n and i in base 2.
Proof.For any integer a ≥ 0, let s 2 (a) be the sum of the digits in the base 2 expansion of a.By [9, Lemma 5.4(a)], n+i i is odd if and only if s 2 (n + i) = s 2 (n) + s 2 (i).
The following statement is obvious: Lemma 1.2.For any non-negative integer m, we don't carry over units while adding m and m + 1 in base 2 if and only if m + 1 is a power of 2.
The following statement will be applied in the proof of Theorem 4.8: Corollary 1.3.For any non-negative integer m, the binomial coefficient −m−2 m is odd if and only if m + 1 is a power of 2.
Proof.By Lemma 1.1, the binomial coefficient −m − 2 m = (−1) m 2m + 1 m is odd if and only if there is we don't carry over units while adding m and m + 1 in base 2. It remains to apply Lemma 1.2.

Integral and modulo 2 Rost projectors
Let F be a field, X a quasi-projective smooth equidimensional variety over F .We write CH(X) for the modulo 2 Chow group of X.The usual (integral) Chow group is denoted by CH(X).We are working mostly with CH(X), but several times we have to use CH(X) (for example, already the definition of a modulo 2 Rost correspondence can not be given on the level of the modulo 2 Chow group).
The both groups are graded.We use the upper indices for the gradation by codimension of cycles and we use the lower indices for the gradation by the dimension of cycles.
For projective X 1 and X 2 , an element ρ ∈ CH(X 1 × X 2 ) (we do not consider the gradation on CH for the moment) can be viewed as a correspondence from X 1 to X 2 ([2, §16.1]).In particular, it gives a homomorphism [2, def. 16 where pr 1 and pr 2 are the two projections of X 1 × X 2 onto X 1 and X 2 , and can be composed with another correspondence ρ ∈ CH(X 2 × X 3 ) [2, def.16. 1.1].The same can be said and defined with CH replaced by CH.
Starting from here, we are constantly assuming that char F = 2. Let ϕ be a non-degenerate quadratic form over F , and let X be the projective quadric ϕ = 0. We set n = dim X = dim ϕ − 2 and we assume that n ≥ 1.
An element ∈ CH n (X × X) is called an (integral) Rost correspondence, if over an algebraic closure F of F one has: with X = X F and a rational point x ∈ X.A Rost projector is a Rost correspondence which is an idempotent with respect to the composition of correspondences.
Remark 2.1.Assume that the quadric X is isotropic, i.e., contains a rational Proof.Let ρ be a modulo 2 Rost projector and let be an integral Rost correspondence representing ρ.The correspondence F is idempotent, therefore, by the Rost nilpotence theorem (see [7, th. 3.1]), r is idempotent for some r; so, r is an integral Rost projector.Since ρ is idempotent as well, r still represents ρ.Lemma 2.4.Let be an integral Rost correspondence and let ρ be a modulo 2 Rost correspondence.Then * on CH 0 (X) and on CH 0 (X) is the identity; also ρ * on CH 0 (X) and on CH 0 (X) is the identity.Moreover, for every i with 0 < i < n, the group * CH i (X) vanishes over F .
Proof.It suffices to prove the statements on .Since CH 0 (X) and CH 0 (X) inject into CH 0 (X F ) and CH 0 (X F ) (see [5, prop. 2.6] or [13] for the statement on CH 0 (X)), it suffices to consider the case where the quadric X has a rational closed point x and = ) = 0 and since [X] generates CH 0 (X) while [x] generates CH 0 (X), we are done with the statements on CH 0 (X) and on CH 0 (X).Since ) for any closed subvariety Z ⊂ X of codimension = 0, n, we are done with the rest.

Steenrod operations
In this section we briefly recall the basic properties of the Steenrod operations on the modulo 2 Chow groups constructed in [1].
Let X be a smooth quasi-projective equidimensional variety over a field F .For every i ≥ 0 there are certain homomorphisms S i : CH * (X) → CH * +i (X) called Steenrod operations; their sum (which is in fact finite because is the total Steenrod operation (we omit the * in the notation of the Chow group to notify that S is not homogeneous).They have the following basic properties (see [1] for the proofs): for any smooth quasi-projective F -scheme X, the total operation S : CH(X) → CH(X) is a ring homomorphism such that for every morphism f : Y → X of smooth quasi-projective F -schemes and for every field extension E/F , the squares are commutative.Moreover, the restriction S i | CH n (X) is 0 for n < i and the map α → α 2 for n = i; finally S 0 is the identity.
Also, the total Steenrod operation satisfies the following Riemann-Roch type formula: (in other words, S modified by c(−T ) this way, commutes with the pushforwards) for any proper f : Y → X and any α ∈ CH(Y ), where f * : CH(Y ) → CH(X) is the push-forward, c is the total Chern class, T X is the tangent bundle of X, and c(−T X ) = c −1 (T X ) (the expression −T X makes sense if one considers T X as the element of K 0 (X)).This formula is proved in [1].Also it follows from the previous formulated properties of S by the general Riemann-Roch theorem of Panin [10].
Lemma 3.1.Assume that X is projective.For any α ∈ CH(X) and for any ρ ∈ CH(X × X), one has where T X is the class in K 0 (X) of the tangent bundle of X.
Proof.Let pr 1 , pr 2 : X × X → X be the first and the second projections.By the Riemann-Roch formula applied to the morphism pr 2 , one has By the projection formula for pr 2 , this gives and since S (as well as c) commutes with the products and the pull-backs, we get

Main theorem
In this section, let ϕ be an anisotropic quadratic form over F , and let X be the projective quadric ϕ = 0 with n = dim X = dim ϕ − 2 ≥ 1.We are assuming that an integral Rost projector (see §2 for the definition) ∈ CH n (X × X) exists for our X and we write ρ ∈ CH n (X × X) for the modulo 2 Rost projector.We write h for the class in CH 1 (X) (as well as in CH 1 (X)) of a hyperplane section of X. Proposition 4.1.One has for every i ≥ 0: Proof.Since h ∈ CH 1 (X), we have S(h) = S 0 (h) + S 1 (h).Since S 0 = id while S 1 on CH 1 (X) is the squaring, S(h Besides, for the tangent bundle of the quadric we have: c(T X ) = (1 + h) n+2 .Therefore, the formula of Lemma 3.1 gives the formula of Proposition 4.1.
Lemma 4.2.Let L/F be a field extension such that the quadric X L is isotropic.Then S i (ρ L ) = 0 for every i > 0.
Proof.By the uniqueness of a modulo 2 Rost projector on an isotropic quadric (Remark 2.1) Y has a rational point over F (X). Therefore there exists a rational morphism X → Y .Let α ∈ CH n (X × X) be the correspondence given by the closure of the graph of this morphism.Then ∆ * ( •α) ∈ CH 0 (X), where ∆ : X → X ×X is the diagonal morphism, is an element of CH 0 (X) of degree 1.This is a contradiction with the fact that the quadric X is anisotropic.Lemma 4.4.Let X be an anisotropic F -quadric such that the Witt index of the quadratic form ϕ F (X) is 1.Then for every α ∈ CH i (X F (X) ), i > 0, the degree of the 0-cycle class h i • α is even.
Proof.It is sufficient to consider the case i = 1.We have ϕ F (X) ψ⊥H for an anisotropic quadratic form ψ over F (X) (where H is a hyperbolic plane).Let X be the quadric ψ = 0 over F (X).There is an isomorphism [5, §2.2] for every α ∈ CH 1 (X) and this integer is even.
Proof.We replace µ by its representative in CH i (X × X) and we mean by h the integral class of a hyperplane section of X in the proof.Since the degree homomorphism deg : CH n (X) → Z is injective ([5, prop.2.6] or [13]) with the image 2Z, it suffices to show that deg(µ * (h i )) is divisible by 4. Let us compute this degree.By definition of µ * , we have µ * (h i ) = pr 2 * µ • pr * 1 (h i ) .Note that the product µ • pr * 1 (h i ) is in CH 0 (X × X) and the square CH 0 (X × X) commutes (the two compositions being the degree homomorphism of the group CH 0 (X × X)).Therefore the degree of µ * (h i ) coincides with the degree of pr 1 * µ•pr * 1 (h i ) .By the projection formula for pr 1 * the latter element coincides with the product h i • pr 1 * (µ).
We are going to check that the degree of this element is divisible by 4. Since the degree does not change under extensions of the base field, it suffices to verify the divisibility relation over F (X).The class pr 1 * (µ) F (X) is divisible by 2 by assumption, therefore the statement follows from Lemmas 4.3 and 4.4.
Putting together Corollary 4.6 and Proposition 4.1, we get Corollary 4.7.For every i > 0, one has: Finally, by Corollary 1.3, computing the binomial coefficient modulo 2, together with Lemma 2.4, computing ρ * (h n ), we get Theorem 4.8.Suppose that the anisotropic quadric X of dimension n possesses a Rost projector.Let ρ be a modulo 2 Rost projector on X.Then for every i with 0 < i < n, one has Since CH 0 (X) is an infinite cyclic group generated by h n , h n in CH 0 (X) is not 0. Therefore we get Corollary 4.9.For every i such that 0 < i < n and n − i + 1 is a power of 2, the element S n−i (ρ * (h i )) (and consequently ρ * (h i )) is non-zero.

Dimensions of quadrics with Rost projectors
The following Theorem is proved in [3].The proof given there makes use of the Steenrod operations in the motivic cohomology constructed by Voevodsky (since Voevodsky has announced that the operations were constructed in any characteristic = 2 only quite recently, the assumption char F = 0 was made in [3]).Here we give an elementary proof.
Theorem 5.1.If X is an anisotropic smooth projective quadric possessing a Rost projector, then dim X + 1 is a power of 2.
Proof.Let us assume that this is not the case.Let r be the largest integer such that n > 2 r −1 where n = dim X.Then Theorem 4.8 applies to i = n−(2 r −1), stating that S n−i ρ * (h i ) = 0. Note that n − i ≥ i.Since the Steenrod operation S i is trivial on CH j (X) with i > j, it follows that n − i = i and therefore S n−i ρ * (h i ) = ρ * (h i ) 2 .Since the element * (h i ) (where is the integral Rost projector) vanishes over F , its square vanishes over F as well.The group CH 0 (X) injects however into CH 0 (X F ), whereby * (h i ) 2 = 0 and thereafter S n−i ρ * (h i ) = 0, giving a contradiction with Corollary 4.9.
Remark 5.2.It turns out that Theorem 5.1 is extremely useful in the theory of quadratic forms.For example, it is the main ingredient of Vishik's proof of the theorem that there is no anisotropic quadratic forms satisfying 2 r < dim ϕ < 2 r + 2 r−1 and [ϕ] ∈ I r (F ) (see [14]).

Rost motives
Let Λ be an associative commutative ring with 1.We set ΛCH = Λ ⊗ Z CH (we still do not need any other Λ but Z and Z/2).
We briefly recall the construction of the category of Grothendieck ΛCHmotives as given in [4].A motive is a triple (X, p, n), where X is a smooth projective equidimensional F -variety, p ∈ ΛCH dim X (X × X) an idempotent correspondence, and n an integer.Sometimes the reduced notations are used: (X, n) for (X, p, n) with p the diagonal class; (X, p) for (X, p, n) with n = 0; and (X) for (X, 0), the motive of the variety X.
For a motive M = (X, p, n) and an integer m, the m-th twist M (m) of M is defined as (X, p, n + m).
The set of morphisms is defined as In particular, every homogeneous correspondence α ∈ CH(X × X ) determines a morphism of every twist of (X, p) to a certain twist of (X , p ).The Chow group ΛCH * (X, p, n) of a motive (X, p, n) is defined as It gives an additive functor of the category of ΛCH-motives to the category of graded abelian groups.For any Λ, there is an evident additive functor of the category of CH-motives to the category of ΛCH-motives (identical on the motives of varieties).In particular, every isomorphism of CH-motives automatically produces an isomorphism of the corresponding ΛCH-motives.This is why bellow we mostly formulate the results only on the integral motives.
We are coming back to the quadratic forms.Definition 6.1.Let be an integral Rost projector on a projective quadric X.We refer to the motive (X, ) as to an (integral) Rost motive.(While the CH-motive given by a modulo 2 Rost projector can be called a modulo 2 Rost motive.)A Rost motive is anisotropic, if the quadric X is so.
Let now π be a Pfister form and let ϕ be a neighbor of π which is minimal, that is, has dimension dim π/2+1.According to [7, 5.2], the projective quadric X given by ϕ possesses an integral Rost projector .Proposition 6.2.Let ϕ be as above.Let be the Rost projector on the quadric X given by a minimal neighbor ϕ of another Pfister form π .The Rost motives (X, ) and (X , ) are isomorphic if and only if the Pfister forms π and π are isomorphic.
Proof.First we assume that (X, ) (X , ).Looking at the degrees of 0cycles on X and on X , we see that ϕ is isotropic if and only if ϕ is isotropic, whereby π is isotropic if and only if π is isotropic.Therefore, the forms π F (π ) and π F (π) are isotropic.Since π and π are Pfister forms, it follows that π π .Conversely, assume that π π .By [7, cor. 3.3], in order to show that (X, ) (X , ), it suffices to construct morphisms of motives (X, ) (X , ) which become mutually inverse isomorphisms over over an algebraic closure F of F (in this case, the initial F -morphisms are isomorphisms by [7, cor. 3.3], although probably not mutually inverse ones).
Since π π , the quadratic forms ϕ F (ϕ) and ϕ F (ϕ ) are isotropic.Therefore there exist rational morphisms X → X and X → X.The closures of their graphs give two correspondences α ∈ CH(X × X ) and β ∈ CH(X × X).
Over F we have: where x ∈ X F and x ∈ X F are closed rational points, while a is an integer (which coincides, in fact, with the degree of the rational morphism).Similarly, We are going to check that the integers a and b are odd.For this we consider the composition Over F this composition gives [X ×x]+ab [x×X].Consequently, by [6, th. 6.4] and Lemma 4.3, the integer ab is odd.
Let us take now with some degree 2 closed points y ∈ X and y ∈ X .Then the two Fmorphisms (X, ) (X , ) given by these α and β become mutually inverse isomorphisms over F .Definition 6.3.The motive (X, ) for X and as in Proposition 6.2 (more precisely, the isomorphism class of motives) is called the Rost motive of the Pfister form π and denoted R(π).Remark 6.4.It is conjectured in [7, conj. 1.6] that every anisotropic Rost motive is the Rost motive of some Pfister form.

Motivic decompositions of excellent quadrics
Theorem 7.1.Let ϕ be a neighbor of a Pfister form π and let ϕ be the complementary form (that is, ϕ is such that the form ϕ⊥ϕ is similar to π).Then where m = (dim ϕ − dim ϕ )/2, X is the quadric defined by ϕ, and X is the quadric defined by ϕ .
We recall that a quadratic form ϕ over F is called excellent, if for every field extension E/F the anisotropic part of the form ϕ E is defined over F .An anisotropic quadratic form is excellent if and only if it is a Pfister neighbor whose complementary form is excellent as well [8, §7].
Let π 0 ⊃ π 1 ⊃ • • • ⊃ π r be a strictly decreasing sequence of embedded Pfister forms.Let ϕ be the quadratic form such that the class [ϕ] of ϕ in the Witt ring of F is the alternating sum while the dimension of ϕ is the alternating sum of the dimensions of the Pfister forms.Clearly, ϕ is excellent.Moreover, every anisotropic excellent quadratic form is similar to a form obtained this way and the Pfister forms are uniquely determined by the initial excellent quadratic form.
Let X be an excellent quadric, that is, the quadratic form ϕ giving X is excellent.As Theorem 7.1 shows, the motive of X is a direct sum of twisted Rost motives.More precisely, Corollary 7.2.Let X be the excellent quadric determined by Pfister forms Example 7.5 (Norm forms, [12, th.17]).Let ϕ be a norm quadratic form, that is, ϕ is a minimal neighbor of a Pfister form π containing a 1-codimensional subform which is similar to a Pfister form π .Then

Chow groups of Rost motives
The following theorem computes the Chow groups of the modulo 2 Rost motive of a Pfister form.Theorem 8.1.Let ρ be the modulo 2 Rost projector on the projective ndimensional quadric X given by an anisotropic minimal Pfister neighbor.Let i be an integer with 0 ≤ i ≤ n.If i + 1 is a power of 2, then the group ρ * CH i (X) is cyclic of order 2 generated by ρ * (h n−i ).Otherwise this group is 0.
Proof.According to Proposition 6.2, we may assume that X is a norm quadric, that is, X contains a 1-codimensional subquadric Y being a Pfister quadric.Let r be the integer such that n = dim X = 2 r − 1.
We proceed by induction on r.Let Y ⊂ Y be a subquadric of dimension 2 r−1 − 2 which is a Pfister quadric.Let X be a norm quadric of dimension 2 r−1 − 1 such that Y ⊂ X ⊂ Y .Let ρ be a modulo 2 Rost projector on X .By Example 7.5, passing from CH-motives to the category of CH-motives, we see that the motive of X is the direct sum of the motive (X, ρ) and the motives (X , ρ , i) (we do not care about the gradations on the Chow groups).
Also the motive of Y decomposes in the direct sum of the motives (X , ρ , i) In The group ρ * CH(X ) is known by induction.In particular, the order of this group is 2 r .It follows that the order of ρ * CH(X) is at most 2 r+1 .Corollary 4.9 gives already r+1 non-zero elements of ρ * CH * (X) living in different dimensions (more precisely, ρ * (h n−2 s +1 ) = 0 for s = 1, . . ., r − 1 by Corollary 4.9 and for s = 0, r by Lemma 2.4) and therefore generating a subgroup of order 2 r+1 .It follows that the order of ρ * CH(X) is precisely 2 r+1 and the non-zero elements we have found generate the group ρ * CH(X).
The integral version of Theorem 8.1 is given by Corollary 8.2.For X as in Theorem 8.1, let be the integral Rost projector on X.Then for every i with 0 ≤ i ≤ n, the group * CH i (X) is a cyclic group generated by * (h n−i ).Moreover, the element * (h n−i ) is • 0, if i + 1 is not a power of 2; • of order 2, if i + 1 is a power of 2 and i ∈ {0, n}; • of the infinite order, if i ∈ {0, n}.
Proof.The statements on CH 0 (X) and on CH 0 (X) are clear.The rest follows from Theorem 8.1, if we show that 2 * CH i (X) = 0 for every i with 0 < i < n.Let L/F be a quadratic extension such that X L is isotropic.Then ( L ) * CH i (X L ) = 0 for such i by [7, cor. 4.2].Therefore, by the transfer argument, 2 * CH i (X) = 0.
Remark 8.3.The result of Corollary 8.2 was announces in [11].A proof has never appeared.

Remark 8 . 4 .
Clearly, Corollary 8.2 describes the Chow group of the Rost motive of an anisotropic Pfister form.Since the motive of any anisotropic excellent quadric is a direct sum of twists of such Rost motives (Corollary 7.2), we have computed the Chow group of an arbitrary anisotropic excellent projective quadric.Note that the answer depends only on the dimension of the quadric.
is a Rost projector.Moreover, this is the unique Rost projector on X ([7, lemma 4.1]).Rost correspondence ρ ∈ CH n (X × X) is a correspondence which can be represented by an integral Rost correspondence.A modulo 2 Rost projector is an idempotent modulo 2 Rost correspondence.