MOTIVIC DECOMPOSITION OF COMPACTIFICATIONS OF CERTAIN GROUP VARIETIES

NIKITA A. KARPENKO AND ALEXANDER S. MERKURJEV

Abstract. Let D be a central simple algebra of prime degree over a field and let E be an SL_{1}(D)-torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Contents

1. Introduction 1 2. K-cohomology 3 3. BGQ spectral sequence 4 4. Specialization 4 5. A source of split motives 5 6. Compactifications of algebraic groups 6 7. Some computations in CH(SL_{1}(D)) 9 8. Rost’s theorem 10 9. The category of D-motives 12 10. Motivic decomposition of compactifications of SL_{1}(D) 14 References 16

1. Introduction

Let p be a prime number. For any integer n ≥ 2, a Rost motive of degree n is a direct summand R of the Chow motive with coefficients in ℤ_{(p)} (the localization of the integers at the prime ideal (p)) of a smooth complete geometrically irreducible variety X over a field F such that for any extension field K/F with a closed point on X_{K} of degree prime to p, the motive R_{K} is isomorphic to the direct sum of Tate motives

ℤ_{(p)} ⊕ ℤ_{(p)}(b) ⊕ ℤ_{(p)}(2b) ⊕ ⋯ ⊕ ℤ_{(p)}((p − 1)b),

Date: February 14, 2014. Extended: February 22, 2014.

Key words and phrases. Central simple algebras, special linear groups, principle homogeneous spaces, compactifications, Chow groups and motives. Mathematical Subject Classification (2010): 20G15; 14C25.

The first author acknowledges a partial support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005; his work has been also supported by the start-up grant of the University of Alberta. The work of the second author has been supported by the NSF grant DMS #1160206 and the Guggenheim Fellowship.

where b = (p^{n−1} − 1)/(p − 1). The isomorphism class of R is determined by X, [19, Proposition 3.4]; R is indecomposable as long as X has no closed points of degree prime to p.

A smooth complete geometrically irreducible variety X over F is a p-generic splitting variety for an element s ∈ H_{ét}^{n}(F, ℤ/pℤ(n − 1)), if s vanishes over a field extension K/F if and only if X has a closed point of degree prime to p over K. A norm variety of s is a p-generic splitting variety of dimension p^{n−1} − 1.

A Rost motive living on a p-generic splitting variety of an element s is determined by s up to isomorphism and called the Rost motive of s. In characteristic 0, any symbol s admits a norm variety possessing a Rost motive. This played an important role in the proof of the Bloch-Kato conjecture (see [31]). It is interesting to understand the complement to the Rost motive in the motive of a norm variety X for a given s; this complement, however, depends on X and is not determined by s anymore.

For p = 2, there are nice norm varieties known as norm quadrics. Their complete motivic decomposition is a classical result due to M. Rost. A norm quadric X can be viewed as a compactification of the affine quadric U given by π = c, where π is a quadratic (n − 1)-fold Pfister form and c ∈ F^{×}. The summands of the complete motivic decomposition of X are given by the degree n Rost motive of X and shifts of the degree n − 1 Rost motive of the projective Pfister quadric π = 0. It is proved in [16, Theorem A.4] that CH(U) = ℤ. In the present paper we extend these results to arbitrary prime p (and n = 3).

For arbitrary p, there are nice norm varieties in small degrees. For n = 2, these are the Severi-Brauer varieties of degree p central simple F -algebras. Any of them admits a degree 2 Rost motive which is simply the total motive of the variety.

The first interesting situation occurs in degree n = 3. Let D be a degree p central division F -algebra, G = SL_{1}(D) the special linear group of D, and E a principle homogeneous space under G. The affine variety E is given by the equation Nrd = c, where Nrd is the reduced norm of D and c ∈ F^{×}. Any smooth compactification of E is a norm variety of the element s := [D]∪(c) ∈ H_{ét}^{3}(F, ℤ/pℤ(2)). It has been shown by N. Semenov in [26] for p = 3 (and char F = 0) that the motive of a certain smooth equivariant compactification of E decomposes in a direct sum, where one of the summands is the Rost motive of s, another summand is a motive ε vanishing over any field extension of F splitting D, and each of the remaining summands is a shift of the motive of the Severi-Brauer variety of D. All these summands (but ε) are indecomposable and ε was expected to be 0.

Another proof of this result (covering arbitrary characteristic) has been provided in [30] along with the claim that ε = 0, but the proof of the claim was incomplete.

In the present paper we prove the following main result (see Theorem 10.3):

Theorem 1.1. Let F be a field, D a central division F -algebra of prime degree p, X a smooth compactification of an SL_{1}(D)-torsor, and M(X) its Chow motive with ℤ_{(p)}- coefficients. Assume that M(X) over the function field of the Severi-Brauer variety S of D is isomorphic to a direct sum of Tate motives. Then M(X) (over F) is isomorphic to the direct sum of the Rost motive of X and several shifts of M(S). This is the unique decomposition of M(X) into a direct sum of indecomposable motives.

We note that the compactification in [26] (for p = 3) has the property required in Theorem 1.1 (see Example 10.6).

In Section 6 we show that the condition that M(X) is split over F (S) is satisfied for all smooth G×G-equivariant compactifications of G = SL_{1}(D). Moreover, we prove that the motive M(X) is split for all smooth equivariant compactifications X of split semisimple groups (see Theorem 6.8).

We also compute the Chow ring of G in arbitrary characteristic as well as the Chow ring of E in characteristic 0 (see Theorem 9.7 and Corollary 10.8):

Theorem 1.2. Let D be a central division algebra of prime degree p and G = SL_{1}(D).

1) There is an element h ∈ CH^{p+1}(G) such that

CH(G) = ℤ · 1 ⊕ (ℤ/pℤ)h ⊕ (ℤ/pℤ)h^{2} ⊕ ⋯ ⊕ (ℤ/pℤ)h^{p−1}.

2) Let E be a nonsplit G-torsor. If char F = 0, then CH(E) = ℤ.

Acknowledgements. We thank Michel Brion for teaching us the theory of equivariant compactifications. We also thank Markus Rost and Kirill Zainoulline for helpful information.

2. K-cohomology

Let X be a smooth variety over F. We write A^{i}(X, K_{n}) for the K-cohomology groups as defined in [25]. In particular, A^{i}(X, K_{i}) is the Chow group CH^{i}(X) of classes of codimension i algebraic cycles on X.

Let G be a simply connected semisimple algebraic group. The group A^{1}(G, K_{2}) is additive in G, i.e., if G and G′ are two simply connected group, then the projections of G×G′ onto G and G′ yield an isomorphism (see [13, Part II, Proposition 7.6 and Theorem 9.3])

A^{1}(G, K_{2}) ⊕ A^{1}(G′, K_{2}) ⟶^{∼} A^{1}(G × G′, K_{2}).

The following lemma readily follows.

Lemma 2.1. 1) The map

A^{1}(G, K_{2}) → A^{1}(G × G, K_{2}) = A^{1}(G, K_{2}) ⊕ A^{1}(G, K_{2})

induced by the product homomorphism G × G → G is equal to (1, 1).

2) The map A^{1}(G, K_{2}) → A^{1}(G, K_{2}) induced by the morphism G → G, x ↦ x^{−1} is equal to −1.

Proof. 1) It suffices to note that the isomoprhism

A^{1}(G × G′, K_{2})⟶^{∼}A^{1}(G, K_{2}) ⊕ A^{1}(G′, K_{2})

inverse to the one mentioned above, is given by the pull-backs with respect to the group embeddings G, G′ ↪ G × G′.

2) The composition of the embedding of varieties G ↪ G × G, g ↦ (g, g^{−1}) with the product map G × G → G is trivial. □

If G is an absolutely simple simply connected group, then A^{1}(G, K_{2}) is an infinite cyclic group with a canonical generator q_{G} (see [13, Part II, §7]).

3. BGQ spectral sequence

Let X be a smooth variety over F. We consider the Brown-Gersten-Quillen coniveau spectral sequence

(3.1)  E_{2}^{s,t} = A^{s}(X, K_{−t}) ⇒ K_{−s−t}(X)

converging to the K-groups of X with the topological filtration [23, §7, Th. 5.4].

Example 3.2. Let G = SL_{n}. By [29, §2], we have CH(G) = ℤ. It follows that all the differentials of the BGQ spectral sequence for G coming to the zero diagonal are trivial.

Lemma 3.3 ([20, Theorem 3.4]). If δ is a nontrivial differential in the spectral sequence (3.1) on the q-th page E_{q}^{∗,∗}, then δ is of finite order and for every prime divisor p of the order of δ, the integer p − 1 divides q − 1.

Let p be a prime integer, D a central division algebra over F of degree p and G = SL_{1}(D). As D is split by a field extension of degree p, it follows from Example 3.2 that all Chow groups CH^{i}(G) are p-periodic for i > 0 and the order of every differential in the BGQ spectral sequence for G coming to the zero diagonal divides p. The edge homomorphism K_{1}(G) → E_{2}^{0,−1} = A^{0}(G, K_{1}) = F^{×} is a surjection split by the pull-back with respect to the structure morphism G → Spec F. Therefore, all the differentials starting at E_{∗}^{0,−1} are trivial.

It follows then from Lemma 3.3 that the only possibly nontrivial differential coming to the terms E_{q}^{i,−i} for q ≥ 2 and i ≤ p + 1 is

∂_{G} : A^{1}(G, K_{2}) = E_{p}^{1,−2} → E_{p}^{p+1,−p−1} = CH^{p+1}(G).

By [29, Theorem 6.1] (see also [22, Theorem 5.1]), K_{0}(G) = ℤ, hence the factors

K_{0}(G)^{(i)}/K_{0}(G)^{(i+1)} = E_{∞}^{i,−i}

of the topological filtration on K_{0}(G) are trivial for i > 0. It follows that the map ∂_{G} is surjective. As the group A^{1}(G, K_{2}) is cyclic with the generator q_{G}, the group CH^{p+1}(G) is cyclic of order dividing p. It is shown in [33, Theorem 4.2] that the differential ∂_{G} is nontrivial. We have proved the following lemma.

Lemma 3.4. If D is a central division algebra, then CH^{p+1}(G) is a cyclic group of order p generated by ∂_{G}(q_{G}). □

4. Specialization

Let A be a discrete valuation ring with residue field F and quotient field L. Let X be a smooth scheme over A and set X = X ⊗_{A} F, X′ = X ⊗_{A} L. By [11, Example 20.3.1], there is a specialization ring homomorphism

σ : CH^{∗}(X′) → CH^{∗}(X).

Example 4.1. Let X be a variety over F, L = F (t) the rational function field. Consider the valuation ring A ⊂ L of the parameter t and X = X ⊗_{F} A. Then X′ = X_{L} and we have a specialization ring homomorphism σ : CH^{∗}(X_{L}) → CH^{∗}(X).

A section of the structure morphism X → Spec A gives two rational points x ∈ X and x′ ∈ X′. By definition of the specialization, σ([x′]) = [x].

Let F be a field of finite characteristic. By [2, Ch. IX, §2, Propositions 5 and 1], there is a complete discretely valued field L of characteristic zero with residue field F. Let A be the valuation ring and D a central simple algebra over F. By [14, Theorem 6.1], there is an Azumaya algebra D over A such that D ≃ D ⊗_{A} F. The algebra D′ = D ⊗_{A} L is a central simple algebra over L. Then we have a specialization homomorphism

σ : CH^{∗}(SL_{1}(D′)) → CH^{∗}(SL_{1}(D))

satisfying σ([e′]) = [e], where e and e′ are the identities of the groups.

5. A source of split motives

We work in the category of Chow motives over a field F, [9, §64]. We write M(X) for the motive (with integral coefficients) of a smooth complete variety X over F.

A motive is split if it is isomorphic to a finite direct sum of Tate motives ℤ(a) (with arbitrary shifts a). Let X be a smooth proper variety such that the motive M(X) is split, i.e., M(X) = ∐_{i} ℤ(a_{i}) for some a_{i}. The generating (Poincaré) polynomial P_{X}(t) of X is defined by

P_{X}(t) = ∑_{i} t^{a_{i}}.

Note that the integer a_{i} is equal to the rank of the (free abelian) Chow group CH^{i}(X).

Example 5.1. Let G be a split semisimple group and B ⊂ G a Borel subgroup. Then

P_{G/B}(t) = ∑_{w∈W} t^{l(w)},

where W is the Weyl group of G and l(w) is the length of w (see [8, §3]).

Proposition 5.2 (P. Brosnan, [4, Theorem 3.3]). Let X be a smooth projective variety over F equipped with an action of the multiplicative group 𝔾_{m}. Then

M(X) = ∐_{i} M(Z_{i})(a_{i}),

where the Z_{i} are the (smooth) connected components of the subscheme of X^{𝔾_{m}} of fixed points and a_{i} ∈ ℤ. Moreover, the integer a_{i} is the dimension of the positive eigenspace of the action of 𝔾_{m} on the tangent space T_{z} of X at an arbitrary closed point z ∈ Z_{i}. The dimension of Z_{i} is the dimension of (T_{z})^{𝔾_{m}}.

Let T be a split torus of dimension n. The choice of a ℤ-basis in the character group T^{∗} allows us to identify T^{∗} with ℤ^{n}. We order ℤ^{n} (and hence T^{∗}) lexicographically.

Suppose T acts on a smooth variety X and let x ∈ X be an T -fixed rational point. Let χ_{1}, χ_{2}, …, χ_{m} be all characters of the representation of T in the tangent space T_{x} of X at x. Write a_{x} for the number of positive (with respect to the ordering) characters among the χ_{i}’s.

Corollary 5.3. Let X be a smooth projective variety over F equipped with an action of a split torus T. If the subscheme X^{T} of T -fixed points in X is a disjoint union of finitely many rational points, the motive of X is split. Moreover,

P_{X}(t) = ∑_{x∈X^{T}} t^{a_{x}}.

Proof. Induction on the dimension of T. □

Example 5.4. Let T be a split torus of dimension n and X a smooth projective toric variety (see [12]). Let σ be a cone of dimension n in the fan of X and {χ_{1}, χ_{2}, …, χ_{n}} a (unique) ℤ-basis of T^{∗} generating the dual cone σ^{∨}. The standard T -invariant affine open set corresponding to σ is V_{σ} := Spec F [σ^{∨}]. The map V_{σ} → 𝔸^{n}, taking x to (χ_{1}(x), χ_{2}(x), …, χ_{n}(x)) is a T -equivariant isomorphism, where t ∈ T acts on the affine space 𝔸^{n} by componentwise multiplication by χ_{i}(t). The only one T -equivariant point x ∈ V_{σ} corresponds to the origin under the isomorphism, so we can identify the tangent space T_{x} with 𝔸^{n}, and the χ_{i}’s are the characters of the representation of T in the tangent space T_{x}. Let a_{σ} be the number of positive χ_{i}’s with respect to a fixed lexicographic order on T^{∗}. Every T -fixed point in X belongs to V_{σ} for a unique σ. It follows that the motive M(X) is split and

P_{X}(t) = ∑_{σ} t^{a_{σ}},

where the sum is taken over all dimension n cones in the fan of X.

6. Compactifications of algebraic groups

A compactification of an affine algebraic group G is a projective variety containing G as a dense open subvariety. A G × G-equivariant compactification of G is a projective variety X equipped with an action of G × G and containing the homogeneous variety G = (G × G)/ diag(G) as an open orbit. Here the group G × G acts on G by the left-right translations.

Let G be a split semisimple group over F. Write G_{ad} for the corresponding adjoint group. The group G_{ad} admits the so-called wonderful G_{ad} × G_{ad}-equivariant compactification X (see [3, §6.1]). Let T ⊂ G be a split maximal torus and T_{ad} the corresponding maximal torus in G_{ad}. The closure X′ of T_{ad} in X is a toric T_{ad}-variety with fan consisting of all Weyl chambers in (T_{ad})_{∗} ⊗ ℝ = T_{∗} ⊗ ℝ and their faces.

Let B be a Borel subgroup of G containing T and B^{−} the opposite Borel subgroup. There is an open B^{−} × B-invariant subscheme X_{0} ⊂ X such that the intersection X′_{0} := X_{0}∩X′ is the standard open T_{ad}-invariant subscheme of the toric variety X′ corresponding to the negative Weyl chamber Ω that is a cone in the fan of X′. Note that the Weyl group W of G acts simply transitively on the set of all Weyl chambers.

A G × G-equivariant compactification X of G is called toroidal if X is normal and the quotient map G → G_{ad} extends to a morphism π : X → X (see [3, §6.2]). The closed subscheme X′ := π^{−1}(X′) of X is a projective toric T -variety. Note that the diagonal subtorus diag(T) ⊂ T × T acts trivially on X′. The fan of X′ is a subdivision of the fan consisting of the Weyl chambers and their faces. The scheme X is smooth if and only if so is X′.

Conversely, if F is a perfect field, given a smooth projective toric T -variety with a W -invariant fan that is a subdivision of the fan consisting of the Weyl chambers and their faces, there is a unique smooth G × G-equivariant toroidal compactification X of G with the toric variety X′ isomorphic to the given one (see [3, §6.2] and [15, §2.3]). By [5] and [7], such a smooth toric variety exists for every split semisimple group G. In other words, the following holds.

Proposition 6.1. Every split semisimple group G over a perfect field admits a smooth G × G-equivariant toroidal compactification. □

Let X be a smooth G×G-equivariant toroidal compactification of G over F. Recall that the toric T -variety X′ is smooth projective. Set X_{0} := π^{−1}(X_{0}) and X′_{0} := π^{−1}(X′_{0}) = X′ ∩ X_{0}. Then the T -invariant subset X′_{0} ⊂ X′ is the union of standard open subschemes V_{σ} of X′ (see Example 5.4) corresponding to all cones σ in the negative Weyl chamber Ω. The subscheme (V_{σ})^{T} reduces to a single rational point if σ is of largest dimension. In particular, the subscheme (X′_{0})^{T} of T -fixed points in X′_{0} is a disjoint union of k rational points, where k is the number of cones of maximal dimension in Ω. It follows that |(X′)^{T} | = k|W |, the number of all cones of maximal dimension in the fan of X′.

Let U and U^{−} be the unipotent radicals of B and B^{−} respectively.

Lemma 6.2 ([3, Proposition 6.2.3]). 1) Every G × G-orbit in X meets X′_{0} along a unique T -orbit.

2) The map

U^{−} × X′_{0} × U → X_{0}, (u, x, v) ↦ uxv^{−1},

is a T × T -equivariant isomorphism.

3) Every closed G × G-orbit in X is isomorphic to G/B × G/B.

Proposition 6.3. The scheme X^{T ×T} is the disjoint union of W x_{0}W over all x_{0} ∈ (X′_{0})^{T} and W x_{0}W is a disjoint union of |W |^{2} rational points.

Proof. Take x ∈ X^{T ×T}. Let x be the image of x under the map π : X → X. Computing dimensions of maximal tori of the stabilizers of points in the wonderful compactification X, we see that x lies in the only closed G × G-orbit O in X (e.g., [10, Lemma 4.2]). By Lemma 6.2(3), applied to the compactification X of G_{ad}, O ≃ G/B × G/B. In view of Lemma 6.2(1), O ∩ X′_{0} is a closed T -orbit in X′_{0} and therefore, reduces to a single rational T -invariant point in X′_{0}. The group W × W acts simply transitively on the set of T ×T -fixed point in G/B ×G/B. It follows that |W xW | = |W |^{2} and W xW intersects X′_{0}. Therefore, W xW intersects X^{T ×T} ∩ X′_{0} = (X′_{0})^{T}, that is the disjoint union of k rational points. Hence x is a rational point, x ∈ W (X′_{0})^{T} W and |W xW | = |W |^{2}.

Note that for a point x_{0} ∈ (X′_{0})^{T}, the G×G-orbit of x_{0} intersects X′_{0} by the T -orbit {x_{0}} in view of Lemma 6.2(1). It follows that different W x_{0}W do not intersect and therefore, X^{T ×T} is the disjoint union of W x_{0}W over all x_{0} ∈ (X′_{0})^{T}. □

Let X be a smooth G × G-equivariant toroidal compactification of a split semisimple group G of rank n. By Proposition 6.3, every T × T -fixed point x in X is of the form x = w_{1}x_{0}w_{2}^{−1}, where w_{1}, w_{2} ∈ W and x_{0} ∈ (X′_{0})^{T}. Recall that X′_{0} is the union of the standard affine open subsets V_{σ} of the toric T -variety X′ over all cones σ of dimension n in the Weyl chamber Ω. Let σ be a (unique) cone in Ω such that x_{0} ∈ V_{σ}.

By Lemma 6.2(2), the map

f : U^{−} × V_{σ} × U → X, (u_{1}, y, u_{2}) ↦ w_{1}u_{1}x_{0}u_{2}^{−1}w_{2}^{−1}

is an open embedding. We have f(1, x_{0}, 1) = x. Thus, f identifies the tangent space T_{x} of x in X with the space 𝔲^{−} ⊕ 𝔞 ⊕ 𝔲, where 𝔲 and 𝔲^{−} are the Lie algebras of U and U^{−} respectively and 𝔞 is the tangent space of V_{σ} at x′. The torus T × T acts linearly on the tangent space T_{x} leaving invariant 𝔲^{−}, 𝔞 and 𝔲. For convenience, we write T × T as S := T_{1} × T_{2} in order to distinguish the components. Let Φ_{1}^{−} and Φ_{2}^{−} be two copies of the set of negative roots in T_{1}^{∗} and T_{2}^{∗} respectively. The set of characters of the Srepresentation 𝔲^{−} (respectively, 𝔲) is w_{1}(Φ_{1}^{−}) (respectively, w_{2}(Φ_{2}^{−})).

Let {χ_{1}, χ_{2}, …, χ_{n}} be a (unique) ℤ-basis of T^{∗} generating the dual cone σ^{∨}. By Example 5.4, the set of characters of the S-representation 𝔞 is

{(w_{1}(χ_{i}), −w_{2}(χ_{i}))}_{i=1}^{n} ⊂ S^{∗} = T_{1}^{∗} ⊕ T_{2}^{∗}.

Let Π_{1} and Π_{2} be (ordered) systems of simple roots in Φ_{1} and Φ_{2} respectively. Consider the lexicographic ordering on S^{∗} = T_{1}^{∗} ⊕ T_{2}^{∗} corresponding to the basis Π_{1} ∪ Π_{2} of S^{∗}. As χ_{i} ≠ 0, we have (w_{1}(χ_{i}), −w_{2}(χ_{i})) > 0 if and only if w_{1}(χ_{i}) > 0. For every w ∈ W, write b(σ, w) for the number of all i such that w(χ_{i}) > 0. Note that the number of positive roots in w(Φ^{−}) is equal to the length l(w) of w. By Corollary 5.3, we have

(6.4)  P_{X}(t) = ∑_{w_{1},w_{2}∈W, σ⊂Ω} t^{l(w_{1})+b(σ,w_{1})+l(w_{2})} = ( ∑_{w∈W, σ⊂Ω} t^{l(w)+b(σ,w)}) · P_{G/B}(t),

as by Example 5.1,

P_{G/B}(t) = ∑_{w∈W} t^{l(w)}.

We have proved the following theorem.

Theorem 6.5. Let X be a smooth G × G-equivariant toroidal compactification of a split semisimple group G. Then the motive M(X) is split into a direct sum of s|W | Tate motives, where s is the number of cones of maximal dimension in the fan of the associated toric variety X′. Moreover,

P_{X}(t) = ( ∑_{w∈W, σ⊂Ω} t^{l(w)+b(σ,w)}) · P_{G/B}(t).

In particular, the motive M(X) is divisible by M(G/B).

Example 6.6. Let G be a semisimple adjoint group and X the wonderful compactification of G. Then the negative Weyl chamber Ω is the cone σ = Ω in the fan of X′. The dual cone σ^{∨} is generated by −Π. Hence b(w, σ) is equal to the number of simple roots α such that w(α) ∈ Φ^{−}.

Example 6.7. Let G = SL_{3}, Π = {α_{1}, α_{2}}. Bisecting each of the six Weyl chambers we get a smooth projective fan with 12 two-dimensional cones. The two cones dual to the ones in the negative Weyl chamber are generated by {−α_{1}, (α_{1}−α_{2})/3} and {−α_{2}, (α_{2}−α_{1})/3} respectively. Let X be the corresponding G × G-equivariant toroidal compactification of G. By (6.4),

P_{X}(t) = (t^{5} + t^{4} + 4t^{3} + 4t^{2} + t + 1)(t^{3} + 2t^{2} + 2t + 1).

Now consider arbitrary (not necessarily toroidal) G × G-equivariant compactifications.

Theorem 6.8. Let X be a smooth G×G-equivariant compactification of a split semisimple group G over F. Then the subscheme X^{T ×T} is a disjoint union of finitely many rational points. In particular, the motive M(X) is split.

Proof. By [3, Proposition 6.2.5], there is a G × G-equivariant toroidal compactification X̃ of G together with a G×G-equivariant morphism φ : X̃ → X. Let x ∈ X^{T ×T}. By Borel’s fixed point theorem, the fiber φ^{−1}(x) has a T × T -fixed point, so the map X̃^{T ×T} → X^{T ×T} is surjective. By Proposition 6.3, X̃^{T ×T} is a disjoint union of finitely many rational points, hence so is X^{T ×T}. □

Example 6.9. Let Y be a smooth H × H-equivariant compactification of the group H = SL_{n} over F. In particular the projective linear group PGL_{n} acts on Y by conjugation. Let D be a central simple F -algebra of degree n and J the corresponding PGL_{n}-torsor. The twist of H by J is the group G = SL_{1}(D), hence the twist X of Y is a smooth G × G-equivariant compactification of G. If E is a G-torsor, one can twist X by E to get a smooth compactification of E. By Theorem 6.8, the motives of these compactifications are split over every splitting field of D.

7. Some computations in CH(SL_{1}(D))

Let D be a central simple algebra of prime degree p over F and G = SL_{1}(D).

Lemma 7.1. Let X be a smooth compactification of G. Then D is split by the residue field of every point in X \ G.

Proof. Let Y be the projective (singular) hypersurface given in the projective space ℙ(D⊕ F) by the equation Nrd = t^{p}, where Nrd is the reduced norm form. The group G is an open subset in Y, so we can identify the function fields F (X) = F (G) = F (Y). Let x ∈ X \ G. As x is smooth in X, there is a regular system of local parameters around x and therefore a valuation v of F (G) over F with residue field F (x). Since Y is projective, v dominates a point y ∈ Y \ G. Over the residue field F (y) the norm form Nrd is isotropic, hence D is split over F (y). Since v dominates y, the field F (y) is contained in F (v) = F (x). Therefore, D is split over F (x). □

Lemma 7.2. If D is a division algebra, then the group CH_{0}(G) = CH^{p^{2}−1}(G) is cyclic of order p generated by the class of the identity e of G.

Proof. The group of R-equivalence classes of points in G(F) is equal to SK_{1}(D) (see [32, Ch. 6]) and hence is trivial by a theorem of Wang. It follows that we have [x] = [e] in CH_{0}(G) for every rational point x ∈ G(F). If x ∈ G is a closed point, then [x′] = [e] in CH_{0}(G_{K}), where K = F (x) and x′ is a rational point of G_{K} over x. Taking the norm homomorphism CH_{0}(G_{K}) → CH_{0}(G) for the finite field extension K/F, we have [x] = [K : F] · [e] in CH_{0}(G). It follows that CH_{0}(G) is a cyclic group generated by [e].

As p · CH_{0}(G) = 0 it suffices to show that [e] ≠ 0 in CH_{0}(G). Let Y be the compactification of G as in the proof of Lemma 7.1 and let Z = Y \ G. As D is a central division algebra, the degree of every closed point of Z is divisible by p by Lemma 7.1.

It follows that the class [e] in CH_{0}(Y) does not belong to the image of the push-forward homomorphism i in the exact sequence

CH_{0}(Z) ⟶^{i} CH_{0}(Y) → CH_{0}(G) → 0.

Therefore, [e] ≠ 0 in CH_{0}(G). □

Consider the morphism s : G × G → G, s(x, y) = xy^{−1}. Note that s is flat as the composition of the automorphism (x, y) ↦ (xy^{−1}, y) of the variety G × G with the projection G × G → G.

Let h = ∂_{G}(q_{G}) ∈ CH^{p+1}(G).

Lemma 7.3. We have s^{∗}(h) = h × 1 − 1 × h in CH^{p+1}(G × G).

Proof. By Lemma 2.1, we have s^{∗}(q_{G}) = q_{G}×1−1×q_{G} in A^{1}(G×G, K_{2}). The differentials ∂_{G} commute with flat pull-back maps, hence we have

s^{∗}(h) = s^{∗}(∂_{G}(q_{G})) = ∂_{G×G}(s^{∗}(q_{G})) = ∂_{G×G}(q_{G} × 1 − 1 × q_{G}) =
∂_{G}(q_{G}) × 1 − 1 × ∂_{G}(q_{G}) = h × 1 − 1 × h.  □

Proposition 7.4. Let c be an integer with h^{p−1} = c[e] in CH^{p^{2}−1}(G). Then

cΔ_{G} = ∑_{i=0}^{p−1} h^{i} × h^{p−1−i},

where Δ_{G} is the class of the diagonal diag(G) in CH^{p^{2}−1}(G × G).

Proof. The diagonal in G × G is the pre-image of e under s. Hence by Lemma 7.3,

cΔ_{G} = cs^{∗}([e]) = s^{∗}(h^{p−1}) = (h × 1 − 1 × h)^{p−1} = ∑_{i=0}^{p−1} h^{i} × h^{p−1−i}

as \binom{p−1}{i} ≡ (−1)^{i} modulo p and ph = 0. □

8. Rost’s theorem

We have proved in Lemma 3.4 that if D is a central division algebra, then ∂_{G}(q_{G}) ≠ 0 in CH^{p+1}(G). This result is strengthened in Theorem 8.2 below.

Lemma 8.1. If there is an element h ∈ CH^{p+1}(G) such that h^{p−1} ≠ 0, then ∂_{G}(q_{G})^{p−1} ≠ 0.

Proof. By Lemma 3.4, h is a multiple of ∂_{G}(q_{G}). □

Theorem 8.2 (M. Rost). Let D be a central division algebra of degree p, G = SL_{1}(D). Then ∂_{G}(q_{G})^{p−1} ≠ 0 in CH^{p^{2}−1}(G) = CH_{0}(G).

Proof. Case 1 : Assume first that char(F) = 0, F contains a primitive p-th root of unity and D is a cyclic algebra, i.e., D = (a, b)_{F} for some a, b ∈ F^{×}.

Let c ∈ F^{×} be an element such that the symbol

u := (a, b, c) ∈ H_{ét}^{3}(F, ℤ/pℤ(3)) ≃ H_{ét}^{3}(F, ℤ/pℤ(2))

is nontrivial modulo p. Consider a norm variety X of u.

Then u defines a basic correspondence in the cokernel of the homomorphism

CH^{p+1}(X) → CH^{p+1}(X × X)

given by the difference of the pull-backs with respect to the projections. A representative in CH^{p+1}(X × X) of the basic correspondence is a special correspondence. Let z ∈ CH^{p+1}(X_{F (X)}) be its pull-back. The modulo p degree

c(X) := deg(z^{p−1}) ∈ ℤ/pℤ

is independent of the choice of the special correspondence. The construction of c(X) is natural with respect to morphisms of norm varieties (see [24]).

It is shown in [24] that there is an X such that c(X) ≠ 0. We claim that c(X′) ≠ 0 for every norm variety X′ of u. As F (X′) splits u and X is p-generic, X has a closed point over F (X′) of degree prime to p, or equivalently, there is a prime correspondence X′ ⇝ X of multiplicity prime to p. Resolving singularities, we get a smooth complete variety X′′ together with the two morphisms f : X′′ → X of degree prime to p and g : X′′ → X′. It follows by [28, Corollary 1.19] that X′′ is a norm variety of u. Moreover, c(X′′) = deg(f)c(X) ≠ 0 in ℤ/pℤ. As c(X′′) = deg(g)c(X′), c(X′) is also nonzero. The claim is proved.

Let X be a smooth compactification of the G-torsor E given by the equation Nrd = t over the rational function field L = F (t) given by a variable t. By the above, since {a, b, t} ≠ 0, we have an element z ∈ CH^{p+1}(X_{L(X)}) such that deg(z^{p−1}) ≠ 0 in ℤ/pℤ. The torsor E is trivial over L(X), i.e. E_{L(X)} ≃ G_{L(X)}. Then the restriction of z to the torsor gives an element y ∈ CH^{p+1}(G_{L(X)}) with y^{p−1} ≠ 0. The field extension L(X)/F is purely transcendental. By Section 4 and Lemma 7.2, every specialization homomorphism σ : CH^{p^{2}−1}(G_{L(X)}) → CH^{p^{2}−1}(G) is an isomorphism taking the class of the identity to the class of the identity. Specializing, we get an element h ∈ CH^{p+1}(G) with h^{p−1} ≠ 0. It follows from Lemma 8.1 that ∂_{G}(q_{G})^{p−1} ≠ 0.

Case 2 : Suppose that char(F) = 0 but F may not contain p-th roots of unity and D is an arbitrary division algebra of degree p (not necessarily cyclic). There is a finite field extension K/F of degree prime to p containing a primitive p-th root of unity and such that the algebra D ⊗_{F} K is cyclic (and still nonsplit). By Case 1, ∂_{G}(q_{G})_{K}^{p−1} ≠ 0 over K. Therefore ∂_{G}(q_{G})^{p−1} ≠ 0.

Case 3 : F is an arbitrary field. Choose a field L of characteristic zero and a central simple algebra D′ of degree p over L as in Section 4 and let G′ = SL_{1}(D′). By Case 2, there is an element h′ ∈ CH^{p+1}(G′) such that (h′)^{p−1} ≠ 0. Applying a specialization σ (see Section 4), we have h^{p−1} ≠ 0 for h = σ(h′). By Lemma 8.1 again, ∂_{G}(q_{G})^{p−1} ≠ 0. □

Let D be a central division algebra of degree p over F and X a smooth compactification of G. Let h̄ ∈ CH^{p+1}(X) be an element such that h̄|_{G} = ∂_{G}(q_{G}) ∈ CH^{p+1}(G). Let i = 0, 1, …, p − 1. The element h̄^{i} defines the following two morphisms of Chow motives:

f_{i} : M(X) → ℤ((p + 1)i), g_{i} : ℤ((p + 1)(p − 1 − i)) → M(X).

Let

R = ℤ ⊕ ℤ(p + 1) ⊕ ℤ(2p + 2) ⊕ ⋯ ⊕ ℤ(p^{2} − 1).

We thus have the following two morphisms:

f : M(X) → R, g : R → M(X).

The composition f ∘ g is c times the identity, where c = deg h̄^{p−1}. As c is prime to p by Theorem 8.2, switching to the Chow motives with coefficients in ℤ_{(p)}, we have a decomposition

(8.3)  M(X) = R ⊕ N

for some motive N.

9. The category of D-motives

Let D be a central simple algebra of prime degree p over F. For a field extension L/F, let N_{i}^{D}(L) be the subgroup of the Milnor K-group K_{i}^{M}(L) generated by the norms from finite field extensions of L that split the algebra D.

Consider the Rost cycle module (see [25]):

L ↦ K_{∗}^{D}(L) := K_{∗}^{M}(L)/N_{∗}^{D}(L),

and the corresponding cohomology theory with the “Chow groups”

CH_{D}^{i}(X) := A^{i}(X, K_{i}^{D}).

Note that CH_{D}^{i}(X) = 0 if D is split over F (x) for all points x ∈ X.

Let S = SB(D) be the Severi-Brauer variety of right ideals of D of dimension p. We have dim S = p − 1.

Lemma 9.1. For a variety X over F, the group CH_{D}(X) is naturally isomorphic to the cokernel of the push-forward homomorphism pr_{∗} : CH(X × S) → CH(X) given by the projection pr : X × S → X.

Proof. The composition

CH(X × S) ⟶^{pr_{∗}} CH(X) → CH_{D}(X)

factors through the trivial group CH_{D}(X × S) and therefore, is zero. This defines a surjective homomorphism

α : Coker(pr_{∗}) → CH_{D}(X).

The inverse map is obtained by showing that the quotient map CH(X) → Coker(pr_{∗}) factors through CH_{D}(X).

The kernel of the homomorphism CH(X) → CH_{D}(X) is generated by [x] with x ∈ X such that the algebra D_{F (x)} is split and by p[x] with arbitrary x ∈ X. The fiber of pr over x has a rational point y in the first case and a degree p closed point y in the second. The generators are equal to pr_{∗}([y]) in both cases. It follows that they vanish in Coker pr_{∗}. □

Let G = SL_{1}(D).

Corollary 9.2. The natural map CH^{i}(G) → CH_{D}^{i}(G) is an isomorphism for all i > 0.

Proof. The algebra D is split over S. More precisely, D_{X} = End_{X}(I^{∨}) for the rank p canonical vector bundle I over S (see [27, Lemma 2.1.4]). By [29, Theorem 4.2], the pullback homomorphism CH^{∗}(S) → CH^{∗}(G×S) is an isomorphism. Therefore, CH^{j}(G×S) = 0 if j > p − 1 = dim(S). □

Let X be a smooth compactification of G. Write X^{k} = X × X × ⋯ × X (k times).

Lemma 9.3. The restriction homomorphism CH_{D}^{∗}(X^{k}) → CH_{D}^{∗}(G^{k}) is an isomorphism.

Proof. Let Z = X^{k} \ G^{k}. By Lemma 7.1, the residue field of every point in Z splits D, hence CH_{D}(Z) = 0. The statement follows from the exactness of the localization sequence

CH_{D}(Z) → CH_{D}(X^{k}) → CH_{D}(G^{k}) → 0.  □

It follows from Lemma 9.3 and Corollary 9.2 that CH_{D}^{i}(X) ≃ CH^{i}(G) for i > 0.

Consider the category of motives of smooth complete varieties over F associated to the cohomology theory CH_{D}^{∗}(X) (see [21]). Write M^{D}(X) for the motive of a smooth complete variety X. We call M^{D}(X) the D-motive of X. Recall that the group of morphisms between M^{D}(X) and M^{D}(Y) for Y of pure dimension d is equal to CH_{D}^{d}(X × Y). Let ℤ^{D} the motive of the point Spec F.

Recall that we write M(X) for the usual Chow motive of X. We have a functor N ↦ N^{D} from the category of Chow motives to the category of D-motives.

Proposition 9.4. Let N be a Chow motive. Then N^{D} = 0 if and only if N is isomorphic to a direct summand of N ⊗ M(S).

Proof. As M^{D}(S) = 0, we have N^{D} = 0 if N is isomorphic to a direct summand of N ⊗ M(S).

Conversely, suppose N^{D} = 0. Let N = (X, ρ), where X is a smooth complete variety of pure dimension d and ρ ∈ CH^{d}(X × X) is a projector. By Lemma 9.1, we have ρ = f_{∗}(θ) for some θ ∈ CH^{d+p−1}(X × (X × S)), where f : X × X × S → X × X is the projection. Then

f_{∗}((ρ ⊗ id_{S}) ∘ θ ∘ ρ) = ρ

and (ρ ⊗ id_{S}) ∘ θ ∘ ρ can be viewed as a morphism N → N ⊗ M(S) splitting on the right the natural morphism N ⊗ M(S) → N. □

The morphisms f and g in Section 8 give rise to the morphisms f^{D} : M^{D}(X) → R^{D} and g^{D} : R^{D} → M^{D}(X) of D-motives.

Proposition 9.5. The morphism f^{D} : M^{D}(X) → R^{D} is an isomorphism in the category of D-motives.

Proof. As CH_{D}^{p^{2}−1}(X × X) ≃ CH_{D}^{p^{2}−1}(G × G) by Lemma 9.3, the composition g^{D} ∘ f^{D} is multiplication by c ∈ ℤ from Proposition 7.4. By Theorem 8.2, c is not divisible by p. Finally, p CH_{D}(G × G) = 0. □

If D is a central division algebra, it follows from Proposition 9.5 and Corollary 9.2 that for every i > 0,

(9.6)  CH^{i}(G) = CH_{D}^{i}(X) = CH_{D}^{i}(R) = \{ \begin{matrix}(ℤ/pℤ)h^{j}, & if i = (p + 1)j ≤ p^{2} − 1; \\ 0, & otherwise, \end{matrix}

where h = ∂_{G}(q_{G}).

We can compute the Chow ring of G.

Theorem 9.7. Let D be a central division algebra of prime degree p, G = SL_{1}(D) and h = ∂_{G}(q_{G}) ∈ CH^{p+1}(G). Then

CH(G) = ℤ · 1 ⊕ (ℤ/pℤ)h ⊕ (ℤ/pℤ)h^{2} ⊕ ⋯ ⊕ (ℤ/pℤ)h^{p−1}.

Proof. If F is a perfect field, G admits a smooth compactification X by Proposition 6.1. The statement follows from (9.6). In general, we proceed as follows.

A variety X over F is called D-complete is there is a compactification \overline{X} of X such that D is split by the residue field of every point in \overline{X} \ X. Note that the restriction map CH(\overline{X} ×U) → CH(X ×U) is an isomorphism for every variety U. By the proof of Lemma 7.1, G is a D-complete variety.

We extend the category of D-motives by adding the motives M^{D}(X) of smooth Dcomplete varieties X. If X and Y are two smooth D-complete varieties with Y equidimensional of dimension d, we define Hom(M^{D}(X), M^{D}(Y)) := CH_{D}^{d}(X×Y). The composition homomorphism

CH_{D}^{d}(X × Y) ⊗ CH_{D}^{r}(Y × Z) → CH_{D}^{r}(X × Z)

is given by

α ⊗ β ↦ p_{13∗}(p_{12}^{∗}(α) · p_{23}^{∗}(β)),

where p_{ij} are the three projections of X × Y × Z on X, Y and Z, and the push-forward map p_{13∗} is defined as the composition

p_{13∗} : CH_{D}^{d+r}(X × Y × Z) ≃ CH_{D}^{d+r}(X × \overline{Y} × Z) → CH_{D}^{r}(X × Z).

Here \overline{Y} is a compactification of Y satisfying the condition in the definition of a D-complete variety and the second map is the push-forward homomorphism for the proper projection X × \overline{Y} × Z → X × Z.

By Proposition 7.4 and Theorem 8.2, the powers of h = ∂_{G}(q_{G}) yield the following decomposition of D-motives (with coefficients in ℤ_{(p)}):

M^{D}(G) ≃ ℤ_{(p)}^{D} ⊕ ℤ_{(p)}^{D}(p + 1) ⊕ ⋯ ⊕ ℤ_{(p)}^{D}(p^{2} − 1).

The result follows as CH^{i}(G) = CH_{D}^{i}(G) for i > 0 by Corollary 9.2. □

10. Motivic decomposition of compactifications of SL_{1}(D)

Let D be a central division F -algebra of degree a power of a prime p and S = SB(D). We work with motives with ℤ_{(p)}-coefficients in this section.

Proposition 10.1. Let X be a connected smooth complete variety over F such that the motive of X is split over every splitting field of D and D is split over F (X). Then the motive of X is a direct sum of shifts of the motive of S.

Proof. Note that the variety X is generically split, that is, its motive is split over F (X). In particular, X satisfies the nilpotence principle, [30, Proposition 3.1]. Therefore, it suffices to prove the result for motives with coefficients in 𝔽_{p}: any lifting of an isomorphism of the motives with coefficients in 𝔽_{p} to the coefficients ℤ_{(p)} will be an isomorphism since it will become an isomorphism over any splitting field of D.

For 𝔽_{p}-coefficients, here is the argument. The (isomorphism class of the) upper motive U(X) is well-defined and, by the arguments as in the proof of [18, Theorem 3.5], the motive of X is a sum of shifts of U(X). Besides, U(X) ≃ U(S), cf. [18, Corollary 2.15]. Finally, U(S) = M(S) because the motive of S is indecomposable, [18, Corollary 2.22]. □

From now on, the degree of the division algebra D is p. Recall that we work with motives with coefficients in ℤ_{(p)}. So, we set

R = ℤ_{(p)} ⊕ ℤ_{(p)}(p + 1) ⊕ ℤ_{(p)}(2p + 2) ⊕ ⋯ ⊕ ℤ_{(p)}(p^{2} − 1)

now.

Theorem 10.2. Let F be a field, D a central division F -algebra of prime degree p, G = SL_{1}(D), X a smooth compactification of G, and M(X) its Chow motive with ℤ_{(p)}- coefficients. Assume that M(X) is split over every splitting field of D (see Example 6.9). Then the motive M(X) (over F) is isomorphic to the direct sum of R and a direct sum of shifts of M(S).

Proof. By (8.3), M(X) = R ⊕ N for a motive N and by Proposition 9.5, N^{D} = 0. It follows from Proposition 9.4 that N is isomorphic to a direct summand of N ⊗ M(S). In its turn, N ⊗ M(S) is a direct summand of M(X × S). In view of Proposition 10.1, M(X × S) is a direct sum of shifts of M(S). By the uniqueness of the decomposition [6, Corollary 35] and indecomposability of M(S) [18, Corollary 2.22], the motive N is a direct sum of shifts of M(S). □

Theorem 10.3. Let E be an SL_{1}(D)-torsor and X a smooth compactification of E such that the motive M(X) is split over every splitting field of D (see Example 6.9). Then X satisfies the nilpotence principle. Besides, the motive M(X) is isomorphic to the direct sum of the Rost motive R of X and a direct sum of shifts of M(S). The above decomposition is the unique decomposition of M(X) into a direct sum of indecomposable motives.

Proof. By saying that X satisfies the nilpotence principle, we mean that it does it for any coefficient ring, or, equivalently, for ℤ-coefficients. However, since the integral motive of X is split over a field extension of degree p, it suffices to check that X satisfies the nilpotence principle for ℤ_{(p)}-coefficients, where we can simply refer to [9, Theorem 92.4] and Theorem 10.2 (applied to X over F (X)).

It follows that it suffices to get the motivic decomposition of Theorem 10.3 for ℤ_{(p)}- coefficients replaced by 𝔽_{p}-coefficients. For 𝔽_{p}-coefficients we use the following modification of [17, Proposition 4.6]:

Lemma 10.4. Let S be a geometrically irreducible variety with the motive satisfying the nilpotence principle and becoming split over an extension of the base field. Let M be a summand of the motive of some smooth complete variety X. Assume that there exists a field extension L/F and an integer i ∈ ℤ such that the change of field homomorphism Ch(X_{F (S)}) → Ch(X_{L(S)}) is surjective and the motive M(S)(i)_{L} is an indecomposable summand of M_{L}. Then M(S)(i) is an indecomposable summand of M.

Proof. It was assumed in [17, Proposition 4.6] that the field extension L(S)/F (S) is purely transcendental. But this assumption was only used to ensure that the change of field homomorphism Ch(X_{F (S)}) → Ch(X_{L(S)}) is surjective. Therefore the old proof works. □

We apply Lemma 10.4 to our S and X (with L = F (X)). First we take M = M(X) and using Theorem 10.2, we extract from M(X) our first copy of shifted M(S). Then we apply Lemma 10.4 again, taking for M the complementary summand ofM(X). Continuing this way, we eventually extract from M(X) the same number of (shifted) copies of M(S) as we have by Theorem 10.2 over F (X). Let R be the remaining summand of M(X). By uniqueness of decomposition, we have R_{F (X)} ≃ R so that R is the Rost motive. It is indecomposable (over F), because the degree of every closed point on X is divisible by p.

The uniqueness of the constructed decomposition follows by [1, Theorem 3.6 of Chapter I], because the endomorphism rings of M(S) and of R are local (see [19, Lemma 3.3]). □

Remark 10.5. If X is an equivariant toroidal compactification of SL_{1}(D), the number of motives M(S) in the decomposition of Theorem 10.3 is equal to s(p − 1)! − 1, where s is the number of cones of maximal dimension in the fan of the associated toric variety (see Theorem 6.5).

Example 10.6. Let X be the (non-toroidal) equivariant compactification of SL_{1}(D) with p = 3 considered in [26]. Since P_{X}(t) = t^{8} + t^{7} + 2t^{6} + 3t^{5} + 4t^{4} + 3t^{3} + 2t^{2} + t + 1, we have

M(X) ≃ R ⊕ M(S)(1) ⊕ M(S)(2) ⊕ M(S)(3) ⊕ M(S)(4) ⊕ M(S)(5).

Example 10.7. Let X be the toroidal equivariant compactification of SL_{1}(D) with p = 3 considered in Example 6.7 in the split case. We have

M(X) ≃ R ⊕ M(S)(1)^{⊕3} ⊕ M(S)(2)^{⊕5} ⊕ M(S)(3)^{⊕7} ⊕ M(S)(4)^{⊕5} ⊕ M(S)(5)^{⊕3}.

Corollary 10.8. Let E be a nonsplit SL_{1}(D)-torsor. Assume that char F = 0. Then CH(E) = ℤ.

Proof. Since p CH^{>0}(E) = 0, it suffices to prove that CH^{>0}(E) = 0 for ℤ-coefficients replaced by ℤ_{(p)}-coefficients. Below CH stands for Chow group with ℤ_{(p)}-coefficients.

We prove that CH(E) = CH_{D}(E) by the argument of Corollary 9.2. It remains to show that CH_{D}^{>0}(E) = 0.

Let X be a compactification of E as in Theorem 10.3. Since CH_{D}(X) surjects onto CH_{D}(E) and CH_{D}(S) = 0, it suffices to check that CH_{D}^{>0}(R) = 0. Actually, we have CH_{D}(R) ≃ CH_{D}(E) (see Section 9). Moreover, the D-motive of R is isomorphic to M^{D}(E).

The Chow group CH^{>0}(R) has been computed in [19, Appendix RM] (the characteristic assumption is needed here). The generators of the torsion part, provided in [19, Proposition SC.21], vanish in CH_{D}(R) by construction. The remaining generators are norms from a degree p splitting field of D so that they vanish in CH_{D}(R), too. Hence CH_{D}^{>0}(R) = 0 as required. □

References

[1] Bass, H. Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[2] Bourbaki, N. Éléments de mathématique. Masson, Paris, 1983. Algèbre commutative. Chapitre 8. Dimension. Chapitre 9. Anneaux locaux noethériens complets. [Commutative algebra. Chapter 8. Dimension. Chapter 9. Complete Noetherian local rings].

[3] Brion, M., and Kumar, S. Frobenius splitting methods in geometry and representation theory, vol. 231 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2005.

[4] Brosnan, P. On motivic decompositions arising from the method of Białynicki-Birula. Invent. Math. 161, 1 (2005), 91–111.

[5] Brylinski, J.-L. Décomposition simpliciale d’un réseau, invariante par un groupe fini d’automorphismes. C. R. Acad. Sci. Paris Sér. A-B 288, 2 (1979), A137–A139.

[6] Chernousov, V., and Merkurjev, A. Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem. Transform. Groups 11, 3 (2006), 371–386.

[7] Colliot-Thélène, J.-L., Harari, D., and Skorobogatov, A. N. Compactification équivariante d’un tore (d’après Brylinski et Künnemann). Expo. Math. 23, 2 (2005), 161–170.

[8] Demazure, M. Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88. Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I.

[9] Elman, R., Karpenko, N., and Merkurjev, A. The Algebraic and Geometric Theory of Quadratic Forms. American Mathematical Society, Providence, RI, 2008.

[10] Evens, S., and Jones, B. On the wonderful compactification. arXiv:0801.0456v1 [mathAG] 3 Jan 2008 .

[11] Fulton, W. Intersection theory. Springer-Verlag, Berlin, 1984.

[12] Fulton, W. Introduction to toric varieties, vol. 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry.

[13] Garibaldi, R., Merkurjev, A., and Serre, J.-P. Cohomological Invariants in Galois Cohomology. American Mathematical Society, Providence, RI, 2003.

[14] Grothendieck, A. Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. In Dix Exposés sur la Cohomologie des Schémas. North-Holland, Amsterdam, 1968, pp. 46–66.

[15] Huruguen, M. Toric varieties and spherical embeddings over an arbitrary field. J. Algebra 342 (2011), 212–234.

[16] Karpenko, N. A. Characterization of minimal Pfister neighbors via Rost projectors. J. Pure Appl. Algebra 160, 2-3 (2001), 195–227.

[17] Karpenko, N. A. Hyperbolicity of orthogonal involutions. Doc. Math. Extra Volume: Andrei A. Suslin’s Sixtieth Birthday (2010), 371–389 (electronic).

[18] Karpenko, N. A. Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties. J. Reine Angew. Math. 677 (2013), 179–198.

[19] Karpenko, N. A., and Merkurjev, A. S. On standard norm varieties. Ann. Sci. Éc. Norm. Supér. (4) 46, 1 (2013), 175–214.

[20] Merkurjev, A. Adams operations and the Brown-Gersten-Quillen spectral sequence. In Quadratic forms, linear algebraic groups, and cohomology, vol. 18 of Dev. Math. Springer, New York, 2010, pp. 305–313.

[21] Nenashev, A., and Zainoulline, K. Oriented cohomology and motivic decompositions of relative cellular spaces. J. Pure Appl. Algebra 205, 2 (2006), 323–340.

[22] Panin, I. A. Splitting principle and K-theory of simply connected semisimple algebraic groups. Algebra i Analiz 10, 1 (1998), 88–131.

[23] Quillen, D. Higher algebraic K-theory. I. 85–147. Lecture Notes in Math., Vol. 341 (1973).

[24] Rost, M. On the basic correspondence of a splitting variety. September-November 2006, 42 pages. Available on the web page of the author.

[25] Rost, M. Chow groups with coefficients. Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

[26] Semenov, N. Motivic decomposition of a compactification of a Merkurjev-Suslin variety. J. Reine Angew. Math. 617 (2008), 153–167.

[27] Shinder, E. On Motives of Algebraic Groups Associated to Division Algebras. ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Northwestern University.

[28] Suslin, A., and Joukhovitski, S. Norm varieties. J. Pure Appl. Algebra 206, 1-2 (2006), 245–276.

[29] Suslin, A. A. K-theory and K-cohomology of certain group varieties. In Algebraic K-theory, vol. 4 of Adv. Soviet Math. Amer. Math. Soc., Providence, RI, 1991, pp. 53–74.

[30] Vishik, A., and Zainoulline, K. Motivic splitting lemma. Doc. Math. 13 (2008), 81–96.

[31] Voevodsky, V. On motivic cohomology with ℤ/l-coefficients. Ann. of Math. (2) 174, 1 (2011), 401–438.

[32] Voskresenskiĭ, V. E. Algebraic groups and their birational invariants, vol. 179 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ].

[33] Yagunov, S. On some differentials in the motivic cohomology spectral sequence. MPIM Preprint 2007-153 (2007).

Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

  E-mail address: karpenko at ualberta.ca

Department of Mathematics, University of California, Los Angeles, CA, USA

  E-mail address: merkurev at math.ucla.edu