INVARIANTS OF DEGREE 3 AND TORSION IN THE CHOW GROUP OF A VERSAL FLAG

ALEXANDER MERKURJEV, ALEXANDER NESHITOV, AND KIRILL ZAINOULLINE

Abstract. We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.

1. Introduction

Let G be a split semisimple linear algebraic group over a field F. The purpose of the present paper is to relate together three different topics: the geometry of twisted G-flag varieties, the theory of cohomological invariants of G and the representation theory of G.

As for the first, let U/G be a classifying space of G in the sense of Totaro, that is U is an open G-invariant subset in some representation of G with U(F) ≠ ∅ and U → U/G is a G-torsor. Consider the generic fiber U^{gen} of U over U/G. It is a G-torsor over the quotient field K of U/G called the versal G-torsor [7, Ch. I, §5]. We denote by X^{gen} the respective flag variety U^{gen}/B over K, where B is a Borel subgroup of G, and call it the versal flag. The variety X^{gen} can be viewed as the ‘most twisted’ form of the ‘most complicated’ G-flag variety and, hence, is the most natural object to study. In particular, understanding its geometry via studying the Chow group CH(X^{gen}) of algebraic cycles modulo the rational equivalence relation, leads to understanding the geometry of all other G-flag varieties.

Recall that the group CH(X) of a twisted flag variety X has been a subject of intensive investigations for decades: starting with fundamental results by Grothendieck, Demazure, Berstein-Gelfand-Gelfand in 70’s describing its free part and numerous recent results by Baek, Karpenko, Peyre, Zhong and many others (including the authors of the present paper) providing information about its torsion part.

Our second ingredient, the theory of cohomological invariants, has been mainly inspired by the works of J.-P. Serre and M. Rost. Given a field extension L/F and a positive integer d we consider the Galois cohomology group H^{d+1}(L, ℚ/ℤ(d)) denoted by H^{d+1}(L, d). Following [7, Ch. II, §1] a degree d cohomological invariant is a natural transformation of functors

2010 Mathematics Subject Classification. 11E72, 14M17, 14F43.

Key words and phrases. cohomological invariant, linear algebraic group, torsor, Galois cohomology, Chow group.

The second author was supported by the Trillium Foundation (Ontario) and the RFBR grant 12-01-33057. The last author was supported by the NSERC Discovery grant 385795-2010 and the Early Researcher Award (Ontario).

a: H^{1}( — , G) → H^{d}( — , d − 1)

on the category of field extensions over F. We denote the group of degree d invariants by Inv^{d}(G, d − 1).

Following [15, §1] invariant a is called normalized if it sends trivial torsor to zero. We denote the subgroup of normalized invariants by Inv^{d}(G, d − 1)_{norm}. Invariant a is called decomposable if it is given by a cup-product with an invariant of degree 2. We denote the subgroup of decomposable invariants by Inv^{3}(G, 2)_{dec}. The factor group Inv^{3}(G, 2)_{norm}/ Inv^{3}(G, 2)_{dec} is denoted by Inv^{3}(G, 2)_{ind} and is called the group of indecomposable invariants. This group has been studied by Garibaldi, Kahn, Levine, Rost, Serre and others in the simply-connected case. In recent work [15] it was shown how to compute it in general using new results on motivic cohomology obtained in [16]. In particular, it was computed for all adjoint split groups in [15] and for split simple groups in [2].

In the present paper we introduce a new subgroup of semi-decomposable invariants Inv^{3}(G, 2)_{sdec} which consists of invariants a ∈ Inv^{3}(G, 2)_{norm} such that for every field extension L/F and a G-torsor Y over L

a(Y) = ∑_{i finite} ϕ_{i} ∪ b_{i}(Y) for some ϕ_{i} ∈ L^{×} and b_{i} ∈ Inv^{2}(G, 1)_{norm}.

This plays an important role in our computations and applications. Observe that by definition Inv^{3}(G, 2)_{dec} ⊆ Inv^{3}(G, 2)_{sdec} ⊆ Inv^{3}(G, 2)_{norm}.

As for the last ingredient, the representation theory, recall that the classical character map identifies the representation ring of G with the subring ℤ[T^{∗}]^{W} of W -invariant elements of the integral group ring ℤ[T^{∗}], where W is the Weyl group which acts naturally on the group of characters T^{∗} of a split maximal torus T of G. Let (Ĩ^{W}) denote the ideal generated by classes of augmented representations of the simply-connected cover G̃ of G, i.e. by augmented W -invariant elements of the group ring of the weight lattice of G̃.

Our main result says that

Theorem. Let G be a split semisimple linear algebraic group over a field F and let X^{gen} denote the associated versal flag. There is a short exact sequence

0 → \frac{Inv^{3}(G,2)_{sdec}}{Inv^{3}(G,2)_{dec}} → Inv^{3}(G, 2)_{ind} → CH^{2}(X^{gen})_{tors} → 0,

together with a group isomorphism \frac{Inv^{3}(G,2)_{sdec}}{Inv^{3}(G,2)_{dec}} ≃ \frac{c_{2}((Ĩ^{W})∩ℤ[T^{∗}])}{c_{2}(ℤ[T^{∗}]^{W})}, where c_{2} is the second Chern class map of [15, §3c].

In addition, if G is simple, then Inv^{3}(G, 2)_{sdec} = Inv^{3}(G, 2)_{dec}, so there is an isomorphism Inv^{3}(G, 2)_{ind} ≃ CH^{2}(X^{gen})_{tors}.

Observe that if G is not simple, then Inv^{3}(G, 2)_{sdec} does not necessarily coincide with Inv^{3}(G, 2)_{dec} (see Example 3.1).

The nature of our result suggests that it should have applications in several directions, e.g. for cohomological invariants and algebraic cycles on twisted flag varieties.

For instance, since the group Inv^{3}(G, 2)_{ind} has been computed for all simple split groups in [15] and [2], it immediately gives computation of the torsion part of CH^{2}(X^{gen}), hence, extending previous results by [13] and [22]. As another straightforward consequence, using the coincidence Inv^{3}(G, 2)_{sdec} = Inv^{3}(G, 2)_{dec} we construct non-trivial cohomological classes for indecomposable central simple algebras, hence, answering questions posed in [8] and [1].

The paper is organized as follows: In section 2 we construct an exact sequence relating the groups of invariants with the torsion part of the Chow group, hence, proving the first part of the theorem. In section 3 we compute this exact sequence case by case for all simple groups, hence, proving the second part. In the last section we discuss applications.

2. Semi-decomposable invariants and the Chow group

Let G be a split semisimple linear algebraic group over a field F. We fix a split maximal torus T of G and a Borel subgroup B containing T. Consider the T -equivariant structure map U → Spec F = pt, where U is the open G-invariant subset from the introduction.

Characteristic maps and classes. By [5] the induced pullback on T -equivariant Chow groups CH_{T} (pt) → CH_{T} (U) is an isomorphism. Since CH_{T} (U) ≃ CH(U/T) ≃ CH(U/B) and CH_{T} (pt) can be identified with the symmetric algebra Sym(T^{∗}) of the group of characters of T, it gives an isomorphism

(1)  𝔠^{CH} : Sym(T^{∗}) ⟶^{≃} CH(U/B).

Similarly, by the homotopy invariance and localization property of the equivariant K-theory [18, Theorems 8 and 11] the induced pull-back on T -equivariant K-groups gives a surjection

𝔠^{K_{0}} : ℤ[T^{∗}] ↠ K_{0}(U/B),

where the integral group ring ℤ[T^{∗}] can be identified with K_{T} (pt) and K_{0}(U/B) ≃ K_{0}(U/T) ≃ K_{T} (U).

Let τ^{i}(X) denote the i-th term of the topological filtration on K_{0} of a smooth variety X and let τ^{i/i+1}, i ≥ 0 denote its i-th subsequent quotient. Let I denote the augmentation ideal of ℤ[T^{∗}].

2.1. Lemma. The map 𝔠^{K_{0}} induces isomorphisms on subsequent quotients

I^{i}/I^{i+1} ⟶^{≃} τ^{i/i+1}(U/B), for 0 ≤ i ≤ 2,

and, its restriction 𝔠^{K_{0}} : I^{2} → τ^{2}(U/B) is surjective.

Proof. By [6, Ex. 15.3.6] the Chern class maps induce isomorphisms c_{i}: τ^{i/i+1}(X) ⟶^{≃} CH^{i}(X) for 0 ≤ i ≤ 2. Since the Chern classes commute with pullbacks and Sym^{i}(T^{∗}) ≃ I^{i}/I^{i+1}, the isomorphisms then follow from (1).

Finally, since I/I^{2} ≃ τ^{1/2}(U/B), 𝔠^{K_{0}}(x) ∈ τ^{2}(U/B) implies that x ∈ I^{2}. □

Consider the natural inclusion of the versal flag ı: X^{gen} = U^{gen}/B ↪ U/B. Since ı is a limit of open embeddings, by the localization property of Chow groups, the induced pullback gives surjections

ı^{CH} : CH^{i}(U/B) ↠ CH^{i}(X^{gen}).

Moreover, the induced pullback in K-theory restricted to τ^{i} also gives surjections

ı^{K_{0}} : τ^{i}(U/B) ↠ τ^{i}(X^{gen}).

Indeed, by definition τ^{i}(X^{gen}) is generated by the classes [O_{Z}] for closed subvarieties Z of X^{gen} with codim Z ⩾ i and each [O_{Z}] is the pullback of the element [OZ̄] in τ^{i}(U/B), where Z̄ is the closure of Z inside U/B.

Let L be a splitting field of the versal torsor U^{gen}. According to [10, Thm. 4.5] composites

Sym(T^{∗}) ⟶^{𝔠^{CH}} CH(U/B) ⟶^{ı^{CH}} CH(X^{gen}) ⟶^{res} CH(X_{L}^{gen}) and

ℤ[T^{∗}] ⟶^{𝔠^{K_{0}}} K_{0}(U/B) ⟶^{ı^{K_{0}}} K_{0}(X^{gen}) ⟶^{res} K_{0}(X_{L}^{gen})

give the classical characteristic maps for the Chow groups and for the K-groups respectively (here we identify the rightmost groups with the Chow group and the K-group of the split flag G/B respectively). Restricting the latter to I^{2} and τ^{2} we obtain the map

𝔠: I^{2} ⟶^{𝔠^{K_{0}}} τ^{2}(U/B) ⟶^{ı^{K_{0}}} τ^{2}(X^{gen}) ⟶^{res} τ^{2}(X_{L}^{gen}) = τ^{2}(G/B).

From this point on, we denote by 𝔠^{CH}, ı^{CH} and by 𝔠^{K_{0}}, ı^{K_{0}} the respective restrictions to Sym^{2}, CH^{2} and I^{2}, τ^{2}.

Let Λ be the weight lattice. Consider the integral group ring ℤ[Λ]. Let Ĩ denote its augmentation ideal. The Weyl group W acts naturally on ℤ[Λ]. Let (Ĩ^{W}) denote the ideal generated by W -invariant elements in Ĩ.

2.2. Lemma. The kernel of the composite I^{2} →^{𝔠^{K_{0}}} τ^{2}(U/B) →^{ı^{K_{0}}} τ^{2}(X^{gen}) is (Ĩ^{W})∩I^{2}.

Proof. By the results of Panin [21], K_{0}(X^{gen}) is the direct sum of K_{0}(A_{i}) for some central simple algebras A_{i} over K. So K_{0}(X^{gen}) is a free abelian group and, hence, the restriction τ^{2}(X^{gen}) → τ^{2}(X_{L}^{gen}) is injective. Therefore, ker(ı^{K_{0}} ∘ 𝔠^{K_{0}}) coincides with the kernel of the characteristic map 𝔠: I^{2} → τ^{2}(G/B). Since 𝔠 factors as I^{2} ↪ Ĩ^{2} → τ^{2}(G/B) and the kernel of the second map is (Ĩ^{W}) ∩ Ĩ^{2} by the theorem of Steinberg [24], we get ker 𝔠 = (Ĩ^{W}) ∩ I^{2} (here we used that ℤ[T^{∗}] ∩ Ĩ^{i} = I^{i}). □

Consider the second Chern class map c_{2} : τ^{2}(U/B) → CH^{2}(U/B).

2.3. Lemma. We have c_{2}(ker ı^{K_{0}}) = ker ı^{CH}.

Proof. Consider the diagram

\xymatrix{τ^{3}(U/B) \ar[r] \ar[d]_{ı^{K_{0}} |_{τ^{3}}} & τ^{2}(U/B) \ar[r]^{c_{2}} \ar[d]^{ı^{K_{0}}} & CH^{2}(U/B) \ar[r] \ar[d]^{ı^{CH}} & 0 \\ τ^{3}(X^{gen}) \ar[r] & τ^{2}(X^{gen}) \ar[r]^{c_{2}} & CH^{2}(X^{gen}) \ar[r] & 0}

Its vertical maps are surjective and the rows are exact by [6, Ex. 15.3.6]. The result then follows by the diagram chase. □

Consider the composite 𝔠_{2} : I^{2} ⟶^{𝔠^{K_{0}}} τ^{2}(U/B) ⟶^{c_{2}} CH^{2}(U/B). Observe that it coincides with the Chern class map defined in [15, §3c].

2.4. Lemma. We have ker ı^{CH} = 𝔠_{2}((Ĩ^{W}) ∩ I^{2}).

Proof. Since 𝔠^{K_{0}} is surjective by lemma 2.1, we have by lemma 2.2 and 2.3

𝔠_{2}((Ĩ^{W}) ∩ I^{2}) = 𝔠_{2}(ker(ı^{K_{0}} ∘ 𝔠^{K_{0}})) = c_{2}(ker ı^{K_{0}}) = ker ı^{CH}.  □

Following [15] we denote

Dec(G) := (𝔠^{CH})^{−1} ∘ 𝔠_{2}(ℤ[T^{∗}]^{W}).

And we set

SDec(G) := (𝔠^{CH})^{−1} ∘ 𝔠_{2}((Ĩ^{W}) ∩ ℤ[T^{∗}]).

Since the action of W on Λ is essential, i.e. Λ^{W} = 0, we have (Ĩ^{W}) ⊆ Ĩ^{2}. Therefore, for any x ∈ (Ĩ^{W}) we have x ≡ x′ mod Ĩ^{3} and, hence, 𝔠_{2}(x) = 𝔠_{2}(x′) for some x′ ∈ ℤ[Λ]^{W}, where ℤ[Λ]^{W} is the subring of W -invariants. So there are inclusions

(2)  Dec(G) ⊆ SDec(G) ⊆ Sym^{2}(T^{∗})^{W} .

2.5. Lemma. We have CH^{2}(X^{gen}) ≃ Sym^{2}(T^{∗})/ SDec(G).

Proof. By (1) and lemma 2.4 we have

CH^{2}(X^{gen}) ≃ CH^{2}(U/B)/𝔠_{2}((Ĩ^{W}) ∩ I^{2}) ≃ Sym^{2}(T^{∗})/ SDec(G).  □

2.6. Corollary. We have CH^{2}(X^{gen})_{tors} ≃ Sym^{2}(T^{∗})^{W} / SDec(G).

Proof. By the lemma it remains to show that

(Sym^{2}(T^{∗})/ SDec(G))_{tors} = Sym^{2}(T^{∗})^{W} / SDec(G).

Indeed, suppose that x ∈ Sym^{2}(T^{∗}) and nx ∈ SDec(G). Then nx lies in Sym^{2}(T^{∗})^{W} by (2). So for every w ∈ W we have n(wx − x) = 0. Since Sym^{2}(T^{∗}) has no torsion, x ∈ Sym^{2}(T^{∗})^{W}. Conversely, let x ∈ Sym^{2}(T^{∗})^{W}. Since the second Chern class map c_{2} : I^{2} → Sym^{2}(T^{∗}) is surjective, there is a preimage y ∈ I^{2} of x. Take y′ = ∑_{w∈W} w·y ∈ ℤ[T^{∗}]^{W} ⊆ (Ĩ^{W})∩ℤ[T^{∗}]. Since c_{2} is W -equivariant and coincides with the composite (𝔠^{CH})^{−1} ∘ 𝔠_{2}, we get (𝔠^{CH})^{−1} ∘ 𝔠_{2}(y′) = |W | · x ∈ SDec(G). □

Cohomological Invariants. For a smooth F -scheme X let H^{3}(2) denote the Zariski sheaf on X associated to a presheaf W ↦ H_{ét}^{3}(W, ℚ/ℤ(2)). The Bloch-Ogus-Gabber theorem (see [4] and [11]) implies that its group of global sections H_{Zar}^{0}(X, H^{3}(2)) is a subgroup in H^{3}(F (X), 2).

Consider the versal G-torsor U^{gen} over the quotient field K of the classifying space U/G. By [3, Thm. A] the map Θ: Inv^{3}(G, 2) → H^{3}(K, 2) defined by Θ(a) := a(U^{gen}) gives an inclusion

Inv^{3}(G, 2) ↪ H_{Zar}^{0}(U/G, H^{3}(2)).

2.7. Lemma. We have a(U^{gen}) ∈ ker[H^{3}(K, 2) → H^{3}(K(X^{gen}), 2)] for any a ∈ Inv^{3}(G, 2)_{norm}.

Proof. Consider the composite q: Spec K(U^{gen}) → U^{gen} → U/G. Observe that the pullback q^{∗} factors as

q^{∗}: H_{Zar}^{0}(U/G, H^{3}(2)) → H^{3}(K(X^{gen}), 2) → H^{3}(K(U^{gen}), 2).

Since U^{gen} → X^{gen} is a B-torsor, K(U^{gen}) is purely transcendental over K(X^{gen}), so the last map of the composite is injective. Since the U^{gen} becomes trivial over K(U^{gen}), we have q^{∗}(a(U^{gen})) = a(U^{gen} ×_{K} K(U^{gen})) = 0. Therefore, a(U^{gen}) ∈ ker[H^{3}(K, 2) → H^{3}(K(X^{gen}), 2)]. □

2.8. Lemma. Let Y → Spec L be a G-torsor and X = Y/B. Let L^{sep} denote the separable closure of L, Γ_{L} its Galois group and X^{sep} = X ×_{L} L^{sep}. Then the Γ_{L} action on Pic X^{sep} is trivial.

Proof. It follows by [19, Prop. 2.2]. □

The Tits map. Consider a short exact sequence of F -group schemes

1 → C → G̃ →^{π} G → 1.

Given a character χ ∈ C^{∗} of the center and a field extension L/F consider the Tits map [25, §4,5]

α_{χ,L}: H^{1}(L, G) ⟶^{∂} H^{2}(L, C) ⟶^{χ_{∗}} H^{2}(L, 𝔾_{m}),

where ∂ is the connecting homomorphism (if C is non-smooth, we replace it by 𝔾_{m} and G by the respective push-out as in [7, II, Example 2.1]). This gives rise to a cohomological invariant β_{χ} of degree two

β_{χ}: Y ↦ α_{χ,L}(Y) for every G-torsor Y ∈ H^{1}(L, G).

[3, Theorem 2.4] shows that the assignment χ → β_{χ} provides an isomorphism C^{∗} → Inv^{2}(G, 1).

For a G-torsor Y over L there is an exact sequence studied in [17], [22] and [7, II, Thm. 8.9]:

A^{1}((Y/B)^{sep}, K_{2})^{Γ} ⟶^{ρ} ker[H^{3}(L, 2) → H^{3}(L(Y/B), 2)] ⟶^{δ_{Y}} CH^{2}(Y/B).

The multiplication map L^{sep} ⊗ CH^{1}(Y/B)^{sep} → A^{1}((Y/B)^{sep}, K_{2}) is an isomorphism. By lemma 2.8 we obtain an exact sequence

(3)  L ⊗ Λ ⟶^{ρ_{Y}} ker[H^{3}(L, 2) → H^{3}(L(Y/B), 2)] ⟶^{δ_{Y}} CH^{2}(Y/B).

According to [17] the map ρ_{Y} acts as follows:

ρ_{Y} (ϕ ⊗ λ) = ϕ ∪ β\overline{λ}(Y), where ϕ ∈ L^{×}, λ ∈ Λ and

\overline{λ} denotes the image of λ in Λ/T^{∗} = C^{∗}.

We define the subgroup Inv^{3}(G, 2)_{sdec} of semi-decomposable invariants as follows:

2.9. Definition. An invariant a ∈ Inv^{3}(G, 2)_{norm} is called semi-decomposable, if there is a finite set b_{i} ∈ Inv^{2}(G, 1)_{norm} such that for every field extension L/F and a torsor Y ∈ H^{1}(L, G) we have

a(Y) = ∑_{i} ϕ_{i} ∪ b_{i}(Y) for some ϕ_{i} ∈ L^{×}.

Observe that by definition, we have

Inv^{3}(G, 2)_{dec} ⊆ Inv^{3}(G, 2)_{sdec} ⊆ Inv^{3}(G, 2)_{norm}

and a ∈ Inv^{3}(G, 2)_{sdec} if and only if a(Y) ∈ im(ρ_{Y}) = ker(δ_{Y}) for every torsor Y.

2.10. Lemma. We have a ∈ Inv^{3}(G, 2)_{sdec} if and only if a(U^{gen}) ∈ ker(δ_{U^{gen}}).

Proof. If a is a semi-decomposable invariant, then a(U^{gen}) = ∑_{χ∈C^{∗}} ϕ_{χ} ∪ β_{χ}(U^{gen}) lies in the image of ρ_{U^{gen}}, hence, δ_{U^{gen}}(a(U^{gen})) = 0. On the other hand, let a be a degree 3 invariant such that δ_{U^{gen}}(a(U^{gen})) = 0 and let Y be a G-torsor over a field extension L/F. We show that δ_{Y} (a(Y)) = 0.

We may assume that L is infinite (replacing L by L(t) if needed). Choose a rational point y ∈ (U/G)_{L} such that Y is isomorphic to the fiber of U → U/G over y. Let R be the completion of the regular local ring O_{(U/G)_{L},y} and let K̂ be its quotient field. The ring R is a regular local ring with residue field L. By the theorem of Grothendieck Y_{R} is a pullback of Y via the projection Spec R → Spec L(y). Then the G-torsors YK̂ and U gen ; K̂ over K̂ are isomorphic. We have

δ_{Y} (a(Y))K̂ = δ_{YK̂} (a(YK̂)) = δ_{U gen ; K̂}(a(U gen ; K̂)) = δ_{U^{gen}}(a(U^{gen}))K̂ = 0.

The restriction CH^{2}(Y/B) → CH^{2}((Y/B)K̂) is injective, since it is split by the specialization map with respect to a system of local parameters of R. Therefore, δ_{Y} (a(Y)) = 0 for every Y, hence, a is semi-decomposable. □

Now we are ready to prove the first part of the main theorem:

2.11. Theorem. The map δ_{U^{gen}} induces a short exact sequence

0 ⟶ \frac{Inv^{3}(G,2)_{sdec}}{Inv^{3}(G,2)_{dec}} ⟶ Inv^{3}(G, 2)_{ind} ⟶^{g} CH^{2}(X^{gen})_{tors} ⟶ 0,

and there is a group isomorphism

\frac{Inv^{3}(G,2)_{sdec}}{Inv^{3}(G,2)_{dec}} ≃ \frac{c_{2}((Ĩ^{W})∩ℤ[T^{∗}])}{c_{2}(ℤ[T^{∗}]^{W})}.

Proof. Consider the following diagram. The rows are exact sequences given by [12, Thm. 1.1] and vertical arrows are pullbacks:

\xymatrix{0 \ar[r] & CH^{2}(U/G) \ar[r] \ar[d] & ℍ_{ét}^{4}(U/G, ℤ(2)) \ar[r] \ar[d] & H_{Zar}^{0}(U/G, H^{3}(2)) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & CH^{2}(U/B) \ar[r] \ar[d]_{ı^{CH}} & ℍ_{ét}^{4}(U/B, ℤ(2)) \ar[r] \ar[d] & H_{Zar}^{0}(U/B, H^{3}(2)) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & CH^{2}(X^{gen}) \ar[r] & ℍ_{ét}^{4}(X^{gen}, ℤ(2)) \ar[r] & H_{Zar}^{0}(X^{gen}, H^{3}(2)) \ar[r] & 0}

Since F (U/B) = K(X^{gen}), lemma 2.7 implies that the composite

Inv^{3}(G, 2)_{norm} → H_{Zar}^{0}(U/G, H^{3}(2)) → H_{Zar}^{0}(U/B, H^{3}(2))

is zero. By the diagram chase there is a homomorphism

Inv^{3}(G, 2)_{norm} → CH^{2}(U/B)/ CH^{2}(U/G).

The map X^{gen} → U/B → U/G factors as X^{gen} → Spec K → U/G, hence the

composite of pullbacks CH^{2}(U/G) → CH^{2}(U/B) ⟶^{ı^{CH}} CH^{2}(X^{gen}) coincides with the composite CH^{2}(U/G) → CH^{2}(Spec K) → CH^{2}(X^{gen}) which is zero. This gives a homomorphism g: Inv^{3}(G, 2)_{norm} → CH^{2}(U/B)/ CH^{2}(U/G) → CH^{2}(X^{gen}) which by the proof of theorem of B. Kahn (see [7, II, §8, 8.1-8.5]) factors through the map δ_{U^{gen}} of (3). By [15, 3.9] the map g also factors through Inv^{3}(G, 2)_{ind} ⟶^{≃} \frac{Sym^{2}(T^{∗})^{W}}{Dec(G)} . So there is a commutative diagram

(4)  [figure: Inv^{3}(G, 2)_{norm}; g; CH^{2}(X^{gen})_{tors}; \frac{Sym^{2}(T^{∗})^{W}}{Dec(G)}; \frac{Sym^{2}(T^{∗})^{W}}{SDec(G)}; ≃; 2.6^{Cor.}] .

The bottom row of (4) gives a short exact sequence

0 → \frac{SDec(G)}{Dec(G)} → \frac{Sym^{2}(T^{∗})^{W}}{Dec(G)} → CH^{2}(X^{gen})_{tors} → 0.

Lemma 2.10 and composite (3) give an exact sequence

0 → Inv^{3}(G, 2)_{sdec} → Inv^{3}(G, 2)_{norm} ⟶^{g} CH^{2}(X^{gen})_{tors}.

Combining these together and factoring modulo Inv^{3}(G, 2)_{dec} we obtain an isomorphism

\frac{Inv^{3}(G,2)_{sdec}}{Inv^{3}(G,2)_{dec}} ≅ \frac{SDec(G)}{Dec(G)}.  □

3. Semi-decomposable invariants vs. decomposable invariants

In this section we prove case by case that the groups of decomposable Inv^{3}(G, 2)_{dec} and semi-decomposable Inv^{3}(G, 2)_{sdec} invariants coincide for all split simple G, hence, proving the second part of our main theorem. More precisely, we show that

Dec(G) = 𝔠_{2}(ℤ[T^{∗}]^{W}) = 𝔠_{2}((Ĩ^{W}) ∩ ℤ[T^{∗}]) = SDec(G) in Sym^{2}(T^{∗})^{W}

(here we denote (𝔠^{CH})^{−1} ∘ 𝔠_{2} simply by 𝔠_{2}). Observe that in the simply connected case Sym^{2}(Λ)^{W} = ℤq, where q corresponds to the normalized Killing form from [9, §1B], and Dec(G) ⊆ SDec(G) ⊆ SDec(G̃) = Dec(G̃).

3.1. Example. If G is not simple, then Dec(G) ≠ SDec(G) in general. Indeed, consider a quadratic form q of degree 4 with trivial discriminant (it corresponds to a SO_{4}-torsor). According to [7, Example 20.3] there is an invariant given by q ↦ α ∪ β ∪ γ, where α is represented by q and ⟨⟨β, γ⟩⟩ = ⟨α⟩q is the 2-Pfister form. By definition this invariant is semi-decomposable (this fact was pointed to us by Vladimir Chernousov). Since it is non-trivial over an algebraic closure of F, it is not decomposable.

3.1. Adjoint groups of type A_{n} (n ≥ 1), B_{n} (n ≥ 2), C_{n} (n ≥ 3, 4 ∤ n), D_{n} (n ≥ 5, 4 ∤ n), E_{6}, E_{7} and special orthogonal groups of type D_{n} (n ≥ 4).

For classical adjoint types we have Inv^{3}(G, 2)_{norm} = Inv^{3}(G, 2)_{dec} by [15, §4b], so we immediately obtain Inv^{3}(G, 2)_{dec} = Inv^{3}(G, 2)_{sdec}. For exceptional types by [7, p.135] and [15, §4b] we have Dec(G) = Dec(G̃) = 6ℤq for E_{6} and Dec(G) = Dec(G̃) = 12ℤq for E_{7}. For special orthogonal groups G = SO_{2n} by [7, §15] we have Dec(SO_{2n}) = Dec(Spin_{2n}) = 2ℤq (here G̃ = Spin_{2n}), hence, Dec(G) = SDec(G).

3.2. Non-adjoint groups of type A_{n−1} (n ≥ 4).

Let p be a prime integer and G = SL_{p^{s}} /μ_{p^{r}} for some integers s ≥ r > 0. If p is odd, we set k = min{r, s − r} and if p = 2 we assume that s ≥ r + 1 and set k = min{r, s − r − 1}. It is shown in [2, §4] that the group Inv^{3}(G, 2)_{ind} is cyclic of order p^{k}. On the other hand, by [13, Example 4.15] if X is the Severi-Brauer variety of a generic algebra A^{gen}, then CH^{2}(X)_{tors} is also a cyclic group of order p^{k}. The canonical morphism X^{gen} → X is an iterated projective bundle, hence, CH^{2}(X^{gen})_{tors} ≃ CH^{2}(X)_{tors} is a cyclic group of order p^{k}. It follows from the exact sequence of theorem 2.11 that Inv^{3}(G, 2)_{sdec} = Inv^{3}(G, 2)_{dec}.

More generally, let G = SL_{n} /μ_{m}, where m | n. Let p^{s} and p^{r} be the highest powers of a prime integer p dividing n and m respectively. Consider the canonical homomorphism H = SL_{p^{s}} /μ_{p^{r}} → G. We claim that it induces an isomorphism between the p-primary component of Inv^{3}(G, 2)_{ind} and the group Inv^{3}(H, 2)_{ind}.

Indeed, let H′ = SL_{n} /μ_{p^{r}}. It follows from [2, Theorem 4.1] that the natural homomorphism Inv^{3}(H′, 2)_{ind} → Inv^{3}(H, 2)_{ind} is an isomorphism. Thus, it suffices to show that the pull-back map for the canonical surjective homomorphism H′ → G with kernel μ_{t}, where t := m/p^{r} is relatively prime to p, induces an isomorphism between the p-primary component of Inv^{3}(G, 2)_{ind} and Inv^{3}(H′, 2)_{ind}. Let Λ ⊂ Λ′ be the character groups of maximal tori of G and H′ respectively. The factor group Λ′/Λ is isomorphic to μ_{t}^{∗} = ℤ/tℤ. Since the functor Λ ↦ \frac{Sym^{2}(Λ)^{W}}{Dec(Λ)} is quadratic in Λ, the kernel and the cokernel of the homomorphism

Inv^{3}(G, 2)_{ind} = \frac{Sym^{2}(Λ)^{W}}{Dec(Λ)} → \frac{Sym^{2}(Λ′)^{W}}{Dec(Λ′)} = Inv^{3}(H′, 2)_{ind}

are killed by t^{2}. As t is relatively prime to p, the claim follows.

Since the p-primary component of CH(X^{gen})_{tors} and the group CH(X_{H}^{gen})_{tors} are isomorphic by [13, Prop. 1.3] (here X_{H}^{gen} denotes the versal flag for H), we obtain that Inv^{3}(G, 2)_{ind} ≃ CH(X^{gen})_{tors} and, therefore, by the exact sequence of theorem 2.11 Inv^{3}(G, 2)_{sdec} = Inv^{3}(G, 2)_{dec}.

3.3. Adjoint groups of type C_{4m} (m ≥ 1).

By [15, §4b] we have Sym^{2}(T^{∗})^{W} = ℤq and Dec(G) = 𝔠_{2}(ℤ[T^{∗}]^{W}) = 2ℤq. We want to show that 𝔠_{2}(x) ∈ 2ℤq for every element x ∈ (Ĩ^{W}) ∩ ℤ[T^{∗}].

Given a weight χ ∈ Λ we denote by W (χ) its W -orbit and we define ê^{χ} := ∑_{λ∈W (χ)}(1−e^{−λ}). By definition, the ideal (Ĩ^{W}) is generated by elements {ê^{ω_{i}}}_{i=1..4m} corresponding to the fundamental weights ω_{i}. An element x can be written as

(5)  x = ∑_{i=1}^{4m} n_{i}ê^{ω_{i}} + δ_{i}ê^{ω_{i}}, where n_{i} ∈ ℤ and δ_{i} ∈ Ĩ.

Similar to [26, §3] consider a ring homomorphism f: ℤ[Λ] → ℤ[Λ/T^{∗}] induced by taking the quotient Λ → Λ/T^{∗} = C^{∗}. We have Λ/T^{∗} ≃ ℤ/2ℤ and ℤ[Λ/T^{∗}] = ℤ[y]/(y^{2} − 2y), where y = f(e^{ω_{1}} − 1). Observe that C^{∗} is W -invariant.

By definition, f(I) = 0, so f(x) = 0. Since ω_{i} ∈ T^{∗} for all even i, f(ê^{ω_{i}}) = y for all odd i and f(δ_{i}) ∈ f(Ĩ) = (y), we get

0 = f(x) = ∑_{i is odd} n_{i}d_{i}y + m_{i}d_{i}y^{2} = ( ∑_{i is odd} n_{i} + 2m_{i})d_{i}y,

10 ALEXANDER MERKURJEV, ALEXANDER NESHITOV, AND KIRILL ZAINOULLINE

where m_{i} ∈ ℤ and d_{i} = 2^{i}\binom{4m}{i} is the cardinality of W (ω_{i}), which implies that (∑_{i is odd} n_{i} + 2m_{i})d_{i} = 0. Dividing this sum by the g.c.d. of all d_{i}’s and taking the result modulo 2 (here one uses the fact \frac{n}{g.c.d.(n,k)} | \binom{n}{k}), we obtain that the coefficient n_{1} in the presentation (5) has to be even.

We now compute 𝔠_{2}(x). Let Λ = ℤe_{1} ⊕ … ⊕ ℤe_{4m}. The root lattice is given by T^{∗} = {∑ a_{i}e_{i} | ∑ a_{i} is even} and

ω_{1} = e_{1}, ω_{2} = e_{1} + e_{2}, ω_{3} = e_{1} + e_{2} + e_{3}, …, ω_{4m} = e_{1} + … + e_{4m}.

By [9, §2] we have 𝔠_{2}(x) = ∑_{i=1}^{4m} n_{i}𝔠_{2}(ê^{ω_{i}}) and 𝔠_{2}(ê^{ω_{i}}) = N(ê^{ω_{i}})q, where

N(∑ a_{j}e^{λ_{j}}) = \frac{1}{2} ∑ a_{j}⟨λ_{j}, α^{∨}⟩^{2} for a fixed long root α.

If we set α = 2e_{4m}, then ⟨λ, α^{∨}⟩ = (λ, e_{4m}) and

N(ê^{ω_{i}}) = \frac{1}{2} ∑_{λ∈W (ω_{i})}⟨λ, α^{∨}⟩^{2} = \frac{1}{2} ∑_{λ∈W (ω_{i})}(λ, e_{4m})^{2} = 2^{i−1}\binom{4m−1}{i−1}

which is even for i ≥ 2 (here we used the fact that the Weyl group acts by permutations and sign changes on {e_{1}, …, e_{4m}}). Since n_{1} is even, we get that 𝔠_{2}(x) ∈ 2ℤq.

3.4. Half-spin and adjoint groups of type D_{2m} (m ≥ 2).

We first treat the half-spin group G = HSpin_{8m}. As in the C_{n}-case all even fundamental weights are in T^{∗} and all odd fundamental weights correspond to a generator of Λ/T^{∗} ≃ ℤ/2ℤ. Therefore, the map f: ℤ[Λ] → ℤ[Λ/T^{∗}] applied to the element x = ∑_{i=1}^{2m} n_{i}ê^{ω_{i}} + δ_{i}ê^{ω_{i}} gives the same equality (∑_{i is odd} n_{i} + 2m_{i})d_{i} = 0, where m_{i} ∈ ℤ, d_{i} = 2^{i}\binom{2m}{i} for i ≤ 2m − 2 and d_{2m−1} = 2^{2m−1}. Dividing by the g.c.d. of d_{i}’s and taking modulo 2 we obtain that n_{1} is even if m > 2 and n_{1} + n_{3} is even if m = 2.

We now compute 𝔠_{2}(x). Take a long root α = e_{2m−1} + e_{2m}. Then (α, α) = 2 and ⟨λ, α^{∨}⟩ = (λ, e_{2m−1}) + (λ, e_{2m}). For i ≤ 2m − 2 we have

N(ê^{ω_{i}}) = ∑_{λ∈W (ω_{i})}(λ, e_{2m})^{2} + (λ, e_{2m})(λ, e_{2m−1}) = ∑_{λ∈W (ω_{i})}(λ, e_{2m})^{2} = 2^{i}\binom{2m−1}{i−1},

and N({ \atop e^{ω_{2m−1}}}) = N({ \atop e^{ω_{2m}}}) = 2^{2m−3} (here we used the fact that W acts by permutations and even sign changes).

Finally, if m > 2, we obtain 𝔠_{2}(x) = ∑_{i} n_{i}N(ê^{ω_{i}})q ∈ 4ℤq, where 4ℤq = Dec(HSpin_{4m}) by [2, §5]. If m = 2, then N(ê^{ω_{4}}) = 2, hence, 𝔠_{2}(x) ∈ 2ℤq, where 2ℤq = Dec(HSpin_{8}) again by [2, §5].

If m > 2, for the adjoint group G = PGO_{8m} by [15, §4] and the respective half-spin case we obtain

4ℤq = Dec(PGO_{8m}) ⊆ SDec(PGO_{8m}) ⊆ SDec(HSpin_{8m}) = 4ℤq.

If G = PGO_{8}, direct computations (see [20]) show that Dec(G) = SDec(G).

4. Applications

Observe that H^{3}(F, ℤ/nℤ(2)) is the n-th torsion part of H^{3}(F, 2) for every n and H^{3}(F, ℤ/nℤ(2)) = H^{3}(F, μ_{n}^{⊗2}) if char(F) does not divide n.

4.1. Type C_{n}. Let G = PGSp_{2n} be the split projective symplectic group. For a field extension L/F, the set H^{1}(L, G) is identified with the set of isomorphism classes of central simple L-algebras A of degree 2n with a symplectic involution σ (see [14, §29]). A decomposable invariant of G takes an algebra with involution (A, σ) to the cup-product ϕ ∪ [A] for a fixed element ϕ ∈ F^{×}. In particular, decomposable invariants of G are independent of the involution.

Suppose that 4 | n. It is shown in [15, Theorem 4.6] that the group of indecomposable invariants Inv^{3}(G, 2)_{ind} is cyclic of order 2. If char(F) ≠ 2, Garibaldi, Parimala and Tignol constructed in [8, Theorem A] a degree 3 cohomological invariant Δ_{2n} of the group G with coefficients in ℤ/2ℤ. They showed that if a ∈ A is a σ-symmetric element of A^{×} and σ′ = Int(a) ∘ σ, then

(6)  Δ_{2n}(A, σ′) = Δ_{2n}(A, σ) + Nrp(a) ∪ [A],

where Nrp is the pfaffian norm. In particular, Δ_{2n} does depend on the involution and therefore, the invariant Δ_{2n} is not decomposable. Hence the the class of Δ_{2n} in Inv^{3}(G, 2)_{ind} is nontrivial.

It follows from (6) that the class Δ_{2n}(A) ∈ \frac{H^{3}(L,ℤ/2ℤ)}{L^{×}∪[A]} of Δ_{2n}(A, σ) depends only on the L-algebra A of degree 2n and exponent 2 but not on the involution. Since Δ_{2n}(A, σ) is not decomposable, it is not semi-decomposable by our main theorem. The latter implies that Δ_{2n}(A) is nontrivial generically, i.e. there is a central simple algebra A of degree 2n over a field extension of F with exponent 2 such that Δ_{2n}(A) ≠ 0. This answers a question raised in [8]. (See [1, Remark 4.10] for the case n = 4.)

4.2. Type A_{n−1}. Let G = SL_{n} /μ_{m}, where n and m are positive integers such that n and m have the same prime divisors and m | n. Given a field extension L/F the natural surjection G → PGL_{n} yields a map

α : H^{1}(L, G) → H^{1}(L, PGL_{n}) ⊂ Br(L)

taking a G-torsor Y over L to the class of a central simple algebra A(Y) of degree n and exponent dividing m. By definition, a decomposable invariant of G is of the form Y ↦ ϕ ∪ [A(Y)] for a fixed ϕ ∈ F^{×}.

The map SL_{m} → SL_{n} taking a matrix M to the tensor product M ⊗ I_{n/m} with the identity matrix, gives rise to a group homomorphism PGL_{m} → G. The induced homomorphism (see [15, Theorem 4.4])

φ : Inv^{3}(G, 2)_{norm} → Inv^{3}(PGL_{m}, 2)_{norm} = F^{×}/F^{×m}

is a splitting of the inclusion homomorphism

F^{×}/F^{×m} = Inv^{3}(G, 2)_{dec} ↪ Inv^{3}(G, 2)_{norm}.

Collecting descriptions of p-primary components of Inv^{3}(G, 2)_{ind} (see 3.2) we get

(7) Inv^{3}(G, 2)_{ind} ≃ \frac{m}{k}ℤq/mℤq, where k = \{ \begin{matrix}gcd(\frac{n}{m}, m), & if \frac{n}{m} is odd; \\ gcd(\frac{n}{2m}, m), & if \frac{n}{m} is even. \end{matrix}

Let Δ_{n,m} be a (unique) invariant in Inv^{3}(G, 2)_{norm} such that its class in Inv^{3}(G, 2)_{ind} corresponds to \frac{m}{k}q + mℤq and φ(Δ_{n,m}) = 0. Note that the order of Δ_{n,m} in Inv^{3}(G, 2)_{norm} is equal to k. Therefore, Δ_{n,m} takes values in H^{3}(−, ℤ/kℤ(2)) ⊂ H^{3}(−, 2).

Fix a G-torsor Y over F and consider the twists ^{Y}G and SL_{1}(A(Y)) by Y of the groups G and SL_{n} respectively. The group F^{×} acts transitively on the fiber over A(Y) of the map α. If ϕ ∈ F^{×}, we write ^{ϕ}Y for the corresponding element in the fiber. By (7) the image of Δ_{n,m} under the natural composition

Inv^{3}(G, 2)_{norm} ≃ Inv^{3}(^{Y}G, 2)_{norm} ⟶ Inv^{3}(SL_{1}(A(Y)), 2)_{norm}

is a \frac{m}{k}-multiple of the Rost invariant. Recall that the Rost invariant takes the class of ϕ in F^{×}/ Nrd(A(Y)^{×}) = H^{1}(F, SL_{1}(A(Y))) to the cup-product ϕ ∪ [A(Y)] ∈ H^{3}(F, 2). So we get

(8)  Δ_{n,m}(^{ϕ}Y) − Δ_{n,m}(Y) ∈ F^{×} ∪ \frac{m}{k}[A(Y)].

Given a central simple L-algebra A of degree n and exponent dividing m, we define an element

Δ_{n,m}(A) ∈ \frac{H^{3}(L,ℤ/kℤ(2))}{L^{×}∪ \frac{m}{k} [A]}

as follows. Choose a G-torsor Y over L with A(Y) ≃ A and set Δ_{n,m}(A) to be the class of Δ_{n,m}(Y) in the factor group. It follows from (8) that Δ_{n,m}(A) is independent of the choice of Y.

4.1. Proposition. Let A be a central simple L-algebra of degree n and exponent dividing m. Then the order of Δ_{n,m}(A) divides k. If A is a generic algebra, then the order of Δ_{n,m}(A) is equal to k.

Proof. If k′ is a proper divisor of k, then the multiple k′Δ_{n,m} is not decomposable. By our theorem k′Δ_{n,m} is not semi-decomposable and, hence, k′Δ_{n,m}(A) ≠ 0. □

4.2. Example. Let A be a central simple F -algebra of degree 2n divisible by 8 and exponent 2. Choose a symplectic involution σ on A. The group PGSp_{2n} is a subgroup of SL_{2n} /μ_{2}, hence, if char(F) ≠ 2, the restriction of the invariant Δ_{2n,2} on PGSp_{2n} is the invariant Δ_{2n}(A, σ) considered in subsection 4.1. It follows that Δ_{2n,2}(A) = Δ_{2n}(A) in the group H^{3}(F, ℤ/2ℤ)/(F^{×} ∪ [A]).

The class Δ_{n,m} is trivial on decomposable algebras:

4.3. Proposition. Let n_{1}, n_{2}, m be positive integers such that m divides n_{1} and n_{2}. Let A_{1} and A_{2} be two central simple algebras over F of degree n_{1} and n_{2} respectively and of exponent dividing m. Then Δ_{n_{1}n_{2},m}(A_{1} ⊗_{F} A_{2}) = 0.

Proof. The tensor product homomorphism SL_{n_{1}} × SL_{n_{2}} → SL_{n_{1}n_{2}} yields a homomorphism

Sym^{2}(T_{n_{1}n_{2}}^{∗}) → Sym^{2}(T_{n_{1}}^{∗}) ⊕ Sym^{2}(T_{n_{2}}^{∗}),

where T_{n_{1}}, T_{n_{2}} and T_{n_{1}n_{2}} are maximal tori of respective groups. The image of the canonical Weyl-invariant generator q_{n_{1}n_{2}} of Sym^{2}(T_{n_{1}n_{2}}^{∗}) is equal to n_{2}q_{n_{1}} + n_{1}q_{n_{2}}. Since n_{1} and n_{2} are divisible by m, the pull-back of the invariant Δ_{n_{1}n_{2},m} under the homomorphism (SL_{n_{1}} /μ_{m}) × (SL_{n_{2}} /μ_{m}) → SL_{n_{1}n_{2}} /μ_{m} is trivial. □

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Alexander Merkurjev, University of California at Los Angeles

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

Alexander Neshitov, Steklov Mathematical Institute, St.Petersburg / University of Ottawa

E-mail address: neshitov@yandex.ru

Kirill Zainoulline, University of Ottawa

E-mail address: kirill@uottawa.ca