UNRAMIFIED DEGREE THREE INVARIANTS OF REDUCTIVE GROUPS

A. MERKURJEV

Abstract. We prove that if G is a reductive group over an algebraically closed field F, then for a prime integer p ≠ char(F), the group of unramified Galois cohomology H_{nr}^{3}(F (BG), ℚ_{p}/ℤ_{p}(2)) is trivial for the classifying space BG of G if p is odd or the commutator subgroup of G is simple.

1. Introduction

The notion of a cohomological invariant of an algebraic group was introduced by J-P. Serre in [6]. Let G be an algebraic group over a field F and M a Galois module over F. A degree d invariant of G assigns to every G-torsor over a field extension K over F an element in the Galois cohomology group H^{d}(K, M), functorially in K. In this paper we consider the cohomology groups H^{d}(K) = H^{d}(K, ℚ/ℤ(d−1)), where ℚ/ℤ(d−1) is defined as the Galois module of (d−1)-twisted roots of unity. The p-part of this module requires special care if p = char(F) > 0. All degree d invariants of G form an abelian group Inv^{d}(G). An invariant is normalized if it takes a trivial torsor to the trivial cohomology class. The group Inv^{d}(G) is the direct sum of the subgroup Inv^{d}(G)_{norm} of normalized invariants and the subgroup of constant invariants isomorphic to H^{d}(F).

The group Inv^{d}(G)_{norm} for small values of d is well understood. The group Inv^{1}(G)_{norm} is trivial if G is connected. There is a canonical isomorphism Inv^{2}(G)_{norm} ≃ Pic(G) for every reductive group G (see [2, Theorem 2.4]). M. Rost proved (see [6, Part 2]) that if G is simple simply connected then the group Inv^{3}(G)_{norm} is cyclic of finite order with a canonical generator called the Rost invariant. The group Inv^{3}(G)_{norm} for an arbitrary semisimple group G was studied in [10].

For a prime integer p, write H^{d}(K, p) and Inv^{d}(G, p) for the p-primary components of H^{d}(K) and Inv^{d}(G) respectively. If v is a discrete valuation of a field extension K/F trivial on F with residue field F (v), then there is defined the residue homomorphism ∂_{v} : H^{d}(K, p) ⟶ H^{d−1}(F (v), p) for every p ≠ char(F). An element a ∈ H^{d}(K, p) is unramified with respect to v if ∂_{v}(a) = 0. We write H_{nr}^{d}(K, p) for the subgroup of all elements unramified with respect to every discrete valuation of K over F. An invariant in Inv^{d}(G, p) is called unramified if all values of the invariant over every K/F belongs to H_{nr}^{d}(K, p). We write Inv_{nr}^{d}(G, p) for the group of all unramified invariants.

Date: November, 2014.

Key words and phrases. Reductive algebraic group; classifying space: unramified cohomology.

The work has been supported by the NSF grant DMS #1160206 and the Guggenheim Fellowship.

Let V be a generically free representation of G. There is a nonempty Ginvariant open subscheme U ⊂ V and a versal G-torsor U ⟶ X for a variety X over F. We think of X as an approximation of the classifying space BG of G. The larger the codimension of V \ U in V the better X approximates BG. Abusing notation, we will write BG for X. Note that the stable birational type of BG is well defined.

The generic fiber of the versal G-torsor is the generic G-torsor over the function field F (BG) of the classifying space. A theorem of Rost and Totaro asserts that the evaluation at the generic G-torsor yields an isomorphism between Inv^{d}(G, p) and the subgroup of H^{d}(F (BG), p) of all elements unramified with respect to the discrete valuations associated with all irreducible divisors of BG. This isomorphism restricts to an isomorphism

Inv_{nr}^{d}(G, p) ⟶^{∼} H_{nr}^{d}(F (BG), p).

A classical question is whether the classifying space BG of an algebraic group G is stably rational. To disprove stable rationality of BG it suffices to show that the map H^{d}(F, p) ⟶ H_{nr}^{d}(F (BG), p) is not surjective for some d and p or, equivalently, to find a non-constant unramified invariant of G. For example, D. Saltman disproved in [14] the Noether Conjecture (that V/G is stably rational for a faithful representation V of a finite group G over an algebraically closed field) by proving that H_{nr}^{2}(F (BG), p) ≠ H^{2}(F, p) for some G and p, i.e., by establishing a non-constant degree 2 invariant of G. E. Peyre found new examples of finite groups with non-constant unramified degree 3 invariants in [12] . Degree 3 unramified invariants of simply connected groups (over arbitrary fields) were studied in [11] (classical groups) and [7] (exceptional groups).

It is still a wide open problem whether there exists a connected algebraic group G over an algebraically closed field F with the classifying space BG that is not stably rational. Connected groups have no non-trivial degree 1 invariants. F. Bogomolov proved in [3, Lemma 5.7] (see also [2, Theorem 5.10]) that connected groups have no non-trivial degree 2 unramified invariants. In [15] and [16], D. Saltman proved that the projective linear group PGL_{n} has no non-trivial degree 3 unramified invariants.

In the present paper, we study unramified degree 3 invariants of an arbitrary (connected) reductive group G over an algebraically closed field, or equivalently, the unramified elements in H^{3}(F (BG)). The language of invariants seems easier to work with. The main result is the following theorem (see Theorems 8.4 and 11.3):

Theorem. Let G be a split reductive group over an algebraically closed field F and p a prime integer different from char(F). Then

Inv_{nr}^{3}(G, p) = H_{nr}^{3}(F (BG), p) = 0

if p is odd or the commutator subgroup of G is (almost) simple.

Let H be the commutator subgroup of a split reductive group G. We have Inv_{nr}^{3}(G, p) = Inv_{nr}^{3}(H, p) (see Proposition 6.1). If H is a simple group, we compare the group Inv^{3}(H) with the group Inv^{3}(H̃^{gen}), where H̃^{gen} is the simply connected cover of H twisted by a generic H-torsor, and use our knowledge of the unramified degree 3 invariants in the simply connected case. The key statement is the injectivity of the homomorphism Inv^{3}(H) ⟶ Inv^{3}(H̃^{gen}) (see Section 8).

In general, when H is semisimple but not necessarily simple, we consider an embedding of H into a reductive group G′ as the commutator subgroup. Then Inv^{3}(G′) is identified with a subgroup of Inv^{3}(H). If G′ is strict, i.e., the center of G′ is a torus, this subgroup is the smallest possible and is independent of the choice of G′. We write Inv_{red}^{3}(H) for this subgroup. It satisfies

Inv_{nr}^{3}(H, p) ⊂ Inv_{red}^{3}(H, p) ⊂ Inv^{3}(H, p)

for every prime p ≠ char(F). The group Inv_{red}^{3}(H, p) is easier to control than Inv_{nr}^{3}(H, p). We show that Inv_{red}^{3}(H, p) = 0 which implies that Inv_{nr}^{3}(H, p) is also trivial. Acknowledgements. The author thanks J-P. Tignol for valuable remarks and the Max Planck Institute (Bonn) for hospitality.

2. Basic definitions and facts

Let F be a field. If d ≥ 1, we write H^{d}(F) for the Galois cohomology group H^{d}(F, ℚ/ℤ(d − 1)), with ℚ/ℤ(d − 1) the direct sum of colim_{n} μ_{n}^{⊗(d−1)}, where μ_{n} is the group of roots of unity of degree n, and the p-component if p = char(F) > 0 (see [6, Part 2, Appendix A]). In particular, H^{1}(F) is the group of (continuous) characters of the absolute Galois group Gal(F_{sep}/F) of F and H^{2}(F) is the Brauer group Br(F). We view H^{d} as a functor from the category Fields_{F} of field extensions of F to the category of abelian groups (or the category Sets of sets).

Let G be a (linear) algebraic group over a field F. The notion of an invariant of G was defined in [6] as follows. Consider the functor

Tors_{G} : Fields_{F} ⟶ Sets

taking a field K to the set Tors_{G}(K) := H^{1}(K, G) of isomorphism classes of (right) G-torsors over Spec K. A degree d cohomological invariant of G is then a morphism of functors

Tors_{G} ⟶ H^{d},

i.e., a functorial in K collection of maps of sets Tors_{G}(K) ⟶ H^{d}(K) for all field extensions K/F. We denote the group of such invariants by Inv^{d}(G).

An invariant I ∈ Inv^{d}(G) is called normalized if I(E) = 0 for a trivial Gtorsor E. The normalized invariants form a subgroup Inv^{d}(G)_{norm} of Inv^{d}(G) and there is a natural isomorphism

Inv^{d}(G) ≃ H^{d}(F) ⊕ Inv^{d}(G)_{norm}.

Example 2.1. Let G be a (connected) reductive group over F. It is shown in [2, Theorem 2.4] that there is an isomorphism

β_{G} : Pic(G) ⟶^{∼} Inv^{2}(G)_{norm}.

Let G be a split reductive group and H the commutator subgroup of G. Let π : H̃ ⟶ H be a simply connected cover with kernel C̃. There are canonical isomorphisms (see [17, §6])

(2.2)  Pic(G) ⟶^{∼} Pic(H) ≃ C̃^{∗} := Hom(C̃, 𝔾_{m}).

Take any character χ ∈ C̃^{∗} and consider the push-out diagram

[figure: 1; C̃; χ; H̃; π; H; 1; 1; _{𝔾_{m}}; H′; H; 1.]

The isomorphism C̃^{∗} ≃ Pic(H) takes a character χ to the class the line bundle L_{χ} on H given by the 𝔾_{m}-torsor H′ ⟶ H in the bottom row of the diagram. For a field extension K/F and an H-torsor E over K, the value of the invariant β_{H}(L_{χ}) is equal to δ([E]) ∈ H^{2}(K, 𝔾_{m}) = Br(K), where [E] is the class of E in H^{1}(K, H) and δ : H^{1}(K, H) ⟶ H^{2}(K, 𝔾_{m}) is the connecting map for the bottom exact sequence in the diagram.

If f : G_{1} ⟶ G_{2} is a homomorphism of algebraic groups over F and E_{1} is a G_{1}-torsor over a field extension K/F, then E_{2} := (E_{1} × G_{2})/G_{1} is the G_{2}-torsor over K which we denote by f_{∗}(E_{1}). If I is a degree d invariant of G_{2}, we define an invariant f^{∗}(I) of G_{1} by f^{∗}(I)(E_{1}) := I(f_{∗}(E_{1})). Thus, we have a group homomorphism

(2.3)  f^{∗} : Inv^{d}(G_{2}) ⟶ Inv^{d}(G_{1})

taking normalized invariants to the normalized ones.

Let G be an algebraic group over a field F and let V be a generically free representation of G. There is a nonempty G-invariant open subscheme U ⊂ V such that U is a G-torsor over a variety which we denote by U/G (see [18, Remark 1.4]). We think of U/G as an approximation of the “classifying space” BG of G and abusing notation write U/G = BG. The space BG is better approximated by U/G if the codimension of V \ U in V is large. For our purposes it suffices to assume that this codimension is at least 3 (see [2]).

Note that by the No-name Lemma, the stable rationality type of BG is uniquely determined by G.

The generic fiber E^{gen} ⟶ Spec(F (BG)) of the projection U ⟶ U/G is called the generic G-torsor. The value of an invariant of G at the generic torsor E^{gen} yields a homomorphism

Inv^{d}(G) ⟶ H^{d}(F (BG)).

Rost proved (see [6, Part 2, Th. 3.3] or [2, Theorem 2.2]) that this map is injective, i.e., every invariant is determined by its value at the generic torsor.

We decompose the group of invariants into a direct sum of primary components:

Inv^{d}(G) = ∐_{p prime} Inv^{d}(G, p).

Let K be a field extension of F. For a prime integer p, write H^{d}(K, p) for the p-primary component of H^{d}(K). Let v be a discrete valuation of K over F with residue field F (v). If char(F) ≠ p, there is the residue map (see [6, Chapter 2])

∂_{v} : H^{d}(K, p) ⟶ H^{d−1}(F (v), p).

An element a ∈ H^{d}(K, p) is unramified with respect to v if ∂_{v}(a) = 0.

A point x of codimension 1 in BG for an algebraic group G yields a discrete valuation v_{x} on the function field F (BG) over F. Write A^{0}(BG, H^{d}, p) for the group of all elements in H^{d}(F (BG), p) that are unramified with respect to v_{x} for all points x of codimension 1 in BG. It is proved in [6, Part 1, Theorem 11.7] that the value of every invariant from Inv^{d}(G, p) at the generic G-torsor E^{gen} belongs to A^{0}(BG, H^{d}, p). Moreover, we have the following theorem (see [6, Part 1, Appendix C]):

Theorem 2.4. Let G be an algebraic group over F and p a prime different from char(F). Then the evaluation of an invariant at the generic G-torsor yields an isomorphism

Inv^{d}(G, p) ⟶^{∼} A^{0}(BG, H^{d}, p).

The inverse isomorphism is defined as follows. Let E be a G-torsor over a field extension K/F and BG = U/G. We have the following canonical morphisms:

Spec K = E/G ⟵^{f} (E × U)/G ⟶^{h} U/G = BG.

Note that the groups H^{d}(K, p) for all d and all field extensions K/F form a cycle module in the sense of Rost (see [13]). In particular, we have flat pull-back homomorphisms

H^{d}(K, p) = A^{0}(Spec K, H^{d}, p) ⟶^{f^{∗}} A^{0}((E × U)/G, H^{d}, p) ⟵^{h^{∗}} A^{0}(BG, H^{d}, p).

The variety (E × U)/G is an open subscheme of the vector bundle (E × V)/G over Spec K. By the homotopy invariance property, the pull-back homomorphism

H^{d}(K, p) = A^{0}(Spec K, H^{d}, p) ⟶ A^{0}((E × V)/G, H^{d}, p)

is an isomorphism. Since the inclusion of (E × U)/G into (E × V)/G is a bijection on points of codimension 1 (by our assumption on the codimension of V \ U in V), the restriction homomorphism

A^{0}((E × V)/G, H^{d}, p) ⟶ A^{0}((E × U)/G, H^{d}, p)

is an isomorphism. It follows that f^{∗} is an isomorphism.

Let a ∈ A^{0}(BG, H^{d}, p). The invariant I ∈ Inv^{d}(G, p) defined by I(E) = (f^{∗})^{−1}h^{∗}(a) is the inverse image of a under the isomorphism in Theorem 2.4.

3. Decomposable invariants

The group of decomposable degree 3 invariants of a semisimple group was defined in [10, §1]. We extend this definition to the class of split reductive groups.

Let G be a split reductive group over F. The ∪-product H^{2}(K) ⊗ K^{×} ⟶ H^{3}(K) for any field extension K/F yields a pairing

Inv^{2}(G)_{norm} ⊗ F^{×} ⟶ Inv^{3}(G)_{norm}.

The subgroup of decomposable invariants Inv^{3}(G)_{dec} is the image of the pairing.

Proposition 3.1. Let G be a split reductive group over F. Then the composition

Pic(G) ⊗ F^{×} ⟶^{∼} Inv^{2}(G)_{norm} ⊗ F^{×} ⟶ Inv^{3}(G)_{dec}

is an isomorphism.

Proof. The surjectivity of the composition follows from the definition. Let H be the commutator subgroup of G. By [10, Theorem 4.2]), the composition is an isomorphism when G is replaced by H. The injectivity of the composition for G follows then from the fact that the map Pic(G) ⟶ Pic(H) in (2.2) is an isomorphism. □

It follows from the proposition that Inv^{3}(G)_{dec} = 0 if Pic(G) = 0 (for example, G is semisimple simply connected) or F is algebraically closed.

We write

Inv^{3}(G)_{ind} := Inv^{3}(G)_{norm}/ Inv^{3}(G)_{dec}.

4. Unramified invariants

Let K/F be a field extension and p a prime integer different from char(F). We write H_{nr}^{d}(K/F, p) for the subgroup of all elements in H^{d}(K, p) that are unramified with respect to all discrete valuations of K over F. A field extension L/K yields a natural homomorphism H^{d}(K) ⟶ H^{d}(L) that takes H_{nr}^{d}(K/F, p) into H_{nr}^{d}(L/F, p) by [6, Part 1, Proposition 8.2].

Let G be an algebraic group over F. An invariant I ∈ Inv^{d}(G, p) is called unramified if for every field extension K/F and every E ∈ Tors_{G}(K), we have I(E) ∈ H_{nr}^{d}(K/F, p). Note that the constant invariants are always unramified. We will write Inv_{nr}^{d}(G, p) for the subgroup of all unramified invariants in Inv^{d}(G, p).

If f : G_{1} ⟶ G_{2} is a group homomorphism, then the map f^{∗} in (2.3) takes Inv_{nr}^{d}(G_{2}, p) into Inv_{nr}^{d}(G_{1}, p).

Proposition 4.1. Let G be an algebraic group over F. An invariant I ∈ Inv^{d}(G, p) is unramified if and only if the value of I at the generic G-torsor in H^{d}(F (BG), p) is unramified. In particular, Inv_{nr}^{d}(G, p) ≃ H_{nr}^{d}(F (BG), p).

Proof. It suffices to show that the inverse of the isomorphism in Theorem 2.4 takes unramified elements to unramified invariants. Let a ∈ H_{nr}^{d}(F (BG), p) ⊂ A^{0}(BG, H^{d}, p). The corresponding invariant I ∈ Inv^{d}(G, p) is defined by I(E) = (f^{∗})^{−1}h^{∗}(a) (see Section 2). Note that h^{∗} takes unramified elements to unramified ones and f^{∗} yields an isomorphism on the unramified elements as the function field of (E × U)/G is a purely transcendental extension of K. It follows that I(E) is unramified for all E, hence the invariant I is unramified. □

Unramified invariants are constant along rational families of torsors. Precisely, if K/F is a purely transcendental field extension and E is a G-torsor over K, then for every invariant I ∈ Inv_{nr}^{d}(G, p) we have

I(E) ∈ Im(H^{d}(F, p) ⟶ H^{d}(K, p)).

Indeed, I(E) ∈ H_{nr}^{d}(K, p) which is the image of H^{d}(F, p) in H^{d}(K, p).

5. Abstract Chern classes

Let A be a lattice (written additively). Consider the symmetric ring S^{∗}(A) over ℤ and the group ring ℤ[A] of A. We use the exponential notation for ℤ[A]: every element can be written as a finite sum ∑_{a∈A} n_{a}e^{a} with n_{a} ∈ ℤ. There are the abstract Chern classes (see [10, 3c])

c_{i} : ℤ[A] ⟶ S^{i}(A), i ≥ 0

satisfying in particular,

c_{1}(∑_{i} e^{a_{i}}) = ∑_{i} a_{i} ∈ A and c_{2}(∑_{i} e^{a_{i}}) = ∑_{i<j} a_{i}a_{j} ∈ S^{2}(A).

The map c_{1} is a homomorphism and

c_{2}(x + y) = c_{2}(x) + c_{2}(y) + c_{1}(x)c_{1}(y)

for all x, y ∈ ℤ[A].

If A is a W -lattice for a group W acting on A, then all the c_{i}’s are W - equivariant. It follows that c_{2} yields a map (not a homomorphism in general) of groups of W -invariant elements:

c_{2}^{W} : ℤ[A]^{W} ⟶ S^{2}(A)^{W} .

The group ℤ[A]^{W} is generated by the elements ∑ e_{i}^{a}, where the a_{i}’s form a W -orbit in A. It follows that the subgroup of S^{2}(A)^{W} generated by the image of c_{2}^{W} is generated by ∑_{i<j} a_{i}a_{j} with the a_{i}’s forming a W -orbit in A and aa′ for a, a′ ∈ A^{W}. The elements of these two types can be viewed as “obvious” elements in S^{2}(A)^{W} which we call decomposable.

Write S^{2}(A)_{dec}^{W} for the subgroup of S^{2}(A)^{W} generated by the decomposable elements, or equivalently, by the image of c_{2}^{W}. Set

S^{2}(A)_{ind}^{W} := S^{2}(A)^{W} /S^{2}(A)_{dec}^{W}.

Note that if A^{W} = 0, the map c_{2}^{W} is a homomorphism and S^{2}(A)_{ind}^{W} is the cokernel of c_{2}^{W}.

Lemma 5.1. Let A_{1} and A_{2} be W_{1}- and W_{2}-lattices respectively. Then there is a canonical isomorphism

S^{2}(A_{1} ⊕ A_{2})_{ind}^{W_{1}×W_{2}} ≃ S^{2}(A_{1})_{ind}^{W_{1}} ⊕ S^{2}(A_{2})_{ind}^{W_{2}}.

Proof. We have

S^{2}(A_{1} ⊕ A_{2})^{W_{1}×W_{2}} ≃ S^{2}(A_{1})^{W_{1}} ⊕ S^{2}(A_{2})^{W_{2}} ⊕ (A_{1}^{W_{1}} ⊗ A_{2}^{W_{2}})

and

ℤ[A_{1} ⊕ A_{2}]^{W_{1}×W_{2}} ≃ ℤ[A_{1}]^{W_{1}} ⊗ ℤ[A_{2}]^{W_{2}}.

The standard formulas on the Chern classes show that c_{1}(ℤ[A_{i}]^{W_{i}}) = A_{i}^{W_{i}} and

S^{2}(A_{1} ⊕ A_{2})_{dec}^{W_{1}×W_{2}} ≃ S^{2}(A_{1})_{dec}^{W_{1}} ⊕ S^{2}(A_{2})_{dec}^{W_{2}} ⊕ (A_{1}^{W_{1}} ⊗ A_{2}^{W_{2}}),

whence the result. □

Lemma 5.2. Let 0 ⟶ A ⟶ B ⟶ C ⟶ 0 be an exact sequence of W - lattices. Suppose that W acts trivially on A and C^{W} = 0. Then

(1) The sequence

0 ⟶ S^{2}(A) ⟶ S^{2}(B)^{W} ⟶ S^{2}(C)^{W}

is exact.

(2) The natural homomorphism S^{2}(B)_{ind}^{W} ⟶ S^{2}(C)_{ind}^{W} is injective.

Proof. The first statement is proved in [5, Lemma 4.9]. Since W acts trivially on A, for every subgroup W′ ⊂ W, we have H^{1}(W′, A) = 0, hence the map B^{W′} ⟶ C^{W′} is surjective. The group ℤ[C]^{W} is generated by elements of the form ∑_{i} e^{c_{i}}, where the c_{i}’s form a W -orbit in C. By the surjectivity above, applied to the stabilizer W′ ⊂ W, this orbit can be lifted to a W - orbit in B. Therefore, the map ℤ[B]^{W} ⟶ ℤ[C]^{W} is surjective. The second statement follows from this, the first statement of the lemma and the fact that S^{2}(A) = S^{2}(A)_{dec}^{W} ⊂ S^{2}(B)_{dec}^{W}. □

6. Degree 3 invariants of split reductive groups

Let G be a split reductive group over F and let H be the commutator subgroup of G. Thus, H is a split semisimple group and the factor group Q := G/H is a split torus.

Proposition 6.1. 1. The restriction maps Inv^{d}(G) ⟶ Inv^{d}(H) and Inv^{d}(G)_{ind} ⟶ Inv^{d}(H)_{ind} are injective.

2. For every prime p ≠ char(F), the restriction map Inv_{nr}^{d}(G, p) ⟶ Inv_{nr}^{d}(H, p) is an isomorphism.

Proof. For a field extension K/F, the map

j : H^{1}(K, H) ⟶ H^{1}(K, G)

is surjective as H^{1}(K, Q) = 1 and the group Q(K) acts transitively on the fibers of j. It follows that the restriction map Inv^{d}(G) ⟶ Inv^{d}(H) is injective. The injectivity of Inv^{d}(G)_{ind} ⟶ Inv^{d}(H)_{ind} follows then from Proposition 3.1.

As Q is a rational variety, the fibers of j are rational families of H-torsors. Since an unramified invariant of H must be constant on the fibers, it defines an invariant of G. This proves the second statement. □

Let G be a split reductive group, T ⊂ G a split maximal torus. By [10, 3d], there is a commutative diagram

\xymatrix{0 \ar[r] & CH^{2}(BG) \ar[d] \ar[r] & \overline{H}_{ét}^{4,2}(BG) \ar[r] \ar[d] & Inv^{3}(G)_{norm} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & CH^{2}(BT) \ar[r] & \overline{H}_{ét}^{4,2}(BT) \ar[r] & Inv^{3}(T)_{norm} \ar[r] & 0}

with the exact rows, where \overline{H}_{ét}^{4,2}(BH) = \overline{H}^{4}(BH, ℤ(2)) for an algebraic group H is the reduced weight two étale motivic cohomology group (see [9, §5]). The group Inv^{3}(T)_{norm} is trivial as T has no nontrivial torsors and CH^{2}(BT) = S^{2}(T^{∗}) by [2, Example A.5], hence the middle term in the bottom row is isomorphic to S^{2}(T^{∗}).

Let N be the normalizer of T in G and W = N/T the Weyl group. The group W acts naturally on BT. Moreover, if w ∈ W, the composition

BT ⟶^{w} BT ⟶^{s} BG,

where s is the natural morphism, coincides with s. Therefore, the image of the middle vertical homomorphism in the diagram

\overline{H}_{ét}^{4,2}(BG) ⟶ \overline{H}_{ét}^{4,2}(BT) = S^{2}(T^{∗})

is contained in the subgroup S^{2}(T^{∗})^{W} of W -invariant elements. By [10, Lemma 3.8], the image of CH^{2}(BG) under this homomorphism is equal to S^{2}(T^{∗})_{dec}^{W}. Therefore, by diagram chase, we have a homomorphism Inv^{3}(G)_{norm} ⟶ S^{2}(T^{∗})_{ind}^{W}. The group of decomposable invariants Inv^{3}(G)_{dec} is in the kernel of this map since Inv^{3}(G)_{dec} vanishes over an algebraic closure of F and the group S^{2}(T^{∗})_{ind}^{W} does not change. Therefore, we have a well-defined homomorphism

α_{G} : Inv^{3}(G)_{ind} ⟶ S^{2}(T^{∗})_{ind}^{W}.

Theorem 6.2. Let G be a split reductive group over F. Then the map α_{G} is injective. If G is semisimple, then α_{G} is an isomorphism.

Proof. The second statement is proved in [10, Theorem 3.9]. The first statement follows from Proposition 6.1(1), the commutativity of the diagram

\xymatrix{Inv^{3}(G)_{ind} \ar[r]^{α_{G}} \ar[d] & S^{2}(T^{∗})_{ind}^{W} \ar[d] \\ Inv^{3}(H)_{ind} \ar[r]^{α_{H}} & S^{2}(S^{∗})_{ind}^{W},}

where H is the commutator subgroup of G and S is a maximal torus of H, and the second statement applied to H. □

Proposition 3.1 and Lemma 5.1 yield the following additivity property.

Corollary 6.3. Let H_{1} and H_{2} be two split semisimple groups. Then there is a canonical isomorphism

Inv^{3}(H_{1} × H_{2}) ≃ Inv^{3}(H_{1}) ⊕ Inv^{3}(H_{2}).

Let H be a split semisimple group over a field F, π : H̃ ⟶ H a simply connected cover, S̃ the pre-image of a split maximal torus S of H, so S̃ is a split maximal torus of H̃. Then S^{2}(S^{∗}) can be viewed with respect to π as a sublattice of S^{2}(S̃^{∗}) of finite index and we have the following commutative diagram

[figure: Inv^{3}(H)_{ind}; α_{H}; ∼; π^{∗}; S^{2}(S^{∗})_{ind}^{W}; Inv^{3}(H̃)_{norm}; Inv^{3}(H̃)_{ind}; αH̃; ∼; S^{2}(S̃^{∗})_{ind}^{W}.]

If H is simple, the group S^{2}(S̃^{∗})^{W} is infinite cyclic with a canonical generator q (see [6, Part 2, §7]). It follows that S^{2}(S^{∗})^{W} is also infinite cyclic with kq a generator for a unique integer k > 0. The invariant R ∈ Inv^{3}(H̃)_{norm} corresponding to the generator q is called the Rost invariant of H̃. It is a generator of the cyclic group Inv^{3}(H̃).

7. Change of groups

In this section we prove the following useful property.

Proposition 7.1. Let p be a prime integer different from char(F), G an algebraic group over F, C ⊂ G a finite central diagonalizable subgroup of order not divisible by p, H = G/C. Then the natural maps Inv^{d}(H, p) ⟶ Inv^{d}(G, p) and Inv_{nr}^{d}(H, p) ⟶ Inv_{nr}^{d}(G, p) are isomorphisms.

Proof. Both functors in the definition of an invariant can be naturally extended to the category C of F -algebras that are finite product of fields, and every invariant extends uniquely to a morphism of extended functors. If K ⟶ L is a morphism in C and M is an étale K-algebra, then L ⊗_{K} M is also an object of the category C.

For any K in C we have an exact sequence

H_{ét}^{1}(K, G) ⟶ H_{ét}^{1}(K, H) ⟶^{δ_{K}} H_{ét}^{2}(K, C)

and the group H_{ét}^{1}(K, C) acts transitively on the fibers of the first map in the sequence.

Proof of injectivity. Let I ∈ Inv^{d}(H, p) be such that f^{∗}(I) = 0, where f : G ⟶ H is the canonical homomorphism. We prove that I = 0. Take any K in C and E ∈ Tors_{H}(K). As an element of the group H_{ét}^{2}(K, C) is a tuple of elements in Br(K) of order prime to p, there is an étale K-algebra L of (constant) finite rank [L : K] prime to p such that δ_{L}(E_{L}) = 0. It follows that E_{L} = f_{∗}(E′) for some E′ ∈ Tors_{G}(L). We have

I(E)_{L} = I(E_{L}) = I(f_{∗}(E′)) = f^{∗}(I)(E′) = 0.

Since [L : K] is prime to p, we have I(E) = 0, i.e., I = 0.

Proof of surjectivity. Let J ∈ Inv^{d}(G, p). We construct an invariant I ∈ Inv^{d}(H, p) such that J = f^{∗}(I). Take any K in C and E ∈ Tors_{H}(K). As above, choose an étale K-algebra L of finite rank prime to p such that δ_{L}(E_{L}) = 0 and an element E′ ∈ Tors_{G}(L) with E_{L} = f_{∗}(E′). We set

I(E) = \frac{1}{[L : K]} cor_{L/F} (J(E′)).

This is independent of the choice of E′. Indeed, if E_{L} = f_{∗}(E′′) for E′′ ∈ Tors_{G}(L), then there exists ν ∈ H_{ét}^{1}(L, C) with E′′ = ν(E′). Choose an Lalgebra P of constant rank [P : L] prime to p such that ν_{P} = 1. It follows that E′′_{P} = E′_{P} and therefore,

[P : L] cor_{L/F} (J(E′′)) = cor_{P/F} (J(E′′_{P})) = cor_{P/F} (J(E′_{P})) = [P : L] cor_{L/F} (J(E′)).

Since [P : L] is prime to p, we have cor_{L/F} (J(E′′)) = cor_{L/F} (J(E′)).

In order to show that the value I(E) is independent of the choice of L, for the two choices L and L′, it suffices to compare the formulas for L and LL′ := L ⊗_{F} L′:

\frac{1}{[L : K]} cor_{L/F} (J(E′)) = \frac{[L′ : K]}{[LL′ : K]} cor_{L/F} (J(E′)) = \frac{1}{[LL′ : K]} cor_{LL′/F} (J(E′_{LL′})).

We have constructed the invariant I ∈ Inv^{d}(H, p). For any K in C and E′ ∈ Tors_{G}(K), by the definition of I, we have f^{∗}(I)(E′) = I(f_{∗}(E)) = J(E′), hence f^{∗}(I) = J. Note that if J is an unramified invariant, I is also unramified since the corestriction map preserves unramified elements by [6, Part 1, Proposition 8.6]. □

8. Degree 3 unramified invariants of simple groups

The following statement was proved in [11] (classical groups) and [7] (exceptional groups).

Proposition 8.1. Let H be an absolutely simple simply connected group over F and p a prime different from char(F).

1. If the Dynkin diagram of H is different from ^{2}A_{n}, n odd, and ^{1}D_{4}, then Inv_{nr}^{3}(H, p)_{norm} = 0.

2. If H is split, then Inv_{nr}^{3}(H, p)_{norm} = 0.

Let H be a semisimple group over F, E an H-torsor over Spec(K) for a field extension K/F. The twist H^{E} := Aut_{H}(E) of H by E is a semisimple group over K. The twisting argument shows that BH^{E} = BH_{K} and there is a canonical isomorphism Inv^{d}(H^{E}) ≃ Inv^{d}(H_{K}). If E^{gen} is a generic H-torsor, we write H^{gen} for H^{E^{gen}}. Let H̃^{gen} ⟶ H^{gen} be a simply connected cover.

Proposition 8.2. Let H be a split simple group. Then the composition

Inv^{3}(H)_{ind} ⟶ Inv^{3}(H^{gen})_{ind} ⟶ Inv^{3}(H̃^{gen})_{ind} = Inv^{3}(H̃^{gen})

is injective.

Proof. The statement is clear if H is a simply connected group. The case of an adjoint group H was considered in [10, Theorem 4.10]. Consider the other split semisimple groups type-by-type. It suffices to restrict to the p-component of Inv^{3}(H) for a prime p.

Type A_{n−1}, n ≥ 2. We have H = SL_{n} /μ_{m} for an integer m dividing n. By Proposition 7.1, we may assume that m = p^{r} for some r. It is shown in [1, Theorem 4.1] and Theorem 6.2 that

Inv^{3}(H)_{ind} ⟶^{∼} S^{2}(S^{∗})_{ind}^{W} ↪ (ℤ/mℤ)q.

On the other hand, an H-torsor yields a central simple algebra of degree n and exponent dividing m. A generic torsor gives an algebra with the exponent exactly m, hence Inv^{3}(H̃^{gen}) = (ℤ/mℤ)R by [6, Part 2, Theorem 11.5].

Type D_{n}, n ≥ 4. We have H = O_{2n}^{+}, the special orthogonal group or H = HSpin_{2n}, the half-spin group if n is even. It is shown in [6, Part 1,Chapter VI] in the case char(F) ≠ 2 that Inv^{3}(O_{2n}^{+})_{ind} = 0. In general, recall that the character group of a maximal split torus S is a free group of rank n. Let x_{1}, x_{2}, …, x_{n} be a basis for S^{∗} such that the Weyl group W acts on the x_{i}’s by permutations and change of signs. The generator of S^{2}(S^{∗})^{W} is the quadratic form q = x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n}^{2}. It is in S^{2}(S^{∗})_{dec}^{W} since c_{2}(∑_{i} e^{x_{i}} + e^{−x_{i}}) = −q. By [10, Theorem 3.9], Inv^{3}(O_{2n}^{+})_{ind} = 0.

Finally, assume that n is even and H = HSpin_{2n}, the half-spin group. It follows from [1, Theorem 5.1] and Theorem 6.2 that

Inv^{3}(H)_{ind} ⟶^{∼} S^{2}(S^{∗})_{ind}^{W} ↪ (ℤ/4ℤ)q

and Inv^{3}(H)_{ind} = 0 if n = 4. On the other hand, an H-torsor yields a central simple algebra of degree 2n. A generic torsor gives a nonsplit algebra. By [6, Part 2, Theorem 15.4], Inv^{3}(H̃^{gen}) = (ℤ/4ℤ)R if n > 4. □

Remark 8.3. The statement fails for semisimple groups that are not simple, see Example 11.2.

Theorem 8.4. Let H be a split simple group over an algebraically closed field F and p a prime integer different from char(F). Then Inv_{nr}^{3}(H, p) = 0.

Proof. Let I ∈ Inv_{nr}^{3}(H, p). Note that since F is algebraically closed, every decomposable invariant is trivial.

The pull-back Ĩ of I under the composition in Proposition 8.2 is an unramified invariant. As H̃^{gen} is an inner form of H̃, by Proposition 8.1, Ĩ = 0 and hence I = 0 by Proposition 8.2 unless the Dynkin diagram of H is D_{4}.

If H is a simply connected group of type D_{4}, then I = 0 by Proposition 8.1. If H is a half-spin group of type D_{4}, then I = 0 by [1, Theorem 5.1]. Finally assume that H is an adjoint group of type D_{4}. By [10, Theorem 4.7], the group Inv^{3}(H) is cyclic of order 2.

Assume that I ≠ 0. The group H̃^{gen} is the spinor group of a central simple algebra A of degree 8 with and orthogonal involution σ of trivial discriminant. Consider the corresponding special orthogonal group Ĥ^{gen} := O^{+}(A, σ) of (A, σ). An Ĥ^{gen}-torsor over a field K is given by a pair (a, x), where a is an invertible σ-symmetric element in A and x ∈ K^{×} such that Nrd(a) = x^{2} and Nrd is the reduced norm map (see [8, 29.27]).

The canonical homomorphism Inv^{3}(H^{gen}) ⟶ Inv^{3}(H̃^{gen}) factors through Inv^{3}(Ĥ^{gen}). By [10, §4, type D_{n}], the pull-back of I in Inv^{3}(Ĥ^{gen}) is the class of the invariant taking a pair (a, x) to the cup-product (x) ∪ [A] ∈ H^{3}(K). This invariant is ramified as it is non-constant when a runs over a subfield of A of dimension n fixed by σ element-wise, a contradiction. □

9. Structure of reductive groups

Let H be a split semisimple group over a field F, S ⊂ H a split maximal torus. Write Λ_{r} ⊂ S^{∗} for the root lattice of H. Let H̃ ⟶ H be a simply connected cover and let S̃ for the inverse image of S, a maximal torus in H̃. Write Λ_{w} for the character group of S̃. This is the weight lattice freely generated by the fundamental weights. We have

Λ_{r} ⊂ S^{∗} ⊂ Λ_{w}.

The center C of H is a finite diagonalizable group with C^{∗} = S^{∗}/Λ_{r}.

Let G be a split reductive group over a field F with the commutator subgroup H. Choose a split maximal T ⊂ G such that T ∩ H = S. The roots of H can be uniquely lifted to T^{∗} (to the roots of G), so the inclusion of Λ_{r} into S^{∗} is lifted to the inclusion of Λ_{r} into T^{∗}. The composition S̃ ⟶ S ⟶ T yields a homomorphism T^{∗} ⟶ Λ_{w} of lattices. Thus, we have the two homomorphisms

(9.1)  Λ_{r} ↪ T^{∗} ⟶^{f} Λ_{w}

with the composition the canonical embedding of Λ_{r} into Λ_{w}. The image of f in (9.1) is equal to S^{∗}. The center Z of G is a diagonalizable group with Z^{∗} = T^{∗}/Λ_{r}. The factor group G/H = T/S is a torus Q with the character lattice Q^{∗} = Ker(f).

We would like to study all split reductive groups with the fixed commutator subgroup H.

Let H be a split semisimple group over F. Fix a split maximal torus S ⊂ H and consider the root system of H relative to S with the root and weight lattices Λ_{r} ⊂ Λ_{w} respectively.

Consider a category Red(H) with objects split reductive groups G over F with the commutator subgroup H. A morphism between G_{1} and G_{2} in this category is a group homomorphism G_{1} ⟶ G_{2} over F that is the identity on H.

Consider another category Lat(H) with objects the diagrams of the form

(9.2)  Λ_{r} ⟶ A ⟶^{f} Λ_{w},

where A is a lattice, Im(f) = S^{∗} and the composition is the embedding of Λ_{r} into Λ_{w}. A morphism in Lat(R) is a morphism between the diagrams which is identity on Λ_{r} and Λ_{w}.

Let G be an object in Red(H). Write Z for the center of G. Then T := S ·Z is a split maximal torus of G. The diagram (9.1) yields then a contravariant functor

ρ : Red(H) ⟶ Lat(H).

Proposition 9.3. For every split semisimple group H, the functor ρ is an equivalence of categories Red(H) and Lat(H)^{op}.

Proof. We construct a functor ε : Lat(H) ⟶ Red(H) as follows. Given the diagram (9.2), let T be a split torus with T^{∗} = A and Z a diagonalizable subgroup of T with Z^{∗} = A/Λ_{r}. We view the torus S as a subgroup of T via the dual surjective homomorphism A ⟶ Im(f) = S^{∗}.

We embed the center C of H into Z via a homomorphism dual to the surjective composition

Z^{∗} = A/Λ_{r} ⟶ Im(f)/Λ_{r} = S^{∗}/Λ_{r} = C^{∗}.

The sequence

0 ⟶ A ⟶^{g} S^{∗} ⊕ (A/Λ_{r}) ⟶^{h} S^{∗}/Λ_{r} ⟶ 0,

where g(a) = (f(a), a + Λ_{r}) and h(x, a + Λ_{r}) = (x − f(a)) + Λ_{r} is exact. It follows that the product homomorphism S × Z ⟶ T is surjective with the kernel C embedded into S × Z via c ↦ (c, c^{−1}), i.e., T ≃ (S × Z)/C.

We set G = (H × Z)/C. The group Z is naturally a subgroup of G which coincides with the center of G. The torus T is a subgroup of G generated by S and Z, hence T is a split maximal torus of G. The natural sequence

0 ⟶ Ker(f) ⟶ A/Λ_{r} ⟶ Im(f)/Λ_{r} ⟶ 0

is exact. It follows that Z/C is a torus dual to Ker(f). Since G/H ≃ Z/C, G is a (smooth connected) reductive group with H the commutator subgroup. The functor ε, by definition, takes the diagram (9.2) to the group G. By construction, both compositions of ρ and ε are isomorphic to the identity functors. □

Let H be a split semisimple group as above. We consider another category Mor(H) with objects homomorphisms h : B ⟶ Λ_{w}/Λ_{r} with B a finitely generated abelian group, Im(h) = S^{∗}/Λ_{r} and torsion free Ker(h). Morphisms are defined in the obvious way. Consider a contravariant functor

ν : Red(H) ⟶ Mor(H)

taking a reductive group G to the composition Z^{∗} ⟶ C^{∗} ↪ Λ_{w}/Λ_{r}, where Z is the center of G. The kernel of this homomorphism is the character lattice of the torus Z/C = G/H and hence has no torsion.

Proposition 9.4. For every split semisimple group H, the functor ν is an equivalence of categories Red(H) and Mor(H)^{op}.

Proof. We construct a functor λ : Mor(H) ⟶ Red(H) as follows. Let h : B ⟶ Λ_{w}/Λ_{r} be an object in Mor(H) and Z a diagonalizable group with Z^{∗} = B. The map h yields an embedding of C into Z and the factor group Z/C is a torus. Set G = (H × Z)/C as in the proof of Proposition 9.3. The factor group G/H is isomorphic to the torus Z/C, hence G is a reductive group with the commutator subgroup H, i.e., G is an object of Red(H). Then Z is the center of G as the group G/Z ≃ H/C is adjoint. We set λ(h) = G. By construction, both compositions of ρ and λ are isomorphic to the identity functors. □

Remark 9.5. It follows from Propositions 9.3 and 9.4 that the categories Lat(H) and Mor(H) are equivalent. An equivalence between the categories can be described directly as follows. If Λ_{r} ⟶ A ⟶^{f} Λ_{w} is an object in Lat(H), then the induced morphism A/Λ_{r} ⟶ Λ_{w}/Λ_{r} is the corresponding object in Mor(H). Conversely, let μ : B ⟶ Λ_{w}/Λ_{r} be an object in Mor(H). Write A for the kernel of the homomorphism

h : S^{∗} ⊕ B ⟶ S^{∗}/Λ_{r}

defined by h(x, b) = (x + Λ_{r}) − μ(b). The corresponding object

Λ_{r} ⟶ A ⟶^{f} Λ_{w}

in Lat(H) is defined as follows. The map f is given by the first projection followed by the inclusion of S^{∗} into Λ_{w} and the inclusion Λ_{r} ⟶ A takes x to (x, 0). Note that W acts on S^{∗} ⊕ B naturally on S^{∗} and trivially on B.

A split reductive group G is called strict if the center Z of G is a torus, i.e., Z^{∗} is a lattice. An object G of Red(H) is strict if G is strict. If B ⟶ Λ_{w}/Λ_{r} is the object ν(G) of Mor(H), then G is strict if and only if B is torsion-free.

A semisimple group is strict if and only if it is adjoint. A strict envelope of a split semisimple group H is a strict object in Red(H).

Example 9.6. The group GL_{n} is a strict envelope of SL_{n}.

Example 9.7. The object G in Red(H) corresponding to the composition S^{∗} ⟶ S^{∗}/Λ_{r} ↪ Λ_{w}/Λ_{r}, viewed as an object of the category Mor(H), is strict. We call such G the standard strict envelope of H. By Remark 9.5, the lattice T^{∗} is the subgroup in S^{∗} ⊕ S^{∗} consisting of all pairs (x, y) such that x − y ∈ Λ_{r}. Note that the Weyl group acts naturally on the first component of S^{∗} ⊕ S^{∗} and trivially on the second.

A strict envelope of H behaves like an “injective resolution” of H.

Lemma 9.8. Let G_{1} and G_{2} be two objects in Red(H). If G_{2} is strict, then there is a morphism G_{1} ⟶ G_{2} in Red(H).

Proof. Let h_{i} : B_{i} ⟶ Λ_{w}/Λ_{r} be the object ν(G_{i}) in Mor(H) for i = 1, 2. By assumption, B_{2} is a free ℤ-module. Therefore, there is a group homomorphism g : B_{2} ⟶ B_{1} such that g ∘ h_{1} = h_{2}, i.e., g is a morphism in Mor(H). By Proposition 9.4, there is a morphism G_{1} ⟶ G_{2} in Red(H) corresponding to g. □

10. Reductive invariants

Let H be a split semisimple group and G is a reductive group with the commutator subgroup H, i.e., G is an object in Red(H). By Proposition 6.1, the map Inv^{d}(G) ⟶ Inv^{d}(H) is injective. We view Inv^{d}(G) as a subgroup of Inv^{d}(H). If G′ is a strict envelope of H, then it follows from Lemma 9.8 that Inv^{d}(G′) ⊂ Inv^{d}(G). Therefore, the subgroup Inv^{d}(G′) is independent of the choice of the strict resolution G′ of G. We write Inv_{red}^{d}(H) for this subgroup and call the invariants in this group the reductive invariants. By Proposition 6.1, for any prime p ≠ char(F) we have

(10.1)  Inv_{nr}^{d}(H, p) ⊂ Inv_{red}^{d}(H, p) ⊂ Inv^{d}(H, p).

Let A be a lattice and q ∈ S^{2}(A). We can view q as an integral quadratic form on the lattice Â dual to A. The polar bilinear form h of q is the image of q under the polar map pol : S^{2}(A) ⟶ A ⊗ A, aa′ ↦ a ⊗ a′ + a′ ⊗ a. The polar form h is symmetric and even, i.e., h(x, x) ∈ 2ℤ for all x ∈ Â. Conversely, if h ∈ A ⊗ A is a symmetric even bilinear form, then q(x) = \frac{1}{2}h(x, x) is an integral quadratic form with the polar form h.

Let {α_{1}, α_{2} … α_{n}} be a set of simple roots of an irreducible root system, {w_{1}, w_{2}, …, w_{n}} the corresponding fundamental weights generating the weight lattice Λ_{w} and W the Weyl group. Let d_{i} be the square of the length of the co-root α_{i}^{∨}. (We assume that the length of the shortest co-root is 1.) Consider the bilinear form

h = ∑_{i=1}^{n} w_{i} ⊗ d_{i}α_{i} = ∑_{i,j} w_{i} ⊗ d_{i}c_{ij}w_{j} ∈ Λ_{w} ⊗ Λ_{w},

where (c_{ij}) is the Cartan matrix (see [4, Chapitre VI]). The matrix (d_{i}c_{ij}) is symmetric with even diagonal terms, hence h is a symmetric even bilinear form. The corresponding quadratic form

q = \frac{1}{2} ∑_{i=1}^{n} d_{i}w_{i}α_{i} ∈ S^{2}(Λ_{w})

is W -invariant by [10, Lemma 3.2]. It follows that the polar form h of q is also W -invariant.

Consider the three embeddings i_{1}, i_{2}, j = i_{1} + i_{2} : Λ_{w} ⟶ Λ_{w}^{2} := Λ_{w} ⊕ Λ_{w} given by x ↦ (x, 0), (0, x), (x, x) respectively, and the two quadratic forms q^{(1)}, q^{(2)} that are the images of q under the maps S^{2}(i_{1}), S^{2}(i_{2}) : S^{2}(Λ_{w}) ⟶ S^{2}(Λ_{w}^{2}) respectively. We let W act on Λ_{w}^{2} naturally on the first summand and trivially on the second.

Let A be the sublattice of Λ_{w}^{2} of all pairs (x, y) such that x − y ∈ Λ_{r}. Note that Im(j) ⊂ A. In particular, S^{2}(j)(q) ∈ S^{2}(A). Moreover, since h ∈ (Λ_{r} ⊗ Λ_{w}) ∩ (Λ_{w} ⊗ Λ_{r}) by [9, Lemma 2.1], we have (i_{k} ⊗ j)(h) ∈ A ⊗ A and (j ⊗ i_{k})(h) ∈ A ⊗ A for k = 1, 2.

Write m : Λ_{w}^{2} ⊗ Λ_{w}^{2} ⟶ S^{2}(Λ_{w}^{2}) for the canonical homomorphism. We have m(i_{k} ⊗ j)(h) ∈ S^{2}(A) and m(j ⊗ i_{k})(h) ∈ S^{2}(A) for k = 1, 2.

Proposition 10.2. We have q^{(1)} − q^{(2)} ∈ S^{2}(A)^{W} with the polar form h^{(1)} − h^{(2)} = (j ⊗ i_{1})(h) − (i_{2} ⊗ j)(h) ∈ A ⊗ A.

Proof. By construction, q^{(1)} − q^{(2)} is W -invariant. We have

q^{(1)} − q^{(2)} = (q^{(1)} + q^{(2)}) − 2q^{(2)}
= q^{(1)} + q^{(2)} − m(i_{2} ⊗ i_{2})(h)
= q^{(1)} + q^{(2)} + m(i_{1} ⊗ i_{2})(h) − m(j ⊗ i_{2})(h)
= S^{2}(j)(q) − m(j ⊗ i_{2})(h) ∈ S^{2}(A).

The second statement follows from the equality j = i_{1} + i_{2}. □

Corollary 10.3. The image of h^{(1)} − h^{(2)} under the map

A ⊗ A ⟶^{p_{1}⊗1} Λ_{w} ⊗ A,

where p_{1} is the first projection, coincides with the image of h under the natural map

Λ_{w} ⊗ Λ_{r} ⟶^{1⊗i_{1}} Λ_{w} ⊗ A.

Proof. The statement follows from Proposition 10.2 and the equalities p_{1} ∘ j = p_{1} ∘ i_{1} = 1 and p_{1} ∘ i_{2} = 0. □

Let H̃ be a split simply connected cover of H with a split maximal torus S̃, thus S̃^{∗} = Λ_{w}. Consider the standard strict envelope G̃ of H̃ (see Example 9.7). The character group T̃^{∗} of the maximal torus T̃ of G̃ coincides with the group A as above. If H̃ is simple, by Proposition 9.7, q̄:= q^{(1)} −q^{(2)} ∈ S^{2}(T̃^{∗})^{W}. The form q̄ maps to q under the natural map S^{2}(T̃^{∗})^{W} ⟶ S^{2}(S̃^{∗})^{W} = S^{2}(Λ_{w})^{W}.

In the general case,

H̃ = H̃_{1} × H̃_{2} × ⋯ × H̃_{s},

with H̃_{j} the simple simply connected components of H̃. The components define a basis q_{1}, q_{2}, …, q_{s} of S^{2}(H̃^{∗})^{W}. Every q_{j} has a lift q̄_{j} ∈ S^{2}(T̃^{∗})^{W} as above. Lemma 5.2 then yields the following statement.

Corollary 10.4. The map S^{2}(T̃^{∗})^{W} ⟶ S^{2}(S̃^{∗})^{W} is surjective and S^{2}(T̃^{∗})_{ind}^{W} ⟶ S^{2}(S̃^{∗})_{ind}^{W} is an isomorphism. In particular, S^{2}(T̃^{∗})_{ind}^{W} is generated by the classes of the forms q̄_{j}.

We will write α_{ij} for the simple roots of the j-th component and w_{ij} for the corresponding fundamental weights, etc.

Let C̃ ⊂ H̃ be a central subgroup and set G := G̃/C̃ and T := T̃/C̃. The character group C̃^{∗} is a factor group of Λ_{w}/Λ_{r}. Consider the composition

(10.5)  S^{2}(T̃^{∗}) ⟶^{pol} T̃^{∗} ⊗ T̃^{∗} ⟶^{p_{1}} Λ_{w} ⊗ T̃^{∗} ⟶ C̃^{∗} ⊗ T̃^{∗}.

Note that S^{2}(T^{∗}) is contained in the kernel of the composition.

By Corollary 10.3, the image of q_{j} under this composition is equal to

∑_{i,j} d_{ij}\overline{w}_{ij} ⊗ (α_{ij}, 0),

where \overline{x} denotes the image of an x ∈ T̃^{∗} in C̃^{∗}.

Let T̃_{j} ⊂ T̃ be a maximal torus of of the j-th simple component of G̃, so that T̃ = T̃_{1} × ⋯ ×T̃_{s}. Let C̃_{j} be the image of the projection C̃ ⟶ T̃_{j}. Then C̃_{j}^{∗} can be viewed as a subgroup of C̃^{∗} and \overline{w}_{ij} ∈ C̃_{j}^{∗}.

Proposition 10.6. Let q := ∑_{j=1}^{s} k_{j}q̄_{j} ∈ S^{2}(T̃) be a linear combination with integer coefficients k_{j}. If q has trivial image under the composition (10.5) (for example, if q ∈ S^{2}(T^{∗})), then the order of \overline{w}_{ij} in C̃_{j}^{∗} divides k_{j}d_{ij} for all i and j.

Proof. We have ∑_{i,j} k_{j}d_{ij}\overline{w}_{ij} ⊗ (α_{ij}, 0) = 0 in C̃^{∗} ⊗ T̃^{∗}. Note that the elements (α_{ij}, 0) form part of a basis of T̃^{∗} (with the complement (w_{ij}, w_{ij})). It follows that k_{j}d_{ij}\overline{w}_{ij} = 0 in C̃_{j}^{∗} for all i and j, whence the result. □

11. Degree 3 unramified invariants of reductive groups

We assume that the base field F is algebraically closed.

Proposition 11.1. Let H be a (split) semisimple group over F with the components of the Dynkin diagram of types A_{m} for some m or E_{6}. Suppose that E_{6}^{sc} does not split off H as a direct factor. Then Inv_{red}^{3}(H, p) = Inv_{nr}^{3}(H, p) = 0 for all odd primes p ≠ char(F).

Proof. Let H̃ ⟶ H be a simply connected cover with kernel C̃ and G̃ be the standard strict envelope of H̃. By Proposition 7.1, replacing C̃ if necessary, we may assume that C̃^{∗} is a p-group. Set G := G̃/C̃. We choose split maximal tori S ⊂ H, S̃ ⊂ H̃, T ⊂ G, T̃ ⊂ G̃ as in Section 10. The group Q := G/H = G̃/H̃ is a torus.

By Proposition 6.1, it suffices to prove that Inv^{3}(G, p) = 0. By Theorem 6.2, we are reduced to proving that S^{2}(T^{∗})_{ind}^{W}{p} = 0.

By Lemma 5.2(1) and Corollary 10.4, the rows of the diagram

[figure: 0; S^{2}(Q^{∗}); S^{2}(T^{∗})^{W}; S^{2}(S^{∗})^{W}; 0; S^{2}(Q^{∗}); S^{2}(T̃^{∗})^{W}; S^{2}(S̃^{∗})^{W}; 0]

are exact.

Let α ∈ S^{2}(T^{∗})_{ind}^{W}{p}. Since p is odd, it sufficient to show that 2α = 0. The element α lifts to a form q ∈ S^{2}(T^{∗})^{W}. Recall that S^{2}(S̃^{∗})^{W} is a free abelian group with basis {q_{j}}. Hence the image of q in S^{2}(S̃^{∗})^{W} is equal to ∑_{j=1}^{s} k_{j}q_{j} for some k_{j} ∈ ℤ. Write q̄ for ∑_{j=1}^{s} k_{j}q̄_{j} ∈ S^{2}(T̃^{∗})^{W}. Therefore, in S^{2}(T̃^{∗})^{W} we have q = q̄ + t for some t ∈ S^{2}(Q^{∗}).

Note that since the Dynkin diagram of H is simply laced all the integer d_{ij} are equal to 1 for all i and j. The images of q and t are trivial under (10.5), hence so is q̄. By Proposition 10.6, the order of \overline{w}_{ij} in C̃_{j}^{∗} divides k_{j} for all i and j.

We claim that the class of 2k_{j}q_{j} is contained in S^{2}(S^{∗})_{dec}^{W} for all j.

Case 1: The j-th simple component G̃_{j} is of type A_{m} for some m, i.e, H̃_{j} = SL_{m+1}. The center of H̃_{j} is μ_{m+1}, hence C̃_{j} = μ_{p^{r}} for some r. The element \overline{w}_{1j} is a generator of C̃^{∗} = ℤ/p^{r}ℤ, hence the order of \overline{w}_{1j} is equal to p^{r}. Therefore, k_{j} is divisible by p^{k}. As p is odd, by [1, 4.2], the form p^{k}q_{j} and hence k_{j}q_{j} belongs to S^{2}(S_{j}^{∗})_{dec}^{W_{j}}. Taking the image of k_{j}q_{j} under the homomorphism S_{j}^{∗} ⟶ S^{∗}, we see that k_{j}q_{j} ∈ S^{2}(S^{∗})_{dec}^{W}.

Case 2: The j-th simple component H̃_{j} is of type E_{6}^{sc}. The center of H̃_{j} is μ_{3}, hence C̃_{j} is a subgroup of μ_{3}. If C̃_{j} = 1, then H̃_{j} is a direct factor of H and hence E_{6}^{sc} is a direct factor of H. This is impossible by the assumption. Therefore, C̃_{j} = μ_{3} (and hence p = 3). The element \overline{w}_{1j} is a generator of C̃^{∗} = ℤ/3ℤ, hence k_{j} is divisible by 3. By [10, §4, type E_{6}], the form 6q_{j} and hence 2k_{j}q_{j} belongs to S^{2}(S_{j}^{∗})_{dec}^{W_{j}}. Taking the image of 2k_{j}q_{j} under the homomorphism S_{j}^{∗} ⟶ S^{∗}, we see that 2k_{j}q_{j} ∈ S^{2}(S^{∗})_{dec}^{W}. The claim is proved.

It follows from the claim that 2α belongs to the kernel of the map S^{2}(T^{∗})_{ind}^{W} ⟶ S^{2}(S^{∗})_{ind}^{W}. By Lemma 5.2, this map is injective, hence 2α = 0. □

Example 11.2. The statement of the proposition is wrong if p = 2. Consider the group H := (SL_{2})^{n}/C̃, where C̃ ⊂ (μ_{2})^{n} consists of all n-tuples with trivial product. Then the group G := (GL_{2})^{n}/C̃ is a strict envelope of H. A G-torsor over a field K is a tuple (Q_{1}, Q_{2}, …, Q_{n}) of quaternion algebras over K such that [Q_{1}]+[Q_{2}]+⋯+[Q_{n}] = 0 in Br(K). Let φ_{i} be the reduced norm quadratic form of Q_{i}. The sum φ of the forms φ_{i} in the Witt ring W (K) of K belongs to the cube of the fundamental ideal of W (K). The Arason invariant of φ in H^{3}(K) yields a degree 3 invariant I of G (see [8, page 431]). The restriction J of I to H belongs to Inv_{red}^{3}(H) = Im(Inv^{3}(G) ⟶ Inv^{3}(H)), and I and J are nontrivial if n ≥ 3. Note that the invariants I and J are ramified. Moreover, the map Inv^{3}(G) ⟶ Inv^{3}(H̃^{gen}) factors through Inv^{3}(G̃^{gen}), where G̃^{gen} is the product of GL_{1}(Q_{i}^{gen}). The group Inv^{3}(G̃^{gen}) is trivial since GL_{1}(Q_{i}^{gen}) have only trivial torsors. It follows that J belong to the kernel of

Inv^{3}(H) ⟶ Inv^{3}(H̃^{gen}),

hence the map in Proposition 8.2 is not injective.

Theorem 11.3. Let G be a (split) reductive group over an algebraically closed field F. Then Inv_{nr}^{3}(G, p) = 0 for every odd prime p ≠ char(F).

Proof. Let H be the commutator subgroup of G. By Proposition 6.1(2), it suffices to prove that Inv_{nr}^{3}(H, p) = 0. Let H̃ ⟶ H be a simply connected cover with kernel C̃. Let C̃′ ⊂ C̃ be a subgroup such that (C̃/C̃′)^{∗} is the 2-component of C̃^{∗}. Since p is odd, by Proposition 7.1, Inv_{nr}^{3}(H, p) = Inv_{nr}^{3}(H̃/C′, p). Replacing H by H̃/C̃′, we may assume that C̃^{∗} has odd order.

Write H̃ as a product of simple simply connected groups H̃_{j} and let C̃_{j} be the center of H̃_{j}. If the order of C̃_{j}^{∗} is a power of 2, the projection C̃ ⟶ C̃_{j} is trivial and therefore, the simply connected group H̃_{j} splits off H as a direct factor. Thus, the simply connected simple groups of types B_{n}, C_{n}, D_{n}, E_{7}, E_{8}, F_{4} and G_{2} split off H, i.e., H = H_{1} × H_{2}, where H_{1} is simply connected and H_{2} satisfies the conditions of Proposition 11.1. By the additivity property Corollary 6.3, Propositions 8.1(2) and 11.1, we have Inv_{nr}^{3}(H, p) = 0. □

References

[1] H. Bermudez and A. Ruozzi, Degree 3 cohomological invariants of groups that are neither simply connected nor adjoint, Preprint (2013).

[2] S. Blinstein and A. Merkurjev, Cohomological invariants of algebraic tori, Algebra Number Theory 7 (2013), no. 7, 1643–1684.

[3] F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485–516, 688.

[4] N. Bourbaki, Éléments de mathématique, Masson, Paris, 1981, Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6].

[5] H. Esnault, B. Kahn, M. Levine, and E. Viehweg, The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc. 11 (1998), no. 1, 73–118.

[6] R. Garibaldi, A. Merkurjev, and J.-P. Serre, Cohomological invariants in Galois cohomology, American Mathematical Society, Providence, RI, 2003.

[7] S. Garibaldi, Unramified cohomology of classifying varieties for exceptional simply connected groups, Trans. Amer. Math. Soc. 358 (2006), no. 1, 359–371 (electronic).

[8] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

[9] A. Merkurjev, Weight two motivic cohomology of classifying spaces for semisimple

groups, Preprint, http://www.math.ucla.edu/^{∼}merkurev/papers/ssinv.pdf (2013).

[10] A. Merkurjev, Degree three cohomological invariants of semisimple groups, To appear in JEMS.

[11] A. Merkurjev, Unramified cohomology of classifying varieties for classical simply connected groups, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 445–476.

[12] E. Peyre, Unramified cohomology of degree 3 and Noether’s problem, Invent. Math. 171 (2008), no. 1, 191–225.

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

[14] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), no. 1, 71–84.

[15] D. J. Saltman, Brauer groups of invariant fields, geometrically negligible classes, an equivariant Chow group, and unramified H^{3}, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Amer. Math. Soc., Providence, RI, 1995, pp. 189–246.

[16] D. J. Saltman, H^{3} and generic matrices, J. Algebra 195 (1997), no. 2, 387–422.

[17] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80.

[18] B. Totaro, The Chow ring of a classifying space, Algebraic K-theory (Seattle, WA, 1997), Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 249–281.

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

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