COHOMOLOGICAL INVARIANTS OF ALGEBRAIC TORI

S. BLINSTEIN AND A. MERKURJEV

Abstract. Let G be an algebraic group over a field F. As defined by Serre, a cohomological invariant of G of degree n with values in ℚ/ℤ(j) is a functorial in K collection of maps of sets H^{1}(K, G) ⟶ H^{n}(K, ℚ/ℤ(j)) for all field extensions K/F. We study the group of degree 3 invariants of an algebraic torus with values in ℚ/ℤ(2). In particular, we compute the group H_{nr}^{3}(F (S), ℚ/ℤ(2)) of unramified cohomology of an algebraic torus S.

1. Introduction

Let G be a (linear) algebraic group. The notion of an invariant of G was defined in [16] as follows. Fix a field F (of arbitrary characteristic) and consider the category Fields_{F} of field extensions of F. Consider the functor

Tors_{G} : Fields_{F} ⟶ Sets

taking a field K to the set H^{1}(K, G) of isomorphism classes of G-torsors over Spec K. Let

H : Fields_{F} ⟶ Abelian Groups

be another functor. An H-invariant of G is then a morphism of functors

i : Tors_{G} ⟶ H,

viewing H with values in Sets, i.e., a functorial in K collection of maps of sets H^{1}(K, G) ⟶ H(K) for all field extensions K/F. We denote the group of H-invariants of G by Inv(G, H).

An invariant i ∈ Inv(G, H) is called normalized if i(I) = 0 for the trivial G-torsor I. The normalized invariants form a subgroup Inv(G, H)_{norm} of Inv(G, H) and there is a natural isomorphism

Inv(G, H) ≃ H(F) ⊕ Inv(G, H)_{norm},

so it is sufficient to study normalized invariants.

Typically, H is a cohomological functor given by Galois cohomology groups with values in a fixed Galois module. Of particular interest to us is the functor H which takes a field K/F to the Galois cohomology group H^{n}(K, ℚ/ℤ(j)), where the coefficients ℚ/ℤ(j) are defined as follows. For a prime integer p different from char(F), the p-component ℚ_{p}/ℤ_{p}(j) of ℚ/ℤ(j) is the colimit over n of the étale sheaves μ_{p^{n}}^{⊗j}, where μ_{m} is the sheaf of m^{th} roots of unity. In the case p = char(F) > 0, ℚ_{p}/ℤ_{p}(j) is defined via logarithmic de Rham-Witt differentials (see §3b).

Key words and phrases. Algebraic tori, cohomological invariants, Galois cohomology 2000 Mathematical Subject Classifications: 12G05, 11E72 .

The work of the second author has been supported by the NSF grant DMS #0652316.

We write Inv^{n}(G, ℚ/ℤ(j)) for the group of cohomological invariants of G of degree n with values in ℚ/ℤ(j).

The second cohomology group H^{2}(K, ℚ/ℤ(1)) is canonically isomorphic to the Brauer group Br(K) of the field K. In §2c we prove (Theorem 2.4) that if G is a connected group (reductive if F is not perfect), then Inv(G, Br)_{norm} ≃ Pic(G). The group Inv^{3}(G, ℚ/ℤ(2))_{norm} for a semisimple simply connected group G has been studied by Rost (see [16]).

An essential object in the study of cohomological invariants is the notion of a classifying torsor: a G-torsor E ⟶ X for some variety X over F such that every G-torsor over an infinite field K/F is isomorphic to the pull-back of E ⟶ X along a K-point of X. If V is generically free representation of G with a nonempty open subset U ⊂ V such that there is a G-torsor π : U ⟶ X, then π is classifying. The generic fiber of π is the generic torsor over Spec F (X). Evaluation at the generic torsor yields a homomorphism

Inv^{n}(G, ℚ/ℤ(j)) ⟶ H^{n}(F (X), ℚ/ℤ(j)),

and in §3 we show that the image of this map is contained in the subgroup H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j))) of H^{n}(F (X), ℚ/ℤ(j)), where H^{n}(ℚ/ℤ(j)) is the Zariski sheaf associated to the presheaf W ↦ H^{n}(W, ℚ/ℤ(j)) of the étale cohomology groups. In fact, the image is contained in the subgroup H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j)))_{bal} of balanced elements, i.e., elements that have the same images under the pullback homomorphisms with respect to the two projections (U × U)/G ⟶ X. Moreover, the balanced elements precisely describe the image and we prove (Theorem 3.3):

Theorem. Let G be an algebraic group over a field F. We assume that G is connected if F is a finite field. Then there is a natural isomorphism Inv^{n}(G, ℚ/ℤ(j)) ≃ H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j)))_{bal}.

At this point it is convenient to make use of a construction due to Totaro [33]: because the Chow groups are homotopy invariant, the groups CH^{n}(X) do not depend on the choice of the representation V and the open set U ⊂ V provided the codimension of V \ U in V is large enough. This leads to the notation CH^{n}(BG), the Chow groups of the so-called classifying space BG, although BG itself is not defined in this paper.

Unfortunately, the étale cohomology groups with values in ℚ_{p}/ℤ_{p}(j), where p = char(F) > 0, are not homotopy invariant. In particular, we cannot use Rost’s theory of cycle modules of Rost [31].

The main result of this paper is the exact sequence in Theorem 4.4 describing degree 3 cohomological invariants of an algebraic torus T. Writing T̂_{sep} for the character lattice of T over a separable closure of F and T^{∘} for the dual torus, we prove:

Theorem. Let T be an algebraic torus a field F. Then there is an exact sequence

0 ⟶ CH^{2}(BT)_{tors} ⟶ H^{1}(F, T^{0}) ⟶^{α}

Inv^{3}(T, ℚ/ℤ(2))_{norm} ⟶ H^{0}(F, S^{2}(T̂_{sep}))/ Dec ⟶ H^{2}(F, T^{0}).

The homomorphism α is given by α(a)(b) = a_{K} ∪b for every a ∈ H^{1}(F, T^{0}) and b ∈ H^{1}(K, T) and every field extension K/F, where the cup-product is defined in (4.5), and Dec is the subgroup of decomposable elements in the symmetric square S^{2}(T̂_{sep}) defined in Appendix A-II.

We prove that the torsion group CH^{2}(BT)_{tors} is finite of exponent 2 and the last homomorphism in the sequence is also of exponent 2. Moreover, if p is an odd prime, the group Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{norm}, which is the p-primary component of Inv^{3}(T, ℚ/ℤ(2))_{norm}, splits canonically into the direct sum of linear invariants (those that induce group homomorphisms from Tors_{T} to H^{3}) and quadratic invariants, i.e., the invariants i such that the function h(a, b) := i(a+b)−i(a)−i(b) is bilinear and h(a, a) = 2i(a) for all a and b. Furthermore, the groups of linear and quadratic invariants with values in ℚ_{p}/ℤ_{p}(2) are canonically isomorphic to H^{1}(F, T^{∘}){p} and H^{0}(F, S^{2}(T̂_{sep}))/ Dec){p}, respectively.

We also prove (Theorem 4.9) that the degree 3 invariants have control over the structure of all invariants. In particular, the group Inv^{3}(T_{K}, ℚ/ℤ(2))_{norm} is trivial for all K/F if and only if T is universally special, i.e., T has no nontrivial torsors over any field K/F, which in particular means T has no nonconstant H-invariants for every functor H.

Our motivation for considering invariants of tori comes from their connection with unramified cohomology (defined in §5). Specifically, this work began as an investigation of a problem posed by Colliot-Thélène in [5, p. 39]: for n prime to char(F) and i ≥ 0, determine the unramified cohomology group H_{nr}^{i}(F (S), μ_{n}^{⊗(i−1)}), where F (S) is the function field of a torus S over F. The connection is provided by Theorem 5.6 where we show that the unramified cohomology of a torus S is calculated by the invariants of an auxiliary torus:

Theorem. Let S be a torus over F and let 1 ⟶ T ⟶ P ⟶ S ⟶ 1 be a flasque resolution of S, i.e., T is flasque and P is quasi-split. Then there is a natural isomorphism

H_{nr}^{n}(F (S), ℚ/ℤ(j)) ≃ Inv^{n}(T, ℚ/ℤ(j)).

In the present paper, F denotes a field of arbitrary characteristic, F_{sep} a separable closure of F, and Γ the absolute Galois group Gal(F_{sep}/F) of F. The word “scheme” over a field F means a separated scheme over F and a “variety” over F is an integral scheme of finite type over F. If X is a scheme over F and L/F is a field extension then we write X_{L} for X ×_{F} Spec L. When L = F_{sep} we write simply X_{sep}.

Acknowledgements: The authors would like to thank Jean-Louis Colliot-Thélène, Skip Garibaldi, Thomas Geisser, David Harari, Bruno Kahn, Kazuya Kato, and Alena Pirutka for helpful discussions.

2. Invariants of algebraic groups

2a. Definitions and basic properties. Let G be a (smooth linear) algebraic group over a field F. Consider the functor

Tors_{G} : Fields_{F} ⟶ Sets

from the category of field extensions of F to the category of sets taking a field K to the set H^{1}(K, G) := H^{1}(Gal(K_{sep}/K), G(K_{sep})) of isomorphism classes of G-torsors over Spec K.

Let H : Fields_{F} ⟶ Abelian Groups be a functor. We also view H as a functor with values in Sets. Following [16], we define an H-invariant of G as a morphism of functors Tors_{G} ⟶ H from the category Fields_{F} to Sets. All the H-invariants of G form the abelian group of invariants Inv(G, H).

An invariant i ∈ Inv(G, H) is called constant if there is an element h ∈ H(F) such that i(I) = h_{K} for every G-torsor I ⟶ Spec K, where h_{K} is the image of h under natural map H(F) ⟶ H(K). The constant invariants form a subgroup Inv(G, H)_{const} of Inv(G, H) isomorphic to H(F). An invariant i ∈ Inv(G, H) is called normalized, if i(I) = 0 for the trivial G-torsor I. The normalized invariants form a subgroup Inv(G, H)_{norm} of Inv(G, H) and we have the decomposition

Inv(G, H) = Inv(G, H)_{const} ⊕ Inv(G, H)_{norm} ≃ H(F) ⊕ Inv(G, H)_{norm},

so it suffices to determine the normalized invariants.

2b. Classifying torsors. Let G be an algebraic group over a field F. A Gtorsor E ⟶ X over a variety X over F is called classifying if for every field extension K/F, with K infinite, and for every G-torsor I ⟶ Spec K, there is a point x : Spec K ⟶ X such that the torsor I is isomorphic to the fiber E(x) of E ⟶ X over x, i.e., I ≃ E(x) := x^{∗}(E) = Spec(K) ×_{X} E. The generic fiber E_{gen} ⟶ Spec F (X) of a classifying torsor is called a generic G-torsor (see [16, Part 1, §5.3]).

If V is generically free representation of G with a nonempty open subset U ⊂ V such that there is a G-torsor π : U ⟶ X, then π is classifying (see [16, Part 1, §5.4]). We will write U/G for X and call π a standard classifying Gtorsor. Standard classifying G-torsors exist: we can embed G into U := GL_{n,F} for some n as a closed subgroup. Then U is an open subset in the affine space M_{n}(F) on which G acts linearly. Note that U(F) ≠ ∅.

We say that a G-variety Y is G-rational if there is an affine space V with a linear G-action such that Y and V have G-isomorphic nonempty open Ginvariant subvarieties. Note that if U ⟶ U/G is a standard classifying Gtorsor, then U is a G-rational variety.

Let E ⟶ X be a classifying G-torsor and let H : Fields_{F} ⟶ Abelian Groups be a functor. Define the map

\begin{matrix}θ : Inv(G, H) & ⟶ H(F (X)) \\ i & ⟼ i(E_{gen}), \end{matrix} ; (2.1)

by sending an invariant to its value at the generic torsor E_{gen}.

Consider the following property of the functor H:

Property 2.1. The map H(K) ⟶ H(K((t))) is injective for any field extension K/F.

The following theorem, due to M. Rost, was proved in [16, Part II, Th. 3.3]. For completeness, we give a slightly modified proof in Appendix A-I.

Theorem 2.2. Let G be an algebraic group over F. If a functor H : Fields_{F} ⟶ Abelian Groups has Property 2.1, then the map θ is injective, i.e., every Hinvariant of G is determined by its value at the generic G-torsor.

2c. The Brauer group invariants. Let G be a connected algebraic group over F. Every cohomological invariant of G of degree 1 is constant by [24, Prop. 31.15]. In this section we study (degree 2) Br-invariants for the Brauer group functor K ⟼ Br(K). We assume that G is reductive if F is not perfect as required by the auxiliary result [32, Prop. 6.10].

Lemma 2.3. For any field extension K/F such that F is algebraically closed in K, the natural map Pic(G) ⟶ Pic(G_{K}) is an isomorphism.

Proof. We may assume that G is reductive by factoring out the unipotent radical in the case that F is perfect. There is an exact sequence (see [8, Th. 1.2])

1 ⟶ C ⟶ G′ ⟶ G ⟶ 1

with C a group of multiplicative type and G′ a reductive group with Pic(G′_{L}) = 0 for any field extension L/F. Let T be the factor group of G′ by the semisimple part. The result follows from the exact sequence [32, Prop. 6.10]

T̂(L) ⟶ Ĉ(L) ⟶ Pic(G_{L}) ⟶ Pic(G′_{L}) = 0

with L = F and K since the groups T̂(F) and Ĉ(F) don’t change when F is replaced by K. □

Since for any G-torsor E ⟶ Spec(K) over a field extension K/F one has [32, Prop. 6.10] the exact sequence

(2.2)  Pic(E) ⟶ Pic(G_{K}) ⟶^{δ} Br(K) ⟶ Br(E),

we obtain the homomorphism

ν : Pic(G) ⟶ Inv(G, Br)_{norm}

which takes an element α ∈ Pic(G) to the invariant that sends a G-torsor E over a field extension K/F to δ(α_{K}).

Theorem 2.4. Let G be a connected algebraic group over F. Assume that G is reductive if F is not perfect. Then the map ν : Pic(G) ⟶ Inv(G, Br)_{norm} is an isomorphism.

Proof. Choose a standard classifying G-torsor U ⟶ U/G. Write K for the function field F (U/G) and let U_{gen} be the generic G-torsor over K. Consider the commutative diagram

\xymatrix{ & Pic(G) \ar[d]_{j} \ar[r]^{ν} & Inv(G, Br)_{norm} \ar[d]_{θ} \\ Pic(U_{gen}) \ar[r] & Pic(G_{K}) \ar[r]^{δ} & Br(K) \ar[r]^{i} & Br(K(U_{gen})),}

where the bottom sequence is (2.2) for the G-torsor U_{gen} ⟶ Spec(K) followed by the injection Br(U_{gen}) ⟶ Br(K(U_{gen})) (see [29, Ch. IV, Cor. 2.6]), and the map θ is evaluation at the generic torsor U_{gen} given in (2.1) and is injective by Theorem 2.2. Since the generic torsor is split over K(U_{gen}), Im(θ) ⊂ Ker(i) = Im(δ). By Lemma 2.3, j is an isomorphism, hence ν is surjective.

Note that U_{gen} is a localization of U, hence Pic(U_{gen}) = 0 as Pic(U) = 0. It follows that ν is injective. □

An algebraic group G over a field F is called universally special if H^{1}(K, G) = {1} for every field extension K/F, i.e., all G-torsors over any field extension of F are trivial.

Corollary 2.5. If the group G is universally special, then Pic(G) = 0.

3. Invariants with values in ℚ/ℤ(j)

In this section we find a description for the group of cohomological invariants with values in ℚ/ℤ(j) by identifying the image of the embedding θ in (2.1).

3a. Balanced elements. Let G be an algebraic group over a field F. We assume that G is connected if F is finite. Let E ⟶ X be a G-torsor such that E(F) ≠ ∅. We write p_{1} and p_{2} for the two projections E^{2}/G := (E ×E)/G ⟶ X.

Lemma 3.1. Let K/F be a field extension and x_{1}, x_{2} ∈ X(K). Then the Gtorsors E(x_{1}) and E(x_{2}) over K are isomorphic if and only if there is a point y ∈ (E^{2}/G)(K) such that p_{1}(y) = x_{1} and p_{2}(y) = x_{2}.

Proof. “⇒”: By construction, we have G-equivariant morphisms f_{i} : E(x_{i}) ⟶ E for i = 1, 2. Choose an isomorphism h : E(x_{1}) ⟶^{∼} E(x_{2}) of G-torsors over K. The morphism (f_{1}, f_{2}h) : E(x_{1}) ⟶ E^{2} yields the required point Spec K = E(x_{1})/G ⟶ E^{2}/G. “⇐”: The pull-back of E ⟶ X with respect to any projection E^{2}/G ⟶ X coincides with the G-torsor E^{2} ⟶ E^{2}/G, hence

E(x_{1}) = x_{1}^{∗}(E) = y^{∗}p_{1}^{∗}(E) ≃ y^{∗}(E^{2}) ≃ y^{∗}p_{2}^{∗}(E) = x_{2}^{∗}(E) = E(x_{2}). □

Let H be a (contravariant) functor from the category of integral schemes over F and dominant morphisms to the category of abelian groups. We have the two maps p_{i}^{∗} : H(X) ⟶ H(E^{2}/G), i = 1, 2. An element h ∈ H(X) is called balanced if p_{1}^{∗}(h) = p_{2}^{∗}(h). We write H(X)_{bal} for the subgroup of balanced elements in H(X). In other words, H(X)_{bal} = h_{0}(H(E^{•}/G)) in the notation of Appendix A-IV.

We can view H as a (covariant) functor Fields_{F} ⟶ Sets taking a field K to H(K) := H(Spec K).

Lemma 3.2. Let h ∈ H(X)_{bal} be a balanced element, K/F a field extension and I a G-torsor over Spec(K). Let x ∈ X(K) be a point such that E(x) ≃ I. Then the element x^{∗}(h) in H(K) does not depend on the choice of x.

Proof. Let x_{1}, x_{2} ∈ X(K) be two points such that E(x_{1}) ≃ E(x_{2}). By Lemma 3.1, there is a point y ∈ (E^{2}/G)(K) such that p_{1}(y) = x_{1} and p_{2}(y) = x_{2}. Therefore

x_{1}^{∗}(h) = y^{∗}(p_{1}^{∗}(h)) = y^{∗}(p_{2}^{∗}(h)) = x_{2}^{∗}(h).  □

It follows from Lemma 3.2 that if the torsor E ⟶ X is classifying with E(F) ≠ ∅, then every element h ∈ H(X)_{bal} determines an H-invariant i_{h} of G as follows. Let I be a G-torsor over a field extension K/F. We claim that there is a point x ∈ X(K) such that E(x) ≃ I. If K is infinite, this follows from the definition of the classifying G-torsor. If K is finite then all G-torsors over K are trivial by [25], as G is connected. Since E(K) ≠ ∅, we can take for x the image in X(K) of any point in E(K). Defining i_{h}(E) = x^{∗}(h) ∈ H(K), we have a group homomorphism

H(X)_{bal} ⟶ Inv(G, H), h ↦ i_{h}.

3b. Cohomology with values in ℚ/ℤ(j). For every integer j ≥ 0, the coefficients ℚ/ℤ(j) are defined as the direct sum over all prime integers p of the objects ℚ_{p}/ℤ_{p}(j) in the derived category of sheaves of abelian groups on the big étale site of Spec F, where

ℚ_{p}/ℤ_{p}(j) = colim_{n} μ_{p^{n}}^{⊗j}

if p ≠ char F, with μ_{p^{n}} the sheaf of (p^{n})^{th} roots of unity, and

ℚ_{p}/ℤ_{p}(j) = colim_{n} W_{n}Ω_{log}^{j}[−j]

if p = char F > 0, with W_{n}Ω_{log}^{j} the sheaf of logarithmic de Rham-Witt differentials (see [23]).

We write H^{m}(X, ℚ/ℤ(j)) for the étale cohomology of a scheme X with values in ℚ/ℤ(j). Then

H^{m}(X, ℚ/ℤ(j)){p} = colim_{n} H^{m}(X, μ_{p^{n}}^{⊗j})

if p ≠ char F and

H^{m}(X, ℚ/ℤ(j)){p} = colim_{n} H^{m−j}(X, W_{n}Ω_{log}^{j})

if p = char F > 0. In the latter case, the group W_{n}Ω_{log}^{j}(F) is canonically isomorphic to K_{j}^{M}(F)/p^{n}K_{j}^{M}(F), where K_{j}^{M}(F) is Milnor’s K-group of F (see [1, Cor. 2.8]), hence by [21] and [16, Part II, Appendix A], H^{s}(F, W_{n}Ω_{log}^{j}) is isomorphic to

H^{s}(F, K_{j}^{M}(F_{sep})/p^{n}K_{j}^{M}(F_{sep})) = \{ \begin{matrix}K_{j}^{M}(F)/p^{n}K_{j}^{M}(F), & if s = 0; \\ H^{2}(F, K_{j}^{M}(F_{sep}))_{p^{n}}, & if s = 1; \\ 0, & otherwise. \end{matrix}

It follows that in the case p = char F > 0, we have

H^{m}(F, ℚ/ℤ(j))\{p\} = \{ \begin{matrix}K_{j}^{M}(F) ⊗ (ℚ_{p}/ℤ_{p}), & if m = j; \\ H^{2}(F, K_{j}^{M}(F_{sep}))\{p\}, & if m = j + 1; \\ 0, & otherwise. \end{matrix}

The motivic complexes ℤ(j), for j = 0, 1, 2, of étale sheaves on a smooth scheme X were defined in [26] and [27] by S. Lichtenbaum. We write H^{∗}(X, ℤ(j)) for the étale cohomology groups of X with values in ℤ(j).

The complex ℤ(0) is equal to the constant sheaf ℤ and ℤ(1) = 𝔾_{m}_{,X}[−1], thus H^{n}(X, ℤ(1)) = H^{n−1}(X, 𝔾_{m}_{,X}). In particular, H^{3}(X, ℤ(1)) = Br(X), the cohomological Brauer group of X. The complex ℤ(2) is concentrated in degrees 1 and 2 and there is a product map ℤ(1) ⊗^{L} ℤ(1) ⟶ ℤ(2).

The exact triangle in the derived category of étale sheaves

ℤ(j) ⟶ ℚ ⊗ ℤ(j) ⟶ ℚ/ℤ(j) ⟶ ℤ(j)[1]

yields the connecting homomorphism

H^{i}(X, ℚ/ℤ(j)) ⟶ H^{i+1}(X, ℤ(j)),

which is an isomorphism if X = Spec(F) for a field F and i > j [22, Lemme 1.1].

Write H^{n}(ℚ/ℤ(j)) for the Zariski sheaf on a smooth scheme X associated to the presheaf U ↦ H^{n}(U, ℚ/ℤ(j)) of étale cohomology groups.

Let G be an algebraic group over F. We assume that G is connected if F is a finite field and write Inv^{n}(G, ℚ/ℤ(j)) for the group of degree n invariants of G for the functor K ↦ H^{n}(K, ℚ/ℤ(j)). Note that Property 2.1 holds for this functor by [16, Part 2, Prop. A.9].

Choose a classifying G-torsor E ⟶ X with E a G-rational variety such that E(F) ≠ ∅. Applying the construction given in §3a to the functor U ↦ H_{Zar}^{0}(U, H^{n}(ℚ/ℤ(j))), we get a homomorphism

φ : H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j)))_{bal} ⟶ Inv^{n}(G, ℚ/ℤ(j)).

Theorem 3.3. Let G be an algebraic group over a field F. We assume that G is connected if F is a finite field. Let E ⟶ X be a classifying G-torsor with E a G-rational variety such that E(F) ≠ ∅. Then the homomorphism φ is an isomorphism.

Proof. Let E_{gen} ⟶ F (X) be the generic fiber of the classifying G-torsor E ⟶ X. Note that since the pull-back of E ⟶ X with respect to any of the two projections E^{2}/G ⟶ X coincides with the G-torsor E^{2} ⟶ E^{2}/G, the pull-backs of the generic G-torsor E_{gen} ⟶ Spec F (X) with respect to the two morphisms Spec F (E^{2}/G) ⟶ Spec F (X) induced by the projections are isomorphic. It follows that for every invariant i ∈ Inv(G, H^{∗}(ℚ/ℤ(j))) we have

p_{1}^{∗}(i(E_{gen})) = i(p_{1}^{∗}(E_{gen})) = i(p_{2}^{∗}(E_{gen})) = p_{2}^{∗}(i(E_{gen}))

in H^{∗}(F (E^{2}/G), ℚ/ℤ(j)), i.e., i(E_{gen}) ∈ H^{∗}(F (X), ℚ/ℤ(j))_{bal}. By Proposition A.9, ∂_{x}(h) = 0 for every point x ∈ X of codimension 1, hence

θ(i) = i(E_{gen}) ∈ H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j)))_{bal}

by Proposition A.10. By Theorem 2.2, θ is injective and by construction, the composition θ ∘ φ is the identity. It follows that φ is an isomorphism. □

Write \overline{H}_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j))) for the factor group of H_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j))) by the natural image of H^{n}(F, ℚ/ℤ(j)).

Corollary 3.4. The isomorphism φ yields an isomorphism

\overline{H}_{Zar}^{0}(X, H^{n}(ℚ/ℤ(j)))_{bal} ⟶^{∼} Inv^{n}(G, ℚ/ℤ(j))_{norm}.

4. Degree 3 invariants of algebraic tori

In this section we prove the main theorem that describes degree 3 invariants of an algebraic torus with values in ℚ/ℤ(2).

4a. Algebraic tori. Let F be a field and Γ = Gal(F_{sep}/F) the absolute Galois group of F. An algebraic torus of dimension n over F is an algebraic group T such that T_{sep} is isomorphic to the product of n copies of the multiplicative group 𝔾_{m} (see [11] or [35]). For an algebraic torus T over a field F, we write T̂_{sep} for the Γ-module of characters Hom(T_{sep}, 𝔾_{m}). The group T̂_{sep} is a Γlattice, i.e., a free abelian group of finite rank with a continuous Γ-action. The contravariant functor T ↦ T̂_{sep} is an anti-equivalence between the category of algebraic tori and the category of Γ-lattices: the torus T and the group T (F) can be reconstructed from the lattice T̂_{sep} by the formulas

T = Spec(F_{sep}[T̂_{sep}])^{Γ},
T (F) = Hom_{Γ}(T̂_{sep}, F_{sep}^{×}) = (T̂_{sep}^{∘} ⊗ F_{sep}^{×})^{Γ},

where T̂_{sep}^{∘} = Hom(T̂_{sep}, ℤ).

We write T̂ for the character group Hom(T, 𝔾_{m}) = (T̂_{sep})^{Γ} and T^{∘} for the dual torus having character lattice T̂_{sep}^{∘}.

A torus T is called quasi-split if T is isomorphic to the group of invertible elements of an étale F -algebra, or equivalently, the Γ-lattice T̂_{sep} is permutation, i.e., T̂_{sep} has a Γ-invariant ℤ-basis. An invertible torus is a direct factor of a quasi-split torus.

A torus T is called flasque (respectively, coflasque) if H^{1}(L, T̂_{sep}^{∘}) = 0 (respectively, H^{1}(L, T̂_{sep}) = 0) for every finite field extension L/K. A flasque resolution of a torus S is an exact sequence of tori 1 ⟶ T ⟶ P ⟶ S ⟶ 1 with T flasque and P quasi-split. By [11, §4], or [35, §4.7], the torus T in the flasque resolution is invertible if and only if S is a direct factor of a rational torus.

4b. Products. Let T be a torus over F and let T̂(i) denote the complex T̂_{sep} ⊗ ℤ(i) of étale sheaves over F for i = 0, 1, 2. Thus, T̂(0) = T̂_{sep} and T̂(1) = (T̂_{sep} ⊗ F_{sep}^{×})[−1] = T^{∘}(F_{sep})[−1].

Let S and T be algebraic tori over F and let i and j be nonnegative integers with i + j ≤ 2. For any smooth variety X over F, we have the product map

(4.1) (Ŝ_{sep} ⊗ T̂_{sep})^{Γ} ⊗ H^{p}(X, Ŝ^{∘}(i)) ⊗ H^{q}(X, T̂^{∘}(j)) ⟶ H^{p+q}(X, ℤ(i + j))

taking a ⊗ b ⊗ c to a ∪ b ∪ c, via the canonical pairings between Ŝ_{sep} and Ŝ_{sep}^{∘}, T̂_{sep} and T̂_{sep}^{∘}, and the product map ℤ(i) ⊗^{L} ℤ(j) ⟶ ℤ(i + j).

Recall that there is an isomorphism H^{n}(F, ℤ(k)) ≃ H^{n−1}(F, ℚ/ℤ(k)) for n > k. In particular, we have the cup-product map

(4.2)  (Ŝ_{sep} ⊗ T̂_{sep})^{Γ} ⊗ H^{p}(F, S) ⊗ H^{q}(F, T) ⟶ H^{3}(F, ℚ/ℤ(2))

if p + q = 2.

If S = T^{∘} is the dual torus, then (Ŝ_{sep} ⊗ T̂_{sep})^{Γ} = End_{Γ}(T̂_{sep}) contains the canonical element 1_{T}. We then have the product map

(4.3)  H^{p}(X, T̂(i)) ⊗ H^{q}(X, T̂^{∘}(j)) ⟶ H^{p+q}(X, ℤ(i + j))

and in particular, the product maps

(4.4)  H^{1}(F, T̂_{sep}) ⊗ H^{1}(F, T) ⟶ H^{2}(F, ℚ/ℤ(1)) = Br(F),
(4.5)  H^{1}(F, T^{∘}) ⊗ H^{1}(F, T) ⟶ H^{3}(F, ℚ/ℤ(2)),
(4.6)  H^{2}(F, T^{∘}) ⊗ H^{0}(F, T) ⟶ H^{3}(F, ℚ/ℤ(2)),

taking a ⊗ b to 1_{T} ∪ a ∪ b.

The Picard group of T is canonically isomorphic to H^{1}(F, T̂_{sep}) [35, §4.3]. The map ν in Theorem 2.4 is given by the cup-product (4.4) with the class of the T -torsor U_{gen} ⟶ Spec(K) in H^{1}(K, T) (see [13, Prop. 2.9] or [3, Lemma 2.6]). Thus, we have proved the following corollary.

Corollary 4.1. Let T be a torus over F. Then the map

H^{1}(F, T̂_{sep}) ⟶ Inv(T, Br)_{norm},

taking an element a to the invariant b ↦ a_{K} ∪ b for b ∈ H^{1}(K, T) and K/F is an isomorphism.

As T is a commutative group, the set H^{1}(K, T) is an abelian group. An invariant i ∈ Inv(T, H) for a functor H is called linear if i_{K} : H^{1}(K, T) ⟶ H(K) is a group homomorphism for every K/F. Clearly, every linear invariant is normalized. By Corollary 4.1, every normalized Br-invariant of a torus is linear. In the next section we will see that a normalized degree 3 invariant of a torus need not be linear.

4c. Main theorem. Let T be a torus over F and choose a standard classifying T -torsor U ⟶ U/T such that the codimension of V \ U in V is at least 3. Such a torsor exits by [14, Lemma 9].

By [32, Prop. 6.10], there is an exact sequence

F_{sep}[U]^{×}/F_{sep}^{×} ⟶ T̂_{sep} ⟶ Pic((U/T)_{sep}) ⟶ Pic(U_{sep}).

The codimension assumption implies that the side terms are trivial, hence the map T̂_{sep} ⟶ Pic((U/T)_{sep}) is an isomorphism. It follows that the classifying T -torsor U ⟶ U/T is universal in the sense of [7] (see Proposition B.1).

Write K_{∗}(F) for the (Quillen) K-groups of F and K_{∗} for the Zariski sheaf associated to the presheaf U ↦ K_{∗}(U). Then the groups H_{Zar}^{n}(U/G, K_{2}) are independent of the choice of the classifying torsor (cf., [14]). So we write H_{Zar}^{n}(BT, K_{2}) for this group (see Appendix A-IV). As T_{sep} is a split torus, by the Künneth formula (see Example A.5),

H_{Zar}^{n}(BT_{sep}, K_{2}) = \{ \begin{matrix}K_{2}(F_{sep}), & if n = 0; \\ Pic((U/G)_{sep}) ⊗ F_{sep}^{×} = T̂_{sep} ⊗ F_{sep}^{×} = T^{∘}(F_{sep}), & if n = 1; \\ CH^{2}((U/G)_{sep}) = S^{2}(T̂_{sep}), & if n = 2. \end{matrix}

Applying the calculation of the K-cohomology groups to the standard classifying T -torsor U^{i} ⟶ U^{i}/G for every i > 0 instead of U ⟶ U/T, by Proposition B.3, we have the exact sequence (4.7)

0 ⟶ H^{1}(F, T^{∘}) ⟶^{α} \overline{H}^{4}(U^{i}/G, ℤ(2)) ⟶ \overline{H}^{4}((U^{i}/G)_{sep}, ℤ(2))^{Γ} ⟶ H^{2}(F, T^{∘}),

where \overline{H}^{4}(U^{i}/G, ℤ(2)) is the factor group of H^{4}(U^{i}/G, ℤ(2)) by H^{4}(F, ℤ(2)), the map α is given by α(a) = q^{∗}(a) ∪ [U^{i}] with q : U^{i}/G ⟶ Spec F the structure morphism, [U^{i}] the class of the T -torsor U^{i} ⟶ U^{i}/G in H^{1}(U^{i}/G, T), and the cup-product is taken for the pairing (B.6).

Taking the sequences (4.7) for all i, we get the exact sequence of cosimplicial groups

0 ⟶ H^{1}(F, T^{∘}) ⟶^{α} \overline{H}^{4}(U^{•}/T, ℤ(2)) ⟶ \overline{H}^{4}((U^{•}/T)_{sep}, ℤ(2))^{Γ} ⟶ H^{2}(F, T^{∘}).

The first and the last cosimplicial groups in the sequence are constant, hence by Lemma A.2, the sequence

(4.8) 0 ⟶ H^{1}(F, T^{∘}) ⟶^{α} \overline{H}^{4}(U/T, ℤ(2))_{bal} ⟶

\overline{H}^{4}((U/T)_{sep}, ℤ(2))_{bal}^{Γ} ⟶ H^{2}(F, T^{∘})

is exact.

The following theorem was proved in [23, Th. 1.1]:

Theorem 4.2. Let X be a smooth variety over F. Then there is an exact sequence

0 ⟶ CH^{2}(X) ⟶ H^{4}(X, ℤ(2)) ⟶ H_{Zar}^{0}(X, H^{3}(ℚ/ℤ(2))) ⟶ 0.

By Theorem 4.2, there is an exact sequence of cosimplicial groups

0 ⟶ CH^{2}(U^{•}/T) ⟶ \overline{H}^{4}(U^{•}/T, ℤ(2)) ⟶ \overline{H}_{Zar}^{0}(U^{•}/T, H^{3}(ℚ/ℤ(2))) ⟶ 0.

As the functor CH^{2} is homotopy invariant, by Lemma A.4, the first group in the sequence is constant. In view of Lemma A.2, and following the notation for the K-cohomology, the sequence (4.9) 0 ⟶ CH^{2}(BT) ⟶ \overline{H}^{4}(U/T, ℤ(2))_{bal} ⟶ \overline{H}_{Zar}^{0}(U/T, H(ℚ/ℤ(2)))_{bal} ⟶ 0

is exact. By Corollary 3.4, the last group in the sequence is canonically isomorphic to Inv(T, H^{3}(ℚ/ℤ(2)))_{norm}.

As the torus T_{sep} is split, all the invariants of T_{sep} are trivial hence the sequence (4.9) over F_{sep} yields an isomorphism

(4.10)  \overline{H}^{4}((U/T)_{sep}, ℤ(2))_{bal} ≃ CH^{2}(BT_{sep}) ≃ S^{2}(T̂_{sep}).

Combining (4.8), (4.9) and (4.10), we get the following diagram with an exact row and column:

\xymatrix{ &  & 0 \ar[d] \\  &  & H^{1}(F, T^{0}) \ar[d]_{α} \ar@{-->}[dr] \\ 0 \ar[r] & CH^{2}(BT) \ar[r] \ar[dr] & \overline{H}^{4}(U/T, ℤ(2))_{bal} \ar[r] \ar[d] & Inv^{3}(T, ℚ/ℤ(2))_{norm} \ar[r] & 0. \\  &  & S^{2}(T̂_{sep})^{Γ} \ar[d] \\  &  & H^{2}(F, T^{0})}

Write Dec = Dec(T̂_{sep}) for the subgroup of decomposable elements in S^{2}(T̂_{sep})^{Γ} (see Appendix A-II).

Lemma 4.3. The image of the homomorphism CH^{2}(BT) ⟶ CH^{2}(BT_{sep})^{Γ} = S^{2}(T̂_{sep})^{Γ} in the diagram coincides with Dec.

Proof. Consider the Grothendieck ring K_{0}(BT) of the category of T -equivariant vector bundles over Spec(F), or equivalently, of the category of finite dimensional representations of T. If T is split, every representation of T is a direct sum of one-dimensional representations. Therefore, there is an isomorphism between the group ring ℤ[T̂] of all formal finite sums ∑_{x∈T̂} a_{x}e^{x} and K_{0}(BT), taking e^{x} with x ∈ T̂ to the class of the 1-dimensional representation given by x. In general, for every torus T, we have K_{0}(BT_{sep}) = ℤ[T̂_{sep}] and K_{0}(BT) = ℤ[T̂_{sep}]^{Γ} = K_{0}(BT_{sep})^{Γ} (see [28, §9]). The group ℤ[T̂_{sep}]^{Γ} is generated by the sums ∑_{i=1}^{n} e^{γ_{i}x}, where γ_{1}, γ_{2}, …, γ_{n} are representatives of the left cosets of an arbitrary open subgroup Γ′ in Γ and x ∈ (T̂_{sep})^{Γ′}.

The equivariant Chern classes were defined in [14, §2.4]. The first Chern class c_{1} : K_{0}(BT_{sep}) ⟶ CH^{1}(BT_{sep}) = T̂_{sep} takes e^{x} to x. In the diagram

[figure: ℤ[T̂_{sep}]^{Γ}; K_{0}(BT); c_{2}; CH^{2}(BT); ℤ[T̂_{sep}]; K_{0}(BT_{sep}); c_{2}; CH^{2}(BT_{sep}); S^{2}(T̂_{sep})]

the second Chern class maps c_{2} are surjective by [15, Lemma C.3]. It follows from the formula c_{2}(a + b) = c_{2}(a) + c_{1}(a)c_{1}(b) + c_{2}(b) that the composition

ℤ[T̂_{sep}]^{Γ} = K_{0}(BT) ⟶ K_{0}(BT_{sep}) ⟶^{c_{2}} CH^{2}(BT_{sep}) = S^{2}(T̂_{sep}) ⟶ S^{2}(T̂_{sep})/(T̂)^{2}

is a homomorphism and its image is generated by the elements (see Appendix A-II)

c_{2}(∑_{i=1}^{n} e^{γ_{i}x}) = ∑_{i<j}(γ_{i}x)(γ_{j}x) = Qtr(x).  □

By the restriction-corestriction argument, the kernel of the homomorphism CH^{2}(BT) ⟶ CH^{2}(BT_{sep})^{Γ} = S^{2}(T̂_{sep})^{Γ} coincides with the torsion subgroup CH^{2}(BT)_{tors} in CH^{2}(BT).

The following theorem describes degree 3 invariants of an algebraic torus with values in ℚ/ℤ(2):

Theorem 4.4. Let T be an algebraic torus a field F. Then there is an exact sequence

0 ⟶ CH^{2}(BT)_{tors} ⟶ H^{1}(F, T^{0}) ⟶^{α}

Inv^{3}(T, ℚ/ℤ(2))_{norm} ⟶ S^{2}(T̂_{sep})^{Γ}/ Dec ⟶ H^{2}(F, T^{0}).

The homomorphism α is given by α(a)(b) = a_{K} ∪ b for every a ∈ H^{1}(F, T^{0}) and b ∈ H^{1}(K, T) and every field extension K/F, where the cup-product is defined in (4.5).

Proof. The exactness of the sequence follows from the diagram before Lemma 4.3. It remains to describe the map α. Take an a ∈ H^{1}(F, T^{0}) and consider the invariant i defined by i(b) = a_{K} ∪ b, where the cup-product is given by (4.5). We need to prove that i = α(a). Choose a standard classifying T - torsor U ⟶ U/T and set K = F (U/T). Let U_{gen} be the generic fiber of the classifying torsor. By Theorem 2.2, it suffices to show that i(U_{gen}) = α(a)(U_{gen}) over K. This follows from the description of the map α in the exact sequence (4.7). □

4d. Torsion in CH^{2}(BT). We investigate the group CH^{2}(BT)_{tors}, the first term of the exact sequence in Theorem 4.4.

Let S be an algebraic torus over F. Using the Gersten resolution, [30, Prop. 5.8] we identify the group H^{0}(S_{sep}, K_{2}) with a subgroup in K_{2}(F_{sep}(S)). Set \overline{H}^{0}(S_{sep}, K_{2}) := H^{0}(S_{sep}, K_{2})/K_{2}(F_{sep}). By [16, Part 2, §5.7], we have an exact sequence

(4.11)  0 ⟶ Ŝ_{sep} ⊗ F_{sep}^{×} ⟶ \overline{H}^{0}(S_{sep}, K_{2}) ⟶^{λ} Λ^{2}Ŝ_{sep} ⟶ 0

of Γ-modules, where λ({e^{x}, e^{y}}) = x ∧ y for x, y ∈ Ŝ_{sep}.

Consider the Γ-homomorphism

\begin{matrix}γ : Λ^{2}Ŝ_{sep} & ⟶ \overline{H}^{0}(S_{sep}, K_{2}) \\ x ∧ y & ⟼ \{e^{x}, e^{y}\} − \{e^{y}, e^{x}\}. \end{matrix}

We have λ ∘ γ = 2 · Id, hence the connecting homomorphism

(4.12)  ∂ : H^{i}(F, Λ^{2}Ŝ_{sep}) ⟶ H^{i+1}(F, Ŝ_{sep} ⊗ F_{sep}^{×})

satisfies 2∂ = 0.

Lemma 4.5. If S is an invertible torus, then sequence of Γ-modules (4.11) is split.

Proof. We can assume that S is quasi-split. Let {x_{1}, x_{2}, …, x_{m}} be a permutation basis for Ŝ_{sep}. Then the elements x_{i} ∧ x_{j} for i < j form a ℤ-basis for Λ^{2}Ŝ_{sep}. The map Λ^{2}Ŝ_{sep} ⟶ \overline{H}^{0}(S_{sep}, K_{2}), taking x_{i} ∧ x_{j} to \{e^{x_{i}}, e^{x_{j}}\} is a splitting for γ. □

Let

1 ⟶ T ⟶ P ⟶ Q ⟶ 1

be a coflasque resolution of T, i.e., P is a quasi-split torus and Q is a coflasque torus (see [11]). The torus P is an open set in the affine space of a separable F -algebra on which T acts linearly. Hence P ⟶ Q is a standard classifying T -torsor. By Theorem 2.2, the natural map

θ : Inv^{3}(T, ℚ/ℤ(2)) ⟶ H^{3}(F (Q), ℚ/ℤ(2))

is injective.

Consider the spectral sequence (B.11) for the variety X = Q. We have H^{1}(Q_{sep}, K_{2}) = 0 by [16, Part 2, Cor. 5.6]. In view of Proposition B.4, we have an injective homomorphism

(4.13)  β : H^{2}(F, \overline{H}^{0}(Q_{sep}, K_{2})) ⟶ \overline{H}^{4}(Q, ℤ(2))

such that the composition of β with the homomorphism

H^{2}(F, Q^{∘}) ⟶ H^{2}(F, \overline{H}^{0}(Q_{sep}, K_{2}))

is given by the cup-product with the class of the identity in H^{0}(Q, Q).

Lemma 4.6. The group CH^{2}(Q) is trivial.

Proof. By [28, Th. 9.1], the Grothendieck group K_{0}(Q) is generated by the classes of the sheaves i_{∗}(P), where P is an invertible sheaf on X_{L}, L/F a finite separable field extension and i : X_{L} ⟶ X is the natural morphism. By definition of a coflasque torus,

Pic(Q_{L}) = H^{1}(L, Q̂_{sep}) = 0.

It follows that every invertible sheaf on Q_{L} is trivial, hence K_{0}(Q) = ℤ · 1. Since the group CH^{2}(Q) is generated by the second Chern classes of vector bundles on Q [15, Lemma C.3], CH^{2}(Q) = 0. □

It follows from Proposition A.10, Theorem 4.2, and Lemma 4.6 that the homomorphism

(4.14)  κ : \overline{H}^{4}(Q, ℤ(2)) ⟶ \overline{H}^{4}(F (Q), ℤ(2)) = \overline{H}^{3}(F (Q), ℚ/ℤ(2))

is injective.

Consider the diagram

[figure: 0; Q^{∘}(F_{sep}); \overline{H}^{0}(Q_{sep}, K_{2}); s; Λ^{2}Q̂_{sep}; t; 0; 0; Q^{∘}(F_{sep}); P^{∘}(F_{sep}); T^{∘}(F_{sep}; 0]

where s is the composition of the natural map \overline{H}^{0}(Q_{sep}, K_{2}) ⟶ \overline{H}^{0}(P_{sep}, K_{2}) and a splitting of P^{∘}(F_{sep}) ⟶ \overline{H}^{0}(P_{sep}, K_{2}) (see Lemma 4.5).

We have the following diagram

\xymatrix{ &  & H^{1}(F, T^{∘}) \ar@{^{(}->}[d]^{∂_{1}} \ar[r]^{α} & Inv^{3}(T, ℚ/ℤ(2)) \ar@{^{(}->}[d]^{θ} \\ H^{1}(F, \overline{H}^{0}(Q_{sep}, K_{2})) \ar[r] & H^{1}(F, Λ^{2}Q̂_{sep}) \ar[r]^{∂} \ar@{-->}[ur]^{t^{∗}} & H^{2}(F, Q^{∘}) \ar[r]^{σ} & \overline{H}^{3}(F (Q), ℚ/ℤ(2))}

where σ is the composition of the maps in (4.13) and (4.14):

H^{2}(F, Q^{0}) ⟶ H^{2}(F, \overline{A}^{0}(Q_{sep}, K_{2})) ⟶^{β} \overline{H}^{4}(Q, ℤ(2)) ⟶^{κ}

\overline{H}^{4}(F (Q), ℤ(2)) = \overline{H}^{3}(F (Q), ℚ/ℤ(2)).

Note that the connecting map ∂_{1} is injective as H^{1}(F, P^{∘}) = 0 since P^{∘} is a quasi-split torus. As 2∂ = 0 in (4.12), we have 2t^{∗} = 0.

The commutativity of the triangle follows from the definition of t^{∗}. We claim that the square in the diagram is anti-commutative. Note that ∂_{2}(ξ) = [P_{gen}], where ∂_{2} : H^{0}(F, Q) ⟶ H^{1}(F, T) is the connecting homomorphism, P_{gen} is the generic fiber of the morphism P ⟶ Q, and ξ ∈ H^{0}(K, Q) is the generic point of Q with K = F (Q). It follows from the description of the maps α and β in (4.7) and (4.13), respectively, and Lemma A.1 that

σ(∂_{1}(a)) = ∂_{1}(a)_{K} ∪ ξ = (−a_{K}) ∪ ∂_{2}(ξ) = (−a_{K}) ∪ [P_{gen}] = −θ(α(a)).

for every a ∈ H^{1}(F, T^{∘}).

The maps β and κ are injective, hence the bottom sequence in the diagram is exact. Thus, we have an exact sequence

H^{1}(F, \overline{H}^{0}(Q_{sep}, K_{2})) ⟶ H^{1}(F, Λ^{2}Q̂_{sep}) ⟶ Ker(α) ⟶ 0

and 2 · Ker(α) = 2 · Im(t^{∗}) = 0. Furthermore, Ker(α) ≃ CH^{2}(BT)_{tors} and the group H^{1}(F, Λ^{2}Q̂_{sep}) is finite.

We have proved:

Theorem 4.7. Let 1 ⟶ T ⟶ P ⟶ Q ⟶ 1 be a coflasque resolution of a torus T. Then there is an exact sequence

H^{1}(F, \overline{H}^{0}(Q_{sep}, K_{2})) ⟶ H^{1}(F, Λ^{2}Q̂_{sep}) ⟶ CH^{2}(BT)_{tors} ⟶ 0.

Moreover, CH^{2}(BT)_{tors} is a finite group satisfying 2 · CH^{2}(BT)_{tors} = 0.

Corollary 4.8. If T^{∘} is a direct factor of a rational torus, or if T is split over a cyclic field extension, then CH^{2}(BT)_{tors} = 0, i.e., the map α in Theorem 4.4 is injective.

Proof. The exact sequence 1 ⟶ Q^{∘} ⟶ P^{∘} ⟶ T^{∘} ⟶ 1 is a flasque resolution of T^{∘}. If T^{∘} is a direct factor of a rational torus, or if T is split over a cyclic field extension, the torus Q^{∘} ,and hence Q, is invertible (see §4a and [35, §4, Th. 3]). By Lemma 4.5, the sequence (4.11) for the torus Q is split, hence the first map in Theorem 4.7 is surjective. □

4e. Special tori. Let T be an algebraic torus over a field F. The tautological invariant of the torus T^{∘} × T is the normalized invariant taking a pair (a, b) ∈ H^{1}(K, T^{∘}) × H^{1}(K, T) to the cup-product a ∪ b ∈ H^{3}(K, ℚ/ℤ(2)) defined in (4.5).

The following theorem shows that if a torus T has only trivial degree 3 normalized invariants with values in ℚ/ℤ(2) universally, i.e., over all field extensions of F, then T has no non-constant invariants at all by the simple reason: every T -torsor over a field is trivial. Note that it follows from Theorem 2.4 that T has no degree 2 normalized invariants with values in ℚ/ℤ(1) universally if and only if T is coflasque.

Theorem 4.9. Let T be an algebraic torus over a field F. Then the following are equivalent:

(1) Inv^{3}(T_{K}, ℚ/ℤ(2))_{norm} = 0 for every field extension K of F.

(2) The tautological invariant of the torus T^{∘} × T is trivial.

(3) The torus T is invertible.

(4) The torus T is universally special.

Proof. (1) ⇒ (2): Let K/F be a field extension and a ∈ H^{1}(K, T^{∘}). By assumption, the degree 3 normalized invariant i = α(a) with values in ℚ/ℤ(2), defined by i(b) = a ∪ b for every b ∈ H^{1}(K, T), is trivial. In other words, the tautological invariant of the torus T^{∘} × T is trivial.

(2) ⇒ (3): The image of the tautological invariant in the group S^{2}(T̂_{sep}^{∘} ⊕ T̂_{sep})^{Γ}/ Dec is represented by the identity 1T̂ in the direct factor (T̂_{sep}^{∘}⊗T̂_{sep})^{Γ} = End_{Γ}(T̂_{sep}) of S^{2}(T̂_{sep}^{∘} ⊕ T̂_{sep})^{Γ} (see Appendix A-II). The projection of Dec on the direct summand (T̂_{sep}^{∘} ⊗ T̂_{sep})^{Γ} is generated by the traces Tr(a ⊗ b) for all open subgroups Γ′ ⊂ Γ and all a ∈ (T̂_{sep}^{∘})^{Γ′} and b ∈ (T̂_{sep})^{Γ′}. Hence 1T̂ = Tr(a_{i} ⊗ b_{i}) for some open subgroups Γ_{i} ⊂ Γ, a_{i} ∈ (T̂^{∘})^{Γ_{i}} and b_{i} ∈ (T̂)^{Γ_{i}}. If P_{i} = ℤ[Γ/Γ_{i}], then a_{i} can be viewed as a Γ-homomorphism T̂ ⟶ P_{i} and b_{i} can be viewed as a Γ-homomorphism P_{i} ⟶ T̂ such that the composition

T̂ ⟶^{(b_{i})} P ⟶^{(a_{i})} T̂,

where P = ∐ P_{i}, is the identity. It follows that T̂ is a direct summand of a permutation Γ-module P and hence T is invertible.

(3) ⇒ (4): Obvious as every invertible torus is universally special.

(4) ⇒ (1): Obvious. □

4f. Linear and quadratic invariants. Let T be a torus over F. By Theorem 4.4, we have a natural homomorphism to the group of linear invariants:

α : H^{1}(F, T^{∘}) ⟶ Inv^{3}(T, ℚ/ℤ(2))_{lin}.

Let S and T be algebraic tori over F. For every field extension K/F, the cup-product (4.2) yields a homomorphism

ε : (T̂_{sep}^{⊗2})^{Γ} ⟶ Inv^{3}(T, ℚ/ℤ(2))

defined by ε(a)(b) = a_{K} ∪ b ∪ b for a ∈ (T̂_{sep}^{⊗2})^{Γ} and b ∈ H^{1}(K, T).

We say that an invariant i ∈ Inv^{3}(T, ℚ/ℤ(2)) is quadratic if the function h(a, b) := i(a + b) − i(a) − i(b) is bilinear and h(a, a) = 2i(a) for all a and b. For example, the tautological invariant of the torus T^{∘} × T in §4e is quadratic. We write Inv^{3}(T, ℚ/ℤ(2))_{quad} for the subgroup of all quadratic invariants in Inv^{3}(T, ℚ/ℤ(2)). The image of ε is contained in Inv^{3}(T, ℚ/ℤ(2))_{quad}.

Write D for the subgroup of (T̂_{sep}^{⊗2})^{Γ} generated by Dec(T̂_{sep}, T̂_{sep}) (the latter defined in Appendix A-II) and the image of 1 − τ, where τ is the exchange automorphism of (T̂_{sep}^{⊗2})^{Γ}.

Lemma 4.10. The composition of ε with the map Inv^{3}(T, ℚ/ℤ(2)) ⟶ S^{2}(T̂_{sep})^{Γ}/ Dec in Theorem 4.4 is induced by the natural homomorphism T̂_{sep}^{⊗2} ⟶ S^{2}(T̂_{sep}). Moreover, the map ε vanishes on D.

Proof. Let U ⟶ U/T =: X be a standard classifying T -torsor as in §4c. Consider the commutative diagram

\xymatrix{(T̂_{sep}^{⊗2})^{Γ} ⊗ H^{1}(X, T)^{⊗2} \ar[d] \ar[r]^{prod} & \overline{H}^{4}(X, ℤ(2))_{bal} \ar[d] \ar[r] & Inv^{3}(T, ℚ/ℤ(2))_{norm} \\ T̂_{sep}^{⊗2} ⊗ H^{1}(X_{sep}, T)^{⊗2} \ar[r]^{prod} \ar[d]_{η}^{≀} & \overline{H}^{4}(X_{sep}, ℤ(2))_{bal} \ar[d]^{≀} \\ T̂_{sep}^{⊗2} ⊗ (T̂_{sep}^{∘})^{⊗2} ⊗ T̂_{sep}^{⊗2} \ar[r]^{κ} & S^{2}(T̂_{sep})^{Γ}/ Dec}

where the product maps are given by (4.1), η identifies H^{1}(X_{sep}, T) = T̂_{sep}^{∘} ⊗ Pic(X_{sep}) with T̂_{sep}^{∘} ⊗ T̂_{sep} and κ is given by the pairing between the first and second factors. Write [U] for the class of the classifying torsor in H^{1}(X, T). The image of [U] in H^{1}(X_{sep}, T_{sep}) = T̂_{sep}^{∘} ⊗ T̂_{sep} = End(T̂_{sep}) is the identity 1T̂_{sep}. Hence for every a ∈ (T̂_{sep}^{⊗2})^{Γ}, the image of a⊗[U]⊗[U] under the diagonal map in the diagram coincides with the canonical image of a in S^{2}(T̂_{sep})^{Γ}/ Dec.

Clearly, ετ = ε, hence ε is trivial on the image of 1−τ. Let Γ′ ⊂ Γ be an open subgroup and a, b ∈ (T̂_{sep})^{Γ′} = CH^{1}(X_{sep})^{Γ′} = CH^{1}(X_{L}), where L = (F_{sep})^{Γ′}. Then Tr(a ⊗ b) is the image of the trace map CH^{2}(X_{L}) ⟶ CH^{2}(X) and therefore, ε(Tr(a ⊗ b)) belongs to the image of an element from CH^{2}(X) = CH^{2}(BT) in the sequence (4.9) and hence is trivial in Inv^{3}(T, ℚ/ℤ(2)). □

Both compositions of the natural map (T̂_{sep}^{⊗2})^{Γ}/D ⟶ S^{2}(T̂_{sep})^{Γ}/ Dec with the map S^{2}(T̂_{sep})^{Γ}/ Dec ⟶ (T̂_{sep}^{⊗2})^{Γ}/D induced by ab ↦ a ⊗ b + b ⊗ a are multiplication by 2. Hence the kernel and the cokernel of (T̂_{sep}^{⊗2})^{Γ}/D ⟶ S^{2}(T̂_{sep})^{Γ}/ Dec have exponent 2. It follows that 2 times the homomorphism S^{2}(T̂_{sep})^{Γ}/ Dec ⟶ H^{2}(F, T^{0}) from Theorem 4.4 is trivial.

Theorem 4.4 also yields:

Theorem 4.11. Let p be an odd prime and let T be an algebraic torus over F. Then

Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{norm} = Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{lin} ⊕ Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{quad}

and there are natural isomorphisms Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{lin} ≃ H^{1}(F, T^{∘}){p} and Inv^{3}(T, ℚ_{p}/ℤ_{p}(2))_{quad} ≃ (S^{2}(T̂_{sep})^{Γ}/ Dec){p}.

Example 4.12. Let X = {x_{1}, x_{2}, …, x_{n}} be a set of n elements with the natural action of the symmetric group S_{n}. A continuous surjective group homomorphism Γ ⟶ S_{n} yields a separable field extension L/F of degree n. Consider the torus T = R_{L/F} (𝔾_{m}_{,L})/ 𝔾_{m}, where R_{L/F} is the Weil restriction (see [35, Ch. 1, §3.12]). Note that the generic maximal torus of the group PGL_{n} is of this form (see §5b). The character lattice T̂_{sep} is the kernel of the augmentation homomorphism ℤ[X] ⟶ ℤ.

The dual torus T^{∘} is the norm one torus R_{L/F}^{(1)} (𝔾_{m}_{,L}). For every field extension K/F, we have:

H^{1}(K, T) = Br(KL/K), H^{1}(K, T^{∘}) = K^{×}/N(KL)^{×},

where KL := K ⊗ L, N is the norm map for the extension KL/K and Br(KL/K) = Ker(Br(K) ⟶ Br(KL)). The pairing

K^{×}/N(KL)^{×} ⊗ Br(KL/K) ⟶ H^{3}(F, ℚ/ℤ(2))

defines linear degree 3 invariants of both T and T^{∘}.

We claim that S^{2}(T̂_{sep})^{Γ}/ Dec = 0 and S^{2}(T̂_{sep}^{∘})^{Γ}/ Dec = 0, i.e., T and T^{∘} have no nontrivial quadratic degree 3 invariants. We have T̂_{sep}^{∘} = ℤ[X]/ℤN_{X}, where N_{X} = ∑ x_{i}. The group S^{2}(T̂_{sep}^{∘})^{Γ} is generated by S := ∑_{i<j} x_{i} · x_{j}. As S ∈ Dec, we have S^{2}(T̂_{sep}^{∘})^{Γ}/ Dec = 0.

Let D = ∑ x_{i}^{2} and E := Qtr(x_{1} − x_{2}) = 2S − (n − 1)D, where the quadratic map Qtr is defined in Appendix A-II. The group S^{2}(T̂_{sep})^{Γ} is generated by E if n is even and by E/2 if n is odd. A computation shows that nE/2 = Qtr(nx_{1}−N_{X}). It follows that the generator of S^{2}(T̂_{sep})^{Γ} belongs to Dec, hence S^{2}(T̂_{sep})^{Γ}/ Dec is trivial.

Note that as the torus T is rational, it follows from Theorem 4.4 and Corollary 4.8 that Inv^{3}(T^{∘}, ℚ/ℤ(2))_{norm} ≃ Br(L/F).

5. Unramified invariants

Let K/F be a field extension and v a discrete valuation of K over F with valuation ring O_{v}. We say that an element a ∈ H^{n}(K, ℚ/ℤ(j)) is unramified with respect to v if a belongs to the image of the map H^{n}(O_{v}, ℚ/ℤ(j)) ⟶ H^{n}(K, ℚ/ℤ(j)) (see [10]). We write H_{nr}^{n}(K, ℚ/ℤ(j)) for the subgroup of the elements in H^{n}(K, ℚ/ℤ(j)) that are unramified with respect to all discrete valuations of K over F. We have a natural homomorphism

(5.1)  H^{n}(F, ℚ/ℤ(j)) ⟶ H_{nr}^{n}(K, ℚ/ℤ(j)).

A dominant morphism of varieties Y ⟶ X yields a homomorphism

H_{nr}^{n}(F (X), ℚ/ℤ(j)) ⟶ H_{nr}^{n}(F (Y), ℚ/ℤ(j)).

Proposition 5.1. Let K/F be a purely transcendental field extension. Then the homomorphism (5.1) is an isomorphism.

Proof. The statement is well known for the p-components if p ≠ char F (see, for example, [10, Prop. 1.2]). It suffices to consider the case K = F (t) and prove the surjectivity of (5.1). The coniveau spectral sequence for the projective line ℙ^{1} (see Appendix (A.1)) yields an exact sequence

H^{n}(ℙ^{1}, ℚ/ℤ(j)) ⟶ H^{n}(K, ℚ/ℤ(j)) ⟶ ∐_{x∈ℙ^{1}} H_{x}^{n+1}(ℙ^{1}, ℚ/ℤ(j))

and, therefore, a surjective homomorphism H^{n}(ℙ^{1}, ℚ/ℤ(j)) ⟶ H_{nr}^{n}(K, ℚ/ℤ(j)). By the projective bundle theorem (classical if p ≠ char(F) and [17, Th. 2.1.11] if p = char(F) > 0), we have

H^{n}(ℙ^{1}, ℚ/ℤ(j)) = H^{n}(F, ℚ/ℤ(j)) ⊕ H^{n−2}(F, ℚ/ℤ(j − 1))t,

where t is a generator of H^{2}(ℙ^{1}, ℤ(1)) = Pic(ℙ^{1}) = ℤ. As t vanishes over the generic point of ℙ^{1}, the result follows. □

Let G be an algebraic group over F. Choose a standard classifying Gtorsor U ⟶ U/G. An invariant i ∈ Inv^{n}(G, ℚ/ℤ(j)) is called unramified if the image of i under θ : Inv^{n}(G, ℚ/ℤ(j)) ⟶ H^{n}(F (U/G), ℚ/ℤ(j)) is unramified. This is independent of the choice of standard classifying torsor. Indeed, if U′ ⟶ U′/G is another standard classifying torsor G-torsor, then (U × V′)/G ⟶ U/G and (V × U′)/G ⟶ U′/G are vector bundles. Hence the field F ((U × U′)/G) is a purely transcendental extension of F (U/G) and F (U′/G) and by Proposition 5.1,

H_{nr}^{n}(F (U/G), ℚ/ℤ(j)) ≃ H_{nr}^{n}(F ((U×U′)/G), ℚ/ℤ(j)) ≃ H_{nr}^{n}(F (U′/G), ℚ/ℤ(j)).

We write H_{nr}^{n}(F (BG), ℚ/ℤ(j)) for this common value and Inv_{nr}^{n}(G, ℚ/ℤ(j)) for the subgroup of unramified invariants. Similarly, we write Br_{nr}(F (BG)) for the unramified Brauer group H_{nr}^{2}(F (BG), ℚ/ℤ(1)).

Proposition 5.2. The map Inv_{nr}^{n}(G, ℚ/ℤ(j)) ⟶ H_{nr}^{n}(F (BG), ℚ/ℤ(j)) induced by θ is an isomorphism.

Proof. By Theorem 3.3, it suffices to show that H_{nr}^{n}(F (U/G), ℚ/ℤ(j)) ⊂ H^{n}(F (U/G), ℚ/ℤ(j))_{bal}. We follow Totaro’s approach (see [16, p. 99]). Consider the open subscheme W of (U^{2}/G) × 𝔸^{1} of all triples (u, u′, t) such that (2 − t)u + (t − 1)u′ ∈ U. We have the projection q : W ⟶ U^{2}/G, the morphisms f : W ⟶ U/G defined by f(u, u′, t) = (2 − t)u + (t − 1)u′, and h_{i} : U^{2}/G ⟶ W defined by h_{i}(u, u′) = (u, u′, i) for i = 1 and 2. The composition f ∘ h_{i} is the projection p_{i} : U^{2}/G ⟶ U/G and q ∘ h_{i} is the identity of U^{2}/G.

Let w_{i} be the generic point of the pre-image of i with respect to the projection W ⟶ 𝔸^{1} and write O_{i} for the local ring of W at w_{i}. The morphisms q, f, and h_{i} yield F -algebra homomorphisms F (U^{2}/G) ⟶ O_{i}, F (U/G) ⟶ O_{i} and O_{i} ⟶ F (U/G). Note that by Proposition A.11, we have H_{nr}^{n}(F (W), ℚ/ℤ(j)) ⊂ H^{n}(O_{i}, ℚ/ℤ(j)). In the commutative diagram

[figure: H_{nr}^{n}(F (U/G), ℚ/ℤ(j)); f^{∗}; H_{nr}^{n}(F (W), ℚ/ℤ(j)); H_{nr}^{n}(F (U^{2}/G), ℚ/ℤ(j)); ∼; q^{∗}; H^{n}(F (U/G), ℚ/ℤ(j)); p_{i}^{∗}; f^{∗}; H^{n}(O_{i}, ℚ/ℤ(j)); h_{i}^{∗}; H^{n}(F (U^{2}/G), ℚ/ℤ(j)); q^{∗}; H^{n}(F (U^{2}/G), ℚ/ℤ(j))]

the top right map q^{∗} is an isomorphism by Proposition 5.1 since the field extension F (W)/F (U^{2}/G) is purely transcendental. It follows that the restriction of p_{i}^{∗} on H_{nr}^{n}(F (U/G), ℚ/ℤ(j)) coincides with (q^{∗})^{−1} ∘ f^{∗} and hence is independent of i. □

5a. Unramified invariants of tori.

Proposition 5.3. If T is a flasque torus, then every invariant in Inv^{n}(G, ℚ/ℤ(j)) is unramified.

Proof. Consider an exact sequence of tori 1 ⟶ T ⟶ P ⟶ Q ⟶ 1 with P quasi-split. Choose a smooth projective compactification X of Q (see [6]). As T is flasque, by [11, Prop. 9], there is a T -torsor E ⟶ X extending the T -torsor P ⟶ Q. The torsor E is classifying and T -rational. Choose an invariant in Inv^{n}(G, ℚ/ℤ(j)) and consider its image a in H^{n}(F (X), ℚ/ℤ(j))_{bal} (see Theorem 3.3). We show that a is unramified with respect to every discrete valuation v on F (X) over F (cf., [5, Prop. 2.1.8]). By Proposition A.9, a is unramified with respect to the discrete valuation associated to every point x ∈ X of codimension 1, i.e., ∂_{x}(a) = 0.

As X is projective, the valuation ring O_{v} of the valuation v dominates a point x ∈ X. It follows from Proposition A.11 that a belongs to the image of H^{n}(O_{X,x}, ℚ/ℤ(j)) ⟶ H^{n}(F (X), ℚ/ℤ(j)). As the local ring O_{X,x} is a subring of O_{v}, a belongs to the image of H^{n}(O_{v}, ℚ/ℤ(j)) ⟶ H^{n}(F (X), ℚ/ℤ(j)) and hence a is unramified with respect to v. □

Let T be a torus over F. By [12, Lemma 0.6], there is an exact sequence of tori 1 ⟶ T ⟶ T′ ⟶ P ⟶ 1 with T′ flasque and P quasi-split. The following theorem computes the unramified invariants of T in terms of the invariants of T′.

Theorem 5.4. There is a natural isomorphism

Inv_{nr}^{n}(T, ℚ/ℤ(j)) ≃ Inv^{n}(T′, ℚ/ℤ(j)).

Proof. Choose an exact sequence 1 ⟶ T′ ⟶ P′ ⟶ S ⟶ 1 with P′ a quasi-split torus. Let S′ is the cokernel of the composition T ⟶ T′ ⟶ P′. We have an exact sequence 1 ⟶ P ⟶ S′ ⟶ S ⟶ 1. As P is quasi-split, the latter exact sequence splits at the generic point of S and therefore, F (S′) is a purely transcendental field extension of F (S). It follows from Propositions 5.1, 5.2, and 5.3 that

Inv_{nr}^{n}(T, ℚ/ℤ(j)) ≃ H_{nr}^{n}(F (S′), ℚ/ℤ(j)) ≃ H_{nr}^{n}(F (S), ℚ/ℤ(j)) ≃

Inv_{nr}^{n}(T′, ℚ/ℤ(j)) = Inv^{n}(T′, ℚ/ℤ(j)). □

The following corollary is essentially equivalent to [12, Prop. 9.5] in the case when F is of zero characteristic.

Corollary 5.5. The isomorphism Inv(T, Br) ⟶^{∼} Pic(T) = H^{1}(F, T̂) identifies Inv_{nr}(T, Br) with the subgroup H^{1}(F, T̂′) of H^{1}(F, T̂) of all elements that are trivial when restricted to all cyclic subgroups of the decomposition group of T.

Proof. The description of H^{1}(F, T̂′) as a subgroup of H^{1}(F, T̂) is given in [12, Prop. 9.5], and this part of the proof is characteristic free. □

In view of Propositions 5.1 and 5.2 we can calculate the group of unramified cohomology for the function field of an arbitrary torus in terms of the invariants of a flasque torus:

Theorem 5.6. Let S be a torus over F and let 1 ⟶ T ⟶ P ⟶ S ⟶ 1 be a flasque resolution of S, i.e., T is flasque and P is quasi-split. Then there is a natural isomorphism

H_{nr}^{n}(F (S), ℚ/ℤ(j)) ≃ Inv^{n}(T, ℚ/ℤ(j)).

Corollary 5.7. A torus S has no nonconstant unramified degree 3 cohomology with values in ℚ/ℤ(2) universally, i.e., H_{nr}^{3}(K(S), ℚ/ℤ(2)) = H^{3}(K, ℚ/ℤ(2)) for any field extension K/F, if and only if S is a direct factor of a rational torus.

Proof. If S is a direct factor of a rational torus, then S has no nonconstant unramified cohomology by Proposition 5.1.

Conversely, let 1 ⟶ T ⟶ P ⟶ S ⟶ 1 be a flasque resolution of S. By Theorem 5.6, Inv^{3}(T_{K}, ℚ/ℤ(2))_{norm} = 0 for every K/F. It follows from Theorem 4.9 that T is invertible and hence S is a factor of a rational torus (see §4a). □

5b. The Brauer invariant for semisimple groups. The following theorem was proved by Bogomolov [2, Lemma 5.7] in characteristic zero:

Theorem 5.8. Let G be a (connected) semisimple group over a field F. Then Inv_{nr}(G, Br) = Inv(G, Br)_{const} = Br(F) and Br_{nr}(F (BG)) = Br(F).

Proof. Let G′ ⟶ G be a simply connected cover of G and C the kernel of G′ ⟶ G. By Theorem 2.4, we have

Inv(G, Br)_{norm} = Pic(G) = Ĉ(F).

As the map Ĉ(F) ⟶ Ĉ(F_{sep}) is injective, we can replace F by F_{sep} and assume that the group G is split.

Consider the variety T of maximal tori in G and the closed subscheme X ⊂ G × T of all pairs (g, T) with g ∈ T. The generic fiber of the projection X ⟶ T is the generic torus T_{gen} of G. Then T_{gen} is a maximal torus of G_{K}, where K := F (T ). We have an exact sequence

1 ⟶ C_{K} ⟶ T′_{gen} ⟶ T_{gen} ⟶ 1,

where T′_{gen} is the generic torus of G′.

The restriction Inv(G, Br)_{norm} ⟶ Inv(T_{gen}, Br)_{norm} can be identified with the connecting homomorphism Ĉ ⟶ H^{1}(K, T̂_{gen}). Note that the decomposition group of T_{gen} coincides with the Weyl group W of G by [34, Th. 1], hence H^{1}(K, T̂_{gen}) ≃ H^{1}(W, T̂_{gen}).

Let w be a Coxeter element in W.^{1} It is the product of reflections with respect to all simple roots (in some order). By [20, Lemma, p. 76], 1 is not an eigenvalue of w on the space of weights T̂′_{gen} ⊗ ℝ. Let W_{0} be the cyclic subgroup in W generated by w. It follows that the first term in the exact sequence

(T̂′_{gen})^{W_{0}} ⟶ Ĉ ⟶ H^{1}(W_{0}, T̂_{gen})

is trivial, i.e., the second map is injective. Hence every nonzero character χ in Ĉ restricts to a nonzero element in H^{1}(W_{0}, T̂_{gen}). It follows that the image of χ in H^{1}(W, T̂_{gen}) is ramified by Theorem 5.5, hence so is χ. □

Appendix A. Generalities

A-I. Proof of Theorem 2.2. Suppose that i(E_{gen}) = 0 for an H-invariant i of G. Let K/F be a field extension and I ⟶ Spec K a G-torsor. We need to show that i(I) = 0 in H(K).

Suppose first that K is infinite. Find a point x ∈ X(K) such that I is isomorphic to the pull-back of the classifying torsor with respect to x. Let x′ be a rational point of X_{K} above x and write O for the local ring O_{X_{K},x′}. The Kalgebra O is a regular local ring with residue field K. Therefore, the completion Ô is isomorphic to K[[t_{1}, t_{2}, …, t_{n}]] over K. Let L be the quotient field of Ô, a field extension of K(X). We have the following diagram of morphisms with a commutative square and three triangles:

\xymatrix{ & Spec K \ar[d] \ar[dr]^{x} \\ Spec L \ar[ur] \ar[r] \ar[d] & _{Spec Ô} \ar[r] \ar[u] \ar[d] & X \\ Spec K(X) \ar[r] & Spec O \ar[ur]}

The pull-back of the classifying torsor E ⟶ X via Spec K(X) ⟶ X is (E_{gen})_{K(X)}. The G-torsor I is the pull-back of E ⟶ X with respect to x. Let Ê be the pull-back of E ⟶ X via Spec Ô ⟶ X. Therefore, I is the pull-back of Ê. By a theorem of Grothendieck [19, Prop. 8.1], Ê is the pull-back of I with respect to Spec Ô ⟶ Spec(K). It follows that I_{L} ≃ (E_{gen})_{L} as torsors over L. Hence the images of i(I) and i(E_{gen}) in H(L) are equal and therefore, i(I)_{L} = 0. By Property 2.1, we have i(I) = 0.

1We owe the idea to use the Coxeter element and the reference below to S. Garibaldi.

If K is finite, we replace F by F ((t)) and K by K((t)). By the first part of the proof, i(I) belongs to the kernel of H(K) ⟶ H(K((t))) and hence is trivial by Property 2.1 again.

A-II. Decomposable elements. Let Γ be a profinite group and A a Γ-lattice. Write A^{Γ} for the subgroup of Γ-invariant elements in A. Let Γ′ ⊂ Γ be an open subgroup and choose representatives γ_{1}, γ_{2}, …, γ_{n} for the left cosets of Γ′ in Γ. We have the trace map Tr : A^{Γ′} ⟶ A^{Γ} defined by Tr(a) = ∑_{i=1}^{n} γ_{i}a.

Let S^{2}(A) be the symmetric square of A. Consider the quadratic trace map Qtr : A^{Γ′} ⟶ S^{2}(A)^{Γ} defined by Qtr(a) = ∑_{i<j}(γ_{i}a)(γ_{j}a). Write Dec(A) for the subgroup of decomposable elements in S^{2}(A)^{Γ} generated by the square (A^{Γ})^{2} of A^{Γ} and the elements Qtr(a) for all open subgroups Γ′ ⊂ Γ and all a ∈ A^{Γ′}.

Let B be another Γ-lattice. We write Dec(A, B) for the subgroup of (A⊗B)^{Γ} generated by elements of the form Tr(a ⊗ b) for all open subgroups Γ′ ⊂ Γ and all a ∈ A^{Γ′}, b ∈ B^{Γ′}.

There is a natural isomorphism

S^{2}(A ⊕ B) ≃ S^{2}(A) ⊕ (A ⊗ B) ⊕ S^{2}(B).

Moreover, the equality

Qtr(a + b) = Qtr(a) + (Tr(a) Tr(b) − Tr(ab)) + Qtr(b)

yields the decomposition

Dec(A ⊕ B) ≃ Dec(A) ⊕ Dec(A, B) ⊕ Dec(B).

A-III. Cup-products. Let 1 ⟶ T ⟶ P ⟶ Q ⟶ 1 be an exact sequence of tori. We consider the connecting maps

∂_{1} : H^{p}(F, T̂(i)) ⟶ H^{p+1}(F, Q̂(i))

for the exact sequence 0 ⟶ Q̂_{sep} ⟶ P̂_{sep} ⟶ T̂_{sep} ⟶ 0 of character Γlattices and

∂_{2} : H^{q}(F, Q̂^{∘}(j)) ⟶ H^{q+1}(F, T̂^{∘}(j))

for the dual sequence of lattices (see notation in §4b).

Lemma A.1. Let a ∈ H^{p}(F, T̂(i)) and b ∈ H^{q}(F, Q̂^{∘}(j)) with i+j ≤ 2. Then ∂_{1}(a) ∪ b = (−1)^{p+1}a ∪ ∂_{2}(b) in H^{p+q+1}(F, ℤ(i + j)), where the cup-product is defined in (4.3).

Proof. By [4, Ch. V, Prop. 4.1], the elements ∂_{1}(1_{T}) and ∂_{2}(1_{Q}) in

H^{1}(F, T̂_{sep}^{∘} ⊗ Q̂_{sep}) = Ext_{Γ}^{1}(T̂_{sep}, Q̂_{sep})

differ by a sign. Write τ for the isomorphism induced by permutation of the factors. By the standard properties of the cup-product, we have

∂_{1}(a) ∪ b = 1_{T} ∪ ∂_{1}(a) ∪ b
= ∂_{1}(1_{T}) ∪ a ∪ b
= (−1)^{pq}τ(∂_{1}(1_{T}) ∪ b ∪ a)
= (−1)^{pq+1}τ(∂_{2}(1_{Q}) ∪ b ∪ a)
= (−1)^{pq+1}τ(1_{Q} ∪ ∂_{2}(b) ∪ a)
= (−1)^{p+1}1_{Q} ∪ a ∪ ∂_{2}(b)
= (−1)^{p+1}a ∪ ∂_{2}(b).  □

A-IV. Cosimplicial abelian groups. Let A^{•} be a cosimplicial abelian group

\xymatrix{A^{0} \ar[r]^{d^{0}} \ar[r]_{d^{1}} & A^{1} \ar[r] \ar[r] \ar[r] & A^{2} \ar[r] \ar[r] \ar[r] \ar[r] & ⋯}

and write h_{∗}(A^{•}) for the homology groups of the associated complex of abelian groups. In particular,

h_{0}(A^{•}) = Ker [(d^{0} − d^{1}) : A^{0} ⟶ A^{1}].

We say that the cosimplicial abelian group A^{•} is constant if for every i, all the coface maps d_{j} : A^{i−1} ⟶ A^{i}, j = 0, 1, … i, are isomorphisms. In this case all the d_{j} are equal as d_{j} = s_{j}^{−1} = d_{j+1}, where the s_{j} are the codegeneracy maps. For a constant cosimplicial abelian group A^{•}, we have h_{0}(A^{•}) = A^{0} and h_{i}(A^{•}) = 0 for all i > 0. We will need the following straightforward statement.

Lemma A.2. Let 0 ⟶ A^{•} ⟶ B^{•} ⟶ C^{•} ⟶ D^{•} be an exact sequence of cosimplicial abelian groups with A^{•} a constant cosimplicial group. Then the sequence of groups 0 ⟶ A^{0} ⟶ h_{0}(B^{•}) ⟶ h_{0}(C^{•}) ⟶ h_{0}(D^{•}) is exact.

Let H be a contravariant functor from the category of schemes over F to the category of abelian groups. We say that H is homotopy invariant if for every vector bundle E ⟶ X over F, the induced map H(X) ⟶ H(E) is an isomorphism.

For an integer d > 0 consider the following property of the functor H:

Property A.3. For every closed subscheme Z of a scheme X with codim_{X}(Z) ≥ d, the natural homomorphism H(X) ⟶ H(X \ Z) is an isomorphism.

Let G be an algebraic group over a field F and choose a standard classifying G-torsor U ⟶ U/G. Let U^{i} denote the product of i copies of U. We have the G-torsors U^{i} ⟶ U^{i}/G.

Consider the cosimplicial abelian group A^{•} = H(U^{•}/G) with A^{i} = H(U^{i+1}/G) and coface maps A^{i−1} ⟶ A^{i} induced by the projections U^{i+1}/G ⟶ U^{i}/G.

Lemma A.4. Let H be a homotopy invariant functor satisfying Property A.3 for some d. Let U ⟶ U/G be a standard classifying G-torsor and U′ an open subset of a G-representation V′.

1. If codim_{V′}(V′ \ U′) ≥ d, then the natural homomorphism H(U/G) ⟶ H((U × U′)/G) is an isomorphism.

2. If codim_{V} (V \ U) ≥ d, then the cosimplicial group H(U^{•}/G) is constant.

Proof. 1. The scheme (U × U′)/G is an open subset of the vector bundle (U × V′)/G over U/G with complement of codimension at least d. The map in question is the composition H(U/G) ⟶ H((U × V′)/G) ⟶ H((U × U′)/G) and both maps in the composition are isomorphisms since H is homotopy invariant and satisfies Property A.3.

2. By the first part of the lemma applied to the G-torsor U^{i} ⟶ U^{i}/G and U′ = U, the map H(U^{i}/G) ⟶ H(U^{i+1}/G) induced by a projection U^{i+1}/G ⟶ U^{i}/G is an isomorphism. □

By Lemma A.4, if H is a homotopy invariant functor satisfying Property A.3 for some d, then the group H(U/G) does not depend on the choice of the representation V and the open set U ⊂ V provided codim_{V} (V \ U) ≥ d. Following [33], we denote this group by H(BG).

Example A.5. The split torus T = (𝔾_{m})^{n} over F acts freely on the product U of n copies of 𝔸^{r+1} \ {0} with U/T ≃ (ℙ^{r})^{n}, i.e., BT is “approximated” by the varieties (ℙ^{r})^{n} if “r >> 0.” We then have CH^{∗}(BT) = S^{∗}(T̂), where S^{∗} represents the symmetric algebra and T̂ is the character group of T (see [14, p. 607]). In particular, Pic(BT) = CH^{1}(BT) = T̂. More generally, by the Künneth formula [15, Prop. 3.7],

H_{Zar}^{∗}(BT, K_{∗}) ≃ CH^{∗}(BT) ⊗ K_{∗}(F) ≃ S^{∗}(T̂) ⊗ K_{∗}(F),

where K_{n}(F) is the Quillen K-group of F and K_{n} is the Zariski sheaf associated to the presheaf U ↦ K_{n}(U).

A-V. Étale cohomology. Let A be a sheaf of abelian groups on the big étale site over F. For a scheme X and a closed subscheme Z ⊂ X we write H_{Z}^{∗}(X, A) for the étale cohomology group of X with support in Z and values in A [29, Ch. III, §1]. Write X^{(i)} for the set of points in X of codimension i. For a point x ∈ X^{(1)} set

H_{x}^{∗}(X, A) = colim_{x∈U} H_{\overline{\{x\}}∩U}^{∗}(U, A),

where the colimit is taken over all open subsets U ⊂ X containing x. If X is a variety, write

∂_{x} : H^{∗}(F (X), A) ⟶ H_{x}^{∗+1}(X, A)

for the residue homomorphisms arising from the coniveau spectral sequence [9, 1.2]

(A.1)  E_{1}^{p,q} = ∐_{x∈X^{(p)}} H_{x}^{p+q}(X, A) ⇒ H^{p+q}(X, A).

Let f : Y ⟶ X be a dominant morphism of varieties over F, y ∈ Y^{(1)}, and x = f(y). If x ∈ X^{(1)}, there is a natural homomorphism f_{y}^{∗} : H_{x}^{∗}(X, A) ⟶ H_{y}^{∗}(Y, A). The following lemma is straightforward.

Lemma A.6. Let f : Y ⟶ X be a dominant morphism of varieties over F, y ∈ Y^{(1)} and x = f(y).

(1) If x is the generic point of X, then the composition

H^{∗}(F (X), A) ⟶^{f^{∗}} H^{∗}(F (Y), A) ⟶^{∂_{y}} H_{y}^{∗+1}(Y, A)

is trivial.

(2) If x ∈ X^{(1)}, the diagram

\xymatrix{H^{∗}(F (X), A) \ar[d]_{f^{∗}} \ar[r]^{∂_{x}} & H_{x}^{∗+1}(X, A) \ar[d]_{f_{y}^{∗}} \\ H^{∗}(F (Y), A) \ar[r]^{∂_{y}} & H_{y}^{∗+1}(Y, A).}

is commutative.

Lemma A.7. Let X be a geometrically irreducible variety, Z ⊂ X a closed subvariety of codimension 1, and x the generic point of Z. Let P be a variety over F such that P (K) is dense in P for every field extension K/F with K infinite, and let y be the generic point of Z × P in Y := X × P. Then the homomorphism f_{y}^{∗} : H_{x}^{∗}(X, A) ⟶ H_{y}^{∗}(Y, A) induced by the projection f : Y ⟶ X is injective.

Proof. Assume first that the field F is infinite. An element α ∈ H_{x}^{∗}(X, A) is represented by an element h ∈ H_{Z∩U}^{∗}(U, A) for a nonempty open set U ⊂ X containing x. If α belongs to the kernel of f_{y}^{∗} : H_{x}^{∗}(X, A) ⟶ H_{y}^{∗}(Y, A), then there is an open subset W ⊂ U × P containing y such that h belongs to the kernel of the composition

g : H_{Z∩U}^{∗}(U, A) ⟶ H_{(Z∩U)×P}^{∗} (U × P, A) ⟶ H_{(Z×P)∩W}^{∗} (W, A).

As F is infinite, by the assumption on P, there is a rational point t ∈ P in the image of the dominant composition (Z × P) ∩ W ↪ Z × P ⟶ P. We have (U × t) ∩ W = U′ × t for an open subset U′ ⊂ U such that x ∈ U′. Composing g with the homomorphism H_{(Z×P)∩W}^{∗} (W, A) ⟶ H_{Z∩U′}^{∗}(U′, A) induced by the morphism (U′, Z ∩U′) ⟶ (W, (Z ×P)∩W), u ↦ (u, t), we see that h belongs to the kernel of the restriction homomorphism H_{Z∩U}^{∗}(U, A) ⟶ H_{Z∩U′}^{∗}(U′, A), hence the image of α in H_{x}^{∗}(X, A) is trivial.

Suppose now that F is a finite field. Choose a prime integer p and an infinite algebraic pro-p-extension L/F. By the first part of the proof, the statement holds for the variety X_{L} over L. By the restriction-corestriction argument, Ker(f_{y}^{∗}) is a p-primary torsion group. Since this holds for every prime p, we have Ker(f_{y}^{∗}) = 0. □

Corollary A.8. Let E ⟶ X be a G-torsor over a geometrically irreducible variety X with E a G-rational variety and consider the first projection p : E^{2}/G ⟶ X. Let x ∈ X and y ∈ E^{2}/G be points of codimension 1 such that p(y) = x. Then the homomorphism p_{y}^{∗} : H_{x}^{∗}(X, A) ⟶ H_{y}^{∗}(E^{2}/G, A) is injective.

Proof. Choose a linear G-space V and a nonempty G-variety U that is Gisomorphic to open subschemes of E and V. We can replace the variety E^{2}/G by (E × U)/G, an open subscheme in the vector bundle (E × V)/G over X. Shrinking X around x we may assume that the vector bundle is trivial, i.e., (E × U)/G is isomorphic to an open subscheme in X × V. The statement then follows from Lemma A.7. □

Proposition A.9. In the conditions of Corollary A.8, let h ∈ H^{∗}(F (X), A)_{bal}. Then ∂_{x}(h) = 0 for every point x ∈ X of codimension 1.

Proof. Let y ∈ E^{2}/G be the point of codimension 1 such that p_{1}(y) = x. As p_{2}(y) is the generic point of X, by Lemma A.6(1), ∂_{y}(h′) = 0, where h′ = p_{1}^{∗}(h) = p_{2}^{∗}(h) in H^{∗}(F (E^{2}/G), A). It follows from Lemma A.6(2) that ∂_{x}(h) is in the kernel of (p_{1})_{y}^{∗} : H_{x}^{∗}(X, A) ⟶ H_{y}^{∗}(E^{2}/G, A) and hence is trivial by Corollary A.8. □

The sheaf H^{∗}(ℚ/ℤ(j)) defined in §3 has a flasque resolution related to the Cousin complex by [9, §2] (for the p-components with p ≠ char F) and [18, Th. 1.4] (for the p-component with p = char F > 0):

0 ⟶ H^{n}(ℚ/ℤ(j)) ⟶ ∐_{x∈X^{(0)}} i_{x∗}H_{x}^{n}(X, ℚ/ℤ(j)) ⟶

∐_{x∈X^{(1)}} i_{x∗}H_{x}^{n+1}(X, ℚ/ℤ(j)) ⟶ ⋯,

where i_{x} : Spec F (x) ⟶ X are the canonical morphisms. In particular, we have:

Proposition A.10. Let X be a smooth variety over F. The sequence

0 ⟶ H_{Zar}^{0}(X, H^{∗}(ℚ/ℤ(j))) ⟶ H^{∗}(F (X), ℚ/ℤ(j)) ⟶^{∂} ∐_{x∈X^{(1)}} H_{x}^{∗+1}(X, ℚ/ℤ(j)),

where ∂ = ∐ ∂_{x}, is exact.

Proposition A.11. Let X be a smooth variety over F and x ∈ X. The sequence

0 ⟶ H^{∗}(O_{X,x}, ℚ/ℤ(j))) ⟶ H^{∗}(F (X), ℚ/ℤ(j)) ⟶^{∂} _{∐_{x′∈X^{(1)}} ; x′∈\overline{\{x\}}}H_{x′}^{∗+1}(X, ℚ/ℤ(j))

is exact.

Appendix B. Spectral sequences

B-I. Hochschild-Serre spectral sequence. Let

A ⟶^{W} B ⟶^{V} C

be additive left exact functors between abelian categories with enough injective objects. If W takes injective objects to V -acyclic ones, there is a spectral sequence

E_{2}^{p,q} = R^{p}V (R^{q}W (A)) ⇒ R^{p+q}(V W)(A)

for every complex A in A bounded from below.

We have exact triangles in the derived category of B:

(B.1)  τ_{≤n}RW (A) ⟶ RW (A) ⟶ τ_{≥n+1}RW (A) ⟶ τ_{≤n}RW (A)[1],
(B.2) τ_{≤n−1}RW (A) ⟶ τ_{≤n}RW (A) ⟶ R^{n}W (A)[−n] ⟶ τ_{≤n−1}RW (A)[1].

The filtration on R^{n}(V W)(A) is defined by

F^{j}R^{n}(V W)(A) = Im(R^{n}V (τ_{≤(n−j)}RW (A)) ⟶ R^{n}V (RW (A)) = R^{n}(V W)(A)).

As τ_{≥n+1}RW (A) is acyclic in degrees ≤ n, the morphism

R^{n}V (τ_{≤n}RW (A)) ⟶ R^{n}V (RW (A)) = R^{n}(V W)(A)

is an isomorphism, in particular, F^{0}R^{n}(V W)(A) = R^{n}(V W)(A).

The edge homomorphism is defined as the composition

R^{n}(V W)(A) ⟶^{∼} R^{n}V (τ_{≤n}RW (A)) ⟶ R^{n}V (R^{n}W (A)[−n]) = V (R^{n}W (A)).

Moreover, the kernel F^{1}R^{n}(V W)(A) of the edge homomorphism is isomorphic to R^{n}V (τ_{≤n−1}RW (A)). We define the morphism d_{n} as the composition

d_{n} : F^{1}R^{n}(V W)(A) ⟶ R^{n}V (R^{n−1}W (A)[−n+1]) = R^{1}V (R^{n−1}W (A)) = E_{2}^{1,n−1}.

B-II. First spectral sequence. Let X be a smooth variety over a field F. We have the functors

Sheaves_{ét}(X) ⟶^{q_{∗}} Sheaves_{ét}(F) ⟶^{V} Ab,

where q_{∗} is the push-forward map for the structure morphism q : X ⟶ Spec(F) and V (M) = H^{0}(F, M).

Consider the Hochschild-Serre spectral sequence

(B.3)  E_{2}^{p,q} = H^{p}(F, H^{q}(X_{sep}, ℤ(2)) ⇒ H^{p+q}(X, ℤ(2)).

Set Δ(i) := Rq_{∗}(ℤ(i)) for i = 1 or 2. Then Δ(i) is the complex of étale sheaves on F concentrated in degrees ≥ 1. The j^{th} term F^{j}H^{n}(X, ℤ(i)) of the filtration on H^{n}(X, ℤ(i)) coincides with the image of the canonical homomorphism

H^{n}(F, τ_{≤(n−j)}Δ(i)) ⟶ H^{n}(F, Δ(i)) = H^{n}(X, ℤ(i)).

Let M be a Γ-lattice viewed as an étale sheaf over F. Note that there are canonical isomorphisms

(B.4)  H^{∗}(F, M^{∘} ⊗ Δ(i)) = Ext_{F}^{∗} (M, Δ(i)) = Ext_{X}^{∗}(q^{∗}M, ℤ(i)),

where M^{∘} := Hom(M, ℤ) is the dual lattice.

Consider also the following product map

ℤ(1)⊗^{L} Δ(1) ⟶ Rq_{∗}(q^{∗}ℤ(1)⊗^{L} ℤ(1)) ⟶ Rq_{∗}(ℤ(1)⊗^{L} ℤ(1)) ⟶ Rq_{∗}(ℤ(2)).

The complex ℤ(1) ⊗^{L} τ_{≤2}Δ(1) is trivial in degrees > 3, hence we have a commutative diagram

[figure: ℤ(1) ⊗^{L} τ_{≤2}Δ(1); prod; τ_{≤3}Rq_{∗}(ℤ(2)); =; τ_{≤3}Δ(2); ℤ(1) ⊗^{L} Δ(1); prod; Rq_{∗}(ℤ(2)); =; Δ(2).]

There are canonical morphisms from (B.2):

h_{2} : τ_{≤2}Δ(1)[2] ⟶ H^{2}(X_{sep}, ℤ(1)),
h_{3} : τ_{≤3}Δ(1)[3] ⟶ H^{3}(X_{sep}, ℤ(2)).

Consider an element

δ ∈ H^{1}(F, M ⊗ F_{sep}^{×}) = Ext_{F}^{1} (M^{∘}, 𝔾_{m}_{,F}) = Ext_{F}^{2} (M^{∘}, ℤ(1)),

and view δ as a morphism

δ : M^{∘} ⟶ ℤ(1)[2]

in D^{+}(Sheaves_{ét}(F)).

The following diagram

\xymatrix{M^{∘} ⊗ Δ(1)[2] \ar[r]^{δ⊗1} & ℤ(1) ⊗^{L} Δ(1)[4] \ar[r]^{prod} & Δ(2)[4] \\ M^{∘} ⊗ τ_{≤2}Δ(1)[2] \ar[d]_{1⊗h_{2}} \ar[u]^{(1⊗i_{2})[2]} \ar[r]^{δ⊗1} & ℤ(1) ⊗^{L} τ_{≤2}Δ(1)[4] \ar[d]_{1⊗h_{2}} \ar[u]^{(1⊗i_{2})[4]} \ar[r]^{prod} & τ_{≤3}Δ(2)[4] \ar[d]_{h_{3}} \ar[u]^{(i_{3})[4]} \\ M^{∘} ⊗ H^{2}(X_{sep}, ℤ(1)) \ar[r]^{δ⊗1} & ℤ(1) ⊗^{L} H^{2}(X_{sep}, ℤ(1))[2] \ar[r]^{prod} & H^{3}(X_{sep}, ℤ(2))[1],}

where i_{2} : τ_{≤2}Δ(1) ⟶ Δ(1) and i_{3} : τ_{≤3}Δ(2) ⟶ Δ(2) are natural morphisms, is commutative.

By (B.4), we have

H^{0}(F, M^{∘} ⊗ Δ(1)[2]) = Ext_{F}^{2} (M, Δ(1)) = Ext_{X}^{2}(q^{∗}M, ℤ(1)).

Furthermore, the diagram above yields a commutative square

\xymatrix{Ext_{X}^{2}(q^{∗}M, ℤ(1)) \ar[d]_{d_{2}} \ar[r]^{q^{∗}(δ) ∪ −} & F^{1}H^{4}(X, ℤ(2)) \ar[d]_{d_{4}} \\ Hom_{Γ}(M, H^{2}(X_{sep}, ℤ(1)) \ar[r]^{j} & H^{1}(F, H^{3}(X_{sep}, ℤ(2))),}

where d_{2} is the edge map coming from the spectral sequence

(B.5)  Ext_{F}^{p} (M, H^{q}(X_{sep}, ℤ(1))) ⇒ Ext_{X}^{p+q}(q^{∗}M, ℤ(1))

and j coincides with the composition

Hom_{Γ}(M, H^{2}(X_{sep}, ℤ(1)) = H^{0}(F, M^{∘} ⊗ H^{2}(X_{sep}, ℤ(1))) ⟶^{δ ∪ −}

H^{1}(F, F_{sep}^{×} ⊗ H^{2}(X_{sep}, ℤ(1))) ⟶^{ρ} H^{1}(F, H^{3}(X_{sep}, ℤ(2))),

with ρ given by the product map.

Now suppose the group H^{2}(X_{sep}, ℤ(1)), which is canonically isomorphic to Pic(X_{sep}), is a lattice. Let M = Pic(X_{sep}) and consider the torus T over F with T̂_{sep} = M. It follows that

δ ∈ H^{1}(F, T^{∘}) = H^{1}(F, T̂_{sep} ⊗ F_{sep}^{×}) = H^{2}(F, T̂_{sep} ⊗ ℤ(1)),

where T^{∘} is the dual torus. Note that δ ∪ 1_{M} = δ, where

1_{M} ∈ H^{0}(F, M^{∘} ⊗ H^{2}(X_{sep}, ℤ(1))) = End_{Γ}(M)

is the identity.

The top map in the last diagram is given by the pairing

(B.6)  H^{1}(X, T^{0}) ⊗ H^{1}(X, T) ⟶ F^{1}H^{4}(X, ℤ(2)),
(B.7)  a ⊗ b ↦ a ∪ b

defined as the cup-product in (4.3),

H^{2}(X, T̂(1)) ⊗ H^{2}(X, T̂^{∘}(1)) ⟶ F^{1}H^{4}(X, ℤ(2)),

if we identify Ext_{X}^{2}(q^{∗}M, ℤ(1)) with H^{2}(X, T̂^{∘}(1)) = H^{1}(X, T).

In this case, the homomorphism

(B.8)  ρ : H^{1}(F, T^{∘}) ⟶ H^{1}(F, H^{3}(X_{sep}, ℤ(2)))

is given by the product homomorphism

T^{∘}(F_{sep}) = F_{sep}^{×} ⊗ T̂_{sep} = F_{sep}^{×} ⊗ Pic(X_{sep}) ⟶ H^{3}(X_{sep}, ℤ(2)).

A T -torsor E ⟶ X is called universal if the class of E in H^{1}(X, T) = Ext_{X}^{2}(q^{∗}M, ℤ(1)) satisfies d_{2}([E]) = 1_{M} (see [7]).

Commutativity of the previous diagram gives:

Proposition B.1. Let X be a smooth variety over F such that Pic(X_{sep}) is a lattice. Let T be the torus over F satisfying T̂_{sep} = Pic(X_{sep}) and let E be a universal T -torsor over X with the class [E] ∈ H^{1}(X, T). Then for every δ ∈ H^{1}(F, T^{∘}), we have

d_{4}(q^{∗}(δ) ∪ [E]) = ρ(δ),

where d_{4} : F^{1}H^{4}(X, ℤ(2)) ⟶ H^{1}(F, H^{3}(X_{sep}, ℤ(2))) is the map induced by the Hochschild-Serre spectral sequence (B.3) and the cup-product is taken for the pairing (B.6).

B-III. Second spectral sequence. We assume that H^{3}(X_{sep}, ℤ(2)) = 0, hence in particular E_{2}^{0,3} = 0 in the spectral sequence (B.3) and so E_{∞}^{2,2} ⊂ E_{2}^{2,2}. Therefore, we have a canonical map

e_{4} : F^{2}H^{4}(X, ℤ(2)) ⟶ E_{∞}^{2,2} ↪ E_{2}^{2,2} = H^{2}(F, H^{2}(X_{sep}, ℤ(2)).

Let N be a Γ-lattice. Consider an element

γ ∈ H^{2}(F, N ⊗ F_{sep}^{×}) = Ext_{F}^{2} (N^{∘}, 𝔾_{m}_{,F}) = Ext_{F}^{3} (N^{∘}, ℤ(1)),

and view γ as a morphism

γ : N^{∘} ⟶ ℤ(1)[3]

in D^{+}(Sheaves_{ét}(F)).

As above, the commutative diagram

\xymatrix{N^{∘} ⊗ Δ(1)[1] \ar[r]^{γ⊗1} & ℤ(1) ⊗^{L} Δ(1)[4] \ar[r]^{prod} & Δ(2)[4] \\ N^{∘} ⊗ τ_{≤1}Δ(1)[1] \ar[d]_{1⊗h_{1}} \ar[u]^{(1⊗i_{1})[1]} \ar[r]^{γ⊗1} & ℤ(1) ⊗^{L} τ_{≤1}Δ(1)[4] \ar[d]_{1⊗h_{1}} \ar[u]^{(1⊗i_{1})[4]} \ar[r]^{prod} & τ_{≤2}Δ(2)[4] \ar[d]_{h_{2}} \ar[u]^{(i_{2})[4]} \\ N^{∘} ⊗ H^{1}(X_{sep}, ℤ(1)) \ar[r]^{γ⊗1} & ℤ(1) ⊗^{L} H^{1}(X_{sep}, ℤ(1))[3] \ar[r]^{prod} & H^{2}(X_{sep}, ℤ(2))[2],}

where i_{1}, i_{2}, h_{1} and h_{2} are defined in a similar fashion as in §B-II, yields a commutative square

\xymatrix{Ext_{X}^{1}(q^{∗}N, ℤ(1)) \ar[d]_{d_{1}} \ar[r]^{q^{∗}(γ) ∪ −} & F^{2}H^{4}(X, ℤ(2)) \ar[d]_{e_{4}} \\ Hom_{Γ}(N, H^{1}(X_{sep}, ℤ(1)) \ar[r]^{k} & H^{2}(F, H^{2}(X_{sep}, ℤ(2))),}

where d_{1} is the edge map coming from the spectral sequence

Ext_{F}^{p} (N, H^{q}(X_{sep}, ℤ(1))) ⇒ Ext_{X}^{p+q}(q^{∗}N, ℤ(1))

and k coincides with the composition

Hom_{Γ}(N, H^{1}(X_{sep}, ℤ(1)) = H^{0}(F, N^{∘} ⊗ H^{1}(X_{sep}, ℤ(1))) ⟶^{γ ∪ −}

H^{2}(F, F_{sep}^{×} ⊗ H^{1}(X_{sep}, ℤ(1))) ⟶ H^{2}(F, H^{2}(X_{sep}, ℤ(2)))

with the last homomorphism given by the product map.

Suppose N is a Γ-lattice in F_{sep}[X]^{×} such that the composition N ↪ F_{sep}[X]^{×} ⟶ F_{sep}[X]^{×}/F_{sep}^{×} is an isomorphism. Consider the torus Q with Q̂_{sep} = N, so that γ ∈ H^{2}(F, Q^{∘}).

Note that γ ∪ i_{N} = γ, where

i_{N} ∈ H^{0}(F, N^{∘} ⊗ H^{1}(X_{sep}, ℤ(1))) = Hom_{Γ}(N, F_{sep}[X]^{×})

is the embedding.

The top map in the previous diagram is given by the pairing

\begin{matrix}H^{2}(X, Q^{0}) ⊗ H^{0}(X, Q) & ⟶ F^{2}H^{4}(X, ℤ(2)) \\ a ⊗ b & ⟼ a ∪ b, \end{matrix} ; (B.9)

defined as the cup-product in (4.3),

H^{3}(X, Q̂(1)) ⊗ H^{1}(X, Q̂^{∘}(1))) ⟶ H^{4}(X, ℤ(2)),

if we identify Ext_{X}^{1}(q^{∗}N, ℤ(1)) with H^{1}(X, Q̂^{∘}(1)) = H^{0}(X, Q).

The inclusion of Q̂_{sep} into F_{sep}[X]^{×} yields a morphism ε : X ⟶ Q that can be viewed as an element of H^{0}(X, Q). Consider the map

(B.10)  μ : H^{2}(F, Q^{∘}) ⟶ H^{2}(F, H^{2}(X_{sep}, ℤ(2)))

given by composition with the product homomorphism

Q^{∘}(F_{sep}) = F_{sep}^{×} ⊗ Q̂_{sep} ⟶ F_{sep}^{×} ⊗ H^{1}(X_{sep}, ℤ(1)) ⟶ H^{2}(X_{sep}, ℤ(2)).

We have proved:

Proposition B.2. Let X be a smooth variety over F such that H^{3}(X_{sep}, ℤ(2)) =

0. Let N be a Γ-lattice in F_{sep}[X]^{×} such that the composition N ↪ F_{sep}[X]^{×} ⟶ F_{sep}[X]^{×}/F_{sep}^{×} is an isomorphism. Let Q be the torus over F satisfying Q̂_{sep} = N. Then for every γ ∈ H^{2}(F, Q^{∘}), we have

e_{4}(q^{∗}(γ) ∪ ε) = μ(γ),

where e_{4} : F^{2}H^{4}(X, ℤ(2)) ⟶ H^{2}(F, H^{2}(X_{sep}, ℤ(2))) is the map induced by the Hochschild-Serre spectral sequence (B.3) and the cup-product is taken for the pairing (B.9).

B-IV. Relative étale cohomology. Let X be a smooth variety over F. Following [23, §3], we define the relative étale cohomology groups as follows. Recall that Δ(i) = Rq_{∗}(ℤ(i)) for i = 1 and 2, where q : X ⟶ Spec(F) is the structure morphism, and let Δ′(i) be the cone of the natural morphism ℤ(i) ⟶ Δ(i) in D_{+}(Sheaves_{ét}(F)). Define H^{∗}(X/F, ℤ(2)) := H^{∗}(F, Δ′(2)). There is an infinite exact sequence

… ⟶ H^{i}(F, ℤ(2)) ⟶ H^{i}(X, ℤ(2)) ⟶ H^{i}(X/F, ℤ(2)) ⟶ H^{i+1}(F, ℤ(2)) ⟶ ⋯

If X has a rational point, we have

H^{i}(X/F, ℤ(2)) = \overline{H}^{i}(X, ℤ(2)) := H^{i}(X, ℤ(2))/H^{i}(F, ℤ(2)).

There is a Hochschild-Serre type spectral sequence [23, §3]

(B.11)  E_{2}^{p,q} = H^{p}(F, H^{q}(X_{sep}/F_{sep}, ℤ(2))) ⇒ H^{p+q}(X/F, ℤ(2)),

and we have by [23, Lemma 3.1] that

H^{q}(X_{sep}/F_{sep}, ℤ(2)) = \{ \begin{matrix}0, & if q ≤ 0; \\ uniquely divisible group, & if q = 1; \\ \overline{H}_{Zar}^{0}(X_{sep}, K_{2}), & if q = 2; \\ H_{Zar}^{1}(X_{sep}, K_{2}), & if q = 3. \end{matrix}

It follows that E_{2}^{p,q} = 0 if q ≤ 1 and p > 0. Comparing the spectral sequences (B.3) and (B.11), by Proposition B.1 we have:

Proposition B.3. Let X be a smooth variety over F such that X(F) ≠ ∅. If H_{Zar}^{0}(X_{sep}, K_{2}) = K_{2}(F_{sep}), then the spectral sequence (B.11) yields an exact sequence

0 ⟶ H^{1}(F, H_{Zar}^{1}(X_{sep}, K_{2})) ⟶^{α} \overline{H}^{4}(X, ℤ(2)) ⟶

\overline{H}^{4}(X_{sep}, ℤ(2))^{Γ} ⟶ H^{2}(F, H_{Zar}^{1}(X_{sep}, K_{2})).

If, moreover, the group Pic(X_{sep}) is a lattice and T is the torus over F such that T̂_{sep} = Pic(X_{sep}), then α(ρ(δ)) = q^{∗}(δ) ∪ [E] for every δ ∈ H^{1}(F, T^{∘}), where ρ is defined in (B.8) and E is a universal T -torsor over X.

Comparing the spectral sequences (B.3) and (B.11), by Proposition B.2 we have:

Proposition B.4. Let X be a smooth variety over F such that X(F) ≠ ∅. If H_{Zar}^{1}(X_{sep}, K_{2}) = 0, then the spectral sequence (B.11) yields an exact sequence

0 ⟶ H^{2}(F, \overline{H}_{Zar}^{0}(X_{sep}, K_{2})) ⟶^{β} \overline{H}^{4}(X, ℤ(2)) ⟶

\overline{H}^{4}(X_{sep}, ℤ(2))^{Γ} ⟶ H^{3}(F, \overline{H}_{Zar}^{0}(X_{sep}, K_{2})).

If N is a Γ-lattice in F_{sep}[X]^{×} such that the composition N ↪ F_{sep}[X]^{×} ⟶ F_{sep}[X]^{×}/F_{sep}^{×} is an isomorphism and Q is the torus over F satisfying Q̂_{sep} = N, then β(μ(γ)) = q^{∗}(γ) ∪ ε for every γ ∈ H^{2}(F, Q^{∘}), where μ is defined in (B.10) and ε ∈ H^{0}(X, Q) is given by the inclusion of Q̂_{sep} into F_{sep}[X]^{×}.

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

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

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

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