THIN SEVERI–BRAUER VARIETIES

MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL

À Jacques Tits, pour son 80^{ème} anniversaire

Abstract. Severi–Brauer varieties are twisted forms of projective spaces (in the sense of Galois cohomology) and are associated in a functorial way to central simple algebras. Similarly quadrics are related to algebras with involution. Since thin projective spaces are finite sets, thin Severi–Brauer varieties are finite sets endowed with a Galois action; they are associated to étale algebras. Similarly, thin quadrics are étale algebras with involution. We discuss embeddings of thin Severi–Brauer varieties and thin quadrics in Severi–Brauer varieties and quadrics as geometric analogues of embeddings of étale algebras into central simple algebras (with or without involution), and consider the geometric counterpart of the Clifford algebra construction.

1. Introduction

Paraphrasing Tits [18], “thin” geometric objects are characterised by the fact that their automorphism groups are the Weyl groups of the groups of automorphisms of the corresponding classical geometric objects.^{1} Thus a thin (n − 1)-dimensional projective space is a finite set of n elements and a 2(n−1)-dimensional thin quadric is a double covering of a set of n elements. Our starting point is to consider twisted forms (in the sense of Galois cohomology) of projective spaces and quadrics. Fixing a base field, a twist is given by a continuous action of the absolute Galois group of the field. Twisted forms of projective spaces over a field F occur as Severi–Brauer varieties associated to central simple algebras over F, and twisted quadrics correspond to central simple algebras with quadratic pair. Similarly, twisted forms of finite sets are associated to algebraic objects, namely étale algebras, by a correspondence known as Grothendieck’s version of Galois theory. Double coverings correspond to étale algebras with involution. Moreover the Grassmannian of maximal linear subspaces of a split quadric of dimension 2(n − 1) corresponds as a geometric object to the Clifford algebra of a central algebra with quadratic pair as algebraic object. We call this Grassmannian the Clifford set of the quadric. The first aim of the paper is to study equivariant embeddings of finite twisted objects into corresponding classical objects. For example a general embedding of a finite set of n elements in a projective space of dimension n − 1 corresponds to the embedding of an étale algebra of rank n in a central simple algebra of degree n (or to the embedding of a torus of rank n − 1 into a twisted form of SL_{n}, see [13, S2]). The second aim is to discuss in detail the geometric side of the Clifford construction, and to relate it to Clifford algebras for étale algebras with involution and for central simple algebras with quadratic pairs. In Section 2 we consider Severi–Brauer varieties, in Section 3 quadrics, and in Section 4 the various types of Clifford constructions, where we introduce a special type of Severi–Brauer variety to describe the geometry of a Clifford algebra. Finally we sketch in Section 5 a discussion of two exceptional phenomena. First we describe consequences of the fact that the Dynkin diagrams A_{3} and D_{3} are equal. This is related on the geometric side with the Klein quadric and on the algebraic side with the resolution of the equation of degree 4. Then we look at triality, that is, outer automorphisms of order 3 of D_{4}. One origin of this paper is the observation of Tits [16, p. 287] that the two phenomena described above also occur over the “field of characteristic 1”. Also, his short geometric proof of the fact that two quaternion algebras whose tensor product is not division share a common quadratic subfield [19] is a beautiful illustration of the power of geometric insights into algebraic questions. Another source of inspiration was [14], where Saltman discusses the embedding of étale algebras with involution into central simple algebras with involution. Pairs consisting of a maximal étale subalgebra of a central simple algebra (and corresponding pairs of algebras with involutions) are also considered in [1] and [13]. Triality in relation with étale algebras is extensively discussed in [9]. We plan to come back to other aspects, in particular to unitary and symplectic analogues, in [10].

The second author is supported in part by the F.R.S.–FNRS (Belgium).

1In [16] Tits says that such objects are defined over the “field of characteristic 1”.

2. Severi–Brauer projective spaces

Γ-projective spaces and central simple algebras. Throughout most of this work, F is an arbitrary field. We denote by F_{s} a separable closure of F and by Γ the absolute Galois group Gal(F_{s}/F), which is a profinite group. A Γ-projective space P of dimension n − 1 over a field F is a projective space ℙ^{n−1}(F_{s}) of dimension n − 1 endowed with the discrete topology and a continuous action of Γ by collineations, denoted by x ↦ ^{γ}x, x ∈ P, γ ∈ Γ. For every integer k ∈ {1, …, n}, the Grassmannian Gr_{k−1}(P) of (k − 1)-dimensional linear varieties in P carries an action of Γ. We view the set:

\overline{P} = \{x ∈ P | ^{γ}x = x for all γ ∈ Γ\} ⊂ P

as a scheme over F. The embedding \overline{P} ⊂ P induces an isomorphism

β: \overline{P} ×_{Spec F} Spec F_{s} ⟶^{∼} P

of Γ-spaces. From now on we shall not distinguish between P and the pair (\overline{P} , β). We have a similar construction for Grassmannians. Continuous actions of Γ are related with Galois cohomology, as we now illustrate in the case of Γ-projective spaces. There is a standard action of Γ on ℙ^{n−1}(F_{s}) induced by the action of Γ on F_{s}. We denote it by x ↦ γ̄x. This standard action induces in turn a Γ-action on the group PGL_{n}(F_{s}) of automorphisms of the projective space ℙ^{n−1}(F_{s}), given by φ ↦ γ̄(φ) = γ̄φγ̄^{−1}, φ ∈ PGL_{n}(F_{s}). A 1-cocycle of Γ with values in PGL_{n} is a continuous map Γ → PGL_{n}(F_{s}), γ ↦ a_{γ}, such that

a_{γδ} = a_{γ} γ̄(a_{δ}) for all γ, δ ∈ Γ.

Two cocycles a_{γ}, a′_{γ} are cohomologous or equivalent if there exists a ∈ PGL_{n}(F_{s}) such that a′_{γ} = aa_{γ} γ̄(a)^{−1}). We denote by H^{1}(Γ, PGL_{n}) the set of equivalence classes of 1-cocycles. Let P = (\overline{P} , β) be any Γ-projective space. The map a_{γ} = βγβ^{−1}γ̄^{−1}, γ ∈ Γ, is an element of PGL_{n}(F_{s}) and defines a 1-cocycle. Moreover one shows that isomorphic Γ-projective spaces lead to cohomologous cocycles. Let Proj_{Γ}^{n−1} be the groupoid^{2} of Γ-projective spaces and let Iso (Proj_{Γ}^{n−1}) be its set of isomorphism classes.

Proposition 2.1. The map (P, β) ↦ a_{γ} induces a bijection

Iso (Proj_{Γ}^{n−1}) ⟶^{∼} H^{1}(Γ, PGL_{n}).

Proof. We describe a map in the opposite direction and refer to [15, p. 161] for details. Given a 1-cocycle (a_{γ})_{γ∈Γ}, we define a Γ-action on P = ℙ^{n−1}(F_{s}) as

x ↦^{γ} x =^{a_{γ} γ̄} x, x ∈ P.  □

The Γ-projective spaces we consider arise from the following construction. Recall that an F -algebra A is central simple of degree n if there exists an isomorphism α: A ⊗_{F} F_{s} ⟶^{∼} End_{F_{s}}(V) for a vector space V of dimension n over F_{s}. Since the group of F_{s}-automorphisms of End_{F_{s}}(V) is isomorphic to PGL_{n}(F_{s}), a 1-cocycle defing a Γ-projective space of dimension n − 1 is also a 1-cocycle defining a central simple algebra of degree n. Thus denoting by CSA_{F}^{n} the groupoid of central simple algebras of degree n, the sets Iso (Proj_{Γ}^{n−1}) and Iso ( CSA_{F}^{n} ) are in bijection, since they are in bijection with H^{1}(Γ, PGL_{n}). There is a canonical way to associate a Γ-projective space to a central simple algebra. Let α: A ⊗_{F} F_{s} ⟶^{∼} End_{F_{s}}(V) as above. Since the n-dimensional right ideals of End_{F_{s}}(V) are of the form Hom_{F_{s}}(V, U) for U ⊂ V a 1-dimensional subspace, any isomorphism A_{s} ≃ End_{F_{s}}(V) defines a one-to-one correspondence between ndimensional right ideals of A_{s} and 1-dimensional subspaces of V. Therefore, we may canonically associate to A a projective space SB(A) over F_{s}, whose (k − 1)- dimensional linear varieties are the nk-dimensional right ideals of A_{s}. The action of the group Γ of F_{s} over F induces a continuous action on A_{s}, hence also on the right ideals of A_{s}, and endows SB(A) with a Γ-projective space structure. We call SB(A) (or its associated scheme \overline{SB(A)} over F) the Severi–Brauer variety of A. Its dimension is n − 1, where n = deg A.

Proposition 2.2. The rule A ↦ SB(A) induces an anti-equivalence

CSA_{F}^{n} ≡ Proj_{Γ}^{n−1}.

Proof. Applying [11, Theorem 1, p. 93] it suffices to show that the functor SB is fully faithfull and essentially surjective. We refer to [6] for details. □

Γ-sets and étale algebras. Finite sets with a continuous left action of Γ (for the discrete topology) are called (finite) Γ-sets. They form a category Set_{Γ} whose morphisms are the Γ-equivariant maps. Finite direct products and direct sums (= disjoint unions) are defined in this category. We denote by |X| the cardinality of any finite set X.

2A groupoid is a category in which all morphisms are isomorphisms.

Following Tits [18] we define a thin Γ-projective space of dimension n − 1 as a Γ-set X of n elements. A thin (k − 1)-dimensional linear subvariety is a Γ-subset of X with k elements. For every integer k, 1 ≤ k ≤ n, the Γ-set

Gr_{k−1}(X) = {(k − 1)-dimensional linear varieties contained in X}
= {k-element subsets of X}

is a thin Γ-Grassmannian. As for Γ-projective spaces over F, we shall not distinguish between a finite Γ-set X and its scheme \overline{X} over F of fixed points of the Γ-action. Like Γ-projective spaces over F, thin Γ-projective spaces are related with algebraic structures. We recall that a finite-dimensional commutative F -algebra L is called étale (over F) if L⊗_{F} F_{s} is isomorphic to the F_{s}-algebra F_{s}^{n} = F_{s}×⋯×F_{s} (n factors) for some n ≥ 1. Étale F -algebras are the direct products of finite separable field extensions of F. We refer to [7, §18.A] for various equivalent characterizations of étale F -algebras. These algebras (with F -algebra homomorphisms) form a category Ét_{F} in which finite direct products and finite direct sums (= tensor products) are defined. For any étale F -algebra L of dimension n, the set of F -algebra homomorphisms

X(L) = Hom_{F -alg}(L, F_{s})

is a Γ-set of n elements since Γ acts on F_{s}. Conversely, if X is a Γ-set of n elements, the F -algebra M(X) of Γ-equivariant maps X → F_{s} is an étale F -algebra of dimension n,

M(X) = {f: X → F_{s} | γ(f(x)) = f(^{γ}x) for γ ∈ Γ, x ∈ X}.

As first observed by Grothendieck, there are canonical isomorphisms

M(X(L)) ≅ L, X(M(X)) ≅ X,

so that the functors M and X define an anti-equivalence of categories^{3}

(2.3)  Set_{Γ} ≡ Ét_{F}

(see [2, Proposition (4.3), p. 25] or [7, (18.4)]), which we refer to as the Grothendieck correspondence. Under this anti-equivalence, the cardinality of Γ-sets corresponds to the dimension of étale F -algebras. The disjoint union ⊔ in Set_{Γ} corresponds to the product × in Ét_{F} and the product × in Set_{Γ} to the tensor product ⊗ in Ét_{F}. For any integer n ≥ 1, we let Ét_{F}^{n} denote the groupoid whose objects are n-dimensional étale F -algebras and whose morphisms are F -algebra isomorphisms, and Set_{Γ}^{n} the groupoid of Γ-sets with n elements. The anti-equivalence (2.3) restricts to an antiequivalence Set_{Γ}^{n} ≡ Ét_{F}^{n}. Isomorphism classes are in bijection with H^{1}(Γ, 𝔖_{n}), where Γ acts trivially on 𝔖_{n} (see [7, 29.9]). The split étale algebra F^{n} corresponds to the Γ-set n of n elements with trivial Γ-action. Étale algebras of dimension 2 are also called quadratic étale algebras.

Embeddings of Γ-sets in Severi–Brauer varieties. For X a Γ-set of n elements and a Γ-projective space P of dimension n − 1, we define a general embedding

ε: X ↪ P

3We let morphisms of Γ-sets act on the right of the arguments (with the exponential notation)

and use the usual function notation for morphisms in the anti-equivalent category of étale algebras.

to be a Γ-equivariant injective map whose image consists of n points in general position, i.e., not contained in a proper subspace^{4}. The map ε then induces a map

Gr_{k−1}ε: Gr_{k−1}(X) ↪ Gr_{k−1}(P) for 1 ≤ k ≤ n,

which carries each k-element subset of X to the linear variety spanned by its image. In the rest of this subsection, we relate F -algebra embeddings of n-dimensional étale F -algebras in a central simple F -algebra A with general embeddings of Γ-sets of n elements in SB(A). We first discuss the case where the étale F -algebra is split.

Proposition 2.4. Let L be a split étale F -algebra of dimension n, and let A be a central simple F -algebra of degree n. For every F -algebra embedding ε: L ↪ A, there is an F -algebra isomorphism φ: A → End_{F} L such that φ ∘ ε: L ↪ End_{F} L is the regular representation, which maps each x ∈ L to multiplication by x.

Proof. Let (e_{i})_{1≤i≤n} be the collection of primitive idempotents of L. We have

A = ε(e_{1})A ⊕ ⋯ ⊕ ε(e_{n})A.

Since the dimension of every right ideal of A is a multiple of n Ind A, we must have

dim ε(e_{i})A = n for 1 ≤ i ≤ n, and Ind A = 1.

Identifying A = End_{F} V for some F -vector space V of dimension n, we may view ε(e_{1}), …, ε(e_{n}) as projections on 1-dimensional subspaces V_{1}, …, V_{n} such that V = V_{1} ⊕ ⋯ ⊕ V_{n}. Any F -vector space isomorphism V → L that maps V_{i} to e_{i}L for all 1 ≤ i ≤ n determines an F -algebra isomorphism φ: A → End_{F} L as required. □

Now, let L be an arbitrary n-dimensional étale F -algebra. Consider an F -algebra embedding into a central simple F -algebra of degree n:

ε: L ↪ A.

For every ξ ∈ X(L) := Hom_{F -alg}(L, F_{s}), set

(2.5)  ε_{∗}(ξ) = {x ∈ A_{s} | (1 ⊗ ξ(ℓ)) · x = (ε(ℓ) ⊗ 1) · x for all ℓ ∈ L}.

Proposition 2.6. The right ideal ε_{∗}(ξ) is n-dimensional, and ε_{∗} is a general embedding

ε_{∗}: X(L) ↪ SB(A).

Proof. Let L_{s} = L ⊗_{F} F_{s}, a split étale F_{s}-algebra, and ε_{s} = ε ⊗ Id_{F_{s}}: L_{s} ↪ A_{s}. By Proposition 2.4, we may identify A_{s} = End_{F_{s}} L_{s} in such a way that ε_{s}: L_{s} ↪ End_{F_{s}} L_{s} is the regular representation. Let (e_{i})_{1≤i≤n} be the collection of primitive idempotents of L_{s}. Using the canonical correspondence Hom_{F -alg}(L, F_{s}) = Hom_{F_{s}-alg}(L_{s}, F_{s}), we may set

X(L) = {ξ_{1}, …, ξ_{n}}

where ξ_{i}: L_{s} → F_{s} maps e_{i} to 1 and e_{j} to 0 for j ≠ i. For f ∈ End_{F_{s}} L_{s}, the condition

(1 ⊗ ξ_{i}(ℓ)) · f = (ε(ℓ) ⊗ 1) ∘ f for all ℓ ∈ L

is equivalent to

(1 ⊗ ξ_{i}(ℓ)) · f = ε_{s}(ℓ) ∘ f for all ℓ ∈ L_{s}.

4Such embeddings are called frames in [18].

For ℓ = e_{1}, …, e_{n}, this condition can be rewritten as

e_{j} · f(x) = 0 and e_{i} · f(x) = f(x) for all x ∈ L_{s}.

Therefore,

ε_{∗}(ξ_{i}) = Hom_{F_{s}}(L_{s}, e_{i}L_{s}),

which shows that dim ε_{∗}(ξ_{i}) = n. Moreover,

ε_{∗}(ξ_{1}) + ⋯ + ε_{∗}(ξ_{n}) = Hom_{F_{s}}(L_{s}, e_{1}L_{s} + ⋯ + e_{n}L_{s}) = End_{F_{s}} L_{s}.

Therefore ε_{∗}(ξ_{1}), …, ε_{∗}(ξ_{n}) are in general position in the projective space SB(A). To complete the proof, we show ε_{∗} is Γ-equivariant. For γ ∈ Γ, we have

γ(ε_{∗}(ξ)) = {x ∈ A_{s} | (1 ⊗ ξ(ℓ)) · γ^{−1}(x) = (ε(ℓ) ⊗ 1) · γ^{−1}(x) for all ℓ ∈ L}.

By applying γ to each side, we see that the condition

(1 ⊗ ξ(ℓ)) · γ^{−1}(x) = (ε(ℓ) ⊗ 1) · γ^{−1}(x)

is equivalent to

(1 ⊗^{γ}ξ(ℓ)) · x = (ε(ℓ) ⊗ 1) · x.

Thus, γ(ε_{∗}(ξ)) = ε_{∗}(^{γ}ξ). □

Conversely, every general embedding X ↪ SB(A) of a Γ-set X of n elements arises from an F -algebra embedding, as we now show. Given n-dimensional right ideals I_{1}, …, I_{n} ⊂ A_{s} such that {I_{1}, …, I_{n}} is stable under Γ and I_{1} + ⋯ + I_{n} = A_{s}, let

L_{s} = {x ∈ A_{s} | xI_{i} ⊂ I_{i} for 1 ≤ i ≤ n}.

The set L_{s} is an F_{s}-subalgebra of A_{s} stable under the action of Γ. Let L = L_{s}^{Γ} be the fixed F -subalgebra.

Proposition 2.7. The F -algebra L is étale of dimension n and {I_{1}, …, I_{n}} is the image of the general embedding ε_{∗}: X(L) ↪ SB(A) induced by the inclusion ε: L ↪ A.

Proof. We may identify A_{s} = M_{n}(F_{s}) in such a way that for 1 ≤ i ≤ n the ideal I_{i} consists of matrices whose nonzero entries are on the i-th row. For x = (x_{ij})_{1≤i,j≤n} we have xI_{j} ⊂ I_{j} if and only if x_{ij} = 0 for i ≠ j. Therefore, L_{s} is the algebra of diagonal matrices. Since L_{s} = L ⊗_{F} F_{s}, it follows that L is an étale F -algebra of dimension n. Under the canonical identification X(L) = Hom_{F_{s}-alg}(L_{s}, F_{s}) we have X(L) = {ξ_{1}, …, ξ_{n}} where ξ_{i}: L_{s} → F_{s} yields the i-th diagonal entry. Matrix computation shows that ε_{∗}(ξ_{i}) = I_{i}. □

Remark 2.8. The definition of the map ε_{∗} is inspired by the correspondence between maximal subfields of a division algebra and closed points on the corresponding Severi–Brauer variety set up by Merkurjev in [12].

3. Quadrics and Algebras with Involution

Γ-quadrics and quadratic pairs. As in the preceding section, let F be an arbitrary field and Γ the Galois group of some separable closure F_{s} of F. We call Γ-quadric in a (2n − 1)-dimensional Γ-projective space P over F any quadric in P defined by a hyperbolic quadratic form and stable under the action of Γ. The Γ-quadrics we consider in the sequel arise from quadratic pairs on central simple algebras, which we now recall from [7, §5].

Let B be an arbitrary F -algebra. An involution^{5} on B is an F -linear anti-automor-phism of B of order 2. The sets of symmetric, skew-symmetric, symmetrized and alternating elements are the following subspaces of B:

Sym(B, σ) = {x ∈ B | σ(x) = x}
Skew(B, σ) = {x ∈ B | σ(x) = −x}
Symd(B, σ) = {x + σ(x) | x ∈ B}
Alt(B, σ)  = {x − σ(x) | x ∈ B}.

Now, let A be a central simple F -algebra of degree 2n. A quadratic pair on A is a pair (σ, f) where σ is an involution on A and f: Sym(A, σ) → F is a linear map, subject to the following conditions:

(1) dim_{F} Sym(A, σ) = n(2n + 1) and Trd_{A} ( Skew(A, σ)) = {0};
(2) f(x + σ(x)) = Trd_{A}(x) for all x ∈ A.

If A = End_{F} V, any quadratic pair (σ, f) on A is attached to a nonsingular quadratic form q on V. The involution σ is the involution adjoint to the quadratic form q and we refer to [7, §5B] for the definition of f. For a central simple algebra A over a field of characteristic not 2 a quadratic pair (σ, f) is determined by the involution σ, which has to be of orthogonal type (see [7, §5B]).

Any quadratic pair (σ, f) on A extends to a quadratic pair (σ_{s}, f_{s}) on A_{s} = A⊗_{F} F_{s}. Since A_{s} is split, we may identify A_{s} = End_{F_{s}}(V) for some 2n-dimensional F_{s}-vector space V. Then (σ_{s}, f_{s}) is adjoint to some hyperbolic quadratic form q on V, which is uniquely determined up to a scalar factor, see [7, (5.11)]. Recall from [7, (6.5)] that a right ideal I ⊂ A_{s} is called isotropic for (σ_{s}, f_{s}) when

σ_{s}(I) · I = {0}  and  f_{s}(I ∩ Sym σ_{s}) = {0}.

Isotropic ideals have the form Hom_{F_{s}}(V, W) where W ⊂ V is a totally isotropic subspace of V for the form q, by [7, (6.6)]. Therefore, we may consider the quadric Q(σ, f) in SB(A) whose points are the 2n-dimensional right ideals in A_{s} that are isotropic for (σ_{s}, f_{s}). The quadric is stable under the action of the Galois group Γ of F_{s}/F and is given by the equation q = 0 in ℙ(V) under the identification SB(A) = ℙ(V). Let QCSA_{F}^{2n} be the groupoid of central simple algebras over F of degree 2n with quadratic pairs and let Quad_{Γ}^{2(n−1)} be the groupoid of Γ-quadrics of dimension 2(n−1) embedded in a Γ-projective space of dimension 2n−1. As in Proposition 2.2, we have:

Proposition 3.1. The rule (A, σ, f) ↦ Q(σ, f) induces an anti-equivalence of categories QCSA_{F}^{2n} ≡ Quad_{Γ}^{2(n−1)}.

The automorphism group of a quadric of dimension 2(n − 1) over F_{s} or of a central simple algebra of degree 2n with a quadratic pair over F_{s} is the projective orthogonal group PGO_{2n}(F_{s}) thus Galois cohomology leads to bijections of the sets Iso (QCSA_{F}^{2n}) and Iso (Quad_{Γ}^{2(n−1)}) with H^{1}(Γ, PGO_{2n}).

5We only consider F -linear involutions, which are sometimes called of the first kind, when semi-linear involutions are allowed.

Thin Γ-quadrics and quadratic étale extensions. A morphism of Γ-sets Y_{0} ⟵^{π} Y is called a Γ-covering if the number of elements in each fiber y_{0}^{π^{−1}} ⊂ Y does not depend on y_{0} ∈ Y_{0}. This number is called the degree of the covering. Of particular importance in the sequel are coverings of degree 2, which are also called double coverings. Each such covering Y_{0} ⟵^{π} Y defines a canonical automorphism Y ⟵^{σ} Y of order 2, which interchanges the elements in each fiber of π. Clearly, this automorphism has no fixed points. We call involution of a Γ-set with an even number of elements any automorphism of order 2 without fixed points. If Y is a Γ-set and Y ⟵^{σ} Y is an involution, the set of orbits

Y/σ = {{y, y^{σ}} | y ∈ Y }

is a Γ-set and the canonical map (Y/σ) ← Y is a double covering. Thus, the notions of double covering and involution of Γ-sets are equivalent. Paraphrasing Tits [18, 7.3] we call a double Γ-covering Y_{0} ⟵^{π} Y with |Y_{0}| = n, resp. a Γ-set with involution (Y, σ) with |Y | = 2n, a thin 2(n − 1)-dimensional Γ-quadric. Putting Y = {a_{1}, b_{1}, a_{2}, b_{2}, …, a_{n}, b_{n}} where a_{i}^{π} = b_{i}^{π} for all i, we view Y_{0} as the set of pairs Y_{0} = {{a_{1}, b_{1}}, {a_{2}, b_{2}}, …, {a_{n}, b_{n}}}. An isotropic linear subvariety of dimension k − 1 is a k-element subset of Y not containing any pair {a_{i}, b_{i}}. Thus, every isotropic linear subvariety is contained in a maximal isotropic linear subvariety, which has dimension n − 1. The maximal isotropic linear subvarieties may be viewed as the sections of the map π; they form a Γ-set C(Y/Y_{0}) (or C(Y, σ)) of 2^{n} elements, called the Clifford Γ-set of Y/Y_{0}, which is studied in more detail in §4.

Étale extensions are defined so as to correspond to Γ-coverings under the antiequivalence (2.3):

Proposition 3.2. For a homomorphism L_{0} ⟶^{ε} L of étale F -algebras, the following conditions are equivalent:

(a) ε endows L with a structure of free L_{0}-module of rank d;

(b) the morphism of Γ-sets X(L_{0}) ⟵^{X(ε)} X(L) is a covering of degree d.

Proof. Let e_{1}, …, e_{r} be the primitive idempotents of L_{0}, so that L_{0} ≅ M_{1}×⋯×M_{r} with M_{i} = e_{i}L_{0} a finite separable field extension of F. Then ε(e_{1}), …, ε(e_{r}) are orthogonal idempotents of L and, letting L_{i} = ε(e_{i})L we have L ≅ L_{1} × ⋯ × L_{r} where each L_{i} is an M_{i}-vector space. Condition (a) holds if and only if

dim_{M_{1}} L_{1} = ⋯ = dim_{M_{r}} L_{r} = d.

To prove (a) and (b) are equivalent, we may extend scalars to F_{s} and therefore assume F = F_{s} = M_{i} for all i. Then (e_{i})_{i=1}^{r} is a base of L_{0}, and we may identify X(L_{0}) with the dual base (e_{i}^{∗})_{i=1}^{r}. The number of elements of X(L) in the fiber above e_{i}^{∗} is the dimension dim_{M_{i}} L_{i}, so (a) and (b) are equivalent. □

When the equivalent conditions of Proposition 3.2 hold, the map ε is injective; identifying L_{0} to a subalgebra of L, we call L an étale extension of L_{0} of degree d. Étale extensions of degree 2 are called quadratic étale extensions. They can also be defined by means of automorphisms, as we now show. Let σ: L → L be an automorphism of order 2 of an étale F -algebra L, and let L_{0} = Sym(L, σ) ⊂ L denote the F -subalgebra of fixed elements, which is necessarily étale.

Proposition 3.3. The following conditions are equivalent:

(a) the inclusion L_{0} ↪ L is a quadratic étale extension of F -algebras;

(b) the automorphism X(σ) is an involution on X(L).

Proof. By the anti-equivalence (2.3) we have

X(L_{0}) = X(L)/ X(σ).

Therefore, (a) is equivalent by Proposition 3.2 to the condition that the canonical map X(L)/ X(σ) ← X(L) is a Γ-covering of degree 2, hence also to (b). □

When the equivalent conditions of Proposition 3.3 hold, the automorphism σ is called an involution of the étale F -algebra L, whose dimension is then necessarily even. Thus, the notions of quadratic étale extension and involution of étale algebras are equivalent. For n ≥ 1 we let QCov_{Γ}^{n} denote the groupoid whose objects are thin Γ-quadrics of dimension 2(n−1), i.e., double coverings Y_{0} ⟵^{π} Y of Γ-sets with |Y_{0}| = n, and whose morphisms are isomorphisms of Γ-coverings. Let QÉtex_{F}^{n} denote the groupoid of quadratic étale extensions L/L_{0} of F -algebras with dim_{F} L_{0} = n. From (2.3) we obtain an anti-equivalence of groupoids

(3.4)  QÉtex_{F}^{n} ≡ QCov_{Γ}^{n}.

Isomorphism classes of these groupoids are in bijection with H^{1}(Γ, 𝔖_{2}^{n} ⋊ 𝔖_{n}) (see [8]). We denote by Y(L/L_{0}) or Y(L, σ) the double covering (i.e., the thin 2(n − 1)- dimensional quadric) attached to the quadratic extension L/L_{0}:

Y(L/L_{0}) = X(L_{0}) ← X(L), Y(L, σ) = X(L)/ X(σ) ← X(L).

The following observation on symmetric and alternating elements in an étale algebra with involution will be used in the next subsection:

Proposition 3.5. For any étale F -algebra of even dimension with involution (L, σ), we have

Sym(L, σ) = Symd(L, σ)  and  Skew(L, σ) = Alt(L, σ).

Proof. The proposition is clear if char F ≠ 2, for then

x = \frac{x}{2} + σ_{L}(\frac{x}{2}) and y = \frac{y}{2} − σ_{L}(\frac{y}{2} ) for x ∈ Sym(L, σ) and y ∈ Skew(L, σ).

For the rest of the proof, we may thus assume char F = 2. The linear map L → Symd(L, σ) that carries ℓ ∈ L to ℓ + σ(ℓ) fits into the exact sequence

0 → Sym(L, σ) → L → Symd(L, σ) → 0.

Since dim_{F} Sym(L, σ) = \frac{1}{2} dim_{F} L, it follows that dim Symd(L, σ) = dim Sym(L, σ). It follows that the obvious inclusion Symd(L, σ) ⊆ Sym(L, σ) is an equality, which proves the first equation. The second equation is the same as the first since char F = 2. □

Embeddings of quadrics and of algebras with involution. Let A be a central simple F -algebra of even degree 2n, and let (σ, f) be a quadratic pair on A. If (L, σ_{L}) is an étale F -algebra of dimension 2n with involution, we define an embedding of algebras with involution

ε: (L, σ_{L}) ↪ (A, σ, f)

to be an F -algebra embedding ε: L ↪ A satisfying the following conditions: (a) ε ∘ σ_{L} = σ ∘ ε, and

(b) f(x) = Trd_{A}(ε(ℓ)x) for all x ∈ Sym(σ) and all ℓ ∈ L such that ℓ+σ_{L}(ℓ) = 1. Note that Proposition 3.5 shows that there exists ℓ ∈ L such that ℓ + σ_{L}(ℓ) = 1, since 1 ∈ Sym(σ_{L}). If ℓ, ℓ′ ∈ L satisfy

ℓ + σ_{L}(ℓ) = ℓ′ + σ_{L}(ℓ′) = 1,

then ℓ − ℓ′ ∈ Skew(σ_{L}), and Proposition 3.5 yields ℓ − ℓ′ ∈ Alt(σ_{L}). If (a) holds, then ε(ℓ) − ε(ℓ′) ∈ Alt(σ), hence, by [7, (2.3)],

Trd_{A}(x(ε(ℓ) − ε(ℓ′))) = 0 for all x ∈ Sym(σ).

Therefore, when (a) holds, condition (b) is equivalent to

(b’) there exists ℓ ∈ L such that ℓ + σ_{L}(ℓ) = 1 and f(x) = Trd_{A}(ε(ℓ)x) for all

x ∈ Sym(σ). It can be further weakened to

(b”) there exists ℓ ∈ L such that f(x) = Trd_{A}(ε(ℓ)x) for all x ∈ Sym(σ_{L}), because this condition on ℓ implies ε(ℓ) + σε(ℓ) = 1, see [7, (5.7)]. Note that condition (b’) (hence also (b”)) holds with ℓ = \frac{1}{2} if char F ≠ 2.

Example 3.6. Let (L, σ_{L}) be an arbitrary étale F -algebra with involution of the first kind. Let L_{0} = Sym(σ_{L}) ⊂ L be the fixed subalgebra under σ_{L}, let T_{0} : L_{0} → F be the trace, and let t: L → F be the quadratic form given by t(x) = T_{0}(σ(x)x) for x ∈ L. Let (σ_{t}, f_{t}) denote the adjoint quadratic pair on End_{F} L. The regular representation is an embedding

(L, σ_{L}) ↪ (End_{F} L, σ_{t}, f_{t}).

Indeed, condition (a) holds because the polar form of t is

b_{t}(x, y) = T_{L_{0}/F} (σ_{L}(x)y + σ_{L}(y)x) for x, y ∈ L,

and we have

b_{t}(xy, z) = b_{t}(y, σ_{L}(x)z) for x, y, z ∈ L.

Condition (b) also is easily checked, see [7, (5.12)].

Proposition 3.7. Use the same notation as in Example 3.6. Suppose L is split and let (A, σ, f) be a central simple F -algebra with quadratic pair. For any embedding

ε: (L, σ_{L}) ↪ (A, σ, f)

there is an isomorphism φ: (A, σ, f) → (End_{F} L, σ_{t}, f_{t}) such that φ ∘ ε: (L, σ_{L}) ↪ (End_{F} L, σ_{t}, f_{t}) is the regular representation.

Proof. Let e_{1}, …, e_{n}, e′_{1}, …, e′_{n} be the collection of primitive idempotents of L, labelled in such a way that e′_{i} = σ_{L}(e_{i}) for i ∈ n. As observed in the proof of Proposition 2.4, we may identify A = End_{F} V and we have a decomposition into 1-dimensional subspaces

V = V_{1} ⊕ ⋯ ⊕ V_{n} ⊕ V′_{1} ⊕ ⋯ ⊕ V′_{n}

such that ε(e_{i}) (resp. ε(e′_{i})) is the projection V → V_{i} (resp. V → V′_{i}). Let q: V → F be a quadratic form such that (σ, f) is adjoint to q, and let b be the polar form of q. Since σ is adjoint to b and σε(e_{i}) = εσ_{L}(e_{i}) = ε(e′_{i}), we have for all i, j ∈ n

b(V_{i}, V_{j}) = b(ε(e_{i})(V_{i}), V_{j}) = b(V_{i}, ε(e′_{i})(V_{j})) = {0}.

Similarly, b(V′_{i}, V′_{j}) = {0} for all i, j ∈ n, and b(V_{i}, V′_{j}) = {0} if i ≠ j. Since b is nonsingular, it follows that b(V_{i}, V′_{i}) = F for all i ∈ n, hence we may find v_{i} ∈ V_{i}, v′_{i} ∈ V′_{i} such that

b(v_{i}, v′_{i}) = 1.

Let S ∈ End_{F} V be the map such that S(v_{i}) = v′_{i} and S(v′_{i}) = v_{i} for all i ∈ n. It is readily verified that for i ∈ n

S ∘ ε(e′_{i}): x ↦ v_{i}b(v_{i}, x) and S ∘ ε(e_{i}): x ↦ v′_{i}b(v′_{i}, x) for all x ∈ V.

Therefore, by [7, (5.11)],

q(v_{i}) = f(S ∘ ε(e′_{i}))  and  q(v′_{i}) = f(S ∘ ε(e_{i})).

If ℓ ∈ L is such that f(x) = Tr(ε(ℓ) ∘ x) for all x ∈ Sym(σ), then

f(S ∘ ε(e′_{i})) = Tr(ε(ℓ) ∘ S ∘ ε(e′_{i})) = Tr(S ∘ ε(e′_{i}ℓ)).

None of the base vectors v_{1}, …, v′_{n} is mapped to a nonzero multiple of itself by S ∘ ε(e′_{i}ℓ), hence Tr(S ∘ ε(e′_{i}ℓ)) = 0 and therefore q(v_{i}) = 0. Likewise, q(v′_{i}) = 0 for i ∈ n. Therefore, mapping v_{i} ↦ e_{i} and v′_{i} ↦ e′_{i} for i ∈ n defines an isometry (V, q) → (L, t), and the induced isomorphism of algebras with quadratic pairs φ: (End_{F} V, σ, f) → (End_{F} L, σ_{t}, f_{t}) is such that φ ∘ ε is the regular representation.

□

We now turn to the geometric counterpart of the notion of embeddings of algebras with involution. Let Q ⊂ P be a Γ-quadric of dimension 2(n − 1) in a Γ-projective space of dimension 2n − 1 over a field F. We let C(Q) denote the Γ-set of (n − 1)- dimensional linear varieties that lie in Q, and call this Γ-set the Clifford Γ-set of Q. If (Y, σ) is a thin Γ-quadric of dimension 2(n − 1), we define an embedding of quadrics

ε: (Y, σ) ↪ Q

as a general embedding ε: Y ↪ P that carries isotropic linear subvarieties of (Y, σ) to isotropic linear subvarieties of P, i.e., to linear subvarieties of Q. In particular, the image of Y lies on Q. Since every isotropic linear subvariety is contained in a maximal isotropic linear subvariety, it suffices to require that the induced map Gr_{n−1}ε: Gr_{n−1}Y ↪ Gr_{n−1}(P) carry the set of maximal isotropic linear subvarieties C(Y, σ) to C(Q). We let C(ε) denote the Γ-equivariant map induced by Gr_{n−1}ε,

C(ε): C(Y, σ) ↪ C(Q).

Let (L, σ_{L}) be an arbitrary étale F -algebra of dimension 2n with involution of the first kind, and let

ε: (L, σ_{L}) ↪ (A, σ, f)

be an embedding in a central simple F -algebra of degree 2n with quadratic pair. Since ε: L ↪ A is an F -algebra embedding, we may consider the general embedding of Proposition 2.6:

ε_{∗}: X(L) ↪ SB(A).

Let Y(L, σ_{L}) be the thin Γ-quadric associated with (L, σ_{L}) and Q(A, σ, f) the Γquadric associated with (A, σ, f).

Proposition 3.8. The map ε_{∗} defines an embedding of quadrics

ε_{∗}: Y(L, σ_{L}) ↪ Q(σ, f).

Proof. As usual, let A_{s} = A ⊗_{F} F_{s}, and let (σ_{s}, f_{s}) be the quadratic pair on A_{s} extending (σ, f). We have to show that for every ω ∈ C(Y(L, σ_{L})) we have Gr_{n−1}ε_{∗}(ω) ∈ C(Q(σ, f)). Let ξ, η ∈ X(L). For x ∈ ε_{∗}(ξ), y ∈ ε_{∗}(η) and ℓ ∈ L we have

(1 ⊗ ξ(ℓ)) · x = (ε(ℓ) ⊗ 1) · x and (1 ⊗ η(ℓ)) · y = (ε(ℓ) ⊗ 1) · y,

hence also

σ_{s}(x) · (ε(ℓ) ⊗ 1) = σ_{s}(x) · (1 ⊗ ξσ_{L}(ℓ)),

and

σ_{s}(x) · y · (1 ⊗ η(ℓ)) = σ_{s}(x) · (ε(ℓ) ⊗ 1) · y = σ_{s}(x) · y · (1 ⊗ ξσ_{L}(ℓ)).

If η ≠ ξσ_{L}, we can find ℓ ∈ L such that η(ℓ) ≠ ξσ_{L}(ℓ), hence the last equation shows that σ_{s}(x) · y = 0. Therefore,

σ_{s}(ε_{∗}(ξ)) · ε_{∗}(η) = {0}, unless η = ξσ_{L}.

In particular, for every ω ∈ C(Y(L, σ_{L})),

(3.9) σ_{s}(⨁_{ξ∈ω} ε_{∗}(ξ))·⨁_{ξ∈ω} ε_{∗}(ξ) = {0}, i.e., σ_{s}(Gr_{n−1}ε_{∗}(ω))·Gr_{n−1}ε_{∗}(ω) = {0}.

Now, let ℓ ∈ L be such that f(x) = Trd(ε(ℓ)x) for all x ∈ Sym(σ). For x ∈ Gr_{n−1}ε_{∗}(ω) ∩ Sym(σ_{s}) we have

(3.10)  f_{s}(x) = Tr((ε(ℓ) ⊗ 1) · x) since x ∈ Sym(σ_{s}),

and

(ε(ℓ) ⊗ 1) · x ∈ Gr_{n−1}ε_{∗}(ω) since x ∈ Gr_{n−1}ε_{∗}(ω).

On the other hand, we have x = σ_{s}(x) ∈ σ_{s}(Gr_{n−1}ε_{∗}(ω)), hence by (3.9),

x · (ε(ℓ) ⊗ 1) · x = 0.

It follows that ((ε(ℓ) ⊗ 1) · x)^{2} = 0, hence Tr((ε(ℓ) ⊗ 1) · x) = 0 and (3.10) shows that f_{s}(x) = 0. Thus, Gr_{n−1}ε_{∗}(ω) is isotropic for all ω ∈ C(Y(L, σ_{L})), and the proposition is proved. □

Conversely, every embedding

ε: (Y, φ) ↪ Q(σ, f)

of a Γ-set Y of 2n elements with an involution φ of the first kind in the quadric associated with a quadratic pair on a central simple algebra A of degree 2n arises from the embedding of the étale F -algebra with involution M(Y, φ) into (A, σ, f), as we now show. Let ε(Y) = {I_{1}, …, I_{n}, I′_{1}, …, I′_{n}}, where the isotropic 2n-dimensional right ideals I_{1}, …, I′_{n} of A_{s} are indexed in such a way that I′_{i} = ε(ξ^{φ}) if ε(ξ) = I_{i}. As in §2, let

L_{s} = {x ∈ A_{s} | x · I_{i} ⊂ I_{i} and x · I′_{i} ⊂ I′_{i} for all i ∈ n}

and L = L_{s}^{Γ}. Proposition 2.7 shows that we may identify X with X(L) in such a way that ε is the general embedding X(L) ↪ SB(A) induced by the inclusion L ↪ A. Let σ_{L} be the involution on L such that X(σ_{L}) = φ.

Proposition 3.11. The inclusion L ↪ A is an embedding (L, σ_{L}) ↪ (A, σ, f).

Proof. The primitive idempotents e_{1}, …, e_{n}, e′_{1}, …, e′_{n} of L_{s} are in one-to-one correspondence with I_{1}, …, I_{n}, I′_{1}, …, I′_{n} in such a way that I_{i} = e_{i} · A_{s} and I′_{i} = e′_{i} · A_{s} for i ∈ n. The indexing of I_{1}, …, I′_{n} is such that σ_{L_{s}} interchanges e_{i} and e′_{i} for i ∈ n. To check that σ restricts to σ_{L}, observe that I′_{i} is the only element among {I_{1}, …, I′_{n}} that does not belong to any ω^{C(ε)} for any ω ∈ C(X, ∗) containing i. Therefore,

σ_{s}(I_{i}) · I_{j} = σ_{s}(I_{i}) · I′_{k} = {0} for all j, k ∈ n with k ≠ i,

hence for all i ∈ n

σ_{s}(e_{i}) · e_{j} = σ_{s}(e_{i}) · e′_{k} = 0 for all j, k ∈ n with k ≠ i.

Similarly, for all i ∈ n,

σ_{s}(e′_{i}) · e_{j} = σ_{s}(e′_{i}) · e′_{k} = 0 for all j, k ∈ n with j ≠ i.

Since e_{1} + ⋯ + e_{n} + e′_{1} + ⋯ + e′_{n} = 1, it follows that for all i ∈ n we have

e′_{i} = (∑_{j∈n} σ_{s}(e_{j}) + σ_{s}(e′_{j})) · e′_{i} = σ_{s}(e_{i}) · e′_{i}

and, on the other hand,

σ_{s}(e_{i}) · (1 − e′_{i}) = σ_{s}(e_{i}) · (∑_{j∈n} e_{j} + ∑_{k∈n∖{i}} e′_{k}) = 0,

hence σ_{s}(e_{i}) = σ_{s}(e_{i}) · e′_{i}. Therefore, σ_{s}(e_{i}) = e′_{i} for i ∈ n. To complete the proof that (L, σ_{L}) ↪ (A, σ, f), it remains to show that for all x ∈ Sym(σ) and all ℓ ∈ L such that ℓ + σ_{L}(ℓ) = 1 we have f(x) = Trd_{A}(ℓx). To show that this equation holds for all x ∈ Sym(σ_{s}) and all ℓ ∈ L_{s} such that ℓ + σ_{s}(ℓ) = 1, it actually suffices to prove it for a single ℓ, as was observed in the equivalence of conditions (b) and (b’) defining an embedding. Let ℓ = ∑_{i∈n} e_{i}. Since ℓ + σ_{s}(ℓ) = 1 we have for all y ∈ A_{s}

Trd_{A_{s}}(ℓ · (y + σ_{s}(y))) = Trd_{A_{s}}((ℓ + σ_{s}(ℓ)) · y) = Trd_{A_{s}}(y).

Since f_{s}(y + σ_{s}(y)) = Trd_{A_{s}}(y), we have

(3.12)  f_{s}(x) = Trd_{A_{s}}(ℓx)  for all x ∈ Symd(σ_{s}).

Let I = ℓ · A_{s} = I_{1} + ⋯ + I_{n} and I′ = (1 − ℓ) · A_{s} = I′_{1} + ⋯ + I′_{n}. These right ideals are isotropic since {I_{1}, …, I_{n}}, {I′_{1}, …, I′_{n}} ∈ C(ε(X, ∗)). For x ∈ I ∩ Sym(σ_{s}) we have f_{s}(x) = 0 and, on the other hand, ℓx ∈ I and xℓx ∈ σ_{s}(I) · I = {0}, hence (ℓx)^{2} = 0. Therefore,

(3.13)  f_{s}(x) = Trd_{A_{s}}(ℓx)  for all x ∈ I ∩ Sym(σ_{s}).

Similarly, for x ∈ I′ ∩ Sym(σ_{s}) we have ℓx = 0, hence

(3.14)  f_{s}(x) = Trd_{A_{s}}(ℓx)  for all x ∈ I′ ∩ Sym(σ_{s}).

Now, for x ∈ Sym(σ_{s}) we have

x = ℓx(1 − ℓ) + (1 − ℓ)xℓ + ℓxℓ + (1 − ℓ)x(1 − ℓ).

Since ℓ ∈ I, 1 − ℓ ∈ I′, and σ_{s}(ℓ) = 1 − ℓ, we have

ℓx(1 − ℓ) ∈ I ∩ Sym(σ_{s}), (1 − ℓ)xℓ ∈ I′ ∩ Sym(σ_{s}),

and

ℓxℓ + (1 − ℓ)x(1 − ℓ) = ℓxℓ + σ_{s}(ℓxℓ) ∈ Symd(σ_{s}).

Therefore,

f_{s}(x) = f_{s}(ℓx(1 − ℓ)) + f_{s}((1 − ℓ)xℓ) + f_{s}(ℓxℓ + (1 − ℓ)x(1 − ℓ)),

and it follows from (3.12), (3.13), and (3.14) that

f_{s}(x) = Trd_{A_{s}}(ℓx).
□

4. Clifford algebras and Γ-sets

As in the previous sections, we let F denote an arbitrary field and Γ the Galois group of a separable closure F_{s} of F.

Clifford constructions. Recall from §3 that for any 2(n − 1)-dimensional thin Γ-quadric Y/Y_{0} we let C(Y/Y_{0}) denote the Γ-set of maximal linear isotropic subvarieties, which can be identified with the 2^{n}-element set of sections of the covering Y_{0} ← Y. We call C(Y/Y_{0}) the Clifford Γ-set of Y/Y_{0}. The canonical involution σ of Y/Y_{0} induces an involution \underline{σ} on C(Y/Y_{0}), which maps every section to its complement in Y. If n = 1, then C(Y/Y_{0}) = Y and \underline{σ} = σ. The following proposition is clear:

Proposition 4.1. If Y/Y_{0} and Z/Z_{0} are thin Γ-quadrics, then (Y ⊔ Z)/(Y_{0} ⊔ Z_{0}) is a thin Γ-quadric and there is a canonical isomorphism

C(Y/Y_{0}) × C(Z/Z_{0}) ⟶^{∼} C(Y ⊔ Z/Y_{0} ⊔ Z_{0})

which maps any pair of isotropic linear varieties (ω, ω′) to ω ⊔ ω′.

We now turn to the analogue of this construction for quadratic étale extensions. Let ε: L_{0} ↪ L be a quadratic étale extension of étale F -algebras, with dim_{F} L_{0} = n. In view of the Grothendieck correspondence there exists an étale algebra C(L/L_{0}) of dimension 2^{n}, with involution \underline{σ}, such that C Y(L/L_{0}) = Y (C(L/L_{0}), \underline{σ}). This étale algebra is constructed explicitly in [8], where it is designated by Ω(L/L_{0}). We briefly recall the construction: the algebra L_{0}^{⊗n} contains an idempotent e_{n}^{L_{0}} that corresponds under the composition of canonical isomorphisms

L_{0}^{⊗n} ⟶^{∼} M(X(L_{0}^{⊗n})) ⟶^{∼} M(X(L_{0})^{n})

to the characteristic function χ of the complement of the “fat diagonal” of X(L_{0})^{n}:

χ(ξ_{1}, …, ξ_{n}) = \begin{cases}1 & if ξ_{1}, …, ξ_{n} are all different; \\ 0 & if ξ_{i} = ξ_{j} for some i, j with i ≠ j. \end{cases}

Then, letting (L^{⊗n})^{𝔖_{n}} denote the subalgebra of L^{⊗n} fixed under the action of 𝔖_{n} by permutation of the factors, we set

C(L/L_{0}) = ε^{⊗n}(e_{n}^{L_{0}}) · (L^{⊗n})^{𝔖_{n}}.

We call the algebra C(L/L_{0}) the Clifford algebra of L/L_{0}. If n = 1, then C(L/L_{0}) = L, and the canonical involution \underline{σ} is the canonical involution of L/L_{0}.

Proposition 4.2. If L/L_{0} and M/M_{0} are quadratic étale extensions, there is a canonical isomorphism

P: C(L × M/L_{0} × M_{0}) ⟶^{∼} C(L/L_{0}) ⊗ C(M/M_{0}),

which corresponds under the Grothendieck correspondence to the canonical isomorphism of Proposition 4.1.

By contrast to Γ-sets, we have an F -linear map

c_{L}: L → C(L/L_{0})

defined by

(4.3) c_{L}(x) = ε^{⊗n}(e_{n}^{L_{0}}) · ((x ⊗ 1 ⊗ ⋯ ⊗ 1) + (1 ⊗ x ⊗ ⋯ ⊗ 1) + (1 ⊗ ⋯ ⊗ 1 ⊗ x))

for x ∈ L.

Proposition 4.4. Let L/L_{0} and M/M_{0} be quadratic étale extensions and let ρ: L× M → L ⊗ M be the linear map given by ρ(x, y) = x ⊗ 1 + 1 ⊗ y. The diagram

C(L × M/L_{0} × M_{0}) ⟶^{P} C(L/L_{0}) ⊗ C(M/M_{0})
c_{L×M} ↑  ↑^{c_{L}⊗c_{M}}
L × M  ⟶_{ρ}  L ⊗ M

is commutative.

Proof. The claim follows easily from the definition of the canonical map P. □

Corollary 4.5. The space c_{L}(L) generates C(L/L_{0}) as an algebra.

Proof. By extending F we may assume that L/L_{0} has a nontrivial decomposition L/L_{0} = M/M_{0} × N/N_{0}. The claim then follows from (4.4) by induction on the dimension. □

Lemma 4.6. Let L_{0} = Sym(σ_{L}) and T_{0} : L_{0} → F be the trace map. The kernel of c_{L}: L → C(L, σ_{L}) is

ker c_{L} = {x ∈ L_{0} | T_{0}(x) = 0}.

Proof. Extend scalars to F_{s} and use induction on dim L. □

Discriminants. Let Q be a 2(n − 1)-dimensional Γ-quadric. The Clifford Γ-set C(Q), consisting of maximal linear subvarieties of Q, decomposes into two classes C_{1}(Q), C_{2}(Q): two linear varieties v_{1}, v_{2} lie in the same class if and only if dim(v_{1} ∩ v_{2}) ≡ n − 1 mod 2, see for example [3, §§II.6]. The equivalence relation is clearly preserved under the action of Γ, hence Γ acts on the set

Δ(Q) := {C^{1}(Q), C^{2}(Q)},

which we call the discriminant of Q. We let δ: C(Q) → Δ(Q) denote the Γequivariant map that carries each linear variety in C(Q) to its class. Likewise, if (Y, σ) is a thin Γ-quadric of dimension 2(n−1), the Clifford Γ-set C(Y, σ) of (n − 1)-dimensional isotropic linear subvarieties decomposes into two classes as for “classical” quadrics, hence we also have a discriminant Δ(Y, σ) and a canonical map δ: C(Y, σ) → Δ(Y, σ). An alternative description of this discriminant is given in [9, Proposition 2.5]: it is canonically isomorphic to the set of orbits under the alternating group 𝔄_{2n} of the set

Σ_{2n}(Y) = {(y_{1}, …, y_{2n}) | Y = {y_{1}, …, y_{2n}}}.

Therefore, Δ(Y, σ) depends only on Y, and not on the involution σ; we may thus use the notation Δ(Y) for Δ(Y, σ).

Proposition 4.7. Let ε: (Y, σ) ↪ Q be an embedding of quadrics. There is an induced isomorphism of Γ-sets Δ(ε): Δ(Y) → Δ(Q) that fits into the following commutative diagram:

C(Y, σ) ⟶^{C(ε)} C(Q)
δ↓  ↓^{δ}
Δ(Y) ⟶^{Δ(ε)} Δ(Q).

Proof. For ω_{1}, ω_{2} ∈ C(Y, σ), we have

|ω_{1} ∩ ω_{2}| = dim(C(ε)(ω_{1}) ∩ C(ε)(ω_{2})),

hence the map C(ε) is compatible with the division of C(Y, σ) and C(Q) into two classes. Therefore, we may define Δ(ε) by mapping the equivalence class of ω ∈ C(Y, σ) to the equivalence class of C(ε)(ω) ∈ C(Q). □

Under the Grothendieck correspondence, the discriminant of Γ-sets yields a discriminant of étale algebras, see [8, §18]. If (L, σ) is an étale algebra with involution, we have a canonical map

δ: Δ(L) → C(L, σ),

so we may consider the Clifford algebra C(L, σ) as an étale extension of Δ(L).

Clifford algebra embeddings. Let (A, σ, f) be a central simple algebra of even degree 2n with quadratic pair over an arbitrary field F. Recall from [7, §8] that the Clifford algebra C(A, σ, f) is defined in such a way that if A = End_{F} V and (σ, f) is induced by some quadratic form q: V → F, then C(A, σ, f) is the even Clifford algebra

C(A, σ, f) = C_{0}(V, q).

The center of C(A, σ, f) is a quadratic étale F -algebra Z(A, σ, f), which is related to the discriminant of (σ, f), see [7, (8.10)], and C(A, σ, f) is an Azumaya algebra of constant rank 2^{n} over Z(A, σ, f). The Clifford algebra C(A, σ, f) is defined as a quotient of the tensor algebra T (A) of the underlying vector space of A, hence there is a canonical F -linear map

c_{A}: A → C(A, σ, f).

In the split case A = End_{F} V, the bilinear polar form of q yields an F -linear isomorphism V → V^{∗} which allows us to identify

V ⊗_{F} V = V ⊗_{F} V^{∗} = End_{F} V;

under this identification, the map c_{A} carries v ⊗ w ∈ V ⊗_{F} V to v · w ∈ C_{0}(V, q). Now, let deg A = 2n, and let

ε: (L, σ_{L}) ↪ (A, σ, f)

be an embedding of an étale F -algebra with involution of the first kind of dimension 2n in (A, σ, f).

Theorem 4.8. There is an F -algebra embedding

(4.9)  C(ε): C(L, σ_{L}) ↪ C(A, σ, f)

such that the following diagram commutes:

(4.10)  \begin{matrix}L & ⟶^{ε} & A \\ ^{c_{L}}↓ & & ↓^{c_{A}} \\ C(L, σ_{L}) & ⟶^{C(ε)} & C(A, σ, f). \end{matrix}

The F -algebra homomorphism C(ε) is uniquely determined by this property. Moreover, the following diagram commutes:

(4.11)  \begin{matrix}C(L, σ_{L}) & ⟶^{C(ε)} & C(A, σ, f) \\ \underline{σ_{L}}↓ & & ↓\underline{σ} \\ C(L, σ_{L}) & ⟶^{C(ε)} & C(A, σ, f). \end{matrix}

Furthermore, there is an F -algebra isomorphism \overline{C(ε)} which makes the following diagram commute:

C(L, σ_{L}) ⟶^{C(ε)} C(A, σ, f)
δ↑  ↑
Δ(L) ⟶^{\overline{C(ε)}} Z(A, σ, f).

The proof uses the following general principle:

Principle 4.12. Let C and C′ be F -algebras, and let U ⊂ C be a subspace which generates C as an F -algebra. Let also f: U → C′ be a linear map. If there exists a field extension K/F and a K-algebra homomorphism f̂_{K}: C ⊗_{F} K → C′ ⊗_{F} K such that f̂_{K}|_{U⊗K} = f⊗Id_{K}, then f extends to an F -algebra homomorphism f̂: C → C′.

Proof. The restriction of f̂_{K} to C is an F -algebra homomorphism C → C′ ⊗_{F} K. Since the restriction of f̂_{K} to U is f, and U generates C, the image of this homomorphism is in fact in C′. □

Proof of Theorem 4.8. Since c(L) generates the algebra C(L, σ_{L}), it is clear that, if it exists, C(ε) is uniquely determined by the commutativity of diagram (4.10). By [7, (8.14)], we have

ker(c_{A} ∘ ε) = {x ∈ L | ε(x) ∈ Sym(σ) and f(ε(x)) = 0}.

Since ε embeds (L, σ_{L}) in (A, σ, f), the condition ε(x) ∈ Sym(σ_{L}) holds if and only if x ∈ L_{0}, and then

f(ε(x)) = Trd_{A}(ε(ℓx)) = T (ℓx) for any ℓ ∈ L such that ℓ + σ_{L}(ℓ) = 1.

Since for x ∈ L_{0} we have

T (ℓx) = T_{0}(ℓx + σ_{L}(ℓx)) = T_{0}(x),

it follows from Lemma 4.6 that ker c_{L} = ker(c_{A} ∘ ε). Therefore, ε induces an injective F -linear map ε′: c_{L}(L) ↪ C(A, σ, f). Since c_{L}(L) generates C(L, σ_{L}), we may apply the general principle above with U = c_{L}(L), C = C(L, σ_{L}), and C′ = C(A, σ, f), taking for K a separable closure F_{s} of F. To prove the existence of C(ε), it suffices to show that ε′_{s} := ε ⊗ Id_{F_{s}} extends to an F_{s}-algebra embedding

C(ε_{s}): C(L_{s}, σ_{L_{s}}) → C(A_{s}, σ_{s}, f_{s}).

By Proposition 3.7, we may assume A_{s} = End_{F_{s}}(L_{s}), (σ_{s}, f_{s}) is the quadratic pair adjoint to the quadratic form t: L_{s} → F_{s} of Example 3.6, and ε_{s} is the regular representation, hence C(A_{s}, σ_{s}, f_{s}) = C_{0}(L_{s}, t). We argue by induction on n. If n = 1 (i.e. L_{s} = F_{s} × F_{s}) the map c_{L} is the identity

c_{L}: L_{s} ⟶^{∼} C(L_{s}, σ_{L_{s}}) = L_{s},

and c_{A} ∘ ε_{s}: L_{s} → C_{0}(L_{s}, t) is an F_{s}-algebra isomorphism, so the result is clear. If n > 1 we may find a nontrivial decomposition (L_{s}, σ_{s}) = (L_{1}, σ_{1}) × (L_{2}, σ_{2}). We then have, by Proposition 4.2,

C(L_{s}, σ_{s}) = C(L_{1}, σ_{1}) ⊗_{F} C(L_{2}, σ_{2}),

and, letting c_{i}: L → C(L_{i}, σ_{i}) for i = 1, 2 denote the canonical F_{s}-linear map,

c_{L}(ℓ_{1}, ℓ_{2}) = c_{1}(ℓ_{1}) ⊗ c_{2}(ℓ_{2}) for ℓ_{1} ∈ L_{1} and ℓ_{2} ∈ L_{2}.

On the other hand, defining quadratic forms t_{1} and t_{2} on L_{1} and L_{2} respectively in the same way as t on L_{s}, we have an orthogonal sum decomposition

(L_{s}, t) = (L_{1}, t_{1}) ⨁^{⊥} (L_{2}, t_{2}),

hence a canonical F -algebra embedding

C_{0}(L_{1}, t_{1}) ⊗_{F} C_{0}(L_{2}, t_{2}) ↪ C_{0}(L_{s}, t).

By induction, we may assume (c∘ε)_{1} : L_{1} → C_{0}(L_{1}, t_{1}) and (c∘ε)_{2} : L_{2} → C_{0}(L_{2}, t_{2}) extend to F_{s}-algebra embeddings

C(ε_{1}): C(L_{1}, σ_{1}) ↪ C_{0}(L_{1}, t_{1})  and  C(ε_{2}): C(L_{2}, σ_{2}) ↪ C_{0}(L_{2}, t_{2}).

Then the composition

C(L_{s}, σ_{L_{s}}) = C(L_{1}, σ_{1})⊗_{F} C(L_{2}, σ_{2}) ⟶^{C(ε_{1})⊗C(ε_{2})} C_{0}(L_{1}, t_{1})⊗_{F} C_{0}(L_{2}, t_{2}) ↪ C_{0}(L_{s}, t)

is an F_{s}-algebra embedding that extends c_{A} ∘ ε_{s}. To prove that diagram (4.11) commutes, it suffices to prove the commutativity of

c_{L}(L) ⟶^{C(ε)} c_{A}(A)
\underline{σ_{L}}↓  ↓\underline{σ}
c_{L}(L) ⟶^{C(ε)} c_{A}(A)

since c_{L}(L) generates C(L, σ_{L}). Commutativity of the latter diagram follows from commutativity of the diagrams

\begin{matrix}L & ⟶^{ε} & A \\ ^{σ_{L}}↓ & & ↓^{σ} \\ L & ⟶^{ε} & A, \end{matrix} \begin{matrix}L & ⟶^{c_{L}} & C(L, σ_{L}) \\ ^{σ_{L}}↓ & & ↓\underline{σ_{L}} \\ L & ⟶^{c_{L}} & C(L, σ_{L}), \end{matrix} and \begin{matrix}A & ⟶^{c_{A}} & C(A, σ, f) \\ ^{σ}↓ & & ↓\underline{σ} \\ A & ⟶^{c_{A}} & C(A, σ, f). \end{matrix}

To complete the proof, it remains to show that C(ε) maps Δ(L) to Z(A, σ, f). It suffices to see that the image of Δ(L) centralizes c_{A}(A) since c_{A}(A) generates C(A, σ, f). Again, we may extend scalars to reduce to the case where L is split, and use induction on dim L. Details are left to the reader. □

Remark 4.13. The situation described in Theorem 4.8 is related to results about Cartan subspaces given in [17]. Assume that F has characteristic different from 2. A Cartan subspace of a central simple algebra A of even degree 2n with an orthogonal involution σ is by definition a subspace of Skew(A, σ_{A}) such that there exists an element a ∈ H which is invertible in A, is separable (i.e., generates an étale F -subalgebra of A) and such that a^{−1}H is an étale commutative subalgebra of A of dimension 2n. Equivalently, if a ∈ Skew(A, σ_{A}) is invertible and separable in A, then H = {x ∈ Skew(A, σ_{A}) | xa = ax} is a Cartan subspace for a. It can be shown that the algebra L_{0} = a^{−1}H is independent of the choice of the element a. Moreover the subalgebra L = L_{0} ⊕ H of A is maximal commutative étale in A and has an involution σ_{L} which is the restriction of σ_{A} to L. Thus the set of skewsymmetric elements in L is a Cartan subspace of A. It follows from [17, Théorème 3] that c_{A}(L) generates a maximal commutative étale subalgebra of C(A, σ_{A}). This algebra is isomorphic to C(L, σ_{L}).

The Severi–Brauer variety of a Clifford algebra. We discuss in this subsection the geometric analogue of Theorem 4.8, showing that an embedding of a thin quadric into a quadric induces an embedding of the corresponding Clifford Γ-sets. Recall that the Severi–Brauer variety of a central simple algebra is a Γ-projective space. To take in account the fact that C(A, σ, f) is an Azumaya algebra of constant rank over the étale F -algebra Z(A, σ, f) we introduce a variant of the Severi–Brauer variety for Azumaya algebras over quadratic étale algebras.^{6} We consider disjoint unions P_{1} ⊔ P_{2} of two projective F -spaces of the same dimension, endowed with a continuous action of a profinite group Γ by collineations that may interchange the two components. We call these unions double Γ-projective spaces. They are equipped with a canonical structure map to a Γ-set Z of two elements

Z := {{P_{1}}, {P_{2}}} ← P_{1} ⊔ P_{2}.

If dim P_{1} = dim P_{2} = n − 1, we may define for every integer k ∈ n two different sets:

gr_{k−1}(P_{1} ⊔ P_{2}) = Gr_{k−1}(P_{1}) ∪ Gr_{k−1}(P_{2}) and

Gr_{k−1}(P_{1} ⊔ P_{2}) = {v_{1} ⊔ v_{2} | v_{1} ∈ Gr_{k−1}(P_{1}), v_{2} ∈ Gr_{k−1}(P_{2})}.

Note that, even though Gr_{k−1}(P_{1}) and Gr_{k−1}(P_{2}) are not Γ-sets since they may be interchanged under the Γ-action, Γ naturally acts on gr_{k−1}(P_{1}⊔P_{2}) and Gr_{k−1}(P_{1}⊔ P_{2}), and gr_{k−1}(P_{1} ⊔ P_{2}) carries a canonical Γ-equivariant map

Z ← gr_{k−1}(P_{1} ⊔ P_{2}).

If Γ_{0} ⊂ Γ is the subgroup fixing Z, we clearly have an isomorphism of Γ_{0}-sets

Gr_{k−1}(P_{1} ⊔ P_{2}) = Gr_{k−1}(P_{1}) × Gr_{k−1}(P_{2}).

If Γ_{0} ≠ Γ, then Gr_{k−1}(P_{1} ⊔ P_{2}) may be identified with the norm of the Γ_{0}-set Gr_{k−1}(P_{1}) (or Gr_{k−1}(P_{2})) following Ferrand’s general definition in [4]. For (Y ⟵^{π} X) a Γ-covering with |Y | = 2 and |X| = 2n, a general embedding

ε: (Y ⟵^{π} X) ↪ (Z ← P_{1} ⊔ P_{2})

is a pair of Γ-equivariant maps ε: X → gr_{0}(P_{1} ⊔ P_{2}), \overline{ε}: Y → Z such that (a) the following diagram commutes:

6For the sake of simplicity we give an ad-hoc construction.

X ⟶^{ε} gr_{0}(P_{1} ⊔ P_{2})
π↓  ↓
Y ⟶^{\overline{ε}}  Z,

(b) \overline{ε} is bijective, and

(c) the image of each fiber of Y is a set of n points in general position in P_{1}

and P_{2} respectively. The embedding ε thus induces Γ-equivariant maps

Gr_{k−1}ε: Gr_{k−1}(X/Y) ↪ gr_{k−1}(P_{1} ⊔ P_{2}) for all k ∈ n.

These maps fit into commutative diagrams

Gr_{k−1}(X/Y) ⟶^{λ^{k}(ε)} gr_{k−1}(P_{1} ⊔ P_{2})
↓  ↓
Y  ⟶^{\overline{ε}}  Z.

Double Γ-projective space arise from the following construction. Let Z be a quadratic étale algebra over an arbitrary field F, and let B be an Azumaya Z-algebra of constant rank. As usual, let F_{s} denote a separable closure of F, let Z_{s} = Z ⊗_{F} F_{s} and B_{s} = B ⊗_{F} F_{s} the F_{s}-algebras obtained by scalar extension, and let Γ be the Galois group of F_{s}/F. If dim_{F} B = 2n^{2}, then B_{s} ≃ M_{n}(F_{s}) × M_{n}(F_{s}); its minimal right ideals have dimension n. We let 𝕊𝔹(B) be the double Γ-projective space whose points are the n-dimensional right ideals of B_{s}, with the natural Γ-action. These ideals are divided into two classes, which correspond to the two primitive idempotents of Z_{s}: to each n-dimensional right ideal I, we associate the primitive idempotent e_{I} ∈ Z_{s} such that e_{I} ·I = I. Alternatively, we associate to I the element I^{δ} ∈ X(Z) such that I^{δ}(e_{I}) = 1. Thus, the canonical structure map of the double Γ-projective space 𝕊𝔹(B) can be identified with

X(Z) ⟵^{δ} 𝕊𝔹(B).

Now, let K ⟶^{i} L be an étale extension of degree n of a quadratic étale F -algebra. (Thus, dim_{F} L = 2n.) By definition, an embedding

ε: (L/K) ↪ (B/Z)

is an F -algebra embedding L ↪ B that restricts to an isomorphism \overline{ε}: K ⟶^{∼} Z.

Proposition 4.14. If, in the situation above, L is a split étale F -algebra, then for every embedding ε: (L/K) ↪ (B/Z) there is an F -algebra isomorphism φ: B ⟶^{∼} End_{K} L whose restriction to Z is \overline{ε}^{−1} such that φ ∘ ε: L ↪ End_{K} L is the regular representation.

Proof. By hypothesis K is split, hence Z also. Using \overline{ε} as an identification, let

K = Z = F × F,

and consider the corresponding decompositions L = L_{1} × L_{2}, B = B_{1} × B_{2}, and ε = (ε_{1}, ε_{2}). By Proposition 2.4, we may find F -algebra isomorphisms φ_{i}: B_{i} ⟶^{∼} End_{F} L_{i}, for i = 1, 2, such that φ_{i} ∘ε_{i}: L_{i} ↪ End_{F} L_{i} is the regular representation. The proposition holds with φ = (φ_{1}, φ_{2}): B → End_{F} L_{1} × End_{F} L_{2} = End_{K} L. □

Now, consider an arbitrary embedding ε: (L/K) ↪ (B/Z) of an étale extension of degree n. For every ξ ∈ X(L) we let as in(2.5)

ε_{∗}(ξ) = {x ∈ B_{s} | (1 ⊗ ξ(ℓ)) · x = (ε(ℓ) ⊗ 1) · x for all ℓ ∈ L}.

Proposition 4.15. The right ideal ε_{∗}(ξ) is n-dimensional, and ε_{∗} defines a general embedding

ε_{∗}: (X(K) ⟵^{X(i)} X(L)) ↪ (X(Z) ← 𝕊𝔹(B)).

Conversely, every general embedding of a Γ-covering of degree n with 2-element base in 𝕊𝔹(B) arises from an embedding of an étale extension in B.

Proof. The proof of the first (resp. second) part is similar to that of Proposition 2.6 (resp. 2.7). □

Proposition 4.15 applies in particular in the situation of Theorem 4.8: for the rest of this subsection, suppose A is a central simple F -algebra of degree 2n with a quadratic pair (σ, f). If (L, σ_{L}) is an étale F -algebra with involution of dimension 2n and

ε: (L, σ_{L}) ↪ (A, σ, f)

is an embedding of algebras with involution, then Theorem 4.8 yields an F -algebra embedding of the étale extension C(L, σ_{L})/Δ(L) into C(A, σ, f)/Z(A, σ, f). By Proposition 4.15 there is a general embedding

(4.16)  C(ε)_{∗}: X C(L, σ_{L}) ↪ 𝕊𝔹(C(A, σ, f)).

On the other hand, the embedding ε also yields by Proposition 3.8 an embedding of quadrics

(4.17)  ε_{∗}: Y(L, σ_{L}) ↪ Q(σ, f).

To complete the picture, we now relate the Clifford Γ-set C(Q(σ, f)) to the Severi– Brauer double projective space 𝕊𝔹(C(A, σ, f)). Recall from [7, §8] that there is a Clifford bimodule B(A, σ, f) over C(A, σ, f), which is equipped with a canonical F -linear map b: A → B(A, σ, f), see [7, §9]. In the split case A = End_{F} V, we have B(A, σ, f) = V ⊗_{F} C_{1}(V, q). The bilinear polar form of q yields an F -linear isomorphism V → V^{∗} which allows us to identify

V ⊗_{F} V = V ⊗_{F} V^{∗} = End_{F} V;

under this identification, the map b carries v⊗w ∈ V ⊗_{F} V to v⊗w ∈ V ⊗_{F} C_{1}(V, q). Since deg A = 2n, we have dim_{F} C(A, σ, f) = 2^{2n−1}. For every 2n-dimensional isotropic right ideal I ⊂ A_{s}, let ψ(I) ⊂ C(A_{s}, σ_{s}, f_{s}) denote the right annihilator of the F -vector space (σ_{s}(I))^{b} ⊂ B(A_{s}, σ_{s}, f_{s}), i.e.,

ψ(I) = {x ∈ C(A_{s}, σ_{s}, f_{s}) | (σ_{s}(I))^{b} · x = 0 for all y ∈ I}.

Proposition 4.18. The map ψ defines an embedding

ψ: (Δ(Q(σ, f)) ← C(Q(σ, f))) ↪ (X(Z(A, σ, f)) ← 𝕊𝔹(C(A, σ, f))).

Proof. We first show dim_{F} ψ(I) = 2^{n−1}. Let A_{s} = End_{F_{s}} V for some 2n-dimensional F_{s}-vector space V. The quadratic pair (σ_{s}, f_{s}) is adjoint to some hyperbolic quadratic form q on V, and I = Hom_{F_{s}}(V, U) for some totally isotropic n-dimensional subspace U ⊂ V. We may therefore identify V with U ⊕ U^{∗}, and q with the quadratic form q(u, φ) = φ(u) for u ∈ U, φ ∈ U^{∗}. Then the full Clifford algebra of q can be identified with the algebra End_{F_{s}}(⋀ U) of F_{s}-linear endomorphisms of the exterior algebra of U, in such a way that the canonical map

V = U ⊕ U^{∗} → C(V, q) = End_{F_{s}}(⋀U)

maps u ∈ U to the left exterior multiplication by u and φ ∈ U^{∗} to its associated derivation d_{φ}, see [7, (8.3)]. Let

⋀0U = ⨁_{i even} ^{⋀i}U ⊂ ⋀U  and  _{⋀1}U = ⨁_{i odd} ^{⋀i}U ⊂ U.

Then

C(A_{s}, σ_{s}, f_{s}) = C_{0}(V, q) = End_{F_{s}}(_{⋀0}U) × End_{F_{s}}(_{⋀1}U)

and

B(A_{s}, σ_{s}, f_{s}) = V ⊗_{F_{s}} C_{1}(V, q) = V ⊗_{F_{s}} (Hom_{F_{s}}(⋀_{0} U, ⋀_{1}U)×Hom_{F_{s}}(⋀_{1} U, ⋀_{0}U)).

Under the identification A_{s} = End_{F_{s}} V = V ⊗_{F_{s}} V we have I = U ⊗_{F_{s}} V, hence

(σ_{s}(I))^{b} = V ⊗_{F_{s}} U ⊂ V ⊗_{F_{s}} C_{1}(V, q).

Therefore, ψ(I) consists of the linear maps f ∈ End_{F_{s}}(⋀ U) such that f(⋀_{0} U) ⊂ ⋀_{0} U, f(⋀_{1} U) ⊂ ⋀_{1} U, and

u ∧ f(x) = 0 for all u ∈ U and all x ∈ ⋀U.

The last condition implies f(x) ∈ ⋀^{n} U, hence

ψ(I) = \begin{cases}Hom_{F_{s}}(⋀_{0} U, ⋀^{n} U) & if n is even, \\ Hom_{F_{s}}(⋀_{1} U, ⋀^{n} U) & if n is odd. \end{cases}

It follows that dim_{F_{s}} ψ(I) = dim_{F_{s}} ⋀_{0} U = dim_{F_{s}} ⋀_{1} U = 2^{n−1}, hence ψ(I) ∈ 𝕊𝔹(C(A, σ, f)). Moreover, for every nonzero φ ∈ U^{∗} we have d_{φ}(⋀^{n} U) ≠ {0}, hence

{v ∈ V | v · f = 0 in C(V, q) for all f ∈ ψ(I)} = U.

It follows that

I = {y ∈ A | (σ(y))^{b} · x = 0 for all x ∈ ψ(I)},

hence the map ψ is injective. It is clear from the definition that the map ψ is Γ-equivariant. To complete the proof, it remains to show that ψ induces an isomorphism

\overline{ψ}: Δ(Q(σ, f)) ⟶^{∼} X(Z(A, σ, f)).

As above, we identify A_{s} = End_{F_{s}} V for some 2n-dimensional F_{s}-vector space V. Let I, I′ ∈ C(Q(σ, f)), so

I = Hom_{F_{s}}(V, U), I′ = Hom_{F_{s}}(V, U′)

for some totally isotropic n-dimensional subspaces U, U′ ⊂ V. By a theorem of Witt, there is an orthogonal transformation g ∈ O(V, q) such that g(U) = U′. This transformation induces an automorphism ĝ of C_{0}(V, q) such that ĝ(ψ(I)) = ψ(I′). If e (resp. e′) is the primitive central idempotent of C_{0}(V, q) such that e·ψ(I) = ψ(I) (resp. e′ · ψ(I′) = ψ(I′)), we therefore have ĝ(e) = e′. Now, we have dim(U ∩ U′) ≡ n mod 2 if and only if g ∈ O^{+}(V, q) by [3, §§ II.6, II.10], and this condition holds if and only if the restriction of ĝ to the center of C_{0}(V, q) is the identity, by [7, (13.2)]. Therefore, e = e′ if and only if the ideals I and I′ belong to the same class of C(Q(σ, f)). This completes the proof. □

Corollary 4.19. If (σ, f) is a hyperbolic quadratic pair on A, then the Clifford algebra C(A, σ, f) decomposes into a direct product of simple F -algebras C(A, σ, f) ≃ C_{1} × C_{2}, and one of the factors C_{1}, C_{2} is split.

Proof. If (σ, f) is hyperbolic, then its discriminant is trivial, hence Z(A, σ, f) ≃ F × F and C(A, σ, f) decomposes into a direct product of simple F -algebras as above. Moreover, there is a maximal isotropic linear variety in Q(σ, f) that is defined over F, hence fixed under the action of Γ. Its image under ψ is a point on 𝕊𝔹(C(A, σ, f)) = SB(C_{1}) ⊔ SB(C_{2}) fixed under the action of Γ. Its existence shows that C_{1} or C_{2} is split. □

Remark 1. This corollary was first proved by D.C. Van Drooge and H.P. Allen, see also Tits [17, Proposition 8].

We may now relate the maps (4.16) and (4.17) associated to an embedding ε of algebras with involution:

Proposition 4.20. The following diagram is commutative:

(4.21)  \begin{matrix}C Y(L, σ_{L}) & ⟶^{C(ε_{∗})} & C(Q(σ, f)) \\ ∥ & & ↓^{ψ} \\ X C(L, σ_{L}) & ⟶^{C(ε)_{∗}} & 𝕊𝔹(C(A, σ, f)). \end{matrix}

Proof. Extending scalars, we reduce to the case where L is split, A = End_{F} L, and ε is the regular representation as in Example 3.6 (see Proposition 3.7). Let ω ∈ C Y(L, σ_{L}), and let e_{1}, …, e_{n}, e′_{1}, …, e′_{n} be the primitive idempotents of L, labelled in such a way that σ(e_{i}) = e′_{i} for i ∈ n and ξ(e_{i}) = 1, ξ(e′_{i}) = 0 for ξ ∈ ω. Under the identification C Y(L, σ_{L}) = XC(L, σ_{L}), the corresponding F -algebra homomorphism ξ_{ω}: C(L, σ_{L}) → F satisfies

ξ_{ω}(c_{L}(e_{i})) = 1 and ξ_{ω}(c_{L}(e′_{i})) = 0 for all i ∈ n.

In C(A, σ, f) = C_{0}(L, t) we have ε(e_{i}) = e_{i}e′_{i} and ε(e′_{i}) = e′_{i}e_{i} for i ∈ n, hence

C(ε)_{∗}(ξ_{ω}) = {x ∈ C_{0}(L, t) | e_{i}e′_{i}x = x and e′_{i}e_{i}x = 0 for all i ∈ n}.

On the other hand, C(ε_{∗})(ω) = Hom_{F} (L, ⨁_{i∈n} e_{i}L), hence

ψ(C(ε_{∗})(ω)) = {x ∈ C_{0}(L, t) | e_{i} · x = 0 for i ∈ n}.

In C(L, t), we have e_{i}^{2} = 0 and e_{i}e′_{i}+e′_{i}e_{i} = 1, hence e_{i}e′_{i}e_{i} = e_{i} for i ∈ n. Therefore, if e′_{i}e_{i}x = 0, then e_{i}x = 0. Conversely, if e_{i}x = 0, then e′_{i}e_{i}x = 0 and

x = (e_{i}e′_{i} + e′_{i}e_{i})x = e_{i}e′_{i}x.

Therefore, ψ(C(ε_{∗})(ω)) = C(ε)_{∗}(ω). □

5. Dynkin diagrams and Severi–Brauer varieties

The projective linear group PGL_{n} has Dynkin diagram of type A_{n−1} and the group 𝔖_{n} is its Weyl group. The special orthogonal group PGO_{2n}^{+}, with Weyl group 𝔖_{2}^{n−1} ⋊ 𝔖_{n}, has Dynkin diagram of type D_{n}. Accordingly we call groupoids occurring in Proposition 2.2 and (2.3) groupoids of type A_{n−1}, and groupoids occurring in (3.4) and Proposition 3.1 groupoids of type D_{n}. Many properties of these groupoids are related to properties of the corresponding Dynkin diagrams. After introducing oriented quadrics and étale algebras, we briefly consider two examples: the equivalence A_{3} = D_{3} and triality.

Special quadrics. A special or oriented Γ-quadric is a pair (Q, ∂), where Q is Γ-quadric (or a thin Γ-quadric) and ∂ is a fixed isomorphism of Γ-sets 2 − ←^{∼} Δ(Q). In particular the Γ-action on Δ(Q) is trivial. There are two possible choices for ∂. A choice is an orientation of Q. If Q is oriented we may use the given isomorphism ∂: 2 − ←^{∼} Δ(Q) to decompose the Clifford Γ-set:

C(Z/Z_{0}) = C_{1}(Z/Z_{0}, ∂) ⊔ C_{2}(Z/Z_{0}, ∂)

where

C_{i}(Z/Z_{0}, ∂) = {ω ∈ C(Z/Z_{0}) | ω^{δ∂} = i} for i = 1, 2.

Similarly, we define oriented extensions of étale F -algebras as pairs (L/L_{0}, ∂) where L/L_{0} is an extension of étale F -algebras and ∂ is an orientation of L, i.e., a fixed isomorphism ∂: Δ(L) ⟶^{∼} F × F. We also have oriented central simple algebras with quadratic pairs, defined as 4-tuples (A, σ, f, ∂), where ∂ is a fixed isomorphism ∂: Z(A, σ, f) ⟶^{∼} F × F. In the corresponding decomposition

C(A, σ, f, ∂) = C_{1}(A, σ, f, ∂) × C_{2}(A, σ, f, ∂),

C_{1}(A, σ, f, ∂) and C_{2}(A, σ, f, ∂) are central simple algebras over F. The classes of morphisms considered in the previous sections can be restricted to preserve the orientations. For example an embedding of oriented quadrics

ε: (Y, σ, ∂_{Y}) ↪ (Q, ∂_{Q})

is an embedding ε: Y ↪ Q such that Δ(ε) ∘ ∂_{Q} = ∂_{Y}. The diagram (4.21) restricts to a commutative diagram

C_{1} Y(L/L_{0}, ∂_{L}) ⟶^{C_{1}(ε_{∗})} C_{1}(Q(σ, f), ∂_{Q})
∥  ↓^{ψ_{1}}
X C_{1}(L/L_{0}, ∂_{L}) ⟶^{C_{1}(ε)_{∗}} SB(C_{1}(A, σ, f, ∂)).

As in 2.2, 2.3, 3.4 and 3.1, we have anti-equivalences of groupoids

OQCSA_{F}^{2n} ≡ OQuad_{Γ}^{2(n−1)} and OQÉtex_{F}^{n} ≡ OQCov_{Γ}^{n},

using the following notations:

(1) OQCSA_{F}^{2n} for the groupoid of oriented central simple algebras over F of degree 2n with quadratic pairs,

(2) OQuad_{Γ}^{2(n−1)} for the groupoid of oriented Γ-quadrics of dimension 2(n − 1) over F,

(3) OQÉtex_{F}^{n} for the groupoid of oriented quadratic étale extensions L/L_{0} of F -algebras with dim_{F} L_{0} = n.

(4) OQCov_{Γ}^{n} for the groupoid of oriented thin Γ-quadrics of dimension 2(n − 1) over F, i.e., double Γ-coverings Y/Y_{0} with |Y_{0}| = n.

Isomorphism classes are in bijection with H^{1}(Γ, PGO_{2n}^{+}), resp. H^{1}(Γ, 𝔖_{2}^{n−1} ⋊ 𝔖_{n}).

The exceptional isomorphism A_{3} = D_{3}. The fact that the Dynkin diagrams A_{3} and D_{3} are identical can be interpreted at the level of the corresponding groupoids. For any 4-element Γ-set X, the Γ-set Gr_{1}(X) of 2-element subsets of X is equipped with a canonical involution σ, which maps each 2-element subset of X to its complement. Thus, it defines a functor Set_{Γ}^{4} → QCov_{Γ}^{3}, which we also designate by Gr_{1}. The sections {S_{1}, S_{2}, S_{3}} ∈ C(Gr_{1}(X), σ) (where S_{1}, S_{2}, S_{3} are 2-element subsets of X such that {σ(S_{1}), σ(S_{2}), σ(S_{3})} ∩ {S_{1}, S_{2}, S_{3}} = ∅) naturally fall into two classes, depending on whether S_{1} ∩ S_{2} ∩ S_{3} is empty or is a 1-element subset of X. Therefore, there is a natural orientation on Gr_{1}(X), and we may view Gr_{1} as a functor Set_{Γ}^{4} → OQCov_{Γ}^{3}. The analogue of Gr_{1} for étale algebras under the Grothendieck correspondence is an “exterior algebra” functor λ^{2} : Ét_{F}^{4} → OQÉtex_{F}^{3} studied in [8]. There is also an “exterior algebra” functor λ^{2} : CSA_{F}^{4} → OQCSA_{F}^{6} discussed in [7, §15D].

Proposition 5.1. The functors Gr_{1} (resp. λ^{2}) and C_{1} induce equivalences:

\xymatrix{Set_{Γ}^{4} \ar[r]^{Gr_{1}} & OQCov_{Γ}^{3}, \ar[l]^{C_{1}} & Ét_{F}^{4} \ar[r]^{λ^{2}} & OQÉtex_{F}^{3} , \ar[l]^{C_{1}}}

\xymatrix{Proj_{Γ}^{3} \ar[r]^{Gr_{1}} & OQuad_{Γ}^{4}, \ar[l]^{C_{1}} & CSA_{F}^{4} \ar[r]^{λ^{2}} & OQCSA_{Γ}^{6}. \ar[l]^{C_{1}}}

Moreover these equivalences are compatible with general embeddings.

Proof. The (equivalent) first two cases are discussed in [8], the third is equivalent to the last and the last follows from [7, (15.32)]. □

Remark 5.2. The equivalence between groupoids of thin objects of type A_{3} and of type D_{3} is induced at the cohomological level by the well-known isomorphism 𝔖_{4} ⟶^{∼} 𝔖_{2}^{2} ⋊ 𝔖_{3}. Classically, for any separable field extension E of degree 4, the subalgebra of λ^{2}E fixed under the involution is the cubic resolvent (see [8]). On the geometric side, the quadric Gr_{1}(P) can be viewed as a 4-dimensional quadric parametrised by a 3-dimensional projective space. This quadric is known as Klein’s quadric.

Triality. All Dynkin diagrams but one admit at most automorphisms of order two, which are related to duality in algebra and geometry. The Dynkin diagram of D_{4}:

\begin{matrix}α_{3} \\ α_{4} \end{matrix} [figure: α_{1}; α_{2}]

is special, in the sense that it admits automorphisms of order 3. Algebraic and geometric objects related to D_{4} are of particular interest as they also usually admit exceptional automorphisms of order 3, which are called trialitarian. This is the case for the four groupoids OQCov_{Γ}^{4}, OQÉtex_{F}^{4} , OQuad_{Γ}^{6} and OQCSA_{F}^{8} of type D_{4}. For (E, ∂_{E}) in OQÉtex_{F}^{4} or in OQCSA_{F}^{8} , the objects C_{1}(E, ∂_{E}) and C_{2}(E, ∂_{E}) belong to QÉtex_{F}^{4} , resp. to QCSA_{F}^{8} (see [9], resp. [7, §35]). Corresponding results hold for objects in OQCov_{Γ}^{4} or OQuad_{Γ}^{6} since these groupoids of quadrics are anti-equivalent to the corresponding groupoids of algebras. Isomorphism classes of these groupoids are in bijection with H^{1}(Γ, PGO_{4}^{+}) or H^{1}(Γ, 𝔖_{2}^{3} ⋊ 𝔖_{4}). The groups PGO_{8}^{+} and 𝔖_{2}^{3} ⋊ 𝔖_{4} admit up to inner automorphisms one outer action of the cyclic group of order 3 (this is well known for PGO_{8}^{+}) and is in [5] for 𝔖_{2}^{3} ⋊ 𝔖_{4}). Thus the cyclic group of order 3 acts on the corresponding sets of isomorphism classes. In fact these automorphisms of order 3 are induced by automorphisms that can already be defined at the level of the groupoids. Let D_{4} be any of the four groupoids above, let D̃_{4} be the corresponding groupoid of unoriented objects, and let F: (E, ∂_{E}) ↦ E be the forgetful functor D_{4} → D̃_{4}.

Theorem 5.3. The functors C_{1}, C_{2} factor through the forgetful functor, i.e., there are functors

C_{1}^{+}, C_{2}^{+} : D_{4} → D_{4}

such that F ∘ C_{i}^{+} = C^{i} for i = 1, 2. These functors satisfy natural equivalences:

(C_{1}^{+})^{3} = Id, (C_{1}^{+})^{2} = C_{2}^{+}

and induce the trialitarian action on the sets of isomorphism classes.

Proof. The claim for thin objects is in [9] (under the hypothesis that char(F) ≠ 2) and the classical case (for algebras) is in [7, §35]. □

Observe that triality is compatible with general embeddings. Triality for thin objects is extensively discussed in [9]. A very nice presentation of classical (geometric and algebraic) triality is in [20].

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Department Mathematik, ETH Zentrum, CH-8092 Zürich, Switzerland

E-mail address: knus@math.ethz.ch

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: jean-pierre.tignol@uclouvain.be