ESSENTIAL p-DIMENSION OF PGL(p^{2})

ALEXANDER S. MERKURJEV

Abstract. Let p be a prime integer and F a field of characteristic different from p. We prove that the essential p-dimension of the group PGL_{F} (p^{2}) is equal to p^{2} + 1. This integer measures complexity of the class of central simple algebras of degree p^{2} over field extensions of F.

1. Introduction

Informally, the essential dimension of an “algebraic structure” over a field F is the smallest number of parameters required to define this structure over a field extension of F (see [1] or [10]). Thus, the essential dimension measures complexity of the structure.

Let p be a prime integer. Essential p-dimension of an “algebraic structure” measures complexity of the structure modulo the “effects of degree prime to p” (see [11]). In practice, the essential p-dimension is easier to compute than the essential dimension.

The formal definition of the essential (p-)dimension is as follows. Let p denote either a prime integer or 0. An integer k is said to be prime to p if k is prime to p when p > 0 and k = 1 when p = 0. Let F be a field. Consider the category Fields/F of field extensions of F and field homomorphisms over F. Let F : Fields/F → Sets be a functor (an “algebraic structure”) and K, E ∈ Fields/F. An element α ∈ F(E) is said to be p-defined over K (and K is called a field of p-definition of α) if there exist a finite field extension E′/E of degree prime to p (so E′ = E if p = 0), a field homomorphism K → E′ over F and an element β ∈ F(K) such that the image of α under the map F(E) → F(E′) coincides with the image of β under the map F(K) → F(E′). The essential p-dimension of α, denoted ed_{p}^{F}(α), is the least transcendence degree tr. deg_{F} (K) over all fields of p-definition K of α. The essential pdimension of the functor F is

ed_{p}(F) = sup{ed_{p}^{F}(α)},

where the supremum is taken over fields E ∈ Fields/F and all α ∈ F(E).

We write ed(F) for ed_{0}(F) and simply call ed(F) the essential dimension of F. Clearly, ed(F) ≥ ed_{p}(F) for all p.

Let G be an algebraic group over F. The essential p-dimension of G is the essential p-dimension of the functor F_{G} : Fields/F → Sets taking a field E to the set of isomorphism classes of all G-torsors (principal homogeneous G-spaces) over Spec(E).

If G = PGL_{n} over F, the functor F_{G} is isomorphic to the functor taking a field E to the set of isomorphism classes of central simple E-algebras of degree n. Let p be a prime integer and let p^{r} be the highest power of p dividing n. Then ed_{p}(PGL_{F} (n)) = ed_{p}(PGL_{F} (p^{r})) [11, Lemma 8.5.5]. Every central simple E-algebra of degree p is cyclic over a finite field extension of degree prime to p, hence ed_{p}(PGL_{F} (p)) = 2 [11, Lemma 8.5.7] as we just need two parameters to define a cyclic algebra. It is shown in [8, Cor. 3.10] and [11, Th. 8.6] that 4 ≤ ed_{p}(PGL_{F} (p^{2})) ≤ p^{2} + 1.

We prove the following:

Theorem 1.1. Let p be a prime integer and F a field of characteristic different from p. Then

ed_{p}(PGL_{F} (p^{2})) = p^{2} + 1.

Corollary 1.2. (Rost) If F is a field of characteristic different from 2, then ed(PGL_{F} (4)) = ed_{2}(PGL_{F} (4)) = 5.

Proof. By Theorem 1.1, we have ed(PGL_{F} (4)) ≥ ed_{2}(PGL_{F} (4)) = 5. On the other hand, ed(PGL_{F} (4)) ≤ 5 by [8]. □

We use the following notation:

X (F) is the character group of the absolute Galois group Gal(F_{sep}/F) of a field F.

Br(F) is the Brauer group of F. For a field extension L/F, we write Br(L/F) for the relative Brauer group Ker(Br(F) → Br(L)).

𝔾_{m} denotes the multiplicative group Spec F [t, t^{−1}] over F.

For a finite separable field extension L/F, we write R_{L/F} for the corestriction operation (see [7, §20.5]). In particular, R_{L/F} (𝔾_{m,L}) is the multiplicative group of L considered as an algebraic group (torus) over F. We write R_{L/F}^{(1)} (𝔾_{m,L}) for the torus of norm 1 elements in L.

If A is a central simple algebra over F, then SB(A) denotes the Severi-Brauer variety of A of reduced rank 1 right ideals in A [7, §1.C].

If p is a prime integer and B is a torsion abelian group, we write B{p} for the p-primary component of B.

In the present paper, the word “scheme” over a field F means a separated scheme of finite type over F and a “variety” over F is an integral scheme over F. If X is a scheme over F and E/F is a field extension, then X(E) = Mor_{F} (Spec(E), X) is the set of points of X over E. We write X_{E} for the scheme X ×_{F} Spec(E) over E.

2. Algebraic tori

2.1. R-trivial homomorphisms of algebraic tori. Let T be an algebraic torus over a field F. As usual, we write T^{∗} for the character group of T over a separable closure F_{sep} of F. The group T^{∗} is a Γ-lattice, where Γ = Gal(F_{sep}/F) is the absolute Galois group of F.

A torus P is quasi-trivial if P^{∗} is a permutation lattice, i.e., there is a Γinvariant ℤ-basis of P^{∗}.

Let E/F be a field extension. Recall that the group of R-equivalence classes T (E)/R is the factor group of T (E) modulo the subgroup RT (E) of all elements that are R-equivalent to 1 (see [2, §5] and [14, Ch. 6]). If P is a quasi-trivial torus, then P (E)/R = 1.

Example 2.1. [2, Prop. 15] Let L/F be a finite Galois field extension and T = R_{L/F}^{(1)} (𝔾_{m,L}) the torus of norm 1 elements in L. Then the subgroup RT (F) is generated by elements of the form σ(u)/u over all σ ∈ Gal(L/F) and u ∈ L^{×}.

Proposition 2.2. Let f : T_{0} → T_{1} be a homomorphism of algebraic tori over F. Let 1 → S_{1} → P_{1} → T_{1} → 1 be an exact sequence with P_{1} a quasi-trivial torus such that for any field extension E/F, the image of P_{1}(E) → T_{1}(E) coincides with RT_{1}(E) (for example, a flasque resolution of T_{1} satisfies this property by [2, Th. 2]). Then the following conditions are equivalent:

(1) For any field extension E/F, the homomorphism T_{0}(E)/R → T_{1}(E)/R induced by f is trivial.

(2) The image of the generic point of T_{0} in T_{1}(F (T_{0}))/R is trivial.

(3) There exists a commutative diagram of homomorphisms of algebraic

tori 1 ⟶ P_{0} ⟶ M_{0} ⟶ T_{0} ⟶ 1

↓  ↓  ^{f}↓
1 ⟶ S_{1} ⟶ P_{1} ⟶ T_{1} ⟶ 1.

with exact rows and a quasi-trivial torus P_{0}.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3): By assumption, there is a point x ∈ P_{1}(F (T_{0})) such that the image of x in T_{1}(F (T_{0})) coincides with the f-image of the generic point of T_{0}. The point x yields a rational morphism g : T_{0} −−→ P_{1} such that the composition T_{0} −−→ P_{1} → T_{1} coincides with f. Let U ⊂ T_{0} be the domain of definition of g. The Γ-lattice of characters T_{0}^{∗} is identified with the factor group F_{sep}[T_{0}]^{×}/F_{sep}^{×}, moreover, T_{0}^{∗} is a sublattice in Λ := F_{sep}[U]^{×}/F_{sep}^{×} and the factor lattice Λ/T_{0}^{∗} is a permutation lattice (see the proof of [2, Prop. 5]). Let M_{0} and P_{0} be the tori with the character lattices Λ and Λ/T_{0}^{∗} respectively. The morphism U → P_{1} yields a homomorphism of Γ-lattices P_{1}^{∗} → Λ and therefore, a homomorphism M_{0} → P_{1} of tori. By construction, the compositions M_{0} → P_{1} → T_{1} coincides with the composition M_{0} → T_{0} → T_{1}.

(3) ⇒ (1): Let E/F be a field extension. The composition M_{0}(E)/R → T_{0}(E)/R → T_{1}(E)/R is trivial as it factors through the trivial group P_{1}(E)/R. The first homomorphism in the composition is surjective since P_{0} is a quasitrivial torus. Hence the homomorphism T_{0}(E)/R → T_{1}(E)/R is trivial. □

We say that a homomorphism of algebraic tori f : T_{0} → T_{1} is R-trivial if f satisfies the equivalent conditions of Proposition 2.2.

Example 2.3. Let p be a prime integer and let (K_{1}/F, σ) and (K′/F, τ) be cyclic field extensions of degree p^{k} (k ≥ 1) and p respectively, L_{1} = K_{1} ⊗_{F} K′ and G_{1} = Gal(L_{1}/F). We assume that L_{1} is a field.

Let K_{0} ⊂ K_{1} be the subfield of degree p over F and set L_{0} = K_{0} ⊗_{F} K′. We write G_{0} for the Galois group of L_{0}/F.

Let T_{0} = R_{L_{0}/F}^{(1)} (𝔾_{m,L_{0}}) and T_{1} = R_{L_{1}/F}^{(1)} (𝔾_{m,L_{1}}) be the norm 1 tori for the extensions L_{0}/F and L_{1}/F respectively. Then T_{0} is a subtorus in T_{1}. We claim that the inclusion homomorphism f : T_{0} → T_{1} is not R-trivial.

For i = 0, 1, there is an exact sequence of G_{i}-modules

(1)  0 → ℤ → ℤ[G_{i}] → T_{i}^{∗} → 0.

If M is a G_{1}-lattice and X a G_{1}-module, we write Êxt_{G_{1}}^{i}(M, X) for the Tate cohomology group Ĥ^{i}(G_{1}, M^{∗} ⊗_{ℤ} X). It follows from (1) that

Êxt_{G_{1}}^{0}(T_{1}^{∗}, T_{0}^{∗}) ≃ Ĥ^{−1}(G_{1}, T_{0}^{∗}) = Ĥ^{−1}(G_{0}, T_{0}^{∗}) ≃ Ĥ^{0}(G_{0}, ℤ) ≃ ℤ/p^{2}ℤ.

Moreover, the class of the map f^{∗} : T_{1}^{∗} → T_{0}^{∗} corresponds to 1 + p^{2}ℤ, so f^{∗} has order p^{2} in Êxt_{G_{1}}^{0}(T_{1}^{∗}, T_{0}^{∗}).

Let P_{1} be the product of two copies of the torus R_{L_{1}/F} (𝔾_{m,L_{1}}) and α : P_{1} → T_{1} the homomorphism taking (u, v) to σ(u)τ(v)/uv. For a field extension E/F, the image of P_{1}(E) in T_{1}(E) coincides with RT_{1}(E) (see Example 2.1). Set S_{1} := Ker(α), so we have an exact sequence of tori

0 → S_{1} → P_{1} → T_{1} → 0.

Suppose the homomorphism f : T_{0} → T_{1} is R-trivial. By Proposition 2.2, there is a diagram

1 ⟶ P_{0} ⟶ M_{0} ⟶ T_{0} ⟶ 1
↓  ↓  ^{f}↓
1 ⟶ S_{1} ⟶ P_{1} ⟶ T_{1} ⟶ 1.

with exact rows and a quasi-trivial torus P_{0}. As P_{1}^{∗} is a free G_{1}-module, the right vertical map in the commutative diagram

Êxt_{G_{1}}^{0}(T_{0}^{∗}, T_{0}^{∗}) ⟶ Êxt_{G_{1}}^{0}(T_{1}^{∗}, T_{0}^{∗})
↓  ↓
Êxt_{G_{1}}^{1}(P_{0}^{∗}, T_{0}^{∗}) ⟶ Êxt_{G_{1}}^{1}(S_{1}^{∗}, T_{0}^{∗})

is an isomorphism. It follows that the image of the identity of T_{0}^{∗} in Êxt_{G_{1}}^{1}(S_{1}^{∗}, T_{0}^{∗})

has order p^{2}. We get a contradiction by showing that the group Êxt_{G_{1}}^{1}(P_{0}^{∗}, T_{0}^{∗}) has exponent p. The G_{1}-lattice P_{0}^{∗} is a direct sum of lattices of the form ℤ[G_{1}/H_{1}], where H_{1} is a subgroup of G_{1}. Let H_{0} be the image of H_{1} under the surjection G_{1} → G_{0}. We have

Êxt_{G_{1}}^{1}(ℤ[G_{1}/H_{1}], T_{0}^{∗}) = Êxt_{H_{1}}^{1}(ℤ, T_{0}^{∗}) = H^{1}(H_{1}, T_{0}^{∗}) = H^{1}(H_{0}, T_{0}^{∗}) = H^{2}(H_{0}, ℤ).

The latter group is isomorphic to the character group of H_{0} that is of exponent p as so is H_{0}.

2.2. Characters, cyclic algebras and tori. Let F be a field and Γ = Gal(F_{sep}/F) the absolute Galois group of F. The character group X (F) of Γ is equal to

Hom_{cont}(Γ, ℚ/ℤ) = H^{1}(F, ℚ/ℤ) ≃ H^{2}(F, ℤ).

For a character χ ∈ X (F) set F (χ) = (F_{sep})^{Ker(χ)}. Then F (χ)/F is a cyclic field extension of degree ord(χ). The Galois group Gal(F (χ)/F ) has a canonical generator σ such that χ(σ̃) = ord(χ)^{−1} + ℤ for any lifting σ̃ of σ to Γ.

Let K/F be cyclic field extension. Choose a character χ ∈ X (F) such that K = F (χ). The cup-product

X (F) ⊗ F^{×} = H^{2}(F, ℤ) ⊗ H^{0}(F, F_{sep}^{×}) → H^{2}(F, F_{sep}^{×}) = Br(F)

takes χ ⊗ a to the class χ ∪ (a) of a cyclic algebra split by K. In fact, every element of Br(K/F) is of the form χ ⊗ a for some a ∈ F^{×}.

Let L be an étale F -algebra of dimension n and S = R_{L/F} (𝔾_{m,L})/ 𝔾_{m}. The exact sequence

1 → 𝔾_{m} → R_{L/F} (𝔾_{m,L}) → S → 1

and Hilbert Theorem 90 yield an isomorphism θ : H^{1}(F, S) →^{∼} Br(L/F). Let α ∈ H^{1}(F, S) and let S_{α} be the corresponding principal homogeneous space of S. As S is an open subscheme of the projective space ℙ_{F} (L), the variety S_{α} is an open subset of the Severi-Brauer variety SB(A) of a central simple F -algebra A_{α} of degree n such that [A_{α}] = θ(α) in Br(L/F). Moreover, S_{α} is trivial if and only if A_{α} is split.

Let χ ∈ X (F) and L = F (χ). Then S ≃ R_{L/F}^{(1)} (𝔾_{m,L}) by Hilbert Theorem 90 and [A_{α}] = χ∪a for some a ∈ F^{×}. Moreover, the principal homogeneous space S_{α} coincides with the fiber S_{a} of the norm homomorphism R_{L/F} (𝔾_{m,L}) → 𝔾_{m} over a.

2.3. Bicyclic algebras and tori. Let χ and η be two characters in X (F) of order n and m respectively. Then the fields K = F (χ) and K′ = F (η) are cyclic extensions of F of degree n and m respectively. Set L = K ⊗_{F} K′, so L is a bicyclic extension of F of degree nm. The group G = Gal(K/F)×Gal(K′/F) acts naturally on L by automorphisms and G is generated by elements σ and τ such that L^{σ} = K′ and L^{τ} = K.

Let I_{G} be the augmentation ideal in the group ring Λ := ℤ[G], i.e., I_{G} = Ker(ε), where ε : Λ → ℤ is defined by ε(ρ) = 1 for all ρ ∈ G. We have:

(2)  Br(L/F) = H^{2}(G, L^{×}) = Ext_{G}^{2}(ℤ, L^{×}) ≃ Ext_{G}^{1}(I_{G}, L^{×}).

Consider the exact sequences of G-modules

(3)  0 → M → Λ^{2} ⟶^{f} I_{G} → 0,

where f(x, y) = (σ − 1)x + (τ − 1)y and M = Ker(f) and

(4)  0 → Λ/ℤN_{G} ⟶^{g} M ⟶^{h} ℤ^{2} → 0,

where N_{G} = ∑_{ρ∈G} ρ ∈ Λ, g(x + ℤN_{G}) = ((τ − 1)x, (1 − σ)x) and h(x, y) = (ε(x)/n, ε(y)/m).

Let T be the torus of norm 1 elements for the extension L/F. We have

(5)  T (F) = Hom_{G}(Λ/ℤN_{G}, L^{×}).

The exact sequences (3) (4), the isomorphisms (2) and (5) and Hilbert Theorem 90 yield a commutative diagram:

\xymatrix{ & Hom_{G}(ℤ^{2}, L^{×}) \ar[dr]^{α} \ar[d]_{h^{∗}} \\ Hom_{G}(Λ^{2}, L^{×}) \ar@{-->}[dr]_{β} \ar[r] & Hom_{G}(M, L^{×}) \ar[r] \ar[d]_{g^{∗}} & Br(L/F) \ar[r] & 0 \\  & T (F) \ar[d] \\  & 0}

It follows that the cokernels of α and β are naturally isomorphic. The image of α : F^{×2} → Br(L/F) is the subgroup of decomposable elements Br_{dec}(L/F) of Br(L/F) generated by χ ∪ (a) and η ∪ (b) with a, b ∈ F^{×}.

The cokernel of β : L^{×2} → T (F) is the group of R-equivalence classes T (F)/R (see Example 2.1). We have proved:

Proposition 2.4. Let L/F is a bicyclic extension and T = R_{L/F}^{(1)} (𝔾_{m,L}). Then there is a natural isomorphism

T (F)/R ≃ Br(L/F)/ Br_{dec}(L/F).

Example 2.5. The torus T = R_{L/F}^{(1)} (𝔾_{m,L}) is not rational if L/F is a bicyclic field extension of degree p^{2} by [14, §4.8]. Moreover, T is not R-trivial generically, i.e., there is a field extension E/F such that T (E)/R ≠ 1. It fact, the image of the generic point of T in T (F (T))/R is not trivial, i.e., the identity map of T is not R-tivial (see Example 2.3).

3. Degree of points of the norm 1 torus for a bicyclic field extension

3.1. Chow groups and push-forward homomorphism. Let X be a scheme over a field F. We write Z(X) for the group of algebraic cycles on X, i.e., the free abelian group generated by points of X. We write CH(X) for the factor group of Z(X) by the subgroup of cycles rationally equivalent to 0 (see [3, §1.3]). The groups Z(X) and CH(X) are graded by the dimension of points. If x ∈ X is a point of dimension i, [x] denotes the class of x in CH_{i}(X).

If X is a variety of dimension d, then the group CH_{d}(X) is infinite cyclic generated by the class of the generic point of X.

Let f : X → Y be a morphism of schemes over F. The push-forward homomorphism f_{∗} : Z(X) → Z(Y) is a graded homomorphism defined by

f_{∗}(x) = \{ \begin{matrix}[F (x) : F (y)] · y, & if [F (x) : F (y)] is finite; \\ 0, & otherwise, \end{matrix}

where x ∈ X and y = f(x). If f is a proper morphism, then f_{∗} factors through the rational equivalence, defining the push-forward homomorphism CH(X) → CH(Y) still denoted by f_{∗} (see [3, §1.4]).

3.2. Degree of a point. Let X be a scheme over a field F, a ∈ X(E) a point over a field extension E/F and {x} the image of a : Spec(E) → X. The dimension of a is the integer dim(a) := dim(x). If f : X → Y is a morphism of varieties over F and a ∈ X(E) for a field extension E/F, we have dim(a) ≥ dim(f(a)). If d = dim(a), we define the class [a] of a in CH_{d}(X) as follows:

[a] := \{ \begin{matrix}[E : F (x)] · [x], & if [E : F (x)] is finite; \\ 0, & otherwise. \end{matrix}

In addition, if X is a variety, the degree of a is the integer deg(a) satisfying [a] = deg(a) · [x] if dim(a) = dim(X) and x is the generic point of X, and deg(a) = 0 otherwise.

If E′/E is a field extension and a ∈ X(E), we write a_{E′} for the image of a in X(E′). If E′/E is finite, we have deg(a_{E′}) = [E′ : E] · deg(a).

If E = F (X) is the function field of X and a ∈ X(E) is the generic point, then deg(a) = 1.

Proposition 3.1. Let f : X → Y be a proper morphism of varieties over F and let a ∈ X(E) be a point over a field extension E/F. Then [f(a)] = f_{∗}([a]) in CH(Y).

Proof. Let {x} be the image of a in X and y = f(x). If one of the field extensions E/F (x) and F (x)/F (y) is infinite, then [f(a)] = 0 and f_{∗}([a]) = 0. We may assume that E is a finite extension of F (y). Then

[f(a)] = [E : F (y)] · [y]
= [E : F (x)]([F (x) : F (y)] · [y])
= [E : F (x)] · f_{∗}([x])
= f_{∗}([a]).  □

If Z is a scheme over F, we write n(Z) for the gcd[F (z) : F] over all closed points z ∈ Z.

Example 3.2. Let T be an algebraic torus over F. We write i(T) for the greatest common divisor of the integers [E : F] over all finite field extensions E/F such that T is isotropic over E. If X is a smooth complete geometrically irreducible variety containing T as an open set, then n(X \ T) = i(T) by [2, Lemme 12] (see also [9, Lemma 5.1]).

We shall need a variant of a push-forward homomorphism for morphisms that are not proper.

Proposition 3.3. Let X be a complete variety over F, U ⊂ X an open subvariety, Z = X \ U and f : U → Y a morphism over F, where Y is a variety of dimension d over F. If n = n(Z_{F (Y)}), then the push-forward homomorphism on cycles f_{∗} : Z(U) → Z(Y), composed with the projection Z(Y) → Z_{d}(Y) = ℤ, gives rise to a well defined homomorphism

f_{⋆} : CH(U) → ℤ/nℤ.

Moreover, for any point a ∈ U(E) over a field extension E/F, one has f_{⋆}([a]) = deg(f(a)) modulo n.

Proof. We define the map f_{⋆} to be trivial on all homogeneous components CH_{i}(U) except i = d, so we just need to define f_{⋆} on CH_{d}(U).

We claim that the image of the push-forward homomorphism

s_{∗} : CH_{d}(Z × Y) → CH_{d}(Y) = ℤ

for the projection s : Z × Y → Y is contained in nℤ. Let u ∈ Z × Y be a point of dimension d. If s(u) is not the generic point of Y, then s_{∗}([u]) = 0. Otherwise, u is a closed point in Z_{F (Y)} ⊂ Z × Y and s_{∗}([u]) coincides with the degree of this closed point and hence is divisible by n. The claim is proven.

The map s_{∗} factors as s_{∗} = q_{∗} ∘ i_{∗}, where i : Z × Y → X × Y is the closed embedding and q : X × Y → Y is the projection. By localization [3, §1.8], CH_{d}(U × Y) is canonically isomorphic to the cokernel of i_{∗}. By the claim, q_{∗} gives rise to a homomorphism CH_{d}(U × Y) → ℤ/nℤ. Composing it with the push-forward homomorphism for the closed embedding (1_{U}, f) : U → U × Y, we get the required homomorphism f_{⋆} : CH_{d}(U) → ℤ/nℤ. The last equality in the statement follows from Proposition 3.1 applied to q. □

Example 3.4. Let T be an algebraic torus over F and n = i(T) (see Example 3.2). Then the structure morphism T → Spec(F) gives rise to a homomorphism CH_{0}(T) → ℤ/nℤ that takes the class of a closed point t ∈ T to [F (t) : F] modulo n.

3.3. Chow groups of tori and Severi-Brauer varieties. Let p be a prime integer and let Z be the product of r copies of the projective space ℙ_{F} (W), where W is a vector space of dimension n > 0 over F. Then

CH(Z) = ℤ[h] := ℤ[h_{1}, h_{2}, …, h_{r}],

with h_{i}^{n} = 0 for all i, where h_{i} is pull-back on Z of the class of a hyperplane on the ith factor of Z. Moreover, ℤ[h] is the factor ring of the polynomial ring on the variables t_{1}, t_{2}, …, t_{r} by the ideal generated by t_{1}^{n}, t_{2}^{n}, …, t_{r}^{n}. Note that the homogeneous ith component ℤ[h]_{i} is trivial if i > r(n−1) and ℤ[h]_{r(n−1)} = ℤh^{n−1}, where h := h_{1}h_{2} ⋯ h_{p}.

Let K/F be a Galois field extension with a cyclic Galois group H of prime order p and let σ be a generator of H. Let V be a vector space of dimension n > 0 over K. Consider the variety X = R_{K/F} (ℙ_{K}(V)) over F. Then X_{K} is the product of p copies of ℙ_{K}(V). The group H acts on the product by cyclic permutation of the factors. We have the graded ring homomorphism

CH(X) → CH(X_{K}) = ℤ[h],

where h = (h_{1}, h_{2}, …, h_{p}).

The group H acts on ℤ[h] permuting cyclically the h_{i}’s. Hence the image of the map CH(X) → ℤ[h] is contained in the subring ℤ[h]^{H} of H-invariant elements, so we have the graded ring homomorphism

CH(X) → ℤ[h]^{H}

(which is in fact an isomorphism). The image of an element α ∈ CH(X) in ℤ[h]^{H} is denoted by ᾱ. For example, if α is the class of the subscheme R_{K/F} (ℙ_{K}(W)) of X, where W is a K-subspace of V of codimension i = 0, 1, …, n − 1, then ᾱ = h^{i}.

Consider the trace homomorphism

tr : ℤ[h] → ℤ[h]^{H}

defined by tr(x) = ∑_{i=0}^{p−1} σ^{i}(x). We write I for the image of tr. Clearly, I is a graded ideal in ℤ[h]^{H}. Note that

(6)  (ℤ[h]^{H})_{j} = \{ \begin{matrix}I_{j}, & if p does not divide j; \\ ℤh^{i} + I_{j}, & if j = pi. \end{matrix}

It follows that ℤ[h]^{H} is generated by I and h^{i}, i = 0, 1, …, n − 1 as an abelian group. Moreover, ph^{j} ∈ I for all j and I_{p(n−1)} = pℤh^{n−1}.

Let A be a central simple algebra over K of degree n and let Y = R_{K/F} (SB(A)), where SB(A) is the Severi-Brauer variety of A over K. The function field E of Y splits A and is linearly disjoint with K/F. Therefore, Y_{E} ≃ X_{E} and we have the ring homomorphism

CH(Y) → CH(Y_{E}) ≃ CH(X_{E}) → ℤ[h]^{H}.

The image of an element α ∈ CH(Y) in ℤ[h]^{H} is denoted by ᾱ.

Proposition 3.5. Let K/F be a cyclic field extension of a prime degree p, let A be a nonsplit central simple K-algebra of degree p and Y = R_{K/F} (SB(A)). Then the image of the map CH(Y) → ℤ[h]^{H} is contained in ℤ + I.

Proof. Consider a more general situation: A is a central simple K-algebra of index p and degree n. Let α ∈ CH(Y). We shall prove in the cases 1 and 2 below that ᾱ ∈ ℤ + I. By (6), we may assume that α ∈ CH^{pi}(Y) for i = 1, 2, …, n − 1. Let a ∈ ℤ be such that ᾱ ≡ ah^{i} modulo I. It suffices to prove that a is divisible by p.

Case 1: i = n − 1. We have ᾱ = bh^{n−1} for some b ≡ a modulo p as I_{p(n−1)} = pℤh^{n−1}. Since h^{n−1} is the class of a rational point of Y over a splitting field and the degree of every closed point of Y is divisible by p, we have b ∈ pℤ. Therefore, a ∈ pℤ.

Case 2: i divides n − 1. Write n − 1 = ij. We have α^{j} ∈ CH^{p(n−1)}(Y) and α^{j} ≡ a^{j}h^{n−1} modulo I. By Case 1, a^{j} and hence a is divisible by p.

Now assume that A is a central division K-algebra of degree p and α ∈ CH^{pi}(Y) with i = 1, 2, …, p−1. We shall prove that ᾱ ∈ I. Write ik +pm = 1 for some integers k and m > 0. The Severi-Brauer variety SB(M_{m}(A)) can be identified with the variety of the reduced rank 1 right A-submodules in the free right A-module A^{m}. The projection to last component A of A^{m} gives rise to a rational morphism SB(M_{m}(A)) → SB(A) that is defined on the complement U of the variety SB(M_{m−1}(A)) embedded into SB(M_{m}(A)) as a closed subvariety via the inclusion A^{m−1} → A^{m}, (a_{1}, …, a_{m−1}) ↦ (a_{1}, …, a_{m−1}, 0). Moreover, the projection U → SB(A) is a vector bundle.

Let Y′ = R_{K/F} (SB(M_{m}(A))) and U′ = R_{K/F} (U). Then U′ is an open subscheme of Y′ and the natural morphism U′ → Y is a vector bundle. Hence we have a surjective homomorphism

CH(Y′) → CH(U′) ≃ CH(Y).

Moreover, the diagram

\begin{matrix}CH(Y′) & ⟶ & CH(Y) \\ ↓ & & ↓ \\ ℤ[h′]^{H} & ⟶ & ℤ[h]^{H} \end{matrix},

where the bottom map takes a monomial h′^{α} to h^{α} if α_{i} < p for all i and to 0 otherwise, is commutative. Lift α to an element α′ ∈ CH^{pi}(Y′). As i divides pm − 1, by Case 2 applied to the algebra M_{m}(A), we have ᾱ′ ∈ I′. Since the bottom map in the diagram takes I′ to I, we have ᾱ ∈ I. □

Let K′/F be a cyclic field extension of degree p and

S = (R_{K′/F}^{(1)} (𝔾_{m,K′}))^{r} ≃ (R_{K′/F} (𝔾_{m,K′})/ 𝔾_{m})^{r}

for some r > 0. We view the variety of the group S as an open subset of Z := ℙ_{F} (K′)^{r}. Hence the restriction gives a surjective ring homomorphism

(ℤ/pℤ)[h] = Ch(Z) → Ch(S),

where h = (h_{1}, h_{2}, …, h_{r}), h_{i}^{p} = 0 for all i, and we write Ch for the Chow groups modulo p. We shall also write h̃_{i} for the image of h_{i} in Ch^{1}(S). The class in Ch^{r(p−1)}(S) of a rational point of S is equal to h̃^{p−1}, where h̃ = h̃_{1}h̃_{2} ⋯ h̃_{r} ∈ Ch^{p}(S). As i(S) = p, we have h̃^{p−1} ≠ 0 by Example 3.4.

Proposition 3.6. The map (ℤ/pℤ)[h] → Ch(S) is a ring isomorphism.

Proof. Suppose that f(h̃_{1}, h̃_{2}, …, h̃_{r}) = 0 for a nonzero homogeneous polynomial f over ℤ/pℤ. Suppose that a monomial h_{1}^{α_{1}} ⋯ h_{r}^{α_{r}} enters f with a nonzero coefficient. Multiplying the equality by h̃_{1}^{β_{1}} ⋯ h̃_{r}^{β_{r}} with β_{i} = p − 1 − α_{i}, we get h̃^{p−1} = 0, a contradiction. □

For an element α in Ch(S) we shall write ᾱ for the corresponding element in (ℤ/pℤ)[h].

Consider the homomorphism f : S × S → S defined by f(x, y) = xy^{−1}. Recall that as i(S) = p, by Example 3.2 and Proposition 3.3, we have the homomorphism

(7)  f_{⋆} : CH_{r(p−1)}(S × S) → ℤ/pℤ.

Lemma 3.7. For any α ∈ Ch^{i}(S) and β ∈ Ch^{j}(S) with i + j = r(p − 1), we have

ᾱ · β̄ = f_{⋆}(α × β)h^{p−1}

in (ℤ/pℤ)[h].

Proof. It suffices to consider the case when α and β are monomials in h̃_{i}. As both sides of the equality commute with products, we may assume that r = 1, i.e., S = R_{K′/F} (𝔾_{m,K′})/ 𝔾_{m}, and α = h̃^{i}, β = h̃^{j}. The cycles α and β are represented by ℙ(U) ∩ S and ℙ(W) ∩ S, where U and W are F -subspaces of K′ of codimension i and j respectively. The fiber of the restriction

f′ : (ℙ(U) ∩ S) × (ℙ(W) ∩ S) → S

of f over a point s of S is isomorphic to ℙ(U ∩ sW) ∩ S. The vector space U ∩sW is one-dimensional for a generic s, hence f′ is a birational isomorphism and f_{⋆}(α × β) = 1 + pℤ. On the other hand, ᾱ · β̄ = h^{i} · h^{j} = h^{p−1}. □

Let L/F be a bicyclic field extension of degree p^{2} and T = R_{L/F}^{(1)} (𝔾_{m,L}). Choose a subfield K of L of degree p over F and let t ∈ K^{×} be an element with N_{K/F} (t) = 1, i.e., t is an F -point of the torus R_{K/F}^{(1)} (𝔾_{m,K}). Write S_{t}

for the fiber of the norm homomorphism T → R_{K/F}^{(1)} (𝔾_{m}) over t. The variety

S_{t} is a principal homogeneous space of the torus S = R_{K/F} (R_{L/K}^{(1)}(𝔾_{m,L})) ≃ R_{K/F} (R_{L/K}(𝔾_{m,L})/ 𝔾_{m,K}).

The variety S_{t} is canonically isomorphic to an open subscheme of the variety Y := R_{K/F} (SB(A_{t})) for a central simple K-algebra A_{t} of degree p (see Section 2.2). Over the function field E of SB(A_{t}) over K, the varieties S_{t} and S become isomorphic to the torus (R_{LE/E}^{(1)}(𝔾_{m,LE}))^{p}, where LE = L ⊗_{K} E, so we can apply the constructions considered above to the torus S_{E} over E. In particular, we have the element ᾱ ∈ (ℤ/pℤ)[h] well defined for any cycle α on S_{t} and S.

Consider the morphism

f : S_{t} × S → S_{t}, f(x, y) = xy^{−1}.

We have defined the homomorphism (see (7)):

f_{⋆} : CH_{p(p−1)}(S_{t} × S) → CH_{p(p−1)}((S_{t})_{E} × S_{E}) → ℤ/pℤ.

Proposition 3.8. Suppose that the principal homogeneous space S_{t} is not trivial. Then f_{⋆}(α ×h̃^{j}) = 0 for any α ∈ Ch^{p(p−j−1)}(S_{t}) and j = 0, 1, …, p−2.

Proof. As S_{t} is not trivial, the algebra A_{t} is not split. We can lift α to a cycle β in Ch(Y). By Proposition 3.5, β̄ belongs to the image Ĩ of the ideal I in (ℤ/pℤ)[h]^{H}. It follows that ᾱ · h^{j} = β̄ · h^{j} ∈ Ĩ_{p(p−1)} = 0. Lemma 3.7 (applied to the field extension E of F and r = p) shows that f_{⋆}(α × h̃^{j}) = 0. □

3.4. A key proposition. Let p be a prime integer, L/F a bicyclic field extension of degree p^{2}, G = Gal(L/F), σ and τ generators of G. Consider the tori T = R_{L/F}^{(1)} (𝔾_{m,L}) of norm 1 elements in L/F and P = R_{L/F} (𝔾_{m,L})/ 𝔾_{m}, both of dimension d := p^{2} −1. The torus T (respectively, P) becomes isotropic over a field extension E/F if and only if E ⊗_{F} L is not a field. It follows that i(T) = i(P) = i(T × P) = p.

Consider the morphisms f and g from T × P to T defined by f(t, v) = t and g(t, v) = tσ(v)/v. By Proposition 3.3 and Example 3.2, f and g give rise to a well defined homomorphisms f_{⋆} and g_{⋆} from CH_{d}(T × P) to ℤ/pℤ.

Proposition 3.9. The maps f_{⋆} and g_{⋆} coincide.

Proof. The torus P is an open subscheme in the projective space ℙ_{F} (L), hence the ring CH(P) is generated by the restriction to P of the class e of a hyperplane in ℙ_{F} (L). Moreover, by the Projective Bundle Theorem [3, Th. 3.3], CH_{d}(T × P) coincides with the sum of subgroups CH_{i}(T) × e^{i} over all i = 0, 1, …, d.

Let β ∈ CH_{i}(T). It suffices to show that f_{⋆}(β × e^{i}) = g_{⋆}(β × e^{i}) for any i = 0, 1, …, d. If i = d, the class e^{i} is represented by the identity point 1 of P. The equality follows from the fact that f and g coincide on T × {1}.

Now assume that i < d. In this case f_{⋆}(β × e^{i}) = 0 and we need to show that g_{⋆}(β × e^{i}) = 0.

Let K be the subfield of σ-invariant elements in L of degree p over F. We have pk + 1 ≤ p^{2} − i ≤ p(k + 1) for some integer k = 0, …, p − 1. Consider a K-linear subspace W of L of K-dimension k such that K ∩ W = 0. Let V be an F -subspace of L of dimension p^{2} − i over F such that

F ⊕ W ⊂ V ⊂ K ⊕ W.

The class of P ∩ ℙ(V) in CH^{i}(P) is equal to e^{i}.

The torus S := R_{K/F} (R_{L/K}^{(1)}(𝔾_{m,L})) is the kernel of the norm homomorphism

T → T_{1} := R_{K/F}^{(1)} (𝔾_{m,K}), so we have an exact sequence

(8)  1 → S → T → T_{1} → 1.

By Hilbert Theorem 90, S ≃ R_{K/F} (R_{L/K}(𝔾_{m,L})/ 𝔾_{m,K}). We view S as an open subscheme of R_{K/F} (ℙ_{K}(L)). The map g factors as follows:

T × P ⟶^{1_{T} ×l} T × S ⟶^{r} T,

where l : P → S is defined by l(v) = v/σ(v) and r(t, s) = ts^{−1}. The image of P ∩ ℙ_{F} (K ⊕ W) under l is the variety S ∩ R_{K/F} (ℙ_{K}(K ⊕ W)) of dimension pk in S ≃ R_{K/F} (R_{L/K}(𝔾_{m,L})/ 𝔾_{m,K}). Hence, if p^{2} − i > pk + 1, then dim(P ∩ ℙ(V)) > pk, but dimension of the image of P ∩ ℙ(V) under l is at most pk, so P ∩ ℙ(V) loses dimension under l, therefore, g_{⋆}(β × e^{i}) = 0.

It remains to consider the case p^{2} − i = pk + 1, k = 1, …, p − 1, i.e., V = F ⊕ W. Since the map P ∩ ℙ(V) → R_{K/F} (ℙ_{K}(K ⊕ W)) given by l is a birational isomorphism, and the class of R_{K/F} (ℙ_{K}(K ⊕ W)) in CH(S) is equal to h^{p−k−1}, where h ∈ CH^{p}(S) is the class given by a K-hyperplane in L, it suffices to show that r_{⋆}(β × h^{p−k−1}) = 0.

Let S_{t} be the fiber of the norm homomorphism T → T_{1} over the generic point t of T_{1}, so S_{t} is a principal homogeneous space of S over the function field F (T_{1}). Denote by

r′ : S_{t} × S → S_{t}

the morphism given by r′(x, s) = xs^{−1}. Thus we have a commutative diagram

S_{t} × S ⟶^{r′} S_{t}
q↓  ↓^{m}
T × S ⟶^{r} T

where m is the canonical morphism and q = m × 1_{S}. It follows that r_{⋆} factors as the composition

CH_{d}(T × S) ⟶^{q^{∗}} CH_{p(p−1)}(S_{t} × S) ⟶^{r′_{⋆}} ℤ/pℤ.

Thus, it suffices to show that r′_{⋆}(α × h^{p−k−1}) = 0 for any α ∈ CH^{pk}(S_{t}). This follows from Proposition 3.8 applied to the torus S over the field F (T_{1}) (with j = p−k −1) if we show that S_{t} is a nontrivial principal homogeneous space of S. Suppose that S_{t} has a point over F (T_{1}). It follows that the exact sequence (8) splits rationally, i.e., the torus T is birationally isomorphic to the product S ×T_{1} and hence is a rational variety. But T is not rational (see Example 2.5), a contradiction. □

3.5. Invariance of the degree under R-equivalence.

Theorem 3.10. Let p be a prime integer, L/F a bicyclic field extension of degree p^{2} and T = R_{L/F}^{(1)} (𝔾_{m,L}). Let M/F be a field extension and let t and t′ be R-equivalent points in T (M). Then deg(t) ≡ deg(t′) modulo p.

Proof. We have t′ = t · σ(u)u^{−1} · τ(v)v^{−1} for some u, v ∈ (LM)^{×} (see Example 2.1). Let t′′ = t · σ(u)u^{−1}. It suffices to prove that deg(t) = deg(t′′) and deg(t′) = deg(t′′) in ℤ/pℤ. We shall prove the first equality (the second being similar). So replacing t′ by t′′ we may assume that t′ = t · σ(u)u^{−1}.

Consider the point w = (t, u) in (T × P)(M) and two morphisms f and g from T × P to T as in Section 3.4. We have f(w) = t and g(w) = t′. By Propositions 3.3 and 3.9, we have in ℤ/pℤ:

deg(t) = deg f(w) = f_{⋆}([w]) = g_{⋆}([w]) = deg g(w) = deg(t′).  □

4. Essential p-dimension of PGL(p^{2})

Let F be a field and p a prime integer different from char(F).

4.1. Central simple algebras and discrete valuations. Let v be a discrete valuation on a field extension E over F, N the residue field, Ê the completion of E. Then N is a field extension of F. Let χ ∈ X (F). Then F (χ)/F is a cyclic field extension of degree ord(χ) with the choice of a generator of Gal(F (χ)/F ). The group X (N) is identified with the character group of the maximal unramified field extension of Ê. For a character χ ∈ X (N), we write χ̂ for the corresponding character in X (Ê).

There is an exact sequence of p-groups [4, Prop. 7.7]:

(9)  0 → Br(N){p} ⟶^{i} Br(Ê){p} ⟶^{∂} X (N){p} → 0.

The first map preserves indices of algebras. For a central simple algebra C over N with C ∈ Br(N){p} let Ĉ be a central simple algebra over Ê of the same degree representing the image of [C] under i. For example, if [C] = χ ∪ (ū) for some χ ∈ X (N){p} and a unit u ∈ Ê, then [Ĉ] = χ̂ ∪ (u).

The choice of a prime element π in Ê provides with a splitting of the sequence (9) by sending a character χ to the class of the cyclic algebra χ̂ ∪ (π). Thus for every central simple algebra A over Ê we can write

[A] = [Ĉ] + (χ̂ ∪ (π))

in Br(Ê) for unique [C] ∈ Br(N){p} and χ = ∂([A]). Moreover (see [5, Th. 5.15(a)] or [13, Prop. 2.4]),

(10)  ind(A) = ord(χ) · ind(C_{N(χ)}).

Let E′/E be a finite field extension and v′ a discrete valuation on E′ extending v with residue field N′. Then for any [A] ∈ Br(E){p} one has

(11)  ∂_{v′}([A]_{E′}) = e · ∂_{v}([A])_{N′},

where e is the ramification index of E′/E [4, Prop. 8.2].

4.2. The functors F_{1} and F_{2}. We define the functors F_{1} and F_{2} from the category Fields/F of field extensions of F to the category Sets as follows. Let E/F be a field extension. Then F_{1}(E) is the set of isomorphism classes of central simple E-algebras of degree p^{2}. Thus, ed_{p}(F_{1}) = ed_{p}(PGL_{F} (p^{2})).

Let S_{2}(E) be the class of pairs (B, K), where B is a central simple algebra of degree p^{2} over E and K is a cyclic étale E-algebra of degree p such that ind(B_{K}) ≤ p. We say that the pairs (B_{1}, K_{1}) and (B_{2}, K_{2}) are equivalent if K_{1} ≃ K_{2} over E and [B_{1}] − [B_{2}] ∈ Br(K_{1}/E) = Br(K_{2}/E). Let F_{2}(E) be the set of equivalence classes in S_{2}(E). We write [B, K] for the class in F_{2}(E) of a pair (B, K).

We say that the class [B, K] is decomposable if [B, K] = [B′, K] with B′ a split algebra.

Let (B, K) ∈ S_{2}(E) with K a field and let χ ∈ X (E) be a character (of order p) such that K = E(χ) (see Section 2.2). As ind(B_{K}) ≤ p, there is a central simple algebra C over the function field E(y) (y is a variable) of degree p^{2} such that

(12)  [C] = [B_{E(y)}] + (χ_{E(y)} ∪ (y))

in Br(E(y)). We have [C] ∈ F_{1}(E(y)) and ∂([C]) = χ, where ∂ is taken with respect to the discrete valuation v on E(y) associated to y (see Section 4.1).

Consider the following condition (∗) on the pair (B, K) in S_{2}(E) and the character χ:

For any finite field extension N/E of degree prime to p, the class [B, K]_{N} is not decomposable and the class of the algebra B_{N} in Br(N) cannot be written in the form [B_{N}] = ρ ∪ (s) for some s ∈ N^{×} and a character ρ ∈ X (N) of order p^{2} such that p · ρ is a multiple of χ_{N}.

Proposition 4.1. Let χ ∈ X (E) be a character of prime order p, K = E(χ), B a central simple algebra of degree p^{2} over E such that (B, K) ∈ S_{2}(E) and (B, K) together with χ satisfy the condition (∗). Then

ed_{p}^{F_{1}}([C]) ≥ ed_{p}^{F_{2}}([B, K]) + 1

for the algebra C defined by (12).

Proof. Let M/E(y) be a finite field extension of degree prime to p, M_{0} ⊂ M a subfield over F and [C_{0}] ∈ F_{1}(M_{0}) such that

(13)  [C_{0}]_{M} = [C]_{M}

in F_{1}(M) and ed_{p}^{F_{1}}([C]) = tr. deg_{F} (M_{0}). We extend the discrete valuation v on E(y) to a discrete valuation v′ on M with ramification index e′ and inertia degree prime to p (see [6, Lemma 1.1]). Thus, the residue field N of v′ is a finite extension of E of degree prime to p. Let v_{0} be the restriction of v′ to M_{0} and N_{0} its residue field. As [N : E] is not divisible by p, it follows from (11) that ∂([C]_{M}) = e′ · χ_{N} ≠ 0. Hence the algebra C_{M} is ramified, i.e., the class of C_{M} does not belong to the image of the map Br(O) → Br(M), where O is the valuation ring of v′. It follows that C_{0} is also ramified, therefore v_{0} is nontrivial and hence v_{0} is a discrete valuation.

Let χ_{0} = ∂([C_{0}]) ∈ X (N_{0}){p} and K_{0} = N_{0}(χ_{0}). Choose a prime element π_{0} in M_{0} and write

(14)  [C_{0}]M̂_{0} = [B̂_{0}] + (χ̂_{0} ∪ (π_{0}))

in Br(M̂_{0}), where B_{0} is a central simple algebra over N_{0} (see Section 4.1). By (10),

(15)  ind(C_{0}) = ord(χ_{0}) · ind(B_{0})_{K_{0}}.

Let e be the ramification index of M/M_{0} and let π be a prime element in M. Write π_{0} = uπ^{e} and y = vπ^{e′} with u and v units in M.

It follows from (13) and (11) that

(16)  e′ · χ_{N} = ∂([C]_{M}) = ∂([C_{0}]_{M}) = e · ∂([C_{0}])_{N} = e · (χ_{0})_{N}.

Recall that e′ is relatively prime to p. It follows that χ_{N} is a multiple of (χ_{0})_{N}. In particular, ord(χ_{0}) is divisible by p.

It follows from (13),(14) and (16) that

(17)  [ ˆ(B_{0})_{N}] + ( ˆ(χ_{0})_{N} ∪ (u)) = [B̂_{N}] + (χ̂_{N} ∪ (v))

in Br(M̂), hence

(18)  [(B_{0})_{N}] + ((χ_{0})_{N} ∪ (ū)) = [B_{N}] + (χ_{N} ∪ (v̄))

in Br(N).

Since ind(C_{0}) ≤ p^{2}, it follows from (10) and (15) that ord(χ_{0}) divides p^{2}. Case 1: ord(χ_{0}) = p^{2}. By (15), ind(B_{0})_{K_{0}} = 1, i.e., B_{0} is split over K_{0}, hence

[B_{0}] = χ_{0} ∪(s_{0}) for some s_{0} ∈ N_{0}^{×}. It follows from (18) that [B_{N}] = (χ_{0})_{N} ∪(s) for some s ∈ N^{×}. If ord(χ_{0})_{N} = p, then (χ_{0})_{N} is a multiple of χ_{N} and hence [B, K]_{N} is decomposable. If ord(χ_{0})_{N} = p^{2}, the character p · (χ_{0})_{N} is a multiple of χ_{N}. In both cases, (B, K) and χ do not satisfy the condition (∗), a contradiction.

Case 2: ord(χ_{0}) = p. Then the characters χ_{N} and (χ_{0})_{N} generate the same subgroup in X (N). It follows that

(19)  K_{0} ⊗_{N_{0}} N ≃ N((χ_{0})_{N}) = N(χ_{N}) ≃ K ⊗_{E} N.

By (15), we have ind(B_{0})_{K_{0}} ≤ p. Therefore, we may assume that deg(B_{0}) = p^{2} and hence (B_{0}, K_{0}) ∈ S_{2}(N_{0}). It follows from (18) that

[B]_{N} − [B_{0}]_{N} ∈ Br(K ⊗_{E} N/N).

By (19), the pairs (B_{N}, K ⊗_{E} N) and ((B_{0})_{N}, K_{0} ⊗_{N_{0}} N) = (B_{0}, K_{0})_{N} are equivalent in S_{2}(N). It follows that the class of [B, K] in F_{2}(E) is p-defined over N_{0}, therefore,

ed_{p}^{F_{1}}([C]) = tr. deg_{F} (M_{0}) ≥ tr. deg_{F} (N_{0}) + 1 ≥ ed_{p}^{F_{2}}([B, K]) + 1. □

4.3. The functor F_{3}. Let E/F be a field extension and let S_{3}(E) be the class of pairs (A, L), where A is a csa of degree p^{2} over E and L is a bicyclic étale E-algebra of dimension p^{2} such that L splits A, i.e., [A] ∈ Br(L/E). We say that the pairs (A_{1}, L_{1}) and (A_{2}, L_{2}) in S_{3}(E) are equivalent if L_{1} ≃ L_{2} and [A_{1}] − [A_{2}] ∈ Br_{dec}(L_{1}/E) = Br_{dec}(L_{2}/E) (see Section 2.3). Let F_{3}(E) be the set of equivalence classes in S_{3}(E). We write [A, L] for the equivalence class of (A, L) in F_{3}(E).

Let L be a bicyclic étale E-algebra of dimension p^{2}. We view the factor group Br(L/E)/ Br_{dec}(L/E) as a subset of F_{3}(E) identifying the class of an algebra A with [A, L].

We say that a class [A, L] is decomposable if [A, L] = [A′, L] with A′ a split algebra.

Let (A, L) ∈ S_{3}(E). Choose characters χ and η in X (E) such that L = E(χ, η) := E(χ)E(η). Let K = E(χ) and K′ = E(η). As ind(A_{K}) ≤ p, there is a central simple algebra B over the function field E(x) (x is a variable) of degree p^{2} such that

(20)  [B] = [A_{E(x)}] + (η_{E(x)} ∪ (x))

in Br(E(x)). We have (B, K(x)) ∈ S_{2}(E(x)) and ∂([B]) = η, where ∂ is taken with respect to the discrete valuation v on E(x) associated to x.

Consider the following condition (∗∗) on the pair (A, L) in S_{3}(E) and the characters χ and η:

For any finite field extension N/E of degree prime to p, the class [A, L]_{N} is not decomposable and the class of the algebra A_{N} in Br(N) cannot be written in the form [A_{N}] = (ρ ∪ (s)) + (ε ∪ (t)) for some s, t ∈ N^{×} and characters ε ∈ X (N) of order p and ρ ∈ X (N) of order p^{2} such that ⟨p · ρ, ε⟩ = ⟨χ_{N}, η_{N}⟩.

Proposition 4.2. Let χ, η ∈ X (E) be linearly independent characters of prime order p, K = E(χ), L = E(χ, η), A a central simple algebra of degree p^{2} over E such that (A, L) ∈ S_{3}(E) and (A, L) with with the characters χ and η satisfy the condition (∗∗). Then

ed_{p}^{F_{2}}([B, K(x)]) ≥ ed_{p}^{F_{3}}([A, L]) + 1

for the algebra B defined by (20).

Proof. Let M/E(x) be a finite field extension of degree prime to p, M_{0} ⊂ M a subfield over F and [B_{0}, R_{0}] ∈ F_{2}(M_{0}) such that

[B_{0}, R_{0}]_{M} = [B, K(x)]_{M}

in F_{2}(M) and ed_{p}^{F_{2}}([B, K(x)]) = tr. deg_{F} (M_{0}). This equality means that

(21)  R := K(x) ⊗_{E(x)} M ≃ R_{0} ⊗_{M_{0}} M and
(22)  [B]_{M} = [B_{0}]_{M} + (χ_{M} ∪ (f))

for some f ∈ M^{×}.

We extend the discrete valuation v on E(x) to a discrete valuation v′ on M with ramification index e′ and inertia degree prime to p (see [6, Lemma 1.1]). Thus, the residue field N of v′ is a finite extension of E of degree prime to p. Let v_{0} be the restriction of v′ to M_{0} and N_{0} its residue field. As [N : E] is not divisible by p, it follows from (11) that ∂([B]_{M}) = e′ · χ_{N} ≠ 0. Hence the algebra B_{M} is ramified. It follows that B_{0} is also ramified, therefore v_{0} is nontrivial and hence v_{0} is a discrete valuation.

As R = KM, the valuation v′ on M extends uniquely to a discrete valuation on R and R/M is unramified.

Let η_{0} = ∂([B_{0}]) ∈ X (N_{0}){p} and K′_{0} = N_{0}(η_{0}). Choose a prime element π_{0} in M_{0} and write

(23)  [B_{0}]M̂_{0} = [Â_{0}] + (η̂_{0} ∪ (π_{0}))

in Br(M̂_{0}), where A_{0} is a central simple algebra over N_{0}. By (10),

(24)  ind(B_{0}) = ord(η_{0}) · ind(A_{0})_{K′_{0}}.

Let e be the ramification index of M/M_{0} and let π be a prime in M. Write π_{0} = uπ^{e}, x = vπ^{e′} and f = wπ^{k} with u, v and w units in M.

It follows from (22) and (11) that

(25) e′ · η_{N} = ∂([B]_{M}) = e · ∂([B_{0}])_{N} + ∂(χ_{M} ∪ (f)) = e · (η_{0})_{N} + k · χ_{N}.

Note that the characters χ_{N} and η_{N} are linearly independent in X (N) since [N : E] is not divisible by p.

As e′ is relatively prime to p, η_{N} belongs to the subgroup of X (N) generated by (η_{0})_{N} and χ_{N}, and η_{0} ≠ 0 since χ_{N} and η_{N} are linearly independent. In particular, p divides ord(η_{0}).

It follows from (22), (23) and (25) that

(26)  [ ˆ(A_{0})_{N}] + (ˆ(η_{0})_{N} ∪ (u)) + (χ̂_{M} ∪ (w)) = [Â_{N}] + (η̂_{N} ∪ (v))

in Br(M̂), hence

(27)  [(A_{0})_{N}] + ((η_{0})_{N} ∪ (ū)) + (χ_{N} ∪ (w̄)) = [A_{N}] + (η_{N} ∪ (v̄))

in Br(N).

Since ind(B_{0}) ≤ p^{2}, it follows from (24) that ord(η_{0}) ≤ p^{2}.

Case 1: ord(η_{0}) = p^{2}. By (24), A_{0} is split over N_{0}(η_{0}), hence [A_{0}] = η_{0} ∪(s_{0}) for some s_{0} ∈ N_{0}^{×}. It follows from (27) that [A_{N}] = ((η_{0})_{N} ∪ (s)) + (χ_{N} ∪ (t)) for some s, t ∈ N^{×}. If ord(η_{0})_{N} = p, by (25), (η_{0})_{N} is contained in ⟨χ_{N}, η_{N}⟩ and hence [A, L]_{N} is decomposable. If ord(η_{0})_{N} = p^{2}, then again by (25), ⟨p · (η_{0})_{N}, χ_{N}⟩ = ⟨χ_{N}, η_{N}⟩. In both cases, (A, L) with the characters χ and η do not satisfy the condition (∗∗), a contradiction.

Case 2: ord(η_{0}) = p. It follows from (25) that (e, p) = 1 and η_{0} belongs to the subgroup generated by χ and η. Hence, by (21), the cyclic extension R_{0}/M_{0} is unramified. Thus, there exists a character χ_{0} ∈ X (N_{0}) with R̂_{0} = M̂_{0}(χ̂_{0}) and (χ_{0})_{N} = χ_{N}.

It follows from (25) that

⟨(χ_{0})_{N}, (η_{0})_{N}⟩ = ⟨χ_{N}, η_{N}⟩

in X (N). Let L_{0} = N_{0}(χ_{0}, η_{0}). Then

(28)  L_{0} ⊗_{N_{0}} N = N((χ_{0})_{N}, (η_{0})_{N}) = N(χ_{N}, η_{N}) = L ⊗_{E} N

is a bicyclic field extension of degree p^{2}, hence so is the extension L_{0}/N_{0}. In particular, χ_{0} and η_{0} generate a subgroup of order p^{2} in X (N_{0}).

Let K_{0} = N_{0}(χ_{0}). It follows from (23) that

[B_{0}]R̂_{0} = [ ˆ(A_{0})_{K_{0}}] + (ˆ(η_{0})_{K_{0}} ∪ (π_{0})).

As (B_{0}, R_{0}) ∈ S_{2}(M_{0}), we have ind(B_{0})_{R_{0}} ≤ p. Since the character (η_{0})_{K_{0}} is nontrivial, it follows from (10) that A_{0} is split by K_{0}((η_{0})_{K_{0}}) = L_{0}. We may then assume that deg(A_{0}) = p^{2} and hence (A_{0}, L_{0}) ∈ S_{3}(N_{0}).

It follows from (27) that [A_{N}] − [(A_{0})_{N}] ∈ Br_{dec}(L ⊗_{E} N/N). By (28), the pairs (A_{N}, L ⊗_{E} N) and ((A_{0})_{N}, L_{0} ⊗_{N_{0}} N) = (A_{0}, L_{0})_{N} are equivalent in S_{3}(N). Then the class [A, L] in F_{3}(E) is p-defined over N_{0}, therefore,

ed_{p}^{F_{2}}([B, K(x)]) = tr. deg_{F} (M_{0}) ≥ tr. deg_{F} (N_{0}) + 1 ≥ ed_{p}^{F_{3}}([A, L]) + 1. □

Let E be a field extension of F and L/E a bicyclic field extension of degree p^{2}. Write T for the torus over E of norm 1 elements for the field extension L/E. Let t ∈ T (E(T)) be the generic point and let [A, L(T)] be the corresponding element in F_{3}(E(T)) via the isomorphism between T (E(T))/R and Br(L(T)/E(T))/ Br_{dec}(L(T)/E(T)) in Proposition 2.4.

Proposition 4.3. ed_{p}^{F_{3}}([A, L(T)]) ≥ p^{2} − 1.

Proof. Let M/E(T) be a field extension of degree prime to p, M_{0} ⊂ M a subfield over F and [A_{0}, L_{0}] ∈ F_{3}(M_{0}) such that [A_{0}, L_{0}]_{M} = [A, L(T)]_{M}. We need to prove that tr. deg_{F} (M_{0}) ≥ p^{2} − 1. Set LM = L ⊗_{E} M. As L_{0} ⊗_{M_{0}} M ≃ LM, we may assume that L_{0} ⊂ LM.

Let T_{0} be the torus over M_{0} of norm 1 elements for the extension L_{0}/M_{0}. We have (T_{0})_{M} ≃ T_{M}. Consider the commutative diagram

T_{0}(M_{0})/R ⟶ T (M)/R
↓  ↓
F_{3}(M_{0}) ⟶ F_{3}(M)

where the vertical injective maps are given by the isomorphisms in Proposition 2.4. The pair [A_{0}, L_{0}] belongs to the image of the left vertical map in the diagram. Hence there exists an element t_{0} ∈ T_{0}(M_{0}) such that (t_{0})_{M} in T_{0}(M) = T (M) is R-equivalent to t_{M}. We have deg(t) = 1, therefore, deg(t_{M}) is not divisible by p as [M : E(T)] is prime to p. By Theorem 3.10, deg((t_{0})_{M}) ≡ deg(t_{M}) modulo p, hence deg((t_{0})_{M}) ≠ 0. It follows that (t_{0})_{M}, viewed as a morphism Spec(M) → T is dominant. Therefore, there is a field homomorphism E(T) → M over E taking t to (t_{0})_{M}. The elements ρ(t) over all ρ ∈ G := Gal(L/E) generate the field L(T) over L. Hence the elements ρ(t_{0})_{M} generate a subfield in LM over L of the transcendence degree dim(T) = p^{2} −1. As t_{0} ∈ L_{0} and L_{0} is normal over M_{0} and hence is G-invariant, the elements ρ(t_{0}) generate a subfield in L_{0} over F of the transcendence degree p^{2} − 1. It follows that tr. deg_{F} (L_{0}) ≥ p^{2} − 1, hence tr. deg_{F} (M_{0}) ≥ p^{2} − 1. □

Remark 4.4. Let L be a bicyclic field extension of degree p^{2} of a field F of arbitrary characteristic and let T = R_{L/F}^{(1)} (𝔾_{m,L}). A similar argument as the one in the proof of Proposition 4.3 shows that ed_{p}(T/R) = p^{2} − 1, where T/R is the functor taking a field E to T (E)/R.

4.4. The main theorem.

Theorem 4.5. Let p be a prime integer and F a field of characteristic different from p. Then

ed_{p}(PGL_{F} (p^{2})) = p^{2} + 1.

Proof. Recall that ed_{p}(PGL_{F} (p^{2})) = ed_{p}(F_{1}). First we prove the inequality ed_{p}(F_{1}) ≥ p^{2} + 1. We may replace F by any field extension. In particular, we may assume that there are linearly independent characters χ, η ∈ X (F) of order p, hence L := F (χ, η)/F is a bicyclic field extension of degree p^{2}. Set K = F (χ) and K′ = F (η). Let T be the norm 1 torus for the extension L/F and set E := F (T). Let [A, LE] be the element of F_{3}(E) corresponding to the generic point t ∈ T (E) via the isomorphism in Proposition 2.4. Consider the pair (B, KE(x)) ∈ S_{2}(E(x)) with [B] = [A_{E(x)}]+(η_{E(x)}∪(x)) in Br(E(x)) and the algebra C of degree p^{2} over E(x, y) with [C] = [B_{E(x,y)}] + (χ_{E(x,y)} ∪ (y)) in Br(E(x, y)).

We claim that the pair (A, LE) in S_{3}(E) and the characters χ_{E} and η_{E} satisfy the condition (∗∗). Indeed, as t ≠ 1 in T (E)/R (see Section 2.3) we have t_{N} ≠ 1 since [N : E] is prime to p and hence [A, LE]_{N} is not decomposable. Now suppose that [A_{N}] = (ρ ∪ (s)) + (ε ∪ (t)) for a field extension N/E of degree prime to p, elements s, t ∈ N^{×} and characters ε ∈ X (N) of order p and ρ ∈ X (N) of order p^{2} such that ⟨p·ρ, ε⟩ = ⟨χ_{N}, η_{N}⟩. Let T_{1} be the norm 1 torus for the field extension L_{1} = N(ρ, ε) over N. By Example 2.3, the inclusion homomorphism T → T_{1} is not R-trivial, i.e, the image of t in T_{1}(E)/R is not trivial. It follows that the image of t in T_{1}(N)/R is also non-trivial. By Proposition 2.4, [A_{N}] does not belong to the kernel of the homomorphism Br(LN/N)/ Br_{dec}(LN/N) → Br(L_{1}/N)/ Br_{dec}(L_{1}/N), a contradiction. The claim is proved.

We claim that the pair (B, KE(x)) in S_{2}(E(x)) and the character χ_{E(x)} satisfy the condition (∗). The same argument as in the previous claim applied to the field E(x) shows that (A_{E(x)}, LE(x)) in S_{3}(E(x)) and the characters χ_{E(x)} and η_{E(x)} satisfy the condition (∗∗). Let N/E(x) be a finite field extension of degree prime to p. As [A_{E(x)}] = [B] − (η_{E(x)} ∪ (x)), the class [B, KE(x)]_{N} is not decomposable. Suppose that [B_{N}] = ρ ∪ (s) for some s ∈ N^{×} and a character ρ ∈ X (N) of order p^{2} such that p · ρ is a multiple of χ_{N}. Then [A_{N}] = (ρ∪(s))−(η_{N} ∪(x)) and we have ⟨p·ρ, η_{N}⟩ = ⟨χ_{N}, η_{N}⟩, a contradiction proving the claim.

By Propositions 4.1, 4.2 and 4.3,

ed_{p}(PGL_{F} (p^{2})) = ed_{p}(F_{1}) ≥ ed_{p}^{F_{1}}([C]) ≥ ed_{p}^{F_{2}}([B, KE(x)]) + 1 ≥

ed_{p}^{F_{3}}([A, LE]) + 2 ≥ (p^{2} − 1) + 2 = p^{2} + 1.

We shall show that ed_{p}(F) ≤ p^{2} + 1. As mentioned in the introduction, this was shown in [8, Cor. 3.10(a)]. For completeness, we give the argument here.

Let F′_{1}(E) be the set of isomorphism classes of central simple E-algebras of degree p^{2} that are crossed products with the group ℤ/pℤ ⊕ ℤ/pℤ. So F′_{1} is a subfunctor of F_{1}. By [12, Th. 1.2], for every [A] ∈ F_{1}(E) there is a finite field extension E′/E of degree prime to p such that [A_{E′}] ∈ F′_{1}(E′). Hence the inclusion of F′_{1} into F_{1} is p-surjective. It follows that ed_{p}(F_{1}) ≤ ed_{p}(F′_{1}) [10, Prop. 1.3]. So it suffices to show that ed(F′_{1}) ≤ p^{2} + 1.

Let E/F be a field extension and [A] ∈ F′_{1}(E). Then [A] ∈ Br(L/E) for a bicyclic field extension L/F of degree p^{2} with Galois group G generated by σ and τ. The exact sequence (3) yields an epimorphism

Hom_{G}(M, L^{×}) → Br(L/E).

Choose a G-homomorphism φ : M → L^{×} corresponding to [A] in Br(L/E). Since rank(M) = p^{2} + 1, the image of φ is contained in L_{0}^{×}, where L_{0} is a Ginvariant subfield of L with tr. deg_{F} (L_{0}) ≤ p^{2} + 1. Note that G acts faithfully on M. Modifying φ by an element in the image of the map Hom_{G}(Λ^{2}, L^{×}) → Hom_{G}(M, L^{×}), we may assume that G acts faithfully on the image of φ and hence on L_{0}. Thus L_{0} is a Galois extension of E_{0} := (L_{0})^{G} with Galois group G and φ defines a central simple E_{0}-algebra A_{0} with [A_{0}] ∈ Br(L_{0}/E_{0}) such that A_{0} ⊗_{F_{0}} E ≃ A. Thus, A is defined over E_{0}, hence

ed^{F′_{1}}([A]) ≤ tr. deg_{F} (E_{0}) = tr. deg_{F} (L_{0}) ≤ p^{2} + 1.  □

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

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