A LOWER BOUND ON THE ESSENTIAL DIMENSION OF PGL_{4} IN CHARACTERISTIC 2

SANGHOON BAEK

Abstract. In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin-Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree 4, we show that ed(PGL_{4}) ≥ 4 over a field of characteristic 2.

1. Introduction

Let F be a field and T : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets. We shall write ed(T ) for the essential dimension of T .

A way to find lower bounds for the essential dimension of T is to give nontrivial cohomological invariants associated to T . More precisely, if there exists a nontrivial cohomological invariant of degree n with values in a torsion Galois module of exponent not divisible by the characteristic of F, then ed(T ) ≥ n. This approach produces lower bounds for various algebraic structures [7, §12], but may not work when the exponent is a multiple of the characteristic of F.

In this paper, we presents a lower bound of the essential dimension based on the cohomological invariants with values in Kato’s cohomology group. As an application, we compute the essential dimension of the functor given by the classes of n-fold quadratic Pfister forms.

In the last section, we provide our main result, ed(PGL_{4}) ≥ 4 over a field of characteristic 2 by relating the result in the previous section and Tignol’s theorem on the second trace form of simple algebras of degree 4.

2. Lower bound via cohomological invariants

Let F be field of characteristic p > 0. For any n ≥ 0, we denote by Ω_{F}^{n} the nth exterior power of the absolute differential module Ω_{F}^{1} . Let d : Ω_{F}^{n} → Ω_{F}^{n+1} be the exterior derivation given by d(ydx_{1} ∧ … ∧ dx_{n}) = dy ∧ dx_{1} ∧ … ∧ dx_{n}.

We write H_{p}^{n+1}(F) for the cokernel of the homomorphism ℘ : Ω_{F}^{n} → Ω_{F}^{n} /dΩ_{F}^{n−1} defined by

℘(y \frac{dx_{1}}{x_{1}} ∧ … ∧ \frac{dx_{n}}{x_{n}}) = (y^{p} − y)\frac{dx_{1}}{x_{1}} ∧ … ∧ \frac{dx_{n}}{x_{n}} .

For a field extension K/F, we have a cohomological functor of degree n + 1

H : Fields/F → Abelian Groups by K ↦ H_{p}^{n+1}(K).

A cohomological invariant of degree n+1 associated to a functor T : Fields/F → Sets is a morphism of functors q : T → H. An invariant q is nontrivial if there exists a field extension K/F containing an algebraic closure F̄ of F and an element t ∈ T (K) such that q(K)(t) ≠ 0 in H(K).

Lemma 2.1. Let F be a field of characteristic p > 0 and let T : Fields/F → Sets be a functor. If there exists a nontrivial invariant of degree n + 1 associated to T , then ed(T ) ≥ n + 1.

Proof. Since essential dimension goes down with field extensions, we may replace F by F̄. Let K/F be a field extension such that there exists t ∈ T (K) with q(K)(t) ≠ 0 in H(K). Assume that ed(T ) ≤ n. Then, there exist an intermediate field E of K/F and t′ ∈ T (E) such that tr. deg_{F} E ≤ n and t = t′_{K}. Therefore, by [8, Chap. II, §4, Exercises 3)(a)] (see also [2, Proposition]), we have H(E) = 0. This leads to a contradiction by the following commutative diagram

T (E) ⟶^{q(E)} H(E)
↓  ↓
T (K) ⟶^{q(K)} H(K)
□

Example 2.2. Let F be a field of characteristic 2. Consider the functor P_{n} : Fields/F → Sets which assigns to a field extension K/F the set of K-isomorphism classes of n-fold quadratic Pfister forms

⟨⟨a_{1}, …, a_{n−1}, a_{n}]] := ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n−1}⟩ ⊗ [1, a_{n}].

Consider the cohomological invariant q : P_{n} → H of degree n defined by

⟨⟨a_{1}, …, a_{n−1}, a_{n}]] ↦ a_{n} \frac{da_{1}}{a_{1}} ∧ … ∧ \frac{da_{n−1}}{a_{n−1}} .

We show that q is nontrivial. Let L = F̄(x_{1}, …, x_{n}) with independent variables x_{1}, …, x_{n}. Then, the form ϕ := ⟨⟨x_{1}, …, x_{n−1}, x_{n}]] is anisotropic over L. Hence, by [5, Theorem (2)], q(L)(ϕ) is nonzero in H(L). By Lemma 2.1, we have ed(P_{n}) ≥ n. Thus, ed(P_{n}) = n.

Example 2.3. As G_{2}-torsors is bijective to P_{3}, we have ed(G_{2}) = 3 over a field of characteristic 2.

3. A lower bound of ed(PGL_{4}) in characteristic 2

Let F be a field of characteristic 2. For any central simple F -algebra A of degree n, there exists a quadratic form ϕ_{A} : A → F given by ϕ_{A}(a) = s_{2}(a), the second coefficient of the reduced characteristic polynomial of A

Prd_{A,a}(x) = x^{n} − s_{1}(a)x^{n−1} + s_{2}(a)x^{n−2} − ⋯ + (−1)^{n}s_{n}(a).

In [9, Théorème 1] Tignol showed that for any central simple F -algebra A of degree 4 there exist 2-fold Pfister form ϕ_{2,A} and 4-fold Pfister form ϕ_{4,A} such that

(1)  ϕ_{A} = [1, 1] + ϕ_{2,A} + ϕ_{4,A} ∈ W_{q}(F).

Moreover, ϕ_{2,A} is a norm form of Q such that [A⊗A] = [Q] in Br(F) (i.e., ϕ_{2,A} = ϕ_{Q}) and ϕ_{4,A} = ⟨⟨a_{1}, a_{2}⟩⟩ ⊗ ϕ_{2,A} for some a_{1}, a_{2} ∈ F^{×}.

By a theorem of Kato [5, Theorem (2)], we have an isomorphism

e_{n+1} : I_{q}^{n+1}(K)/I_{q}^{n+2}(K) → H_{2}^{n+1}(K)

for any field extension K/F. Therefore, the quadratic form ϕ_{2,A} (resp. ϕ_{4,A}) in (1) defines a cohomological invariant of degree 2 (resp. 4) associated to PGL_{4}-torsors:

(2)  q : PGL_{4}-torsors → H given by A ↦ e_{2}(ϕ_{2,A}) (resp. e_{4}(ϕ_{4,A})).

We recall parametrizations of central simple F -algebras of degree 2 and 4. For a field extension K/F, a ∈ K^{×}, and b ∈ K, we write Q := (a, b]_{K} for the quaternion algebra K ⊕Ki⊕Kj ⊕Kji with the multiplication i^{2} = a, j^{2} +j = b, and ji = ij +i. By a simple calculation, we have ϕ_{Q} = ⟨⟨a, b]].

For a ∈ K, we write K_{a} for the quadratic étale K-algebra K[t]/(t^{2} + t + a). Let L be a biquadratic étale K-algebra which is generated by two elements α and β subject to the relations α^{2} + α = a, β^{2} + β = b, αβ = βα for some a, b ∈ K. We shall denote such an algebra by K_{a,b}. The automorphism group of K_{a,b}/K is generated the automorphisms σ and τ with the following relations:

σ(α) = α, σ(β) = β + 1, τ(β) = β, τ(α) = α + 1.

Let u ∈ K_{a}^{×}, v ∈ K_{b}^{×}, and w ∈ K_{a+b}^{×} satisfying N_{K_{a}/K}(u)·N_{K_{b}/K}(v) = N_{K_{a+b}/K}(w). Consider the algebra K_{a,b} ⊕K_{a,b} ·λ⊕K_{a,b} ·δ ⊕K_{a,b} ·λδ over K with the multiplication

λ^{2} = u, δ^{2} = v, (λδ)^{2} = w, λc = σ(c)λ, δc = τ(c)δ

for all c ∈ K_{a,b}. We write (K_{a,b}, u, v, w) for such an algebra. By a theorem of Albert, every central simple K-algebra A of degree 4 is isomorphic to for some (K_{a,b}, u, v, w).

Proposition 3.1. Let F be a field of characteristic 2. Then, ed(PGL_{4}) ≥ 4.

Proof. We first show that the cohomological invariant of degree 4 in (2) is nontrivial. To do this, we slightly modify a generic abelian crossed product algebra given in [9,

4. Exemple] and [1]. Let K = F̄(x, y) with independent variables x, y. Consider a division algebra A := (K_{x,y}, u, v, w) over K such that ind(A) = exp(A) = 4 for some u ∈ K_{x}^{×}, v ∈ K_{y}^{×}, w ∈ K_{x,y}^{×}. By a simple modification, we may assume that u = f + α and v = g + β with nonzero f, g ∈ K.

Let s and t be independent variables over K. Extend the automorphisms σ and τ of Gal(K_{x,y}/K) to the automorphisms of Gal(K_{x,y}(s, t)/K(s, t)) by fixing s and t. Let B := (K_{x,y}(s, t), s, t, st) = (s, x]_{K(s,t)} ⊗ (t, y]_{K(s,t)} be a biquaternion algebra over K(s, t). Then, A_{K(s,t)} ⊗ B is Brauer equivalent to a division K(s, t)-algebra D := (K_{x,y}(s, t), us, vt, wst) such that ind(D) = exp(D) = 4. In Br(K(s, t)), we have

[D ⊗ D] = [A_{K(s,t)} ⊗ A_{K(s,t)}] = [(N_{K_{y}/K}(v), x]_{K(s,t)}],

thus, we obtain an anisotropic form ϕ_{2,D} = ⟨⟨N_{K_{y}/K}(v), x]] = ⟨⟨g^{2} + g + y, x]] over K(s, t) in the decomposition of ϕ_{D}.

Since ϕ_{D}(λ) = s and ϕ_{D}(δ) = t, it follows from [9, Lemme] and [6, Theorem 2.10] that ϕ_{4,D} = ⟨⟨s, t, g^{2} + g + y, x]]. As s and t are independent variables over K and ϕ_{2,D} is anisotropic over F̄(x, y, s, t), the form ϕ_{4,D} is anisotropic over F̄(x, y, s, t). Therefore, by [5, Theorem (2)], the invariant q(F̄(x, y, s, t))(ϕ_{4,D}) in (2) is nonzero in H(F̄(x, y, s, t)). Finally, by Lemma 2.1 we obtain ed(PGL_{4}) ≥ 4. □

Remark 3.2. As ed(PGL_{4}) ≤ 5, we have ed(PGL_{4}) is either 4 or 5. When char(F) ≠ 2, it was shown by Rost that ed(PGL_{4}) = 5.

Acknowledgments. This work was partially supported by an internal fund from KAIST, TJ Park Junior Faculty Fellowship of POSCO TJ Park Foundation, and National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A1A1010171).

References

[1] S. A. Amitsur, D. Saltman, Generic Abelian crossed products and p-algebras, J. Algebra 51 (1978), 76–87.

[2] J. K. Arason and R. Baeza, La dimension cohomologique des corps de type C_{r} en caractéristique p, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 125–126.

[3] S. Garibaldi, A. Merkurjev, and J.-P. Serre, Cohomological invariants in Galois cohomology, American Mathematical Society, Providence, RI, (2003).

[4] K. Kato, Galois cohomology of complete discrete valuation fields, Lecture Notes in Math. 67, Springer-Verlag, Berlin (1982), 215–238.

[5] K. Kato, Symmetric bilinear forms, quadratic forms and Milnor K-theory in characteristic two, Invent. Math. 66 (1982), no. 3, 493–510.

[6] M. Kneser, M.-A. Knus, M. Ojanguren, R. Parimala, and R. Sridharan, Composition of quaternary quadratic forms, Compositio Math. 60 (1986), 133–150.

[7] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

[8] J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer-Verlag, Berlin (2002).

[9] J. -P. Tignol, La forme seconde trace d’une algèbre simple centrale de degré 4 de caractéristique 2, C. R. Math. Acad. Sci. Paris 342 (2006), no. 2, 89–92.

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

E-mail address: sanghoonbaek@kaist.ac.kr