COMPUTATION OF SOME ESSENTIAL DIMENSIONS

MARKUS ROST

Abstract

In these notes we show that the essential dimension (in the sense of [5]) of PGL_{4} is equal to 5.

Along the way we discuss (in a rather unsystematic manner) generalities on essential dimension and degree formulas.

Contents

Abstract 1 1. Notations and conventions 2 2. Places 2 3. Picard groups 2 4. Ramification, Specialization, and Essential Dimension 3 5. Side remarks 4 6. The class Θ 5 7. A side remark 7 8. The invariant ρ 8 9. The invariant Q 12 10. The invariant Q̂ 13 11. The functor M_{4} 13 12. Computation of ed(PSO_{6}) 15 13. Presentations of M and degree formulas 17 14. Complements 18 References 19

Most of the text dates back to August 2000. I am thankful to V. Chernousov for comments. There is now the preprint “Essential p-dimension of PGL(p^{2})” by A. Merkurjev

(Nov. 2008, http://www.math.uni-bielefeld.de/LAG/man/313.html).

Merkurjev also hinted to a serious gap in the proof of Lemma 11.3. In December 2008 I added Lemma 14.1 and complemented the proofs of Lemma 11.3 and Lemma 12.3.

Date: Dec 13, 2008.

1. Notations and conventions

We work over a ground field k. A k-variety is a separated scheme of finite type over k. Let F/k be a finitely generated field extension. By a model of F/k we understand an irreducible k-variety X together with an isomorphism k(X) ≃ F.

From section 6 on we assume all fields to be of characteristic ≠ 2. From section 11 on we assume all fields to contain a square root of −1.

2. Places

The natural frame work for many of our considerations is the category of fields over k with the k-places as morphisms. In this section we recall some basic notions.

For a valuation v on a field F we use the (mostly standard) notations

𝔪_{v} ⊂ O_{v} ⊂ F, κ_{v} = O_{v}/𝔪_{v}, U_{v} = O_{v}^{∗} ⊂ F^{∗}, U_{v}^{[1]} = 1 + 𝔪_{v} ⊂ U_{v}

for the valuation ring and its maximal ideal, for the residue field, for the group of units, and for the group of 1-units, respectively. Valuations on F with the same valuation ring will be identified. If k is a subfield of F, then by a valuation on F/k (or by a k-valuation of F) we understand a valuation v with k ⊂ O_{v}. We write V(F/k) for the set of all k-valuations on F. If F/k is a finitely generated field extension, then there is a natural identification

V(F/k) = \underleftarrow{lim} X

where X runs through the proper models of F/k.

Let E, F be field extensions of k. By a k-place φ: F ⇝ E we understand a pair (v_{φ}, α_{φ}) where v_{φ} is a valuation on F/k and α_{φ}: κ_{v_{φ}} → E is a k-homomorphism. We also use the more geometric notation f: Spec E ⇝ Spec F for k-places and write (v_{f}, α_{f}) for the corresponding pair. A k-place f = (v_{f}, α_{f}) is given by a (uniquely determined) family of k-morphisms

f_{X}: Spec E → X

with X running through the proper models of F/k and with f_{X} = g ∘ f_{X′} for every morphism g: X′ → X of models of F/k. For any X there exist a proper model Y of E such that f_{X} extends to a (uniquely determined) k-morphism

f_{Y,X}: Y → X.

Passing to the limits we obtain a map

f^{∗}: V(E/k) → V(F/k).

This map sends a valuation v on E to the composite valuation of the valuations v_{f} and v|κ_{v_{f}}.

Let d ≥ 0 and let tr. deg(E/k) ≤ d, tr. deg(F/k) ≤ d. For a place f: Spec E ⇝ Spec F we define its d-degree deg_{d}(f) by deg_{d}(f) = [E : F] if f is an inclusion of fields of transcendece degre d, and put deg_{d}(f) = 0 otherwise.

3. Picard groups

Let A be an abelian group. For a finitely generated field extension F/k we put

P(F/k, A) = \underrightarrow{lim}_{X}(Pic(X) ⊗ A)

where X runs through the proper models of F/k. For a k-place f: Spec E ⇝ Spec F the maps f_{Y,X} define a pullback map f^{∗}: P(F/k, A) → P(E/k, A).

One has

P(k(t)/k, A) = Pic(P^{1}) ⊗ A = A.

Let d = tr. deg(F/k) and let u_{i} ∈ P(F/k, Z/n), i = 1, …, d. Then we define

e(u_{1}, …, u_{d}) ∈ Z/n

as follows. Choose a proper model X of F/k and line bundles L_{i} on X which represent the u_{i} and consider the vector bundle V = L_{1} ⊕ ⋯ ⊕ L_{d}. Let

ε: CH_{d}(V) ≃ CH_{0}(X) ⟶^{deg} Z

where the first map is given by homotopy invariance and the second map is the degree map for 0-cycles. We put

e(u_{1}, …, u_{d}) = ε([zero section]) (mod n).

This number does not depend on the choices made and is multi-linear in the u_{i}. Reference ??? Probably in [2].

Remark: For smooth X, the numbers e(u_{1}, …, u_{d}) are just given by intersecting divisors. This is all we need in the current version of this text, where we make free use of resolution of singularities. In a future version we plan to work with arbitrary varieties and then it will be necessary to have the numbers e(u_{1}, …, u_{d}) also for non-smooth X.

4. Ramification, Specialization, and Essential Dimension

Let x ∈ K_{1}F/n = F^{∗}/(F^{∗})^{n}.

If v is a valuation on F, we say that x is unramified in v if x is in the subgroup U_{v}/(U_{v})^{n} ⊂ F^{∗}/(F^{∗})^{n}. In this case we define the specialization

x(v) ∈ K_{1}κ_{v}/n

of x in v as the image of x under U_{v}/(U_{v})^{n} → κ_{v}^{∗}/(κ_{v}^{∗})^{n}.

If f: Spec E ⇝ Spec F is a place, we say that x is unramified in f, if x is unramified in v_{f}. In this case we put

f^{∗}(x) = (α_{f})_{∗}(x(v_{f})) ∈ K_{1}E/n.

We extend these standard considerations to the Milnor K-ring. If v is a valuation on F we define its Milnor K-ring by

K_{∗}^{M}(v) = K_{∗}^{M}F / (1 + 𝔪_{v}) · K_{∗}^{M}F.

In the case of discrete valuations of rank 1 this ring has been considered in [1], [3, remark at the end of p. 323], [6]. In any case there is a natural injection

K_{∗}^{M}κ_{v} → K_{∗}^{M}(v),
\{ū_{1}, …, ū_{n}\} ↦ \overline{\{u_{1}, …, u_{n}\}}.

Let A be an abelian group. For x ∈ (K_{∗}^{M}F) ⊗ A we denote by x(v) its image in K_{∗}^{M}(v). If x(v) belongs to the subgroup K_{∗}^{M}κ_{v}, we say that x is unramified in v and call x(v) its specialization. These notions extend to places f: Spec E ⇝ Spec F in an obvious way.

In the following we consider various covariant functors F ↦ M(F) from the category of fields F/k to sets. These functors will be subfunctors of F ↦ (K_{∗}^{M}F)⊗A for an appropriate abelian group A. They have the following property: If x ∈ M(F) is unramified in v (as an element of (K_{∗}^{M}F) ⊗ A), then x(v) is in M(κ_{v}). A pair (F, x) with x ∈ M(F) is called versal for M, if for any E/k and y ∈ M(E) there exists a place f: Spec E ⇝ Spec F such that x is unramified in f and y = f^{∗}(x). (This definition is tentative.) The essential dimension of M is the minimum of the transcendence degrees tr. deg(F/k) for versal pairs (F, x).

Later we consider also functors of the form M(F) = H^{1}(F, G) where G is linear algebraic group over k. For the notion of essential dimension of these functors, see [5].

5. Side remarks

The material of this section will not be used in later sections.

Problem. For a linear algebraic group G over k let M_{G}(F) = H^{1}(F, G). Give a neat definition of M_{G}(v), in analogy with K_{∗}^{M}(v). Describe M_{G}(v) using Bruhat-Tits theory.

Here is a further type of functors for which the notion of essential dimension is meaningful (these will not be considered later). Let u ∈ K_{n}^{M}k/p and define M_{u}(F) ⊂ {∗} to be nonempty if and only if u_{F} = 0. In this case ed(M_{u}) should be defined as the minimal transcendence degree of a generic splitting field of u. Recent considerations show that for a nontrivial symbol u one may expect ed(M_{u}) = p^{n}−1. This can be proven for p = 2 or n ≤ 3. In general one does not even know whether ed(M_{u}) < ∞.

I don’t know a good definition of functors on fields which is appropriate for the notion of essential dimension and covers all known examples. One feature appearing in all examples is the existence of a pair of morphisms X_{1} \overrightarrow{→} X_{0} such that the set of all F -rational points X_{0}(F) parametrizes all elements of M(F) (let’s say by a function x ↦ α(x)) and such that if α(x) = α(x′) then there exist y ∈ X_{1}(F) mapping to (x, x′). Moreover, for any z ∈ M(F) and any open subset U ⊂ X_{0} one may find x ∈ U(F) with α(x) = z.

In some cases one can compute essential dimensions by ramification methods. For instance, one concludes ed(PGL_{2}) ≥ 2 from the fact that the quaternion algebra Q(s, t) over k((s))((t)) is doubly ramified. One may try to define a notion of “essential valuation dimension” of M related to ramifications over complete valuation rings. Here is a tentative definition. Let F/k be a field extension, let F_{n} = F ((t_{1})) ⋯ ((t_{n})), and let v_{n} be the valuation of F_{n}/F. Let us say that x ∈ M(F_{n}) is totally ramified, if for any subfield F ⊂ E ⊂ F_{n} such that x is in the image of M(E) → M(F_{n}), the rank of v_{n}|E is n. Let us define evd(M) as the maximal n for which there exist F and a totally ramified element x ∈ M(F_{n}).

Certainly one has evd ≤ ed. Here is an example with evd(M) < ed(M) (without proof): Let p be a prime with char k ≠ p, let l/k be a field extension of degree p and let

M(F) = N_{F ⊗l/F} (K_{1}(F ⊗ l)/p) ⊂ K_{1}F/p

be the “group of norms from l/k in K_{1}/p”. One finds ed(M) = p−1 and evd(M) = 1.

Other computations are evd(PGL_{2}) = 2 and, at least if char k ≠ 2 and −1 is a square, evd(PGL_{4}) = 4.

Problem. Give a neat definition of “essential valuation dimension” (or whatever you want to name it).

6. The class Θ

For a field F/k let

M_{0}(F) = { (x_{1}, x_{2}) ∈ K_{1}F/2 ⊕ K_{1}F/2 | x_{1}x_{2} = 0 }.

Thus an element x = (x_{1}, x_{2}) of M_{0}(F) is given by a pair of elements a, b ∈ F^{∗} such that the quaternion algebra Q(a, b) is split.

Elements in K_{1}F/2 will be denoted by {a}, a ∈ F^{∗} and {a, b} ∈ K_{2}F/2 denotes the product of {a}, {b}.

Proposition 6.1. For finitely generated fields F/k and for elements x ∈ M_{0}(F) there exist unique elements Θ(x) ∈ P(F/k, Z/2) such that:

• (Functoriality) If f: Spec E ⇝ Spec F is a k-place, and if x ∈ M_{0}(F) is unramified in f, then

f^{∗}(Θ(x)) = Θ(f^{∗}(x)).

• (Normalization) Let F_{u} = k(t) and x_{u} = ({t}, {1 − t}). Then

Θ(x_{u}) ∈ P(F_{u}/k, Z/2) = Z/2

is the nontrivial element.

We denote the generator of P(F_{u}/k, Z/2) by [∗].

Proof of uniqueness of Θ. If F is finite, then P(F/k, Z/2) = 0. Hence Θ(x) = 0 in this case and we may assume that F is infinite. Then for x = ({a}, {b}) ∈ M_{0}(F) there exist u, v ∈ F^{∗} with b = u^{2}(1 − av^{2}). Let f: Spec F ⇝ Spec F_{u} be the place with f^{∗}(t) = av^{2}. Then x_{u} is unramified in f and f^{∗}(x_{u}) = x. By functoriality one must have Θ(x) = f^{∗}([∗]). □

Along the way have proved that (F_{u}, k_{u}) is versal for M_{0}, at least for infinite k.

Lemma 6.2. Let X be a smooth proper model of F/k and let

f, f′: X → P^{1}

be morphisms with f^{∗}(x_{u}) = f′^{∗}(x_{u}). Then the two maps

f^{∗}, f′^{∗}: Pic(P^{1})/2 → Pic(X)/2

coincide.

Proof. Put x = (x_{1}, x_{2}) = f^{∗}(x_{u}) = f′^{∗}(x_{u}). For the divisors of the components of x we have

div(x_{1}) = f^{∗}[0] − f^{∗}[∞]
div(x_{2}) = f^{∗}[1] − f^{∗}[∞]

in ⨁_{z∈X^{(1)}} Z/2 and similarly for f′. Hence

f^{∗}[∞] = ∑_{z∈X^{(1)}} ; ∂_{z}(x_{1})=∂_{z}(x_{2})≠0 [z] ∈ ⨁_{z∈X^{(1)}} Z/2

where ∂_{z}: K_{1}F/2 → K_{0}κ(z)/2 is the residue map at z. This expresses f^{∗}[∞] entirely in terms of x, and by the same argument for f′ we get f^{∗}[∞] = f′^{∗}[∞]. □

To prove the existence of the class Θ, we have to show that for any F and x = (x_{1}, x_{2}) ∈ M_{0}(F) and any two places f, f′: Spec F ⇝ Spec F_{u} with x = f^{∗}(x_{u}) = f′^{∗}(x_{u}) one has f^{∗}([∗]) = f′^{∗}([∗]). Assuming resolution of singularities, this follows from Lemma 6.2, by extending f, f′ to morphisms X → P^{1} on a smooth model of F/k.

I am pretty sure that one can avoid here resolution of singularities by using instead canonical flatening [4]. Anyway, there is a simpler direct way by investigating the possible choices f, f′ more closely.

Lemma 6.3. Let t, t′ ∈ F^{∗} with t ≠ t′ and assume {t} = {t′} and {1−t} = {1−t′} in K_{1}F/2. Then there exist α, β ∈ F^{∗} with 1 ≠ α^{2} ≠ β^{2} ≠ 1 such that

t = \frac{1 − β^{2}}{α^{2} − β^{2}} , t′ = α^{2} \frac{1 − β^{2}}{α^{2} − β^{2}} .

Proof. By assumption we have t′ = tα^{2} and 1 − t′ = (1 − t)β^{2} for some α, β ∈ F^{∗}. Hence 1 − tα^{2} = (1 − t)β^{2} and the claim is immediate. □

Let P → P^{2} be the blow up in the 4 points [0, 0, 1], [0, 1, 0], [1, 0, 0], and [1, 1, 1]. Let further P̃ → P^{2} be the blow up in the 7 points [0, 0, 1], [0, 1, 0], [1, 0, 0], and [±1, ±1, 1].

Lemma 6.4. The rational maps

P^{2} ⟶^{g} P^{2} ⟶^{h} P^{1} × P^{1},

g([α, β, 1]) = [α^{2}, β^{2}, 1],

h([a, b, 1]) = ([1 − b, a − b], [a(1 − b), a − b])

extend to everywhere defined morphisms

P̃ ⟶^{ḡ} P ⟶^{h̄} P^{1} × P^{1}.

Proof. The verification is left to the reader. □

Let π, π′: P̃ → P^{1} be given by h̄ ∘ ḡ followed by the projections. Note that π^{∗}(x_{u}) = π′^{∗}(x_{u}) ∈ M_{0}(k(P̃))/2. By Lemma 6.2 we find that the two maps

π^{∗}, π′^{∗}: Pic(P^{1})/2 → Pic(P̃)/2

coincide. (Of course one may check this also directly).

Proof of existence of Θ. We have to show that for any F and x = (x_{1}, x_{2}) ∈ M_{0}(F) and any two places f, f′: Spec F ⇝ Spec F_{u} with x = f^{∗}(x_{u}) = f′^{∗}(x_{u}) one has f^{∗}([∗]) = f′^{∗}([∗]).

By Lemma 6.3 there exist a morphism f̂: Spec F → P̃ such that f = π ∘ f̂ and f′ = π′ ∘ f̂. The claim follows now from π^{∗} = π′^{∗} on Pic(P^{1})/2. □

The proof of Proposition 6.1 is now complete. The functoriality of Θ can also be described in the ramified situation:

Lemma 6.5. If f: Spec E ⇝ Spec F is a k-place, and if x ∈ M_{0}(F) is ramified in f, then f^{∗}(Θ(x)) = 0.

Proof. Indeed, let g: Spec F ⇝ Spec F_{u} be a place with x = g^{∗}(x_{u}). If x is ramified in f, then x_{u} is ramified in g ∘ f and therefore g ∘ f must map to one of 0, 1, ∞. But then (g ∘ f)^{∗}([∗]) = 0. □

The functor M_{0} can be described in a more symmetric way as follows. For a field F/k let

M′_{0}(F) = { (x_{1}, x_{2}, x_{3}) ∈ (K_{1}F/2)^{3} | x_{1} + x_{2} + x_{3} = {−1}, x_{i}x_{j} = 0 for i ≠ j }.

Then each of the projections M′_{0}(F) → M_{0}(F), (x_{1}, x_{2}, x_{3}) ↦ (x_{i}, x_{j}), i ≠ j, is a bijection. If v is a valuation of rank 1 and if x = (x_{1}, x_{2}, x_{3}) is ramified in v, then exactly one of the x_{i} is unramified in v and for this component one has x_{i}(v) = 0.

Let Σ(F) be the set of all (x_{1}, x_{2}, x_{3}) ∈ M′_{0}(F) with x_{i} = 0 for at least one i. If f: Spec E ⇝ Spec F is a k-place, and if x ∈ Σ(F) is unramified in f, then f^{∗}(x) ∈ Σ(E).

These remarks and Lemma 6.5 suggest the following definition. Let M̄_{0}(F) be the quotient of M′_{0}(F) by collapsing the set Σ(F) to a point (denoted by 0). Define the map

f^{∗}: M̄_{0}(F) → M̄_{0}(E)

on the unramified elements of M_{0}(F) as before (and passing to the quotient) and sending all other elements to 0. Then we have

Proposition 6.6. For finitely generated fields F/k and for elements x ∈ M̄_{0}(F) there exist unique elements Θ̄(x) ∈ P(F/k, Z/2) such that:

• (Functoriality) If f: Spec E ⇝ Spec F is a k-place, then

f^{∗}(Θ̄(x)) = Θ̄(f^{∗}(x)).

• (Normalization) Let F_{u} = k(t) and x_{u} = ({t}, {1 − t}). Then

Θ̄(x_{u}) ∈ P(F_{u}/k, Z/2) = Z/2

is the nontrivial element.

□

7. A side remark

In the later sections we will meet the following construction. Let x ∈ K_{1}F/2 and let X be a proper smooth model of F/k. We choose a function a ∈ F^{∗} with x = {a} and write

div(a) = A + 2V

where A is a divisor with odd multiplicities (the latter means A ∈ ⨁_{z∈X^{(1)}}(1+2Z)). At this point we just want give some comments on this situation. Let π: Y → X be the normal closure of X in F [t]/(t^{2} − a). Then π is etale of

degree 2 outside its locus of ramification Δ. Since A has odd multiplicities, one has supp(A) ⊂ Δ. Further, π defines a μ_{2}-torsor over X \ Δ and therefore a line bundle L on X \ Δ via μ_{2} → G_{m}. The class of this line bundle and the class of V in Pic(X \ Δ) = CH^{1}(X \ Δ) coincide.

The situation can be made more clean as follows. Assume that A is a divisor with all multiplicities equal to 1 and that A is a smooth divisor with normal crossings (this can be arranged using resolution of singularities). After blowing up the crossings, we may even assume that A is a smooth subvariety of codimension 1 (with no crossings). Then π is flat and the class of V in Pic(X) is given by the class of the line bundle L̄ = π_{∗}(O_{Y})/O_{X}.

In the following we will often use resolution of singularities in order to talk about the divisor V. Very probably this can be replaced by using flatening theorems [4]. One arranges that π is flat and then works with the line bundle L̄ instead of V.

8. The invariant ρ

Let z_{1}, z_{2} ∈ K_{1}k/2 be fixed elements. We denote by k_{1}, k_{2} the corresponding quadratic extensions of k. Further let K = k_{1} ⊗k_{2} and let k_{3} be the third quadratic subextension of K/k. We define I = I(z_{1}, z_{2}) ⊂ K_{0}k = Z as the subgroup generated by the norms from the k_{i}. Thus I = 2Z if K is a field, and I = Z otherwise.

For a field F/k let

M_{1}(F) ⊂ (K_{1}F/2)^{4},

M_{1}(F) = { (x_{1}, y_{1}, x_{2}, y_{2}) | x_{1}x_{2} = y_{1}y_{2} = 0, x_{i} + y_{i} + z_{i} = 0 for i = 1, 2 }.

Our aim is to define for fields F/k with tr. deg(F/k) ≤ 2 and for ω ∈ M_{1}(F) an invariant ρ(ω) ∈ Z/I.

We study a versal parameter space for elements in M_{1} in some detail. Let c, d ∈ k^{∗} with z_{1} = {c} and z_{2} = {d}.

Let

T = T (z_{1}, z_{2}) ⊂ P^{1} × P^{2},
T = { ([s, t], [x, y, z]) | x^{2}s − y^{2}tc − z^{2}(s − t)d = 0 }

Lemma 8.1. T is a smooth proper irreducible surface. The tupel

ω_{T} = ({t/s}, {ct/s}, {1 − (t/s)}, {d(1 − (t/s))})

is an element of M_{1}(k(T)). For any F/k and any ω ∈ M_{1}(F) there is a k-place f: Spec F ⇝ Spec k(T) with x = f^{∗}(ω_{T}).

Proof. The verification is left to the reader. □

Let further T̃ = T̃(z_{1}, z_{2}) → T be the blow up in the 3 points P_{1} = ([1, 1], [0, 0, 1]), P_{2} = ([1, 0], [0, 1, 0]), P_{3} = ([0, 1], [1, 0, 0]). Lemma 8.1 remains valid with T replaced by T̃.

Lemma 8.2. There exist smooth 1-dimensional closed subvarieties D_{1}, D_{2}, D_{3} ⊂ T̃ such that:

• There are the following equalities of (mod 2)-divisors

div({t/s}) = D_{2} + D_{3},
div({1 − (t/s)}) = D_{1} + D_{3}.

• There is a k-morphism D_{i} → Spec k_{i} for i = 1, 2, 3.

• The D_{i} are pairwise disjoint.

• For the self intersection number of D_{i} one has D_{i} · D_{i} ≡ 4 mod 8.

Proof. First compute the divisors of {t/s} and {1−(t/s)} on T. Consider the three divisors

D̄_{2} = {t = 0},
D̄_{3} = {s = 0},
D̄_{1} = {t = s}.

One has

div_{T} ({t/s}) = D̄_{2} − D̄_{3},
div_{T} ({1 − (t/s)}) = D̄_{1} − D̄_{3}.

Each of the divisors D̄_{i} consists geometrically of two lines. Their intersection consists of one point P_{i} at which they meet transversally. The two lines of D̄_{i} are defined over k_{i} and permuted by the Galois action of k_{i}/k. Let D_{i} ⊂ T̃ be the proper transforms of the D̄_{i}. After the blow up, the two lines will be separated, and the D_{i} are smooth. The preimage of D̄_{i} under the blow up is D_{i} + 2E_{i} where E_{i} is the exceptional fiber over the intersection point P_{i}. To compute the self intersection number of D_{i}, note first that D̄_{i}·D̄_{i} = 0, since D̄_{i} is the preimage of a point under the projection T → P^{1}. Thus (D_{i})^{2} = (D̄_{i}−2E_{i})^{2} = D̄_{i}^{2}−4D̄_{i}·E_{i}+4E_{i}^{2} = 0−0−4 = −4. □

In the following we make free use of resolution of singularities in dimension 2 (for simplicity).

Let tr. deg(F/k) = 2, let ω = (x_{1}, y_{1}, x_{2}, y_{2}) ∈ M_{1}(F), and choose a_{1}, a_{2} ∈ F^{∗} with x_{1} = {a_{1}} and x_{2} = {a_{2}}. Let X be a smooth proper model of F/k. We say that X is ω-regular if there exist integral divisors C_{i}, V, W ⊂ X such that:

div(a_{1}) = C_{2} + C_{3} + 2V
div(a_{2}) = C_{1} + C_{3} + 2W.

and such that there exist morphisms supp(C_{i}) → Spec k_{i}.

ω-regular models exist: By resolution of singularities we find X such that there exist a morphism f: X → T̃ with ω = f^{∗}(ω_{T}). Then we may take C_{i} = f^{∗}(D_{i}).

Here we use the pull back maps for the cycle complexes as defined in [6]. For a morphism f: X → Y with Y smooth there exist in particular pull back maps fitting into a commutative diagram

k(X)^{∗} ⟶^{div} ∐_{x∈X^{(1)}} Z
f^{∗}↑  ^{f^{∗}}↑
k(Y)^{∗} ⟶^{div} ∐_{y∈Y^{(1)}} Z

The maps f^{∗} depend in general on the choice of a coordination of the tangent bundle of Y, see [6, Section 12].

Note also that if X′ → X is a smooth proper model F/k lying over an ω-regular model X, then X′ is ω-regular as well. For that one may just take the preimages of the corresponding divisors.

Given an ω-regular model X we put

ρ(ω) = V · W mod I

This class does not depend on the choice of the C_{i}, V, W. Namely let C′_{i}, V′, W′ be another choice. Then V and V′ differ by a sum of divisors which are defined over one of k_{2}, k_{3}. Hence every component of the intersection of V′ − V with any divisor will be defined over one of k_{2}, k_{3} and therefore of even degree (if k_{2}, k_{3} are fields). Similarly for W and W′.

It follows also that ρ(ω) does not depend on the choice of X. Namely using resolution of singularities, any two models are covered by a smooth model.

If the C_{i} are additionally pairwise disjoint, we have

2V · 2W = (C_{2} + C_{3}) · (C_{1} + C_{3}) = C_{3}^{2}

and therefore

ρ(ω) = \frac{C_{3}^{2}}{4} mod I

By Lemma 8.2 this shows that ρ(ω_{T}) = 1 mod I. Hence ρ(ω_{T}) is nontrivial if K is a field.

Proposition 8.3 (Degree formula). Let tr. deg(E/k) ≤ 2, tr. deg(F/k) ≤ 2, let

f: Spec E ⇝ Spec F

be a k-place, and let ω ∈ M_{1}(F) be unramified in f. Then

ρ(f^{∗}ω) = deg_{2}(f)ρ(ω) mod I.

Proof. The intersection number of the pullback of divisors V_{j} under a generically finite map f is the intersection number of the V_{j} times the degree of f. □

From the nontriviality of ρ(ω_{T}) one finds:

Corollary 8.4. If K is a field, then ω_{T} is not defined over a subfield of k(T) of transcendence degree < 2. □

Corollary 8.5. If K is a field, then ed(M_{1}) = 2. □

The invariant ρ has the following symmetry:

Lemma 8.6. ρ(x_{1}, y_{1}, x_{2}, y_{2}) = ρ(y_{1}, x_{1}, y_{2}, x_{2}).

Proof. Let

τ: M_{1}(F) → M_{1}(F),
(x_{1}, y_{1}, x_{2}, y_{2}) ↦ (y_{1}, x_{1}, y_{2}, x_{2}).

τ is an automorphism. There exist a place

τ̄: Spec k(T) ⇝ Spec k(T)

with τ̄^{∗}(τ(ω_{T})) = ω_{T}. Assume that K is a field. Since ρ(ω_{T}) ≠ 0, the degree formula shows that ρ(τ(ω_{T})) ≠ 0. Thus in Z/2 we must have ρ(ω_{T}) = ρ(τ(ω_{T})).

□

The degree formula shows also that τ̄ is of odd degree. One may choose τ̄ as an automorphism of k(T).

One may check the symmetry also directly: If x_{1} = {a_{1}} and x_{2} = {a_{2}}, then y_{1} = {b_{1}} and y_{2} = {b_{2}} with b_{1} = ca_{1} and b_{2} = da_{2}. With these choices one has div(b_{i}) = div(a_{i}).

The following proposition means that ρ(x_{1}, y_{1}, x_{2}, y_{2}) is already determined by (y_{1}, x_{1}, y_{2} + x_{2}).

Proposition 8.7. Let ω = (x_{1}, y_{1}, x_{2}, y_{2}) ∈ M_{1}(F) and let w ∈ K_{1}F/2 with x_{1}w = y_{1}w = 0. Then ω_{w} = (x_{1}, y_{1}, x_{2} +w, y_{2} +w) is in M_{1}(F) and ρ(ω_{w}) = ρ(ω).

Proof. We assume tr. deg(F/k) = 2.

Again let c ∈ k^{∗} with z_{1} = {c}.

We have ω ∈ M_{1}(F), x_{1}w = 0, and z_{1}w = 0. Therefore there exist a smooth proper model X of F/k such that there are morphisms f: X → T̃, g, h: X → P^{1} with f^{∗}(ω_{T}) = ω, g^{∗}(x_{u}) = (x_{1}, w), h^{∗}({1 − ct^{2}}) = w.

Moreover we may assume that x_{1}, x_{2}, and w are unramified outside a smooth divisor H with normal crossings. For n, m, l ∈ Z/2 let H(n, m, l) ⊂ H be the subdivisor where x_{1}, x_{2}, w has ramification index n, m, l, repectively.

Lemma 8.8. The 5 sets H(0, 1, 0) ∪ H(0, 0, 1) ∪ H(0, 1, 1), H(1, 0, 0), H(1, 0, 1), H(1, 1, 0), H(1, 1, 1) are pairwise disjoint.

Proof. We have

H(1, 0, 0) ∪ H(1, 0, 1) = f^{∗}(D_{2}),
H(1, 1, 0) ∪ H(1, 1, 1) = f^{∗}(D_{3}),
H(0, 1, 0) ∪ H(0, 1, 1) = f^{∗}(D_{1}).

Hence these three sets are pairwise disjoint (see Lemma 8.2).

We have

H(1, 0, 0) ∪ H(1, 1, 0) = g^{∗}([0]),
H(1, 0, 1) ∪ H(1, 1, 1) = g^{∗}([∞]),
H(0, 0, 1) ∪ H(0, 1, 1) = g^{∗}([1]).

Hence these three sets are pairwise disjoint.

The claim is immediate. □

Lemma 8.9. There exist morphisms H(1, 0, 1) → Spec K, H(1, 1, 1) → Spec K.

Proof. H(1, 0, 1) ⊂ f^{∗}(D_{2}) maps to Spec k_{2} and H(1, 1, 1) ⊂ f^{∗}(D_{3}) maps to Spec k_{3} (see Lemma 8.2). Furthermore div(w) = h^{∗}({1−ct^{2} = 0}) maps to Spec k_{1}. Thus any of H(?,?, 1) maps to Spec k_{1}. □

To conclude let a_{1}, a_{2}, b ∈ F^{∗} with x_{1} = {a_{1}}, x_{2} = {a_{2}}, and w = {b}.

Then we have integrally

(1)  div(a_{1}) = [H(1, 0, 0) + H(1, 0, 1)] + [H(1, 1, 0) + H(1, 1, 1)] + 2V
(2)  div(a_{2}) = [H(0, 1, 0) + H(0, 1, 1)] + [H(1, 1, 0) + H(1, 1, 1)] + 2W.
(3)  div(b) = H(1, 0, 1) + H(1, 1, 1) + H(0, 1, 1) + H(0, 0, 1) + 2U.

for some divisors V, W, U.

We have

ρ(ω_{w}) − ρ(ω) = V · U mod I.

Further, by Lemma 8.8, one has

2V · 2U = H(1, 0, 1)^{2} + H(1, 1, 1)^{2}.

Again by Lemma 8.8 and by Equation (3) one has

H(1, 0, 1)^{2} = −H(1, 0, 1) · [H(1, 1, 1) + H(0, 1, 1) + H(0, 0, 1) + 2U]
= −2H(1, 0, 1) · U
≡ 0 mod 8,
H(1, 1, 1)^{2} = −H(1, 1, 1) · [H(1, 0, 1) + H(0, 1, 1) + H(0, 0, 1) + 2U]
= −2H(1, 1, 1) · U
≡ 0 mod 8.

For this note also that by Lemma 8.9 one has H(1,?, 1) · Y ≡ 0 mod 4 for all divisors Y (if K is a field). □

9. The invariant Q

In the following we make use of resolution of singularities in dimension 3 (probably this can be avoided).

For a field F/k let

M_{2}(F) ⊂ (K_{1}F/2)^{6},
M_{2}(F) = { (x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}) | x_{1}x_{2} = y_{1}y_{2} = z_{1}z_{2} = 0,
x_{i} + y_{i} + z_{i} = 0 for i = 1, 2 }.

Let z̄_{1} = {t}, z̄_{2} = {1 − t} ∈ K_{1}k(t)/2 and let T = T (z̄_{1}, z̄_{2}) and T̃ = T̃(z̄_{1}, z̄_{2}). T̃ is a 2-dimensional variety over k(t).

Lemma 9.1. ed(M_{2}) ≤ 3.

Proof. Let F̄ be the function field of the variety T̃. Then (F̄, σ̄) with σ̄ = (ω_{T} , z̄_{1}, z̄_{2}) is versal. □

Let T̄ → P^{1} be a proper variety with generic fibre T̃.

Let tr. deg(F/k) = 3 and let σ = (x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}) ∈ M_{2}(F).

Choose a_{1}, a_{2} ∈ F^{∗} with x_{1} = {a_{1}} and x_{2} = {a_{2}}.

Let X be a smooth proper model of F/k such that there exist a morphism f: X → T̄ with f^{∗}(σ̄) = σ. Write

div(a_{1}) = A_{1} + 2V
div(a_{2}) = A_{2} + 2W.

for the integral divisors on X. Here we assume that the A_{i} are divisors with odd multiplicities.

We define Q(σ) ∈ Z/2 by

Q(σ) = Q(X, f, σ) = V · W · f̄^{∗}([∗]) mod 2

where f̄: X ⟶^{f} T̄ → P^{1}.

If we represent f̄^{∗}([∗]) by the generic fibre of X → P^{1}, we see that

(4)  Q(X, f, σ) = ρ((x_{1}, y_{1}, x_{2}, y_{2}))

where ρ is defined with respect to the ground field k(P^{1}) and to z_{1} = {t}, z_{2} = {1− t}. Note that z_{1}, z_{2} are linearly independent square classes and so I(z_{1}, z_{2}) = 2Z.

Equation (4) shows that Q(X′, f, σ) = Q(X, f, σ) for any X′ → X. Thus Q(X, f, σ) does not depend on the choice of X. It does not depend on the choice of f as well, since for X large enough we have f̄^{∗}([∗]) = f̄′^{∗}([∗]) in Pic(X)/2, see section 6.

Proposition 9.2 (Degree formula). Let tr. deg(E/k) ≤ 3, tr. deg(F/k) ≤ 3, let

f: Spec E ⇝ Spec F

be a k-place, and let σ ∈ M_{2}(F) be unramified in f. Then

Q(f^{∗}σ) = deg_{3}(f)Q(σ).
□

Lemma 9.3. Q(σ̄) ≠ 0

Proof. This follows from ρ(ω_{T}) ≠ 0. □

Corollary 9.4. σ̄ is not defined over a subfield of F̄ of transcendence degree < 3. □

Corollary 9.5. ed(M_{2}) = 3. □

Lemma 9.6. Q(σ) is invariant under the permutations x_{1} ↔ x_{2}, y_{1} ↔ y_{2}, z_{1} ↔ z_{2} and x_{i} ↦ y_{i} ↦ z_{i} ↦ x_{i}.

Proof. Use the same argument as in the proof of Lemma 8.6. □

10. The invariant Q̂

For a field F/k let

M_{3}(F) ⊂ (K_{1}F/2)^{3} ⊕ K_{2}F/2,

M_{3}(F) = { (x_{1}, x_{2}, x_{3}, u) | x_{1} + x_{2} + x_{3} = 0, u ∈ x_{i} · K_{1}F/2 for i = 1, 2, 3 }.

We have a map

φ: M_{2}(F) → M_{3}(F),
φ(x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}) = (x_{1}, y_{1}, z_{1}, y_{1}x_{2}).

Lemma 10.1. The map φ is surjective. One has

φ(x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}) = φ(x_{1}, y_{1}, z_{1}, x′_{2}, y′_{2}, z′_{2})

if and only if there exist w ∈ K_{1}F/2 and u ∈ K_{1}F/2 with x_{1}w = y_{1}w = 0, y_{1}u = z_{1}u = 0 and x′_{2} = x_{2} + w, y′_{2} = y_{2} + w + u, z′_{2} = z_{2} + u.

Proof. Usual biquadratic games. □

Corollary 10.2. The pair (F̄, φ(σ̄)) is versal for M_{3}. □

Let tr. deg(F/k) = 3 and σ̂ ∈ M_{3}(F). We put

Q̂(σ̂) = Q(σ) ∈ Z/2

where σ ∈ M_{3}(F) is any element with φ(σ) = σ̂. By Proposition 8.7, Equation (4), Lemma 9.6, and Lemma 10.1 this gives a welldefined invariant.

It is nontrivial on the generic element and obeys a degree formula. From that we may conclude ed(M_{3}) = 3 and

Corollary 10.3. φ(σ̄) is not defined over a subfield of F̄ of transcendence degree < 3. □

11. The functor M_{4}

We consider triples Φ = (D, φ, ψ) where D is a quaternion algebra, and where φ, ψ are skew-hermitian forms over D of dimension 2 and 1, respectively, with det(φ ⊥ ψ) = 1. We say that two such triples (D, φ, ψ), (D′, φ′, ψ′) are similar, if there exist an isomorphism α: D → D′ such that φ is similar to α^{∗}φ′ and ψ is similar to α^{∗}ψ′.

For a field F/k let M_{4}(F) be the set of similarity classes of such triples over F. Let (F̂, σ̂) be a versal pair for M_{3} with tr. deg(F̂/k) = 3. Write σ̂ = (x_{1}, x_{2}, x_{3}, u).

Let D be a quaternion algebra representing u and choose d_{i} ∈ D with Trd(d_{i}) = 0 and {Nrd(d_{i})} = x_{i}. Then Φ̂ = (D, ⟨d_{1}, sd_{2}⟩, ⟨d_{3}⟩) defines an element [Φ̂] of M_{4}(F̂(s)).

Lemma 11.1. (F̂(s), [Φ̂]) is a versal pair for M_{4}.

Proof. First note in general that, if d, d′ ∈ D are trace zero elements with the property {Nrd(d)} = {Nrd(d′)}, then the skew-hermitian forms ⟨d⟩, ⟨d′⟩ are similar. (This follows from Skolem-Noether).

By diagonalization, any Φ′ over any F′ can be written as (D′, ⟨d′_{1}, d′_{2}⟩, ⟨d′_{3}⟩) with d′_{i} ∈ D′, Trd(d′_{i}) = 0, Nrd(d′_{1}) Nrd(d′_{2}) Nrd(d′_{3}) = 1. Then

σ_{Φ′} = ({Nrd(d′_{1})}, {Nrd(d′_{2})}, {Nrd(d′_{3})}, [D])

is an element of M_{3}(F′). It follows that there exist a place f: Spec F′ ⇝ Spec F̂ with f^{∗}(σ̄) = σ_{Φ′}. Then f^{∗}D = D′, and there exist c_{i} ∈ F′^{∗} with ⟨c_{i}f^{∗}d_{i}⟩ ≃ ⟨d_{i}⟩ (≃ denoting isomorphism). Extend the place f to f: Spec F′ ⇝ Spec F̂(s) by f^{∗}(s) = c_{1}^{−1}c_{2}. Then (∼ denoting similarity)

f^{∗}(Φ̂) = (f^{∗}D, ⟨f^{∗}d_{1}, c_{1}^{−1}c_{2}f^{∗}d_{2}⟩, ⟨f^{∗}d_{3}⟩) ∼ (f^{∗}D, ⟨c_{1}f^{∗}d_{1}, c_{2}f^{∗}d_{2}⟩, ⟨c_{3}f^{∗}d_{3}⟩)

is similar to Φ′. □

Corollary 11.2. ed(M_{4}) ≤ 4.

Lemma 11.3. [Φ̂] is not defined over a subfield of F̂(s) of transcendence degree 3.

Proof. Let F′ ⊂ F̂(s) be of trancendence degree 3 and let Φ′ = (D′, ⟨d′_{1}, d′_{2}⟩, ⟨d′_{3}⟩) be a triple defined over F′ with Φ′_{F̂(s)} ∼ Φ̂. Let v be the valuation on F̂((s))/F̂. Since ⟨d_{1}, sd_{2}⟩ is ramified in v (because the Nrd(d_{i}) are not squares), the valuation v cannot be trivial on F′. Then the residue class field κ′ of v|F′ is a subfield of F̂ of transcendence degree (at most) 2. Note that D is unramified.

The proof of the following claim (added in Dec. 2008) had been missing in the version from 2000.

Claim: D′ is unramified.

Proof of the claim. We have

D_{F̂(s)} ≃ D′_{F̂(s)}

(with D defined over F̂ and D′ defined over F′) and with respect to an isomorphism

f: D′_{F̂(s)} → D_{F̂(s)}

one has

⟨d_{1}, sd_{2}⟩_{F̂(s)} ∼ ⟨f(d′_{1}), f(d′_{2})⟩_{F̂(s)}

The elements d_{1}, d_{2} are defined over F̂. Write

f(d′_{i}) = s^{n_{i}}d′′_{i}

with invertible d′′_{i} ∈ D_{F̂[[s]]}. The form ⟨d_{1}, sd_{2}⟩ is ramified. Therefore the exponents n_{1}, n_{2} can’t have the same parity and it follows that the residue forms ⟨d_{1}⟩, ⟨d_{2}⟩ coincide with the residue forms ⟨d̄′′_{1}⟩, ⟨d̄′′_{2}⟩, up to permutation and similarity. The similiarity class of a 1-dimensional D-skew-hermitian form ⟨x⟩ is determined by the square class of Nrd(x). Note that Nrd(d′_{i}) has in F̂(s) the same square class as Nrd(d′′_{i}). It follows that in F̂((s)) the square classes of Nrd(d_{1}), Nrd(d_{2}) coincide with the square classes Nrd(d′_{1}), Nrd(d′_{2}), up to permutation.

Now, if D′ would be ramified, one would have by Lemma 14.1 over F̂((s)):

[D] = (Nrd(d′_{1}), Nrd(d′_{2}))
= (Nrd(d_{1}), Nrd(d_{2}))

Hence

[D] = (Nrd(d_{1}), Nrd(d_{2}))

over F̂.

This would mean that the versal pair (F̂, σ̂) for M_{3} would have the form

σ̂ = (x_{1}, x_{2}, x_{3}, x_{1}x_{2})

But for K = k(u, v) the element

σ = ({u}, {v}, {uv}, 0) ∈ M_{3}(K)

is not of this form.

This ends the proof of the claim.

By standard ramification theory for quadratic forms, the residues of a form up to similarity are well defined, up to a permutation of the first and second residue form. It follows that

Φ′ ∼ Φ̃′ = (D̃′, ⟨d̃′_{1}, sd̃′_{2}⟩, ⟨d̃′_{3}⟩)

with D̃′ and d̃′_{i} defined and regular over the ring of v|F′. Taking residues for Φ̂ and Φ̃′, we see that the quadruple (D, ⟨d_{1}⟩, ⟨d_{2}⟩, ⟨d_{3}⟩) is similar to \overline{(D̃′, ⟨d̃′_{1}⟩, ⟨d̃′_{2}⟩, ⟨d̃′_{3}⟩)} or to \overline{(D̃′, ⟨d̃′_{2}⟩, ⟨d̃′_{1}⟩, ⟨d̃′_{3}⟩)}.

Since these quadruples are defined over κ′, we have a contradiction to Corollary 10.3. □

12. Computation of ed(PSO_{6})

Finally let M_{5}(F) = H^{1}(F, PSO_{6}). Then M_{5}(F) consists of similarity classes of pairs (D, ρ), where D is a quaternion algebra, and where ρ is a skew-hermitian forms over D of dimension 3 with det(ρ) = 1.

Let (E, [Φ̂]) be a versal pair for M_{4} with Φ̂ = (D, ⟨d_{1}, d_{2}⟩, ⟨d_{3}⟩) and with tr. deg(E/k) = 4, see Lemma 11.1. Then x = [(D, ⟨d_{1}, d_{2}, sd_{3}⟩)] is an element of M_{5}(E(s)).

Lemma 12.1. (E(s), x) is a versal pair for M_{5}.

Proof. Similar as for Lemma 11.1. □

Corollary 12.2. ed(M_{5}) ≤ 5.

Lemma 12.3. x is not defined over a subfield of E(s) of transcendence degree 4.

Proof. Similar as for Lemma 11.3, now using Lemma 11.3 instead of Corollary 10.3.

Added in Dec. 2008:

Consider the versal pair for M_{5} in Lemma 12.1 given by

Ψ = (D, ⟨d_{1}, d_{2}, sd_{3}⟩)

over E(s).

Let E′ ⊂ E(s) be of trancendence degree 4 and let

Ψ′ = (D′, ⟨d′_{1}, d′_{2}, d′_{3}⟩)

be a triple defined over E′ with

Ψ′_{E(s)} ∼ Ψ

Let v be the valuation on E((s))/E. Since ⟨d_{1}, d_{2}, sd_{3}⟩ is ramified in v (because the Nrd(d_{i}) are not squares), the valuation v cannot be trivial on E′. Then the residue class field κ′ of v|E′ is a subfield of E of transcendence degree (at most) 3. Note that D is unramified.

Claim: D′ is unramified.

Proof of the claim. We have

D_{E(s)} ≃ D′_{E(s)}

(with D defined over E and D′ defined over E′) and with respect to an isomorphism

f: D′_{E(s)} → D_{E(s)}

one has

⟨d_{1}, d_{2}, sd_{3}⟩_{E(s)} ∼ ⟨f(d′_{1}), f(d′_{2}), f(d′_{3})⟩_{E(s)}

The elements d_{1}, d_{2}, d_{3} are defined over E. Write

f(d′_{i}) = s^{n_{i}}d′′_{i}

with invertible d′′_{i} ∈ D_{E[[s]]}. The form ⟨d_{1}, d_{2}, sd_{3}⟩ is ramified. Therefore the exponents n_{1}, n_{2}, n_{3} can’t have the same parity. Suppose that n_{1} and n_{2} have the same parity. Then the residue form ⟨d_{1}, d_{2}⟩ coincides with the residue form ⟨d̄′′_{1}, d̄′′_{2}⟩ up to similarity:

⟨d_{1}, d_{2}⟩ ∼ ⟨d̄′′_{1}, d̄′′_{2}⟩_{E}

Now, if D′ would be ramified, one would have by Lemma 14.1 over E((s)):

[D] = (Nrd(d′_{1}), Nrd(d′_{2}))
= (Nrd(d′′_{1}), Nrd(d′′_{2}))

Taking residues one gets

[D] = (Nrd(d̄′′_{1}), Nrd(d̄′′_{2}))

over E.

Therefore the versal pair (E, [Φ̂]) for M_{4} would have the form

Φ̂ = (D, ⟨e_{1}, e_{2}⟩, ⟨e_{3}⟩)

with a_{i} = e_{i}^{2} and a_{1}a_{2}a_{3} = 1 and

[D] = (a_{1}, a_{3})

This would mean that for any field K and any element of M_{4}(K) given by

Φ = (D, φ, ψ)

there exist a 1-dimensional subform ρ of φ such that

[D] = (det(ρ), det(φ))

If D is split, this would mean that for any 4-dimensional (usual) quadratic form φ there exist a 2-dimensional quadratic subform ρ of φ such that det(φ) is a norm from the quadratic extension given by ρ. But then det(φ) would be a similarity factor of φ.

However for a 4-dimensional quadratic form of the form

φ = ⟨w, u, v, uv⟩

the determinant is a similarity factor if and only if the Pfister form

⟨⟨u, v, w⟩⟩

is split. This is not the case over the field k(u, v, w).

This ends the proof of the claim.

The rest of the proof is similar as for Lemma 11.3 □

Corollary 12.4. ed(PGL_{4}) = ed(P SO_{6}) = ed(M_{5}) = 5. □

13. Presentations of M and degree formulas

In the following we discuss some general aspects about essential dimensions and “degree formulas”.

Definition 13.1 (tentative). A presentation of M consists of a pair of morphisms

X_{1} ⟶^{π_{0}} ; ⟶_{π_{1}} X_{0}

of k-varieties and a function α on X_{0} with α(x) ∈ M(κ(x)) such that:

• Let x ∈ X_{0} and let v be a valuation on κ(x) with center y ∈ X_{0}. Then α(x) is unramified in v and for its specialization one has α(v) = α(y)_{κ(y)}.

• For any F/k and β ∈ M(F) and any open dense subvariety U ⊂ X_{0} there exists f: Spec F → U with β = f^{∗}(α).

• For every y ∈ X_{1} one has π_{0}^{∗}(α(π_{0}(y)) = π_{1}^{∗}(α(π_{1}(y)) in M(κ(y)).

• For any F/k and any two morphisms f_{0}, f_{1} : Spec F → X_{0} with f_{0}^{∗}(α) = f_{1}^{∗}(α) there exists f: Spec F → X_{1} with f_{i} = π_{i} ∘ f.

Example. Let G ⊂ GL_{n} be a linear algebraic group over k and let M_{G}(F) = H^{1}(F, G). There is natural presentation of M_{G} with X_{0} = GL_{n} /G and X_{1} = GL_{n} × GL_{n} /G.

Example. In section 6 we have seen that π, π′: P̃ → P^{1} is a presentation of M_{0}.

Exercise. Describe presentations of the functors M_{1}, M_{2}, … of the preceding sections.

Let π_{0}, π_{1} : X_{1} \overrightarrow{→} X_{0}, α be a presentation of M with X_{0} irreducible of dimension d. Choose a completion π̄_{0}, π̄_{1} : X̄_{1} \overrightarrow{→} X̄_{0} and consider

δ = (π̄_{1})_{∗} − (π̄_{0})_{∗}: CH_{d}(X̄_{1}) → CH_{d}(X̄_{0}) = Z.

Suppose that im δ ⊂ nZ. Then for F/k with tr. deg(F/k) ≤ d and β ∈ M(F) we have a invariant

Q(β) ∈ Z/n

defined by Q(β) = 0 if tr. deg(F/k) < d and otherwise by Q(β) = f_{∗}([X]) if X is a proper modell of F/k and f: X → X_{0} is a morphism with β = f^{∗}(α). This invariant obeys the degree formula

Q(f^{∗}β) = deg_{d}(f)Q(β).

These considerations seem to provide a natural frame work for a systematic treatment of degree formulas in the context of these notes.

Example. For the presentation π, π′: P̃ → P^{1} of M_{0} in section 6 one finds n = 2.

14. Complements

Recall that we work in characteristic different from 2 and that −1 is a square.

Lemma 14.1. Let R be a complete discrete valuation ring with fraction field K. Let E be a quaternion algebra over K which is ramified with respect to R. Let e_{i} ∈ E (i = 1, 2, 3) with Trd(e_{i}) = 0 and

Nrd(e_{1}) Nrd(e_{2}) Nrd(e_{3}) = 1

Then

[E] = (Nrd(e_{i}), Nrd(e_{j}))

for i ≠ j.

Proof. First note that

(Nrd(e_{i}), Nrd(e_{j}))

is independent of the choices of i, j. This follows from the product relation and from (a, a) = 0.

Let π be a prime element of R and denote by κ = R/πR the residue class field of R. For a ∈ R denote by ā ∈ κ its residue.

Since E is ramfied there exists a, b ∈ R^{×} such that

[E] = (a, πb)

and such that the square class (ā) is nontrivial.

Let 1, X, Y, XY be a basis of E with X^{2} = a, Y^{2} = πb and XY + Y X = 0. Then

e_{i} = π^{n_{i}}(Xα_{i} + Y β_{i} + XY γ_{i})

with n_{i} ∈ Z and α_{i}, β_{i}, γ_{i} ∈ R such that

(ᾱ_{i}, β̄_{i}, γ̄_{i}) ≠ 0

in κ^{3} for i = 1, 2, 3.

One now analyzes the product relation Nrd(e_{1}) Nrd(e_{2}) Nrd(e_{3}) = 1. One has

(5)  − Nrd(e_{i}) = π^{2n_{i}}(aα_{i}^{2} + πb(β_{i}^{2} − aγ_{i}^{2}))

Suppose ᾱ_{i} = 0 for some i. Then β̄_{i} ≠ 0 or γ̄_{i} ≠ 0 and since ā is not a square, it follows that (β_{i}^{2} − aγ_{i}^{2}) is a unit of R. Write α_{i} = πα′_{i} with α′_{i} ∈ R. Then one has

(6)  − Nrd(e_{i}) = π^{2n_{i}+1}(πaα_{i}^{′2} + b(β_{i}^{2} − aγ_{i}^{2}))

with the second factor a R-unit.

Suppose ᾱ_{i} = 0 for exactly one or for all 3 of the indices i = 1, 2, 3. Then (5) and (6) show that

1 = ∏_{i=1}^{3} Nrd(e_{i}) = π^{m} · unit

with m odd, a contradiction.

Suppose ᾱ_{i} ≠ 0 for i = 1, 2, 3. Then n_{1} + n_{2} + n_{3} = 0 and

−1 = −\overline{Nrd(e_{1}) Nrd(e_{2}) Nrd(e_{3})} = ∏_{i=1}^{3}(ᾱ_{i})^{2}ā = ā^{3} ∏_{i=1}^{3}(ᾱ_{i})^{2}

Hence ā would be a square, a contradiction.

Suppose ᾱ_{1} ≠ 0 and ᾱ_{i} = 0 for i = 2, 3. Then

− Nrd(e_{1}) = π^{2n_{i}}aα_{1}^{2}U
− Nrd(e_{2}) = π^{2n_{2}+1}b(β_{2}^{2} − aγ_{2}^{2})V

with U, V ∈ R such that Ū = V̄ = 1. Since R is complete, U and V are squares. One finds

(Nrd(e_{1}), Nrd(e_{2})) = (a, πb(β_{2}^{2} − aγ_{2}^{2}))
= [E] + (a, β_{2}^{2} − aγ_{2}^{2})
= [E]
□

References

[1] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) (H. Bass, ed.), Lecture Notes in Mathematics, vol. 342, Springer, Berlin, 1973, pp. 349–446.

[2] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984.

[3] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), 318–344.

[4] M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89.

[5] Z. Reichstein, On the notion of essential dimension for algebraic groups, preprint, 1998, http://ucs.orst.edu/˜ reichstz/.

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

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://math.ohio-state.edu/~rost