ON THE FIRST STEENROD SQUARE FOR CHOW GROUPS

OLIVIER HAUTION

Abstract. We construct a weak version of the homological first Steenrod square, a natural transformation from the modulo two Chow group to the Chow group modulo two and two-torsion. No assumption is made on the characteristic of the base field. As an application, we generalize a theorem of Nikita Karpenko on the parity of the first Witt index of quadratic forms to the case of a base field of characteristic two.

1. Introduction

Good progress has been made lately towards a uniform treatment of the theory of (non-degenerate) quadratic forms, regardless of the fact the base field has characteristic two or not. This approach is developed in the book [EKM08]. The main obstruction that remains, in order to accomplish this program, is that the Steenrod operations modulo two are not available when the base field has characteristic two. Indeed several constructions of the Steenrod operations for Chow groups modulo a prime number p are known, but none of them works over a field of characteristic p.

In this article we construct a weak version of the first homological Steenrod operation on modulo two Chow groups, over a field of arbitrary characteristic. More precisely, if Ch denotes the modulo two Chow group, and C̃h the Chow group modulo its torsion subgroup, tensored with ℤ/2, we construct group homomorphisms

Sq_{1}^{X}: Ch(X) → C̃h(X),

for all separated schemes X of finite type over a field. These morphisms commute with proper push-forwards, scalars extension and external product. They satisfy the formula

Sq_{1}^{X}[X] = c_{1}(T_{X})

when X is a smooth variety with tangent bundle T_{X}.

Date: October 9, 2009.

2000 Mathematics Subject Classification. 14C40; 11E04.

Key words and phrases. Steenrod operations, Riemann-Roch theorem, Chow groups, Witt indices.

Using the operation constructed here, we prove a statement about the parity of the first Witt index of a quadratic form over an arbitrary field. When the base field is of characteristic different from two, this result is a particular case of a more precise statement known as Hoffmann’s Conjecture, [EKM08, Proposition 79.4]. But since we only construct the first Steenrod square, we only get a partial statement.

This work is part of my Ph.D. thesis at the university of Paris 6 under the direction of Nikita Karpenko. I am very grateful to him for introducing me to the subject, raising the question studied here, and guiding me during this work.

2. Notations

A variety is a separated scheme of finite type over a field, which need not be irreducible, reduced nor quasi-projective. A regular variety is a variety whose local rings are all regular local rings.

Grothendieck group of coherent sheaves. Let X be a variety. We shall write K_{0}(X) for the Grothendieck group of coherent O_{X}-sheaves. Such a sheaf F has a class [F] ∈ K_{0}(X), and if f: X → Y is a proper morphism of varieties, there is a push-forward map

f_{∗}: K_{0}(X) → K_{0}(Y) , [F] ↦ ∑_{i}(−1)^{i}[R^{i}f_{∗}(F)].

If f: X → Y is flat, there is a pull-back f^{∗}: K_{0}(Y) → K_{0}(X) induced by the tensor product with O_{X} over O_{Y}.

Topological filtration. The group K_{0}(X) is endowed with a topological filtration 0 = K_{0}(X)_{(−1)} ⊂ K_{0}(X)_{(0)} ⊂ ⋯ ⊂ K_{0}(X)_{(dim X)} = K_{0}(X). For every integer d, K_{0}(X)_{(d)} is the subgroup of K_{0}(X) generated by classes of coherent O_{X}-sheaves supported in dimension ≤ d. This filtration is compatible with proper push-forwards. The associated graded group shall be denoted by gr_{•} K_{0}(X).

External product. Let X and Y be varieties over a common field, and p_{X}: X × Y → X, p_{Y} : X × Y → Y be the two projections. There is an external product

− ⊠ −: K_{0}(X) ⊗ K_{0}(Y) → K_{0}(X × Y)

induced by the association

(E, F) ↦ (p_{X}^{∗}E) ⊗_{O_{X×Y}} (p_{Y} ^{∗}F).

For any integers m and n, we have

K_{0}(X)_{(m)} ⊠ K_{0}(Y)_{(n)} ⊂ K_{0}(X × Y)_{(m+n)}.

Comparison with Chow groups. There is a morphism of graded abelian groups

φ_{X}: CH_{•}(X) → gr_{•} K_{0}(X) , [Z] ↦ [O_{Z}] mod K_{0}(X)_{(dim Z−1)}

which commutes with proper push-forwards (see [Ful98, Example 15.1.5]). Let × denote the external product for Chow groups. If X and Y are varieties over a common field, and x ∈ K_{0}(X), y ∈ K_{0}(Y), we have

(1)  φ_{X×Y} (x × y) = φ_{X}(x) ⊠ φ_{Y} (y).

Scalars extension. For a variety X over a field F, and L/F a field extension, we shall write X_{L} for the variety X ×_{Spec(F)} Spec(L) over the field L. The flat morphism X_{L} → X induces pull-backs

x ↦ x_{L}, CH(X) → CH(X_{L}) and K_{0}(X) → K_{0}(X_{L}),

which satisfy, for all x ∈ K_{0}(X)

(2)  φ_{X_{L}}(x_{L}) = φ_{X}(x)_{L}.

3. Singular Grothendieck-Riemann-Roch Theorem

For a variety X, we write CH(X)_{ℚ} for CH(X) ⊗_{ℤ} ℚ. An element x ∈ CH(X) has an image x ⊗ 1 ∈ CH(X)_{ℚ}.

Theorem 3.1. For all varieties X there is a homomorphism

τ^{X}: K_{0}(X) → CH(X)_{ℚ}

with the following properties.

(a) If f: X → Y is a proper morphism of varieties, we have

f_{∗} ∘ τ^{X} = τ^{Y} ∘ f_{∗}.

(b) For an integral variety X, we have

τ^{X}[O_{X}] = [X] mod CH_{<dim X}(X)_{ℚ}.

(c) If u: U → X is an open embedding of quasi-projective varieties, then

τ^{U} ∘ u^{∗} = u^{∗} ∘ τ^{X}.

(d) If X is a variety over a field F, L/F a field extension, then for all x ∈ K_{0}(X)

we have in CH(X_{L})_{ℚ}

τ^{X}(x)_{L} = τ^{X_{L}}(x_{L}).

(e) Let i: X ↪ M be a regular closed embedding, with M a smooth variety. Let N be the normal bundle of i and T_{M} the tangent bundle of M. Then, writing Td for the Todd class, we have

τ^{X}[O_{X}] = Td(i^{∗}[T_{M}] − [N]).

Proof. The existence of the map τ^{X} satisfying (a) and (b) is proven in [Ful98, Theorem 18.3]. Applying loc.cit., (4) with f: X → point the structure morphism, and α = 1 ∈ K_{0}(point), we obtain (e). Putting f = u: U → X, we get (c).

We now prove (d). We first assume that X is quasi-projective over F, and choose a closed embedding i: X ↪ M into a smooth F -variety M. We can assume that x = [F] for some coherent O_{X}-sheaf F, and take a resolution E_{•} → i_{∗}F → 0 by locally free O_{M}-sheaves. Then by [Ful98, Theorem 18.3, (3)]

τ^{X}(x) = ch_{X}^{M}(E_{•}) ∩ Td(T_{M}),

where Td(T_{M}) is the Todd class of tangent bundle of M, and ch_{X}^{M}(E_{•})∩−: CH(M)_{ℚ} → CH(X)_{ℚ} the localized Chern character.

We have a closed embedding i_{L}: X_{L} ↪ M_{L}, with M_{L} smooth over L. Let f: X_{L} → X, g: M_{L} → M be the morphisms induced by the field extension L/F. Then x_{L} = [f^{∗}F], we have an isomorphism g^{∗} ∘ i_{∗}F ≃ (i_{L})_{∗} ∘ f^{∗}F, and a resolution g^{∗}E_{•} → (i_{L})_{∗} ∘ f^{∗}F → 0. The tangent bundle T_{M_{L}} of M_{L} is isomorphic to g^{∗}T_{M} = (T_{M})_{L}, therefore using [Ful98, Theorem 18.1], we have

τ^{X_{L}}(x_{L}) = ch_{X_{L}}^{M_{L}}(g^{∗}E_{•}) ∩ Td(g^{∗}T_{M}) = f^{∗}( ch_{X}^{M}(E_{•}) ∩ Td(T_{M})) = τ^{X}(x)_{L}.

Now we drop the assumption that X is quasi-projective over F. Let p: X′ → X be a Chow envelope. The map p_{∗}: K_{0}(X′) → K_{0}(X) is surjective, let y be an antecedent of x. Then we have, by compatibility with proper push-forwards

τ^{X}(x) = τ^{X} ∘ p_{∗}(y) = p_{∗} ∘ τ^{X′}(y).

The fiber product p_{L}: X′_{L} → X_{L} is also a Chow envelope by [Ful98, Lemma 18.3, (2)], and

τ^{X_{L}}(x_{L}) = τ^{X_{L}}(p_{∗}(y)_{L}) = τ^{X_{L}} ∘ (p_{L})_{∗}(y_{L}) = p_{∗} ∘ τ^{X′^{L}}(y_{L}).

The latter is equal to p_{∗}∘τ^{X′}(y)_{L} by the quasi-projective case, and (d) follows. □

The morphism τ^{X} is the homological Chern character. Individual components

τ_{k}^{X}: K_{0}(X) → CH_{k}(X)_{ℚ}

are defined by composing with the projections CH(X)_{ℚ} → CH_{k}(X)_{ℚ}.

4. A 2-integrality property of the homological Chern character

The result obtained in this section, Corollary 4.4, can be thought of as an algebraic analog of [Ada61, Theorem 1] (with r = 1).

Let X be a variety. We denote by CH(X)_{ℤ} ⊂ CH(X)_{ℚ} the Chow group of X modulo its torsion subgroup, and view it as the image of the map CH(X) → CH(X)_{ℚ} given by x ↦ x ⊗ 1.

Lemma 4.1. Let X be a regular, connected and quasi-projective variety of dimension d. Then

2 · τ_{d−1}^{X}[O_{X}] ∈ CH_{d−1}(X)_{ℤ}.

Proof. Since X is quasi-projective, we can find a smooth connected variety M and a closed embedding j: X ↪ M. As is any closed embedding of regular varieties, j is a regular closed embedding ([Bou07, Proposition 2, §5, N^{∘}3, p.65]). Letting N be its normal bundle and T_{M} the tangent bundle of M, we can apply Theorem 3.1, (e). Using [Ful98, Example 3.2.4] asserting that the first term of the Todd class is the half of the first Chern class c_{1}, we obtain in CH(X)_{ℚ} mod CH_{dim X−2}(X)_{ℚ} the congruence

τ^{X}[O_{X}] = Td(j^{∗}T_{M} − N) = ( id +c_{1}(j^{∗}T_{M}) ⊗ 2^{−1}) · ( id +c_{1}(N) ⊗ 2^{−1})^{−1}[X]
= [X] + c_{1}(j^{∗}T_{M} − N) ⊗ 2^{−1}.

The statement follows. □

Lemma 4.2. Let X be a normal, connected and quasi-projective variety of dimension d. Then

2 · τ_{d−1}^{X}[O_{X}] ∈ CH_{d−1}(X)_{ℤ}.

Proof. The set S of points x ∈ X such that O_{X,x} is not a regular local ring is closed in X ([Bou07, Corollaire 4, §7, N^{∘}9, p.102]). We consider S as a closed subscheme of X, by endowing it with the reduced scheme structure. Since X is normal, the subscheme S has codimension at least two in X.

Let u: U → X be the open complement of S in X. The variety U is regular, connected and quasi-projective, hence Lemma 4.1 applies to U: we find an integral cycle y_{U} ∈ CH_{d−1}(U) satisfying 2 · τ_{d−1}^{U}[O_{U}] = y_{U} ⊗ 1. We have the localization sequence

CH(S) → CH(X) ⟶^{u^{∗}} CH(U) → 0.

Let y ∈ CH_{d−1}(X) be such that u^{∗}(y) = y_{U}. By Theorem 3.1, (c), we have

u^{∗}(2 · τ_{d−1}^{X}[O_{X}] − y ⊗ 1) = 2 · τ_{d−1}^{U}[O_{U}] − y_{U} ⊗ 1 = 0,

hence 2 · τ_{d−1}^{X}[O_{X}] − y ⊗ 1 belongs to the image of CH(S)_{ℚ} → CH(X)_{ℚ}, which is contained in CH_{<d−1}(X)_{ℚ}. But 2 · τ_{d−1}^{X}[O_{X}] − y ⊗ 1 belongs to CH_{d−1}(X)_{ℚ}, hence it is zero. □

Proposition 4.3. Let X be an integral variety of dimension d. Then

2 · τ_{d−1}^{X}[O_{X}] ∈ CH_{d−1}(X)_{ℤ}.

Proof. First, using Chow’s Lemma [Gro61, II, Théorème 5.6.1], choose a projective birational morphism e: X′ → X, with X′ quasi-projective and integral. Let n: X̃ → X′ be the normalization of X′. It is a finite birational morphism ([Bou06, Théorème 2, Chapitre V, §3, N^{∘}2, p.59]), with X̃ normal, connected, and quasiprojective. Setting p = e ∘ n, we have p_{∗}[OX̃] = [O_{X}] + i_{∗}(z) for some closed subvariety i: Z ↪ X, with dim Z < d, and some element z ∈ K_{0}(Z). Then

2 · τ_{d−1}^{X}[O_{X}] = 2 · p_{∗} ∘ τ X̃ ; d−1[OX̃] − 2 · i_{∗} ∘ τ_{d−1}^{Z}(z).

Since dim Z ≤ d − 1, we have by Theorem 3.1, (b), τ_{d−1}^{Z}(z) ∈ CH_{d−1}(Z)_{ℤ}, and we conclude by applying Lemma 4.2 to X̃. □

Corollary 4.4. Let X be a variety, and x ∈ K_{0}(X)_{(k)}. Then

2 · τ_{k−1}^{X}(x) ∈ CH_{k−1}(X)_{ℤ}.

5. The first homological Steenrod square

We define functors from the category of varieties and proper morphisms to the category of graded abelian groups by setting, for every variety X, Ch(X) = ℤ/2 ⊗ CH(X) and

C̃h(X) = CH(X)/(2-torsion + 2 CH(X)) = ℤ/2 ⊗ CH(X)_{ℤ}.

There is a natural surjective map

Ch(X) → C̃h(X), x ↦ x̃.

Theorem 5.1. The association

x ↦ 2 · τ_{k−1}^{X}(x)

induces a natural transformation of functors from the category of varieties and proper morphisms to the category of graded abelian groups

Sq_{1} : Ch_{•} → C̃h_{•−1} .

Proof. Let X be a variety. From Corollary 4.4 we get a homomorphism of abelian groups, for integer k

s_{k}: K_{0}(X)_{(k)} → CH_{k−1}(X)_{ℤ} → C̃h_{k−1}(X),

Now take x ∈ K_{0}(X)_{(k−1)} ⊂ K_{0}(X)_{(k)}. By Theorem 3.1, (b), we have 2 · τ_{k−1}^{X}(x) ∈ 2 · CH_{k−1}(X)_{ℤ}, hence s_{k}(x) = 0 ∈ C̃h_{k−1}(X).

This gives a homomorphism of graded abelian groups

ℤ/2 ⊗ gr_{•} K_{0}(X) → C̃h_{•−1}(X).

The fact that it commutes with push-forwards along proper morphisms follows from Theorem 3.1, (a). It only remains to compose with the natural transformation id_{ℤ/2} ⊗φ: Ch_{•} → ℤ/2 ⊗ gr_{•} K_{0}. □

For a variety X, we write Sq_{1}^{X}: Ch_{•}(X) → C̃h_{•}(X) for the morphism given by Theorem 5.1.

Remark 5.2. One can expect the map Sq_{1} to lift to a morphism

Ch_{•} → Ch_{•−1} .

However one can not expect that Sq_{1} descends to a morphism

C̃h_{•} → C̃h_{•−1},

as suggested by the following example.

Let X be an anisotropic projective 3-dimensional quadric over a field F of characteristic not two, defined by a quadratic form of type ⟨⟨a, b⟩⟩ ⊥ c. In this case, by [Kar90, Theorem 5.3] there is an element l_{0} ∈ K_{0}(X)_{(1)} such that l_{0} ∉ K_{0}(X)_{(0)} and 2l_{0} ∈ K_{0}(X)_{(0)}. Note that φ_{X}: CH_{•}(X) → gr_{•} K_{0}(X) is an isomorphism : this is a general fact concerning smooth varieties in codimension 0, 1 and 2 ([Ful98, Example 15.3.6]); in codimension 3 this follows from the fact that CH_{0}(X) ≃ ℤ ([EKM08, Corollary 71.4]) and the fact that φ_{X} has torsion kernel ([Ful98, Example 15.3.6]). Let x ∈ Ch_{1}(X) be the antecedent of l_{0} mod K_{0}(X)_{(0)} under φ_{X}, and y ∈ Ch_{0}(X) the class of a point of degree two. We have φ_{X}(y) = 2l_{0}, and using Theorem 3.1, (b)

τ^{X}(l_{0}) = 2^{−1}τ^{X}(2l_{0}) = y ⊗ 2^{−1}

hence Sq_{1}^{X}(x) = y, which is non-zero in C̃h_{0}(X). On the other hand x is zero in C̃h_{1}(X).

Proposition 5.3. Let X and Y be two varieties over the same field, x ∈ CH_{n}(X), and y ∈ CH_{m}(Y). We have

Sq_{1}^{X×Y} (x × y) = x̃ × Sq_{1}^{Y} (y) + Sq_{1}^{X}(x) × ỹ.

Proof. We know by Theorem 3.1, (b) that τ_{k}^{X}(x) = 0 and τ_{l}^{Y} (y) for k > n and l > m. Then, using [Ful98, Example 18.3.1] and (1), we compute

2 · τ_{m+n−1}^{X×Y} ∘ φ_{X×Y} (x × y)
=2 · τ_{m+n−1}^{X×Y}(φ_{X}(x) ⊠ φ_{Y} (y))
=τ_{m}^{X} ∘ φ_{X}(x) × 2 · τ_{n−1}^{Y} ∘ φ_{Y} (y) + 2 · τ_{m−1}^{X} ∘ φ_{X}(x) × τ_{n}^{Y} ∘ φ_{Y} (y)
=(x ⊗ 1) × 2 · τ_{n−1}^{Y} ∘ φ_{Y} (y) + 2 · τ_{m−1}^{X} ∘ φ_{X}(x) × (y ⊗ 1).  □

Proposition 5.4. Let X be a smooth variety, with tangent bundle T_{X}. Then

Sq_{1}^{X}[X] = { \atop c_{1}(T_{X})}.

Proof. This follows from the proof of Lemma 4.1. □

Remark 5.5 (cohomological operation). For a smooth variety X, define

Sq_{X}^{1} : Ch^{•}(X) → C̃h^{•+1}(X), x ↦ Sq_{1}^{X}(x) + { \atop c_{1}(T_{X})} · x̃.

One can prove that Sq^{1} commutes with pull-backs along arbitrary morphisms of smooth varieties. The formula Sq_{X}^{1}(x·y) = x̃·Sq_{X}^{1}(y)+Sq_{X}^{1}(x)·ỹ then follows from Proposition 5.3. Using the fact that the group CH^{1}(X) is generated by regularly embedded subvarieties when X is smooth, we get the formula Sq_{X}^{1}(x) = x̃^{2} for x ∈ Ch^{1}(X).

In order to prove Theorem 6.2 below, it would suffice to use this cohomological operation, which can also be constructed directly using a simpler form of the Riemann-Roch theorem instead of Theorem 3.1. However it is not clear how to reconstruct the homological operation on singular varieties from the cohomological one.

We now consider a variety X over a field F, and a field extension \overline{F} /F such that variety \overline{X} = X\overline{F} has a torsion-free Chow group. Examples of such varieties include those X such that \overline{X} is cellular. In particular X could be a projective homogeneous variety under a semi-simple algebraic group, and \overline{F} /F an algebraic closure.

Since Sq_{1} commutes with field extensions by (2) and Theorem 3.1, (d), we have a commutative diagram, where vertical arrows are pull-backs along the flat morphism \overline{X} → X,

\xymatrix{Ch(X) \ar[r]^{^{Sq^{X}^{1}}} \ar[d] & C̃h(X) \ar[d] \\ Ch(\overline{X}) \ar[r]_{_{Sq\overline{X} ; 1}} & C̃h(\overline{X}).}

Note that Ch(\overline{X}) ≃ C̃h(\overline{X}), hence the operation Sq\overline{X} ; 1 is an endomorphism of Ch(\overline{X}) which preserves rationality, i.e. induces an endomorphism of the image of the map Ch(X) → Ch(\overline{X}).

6. Parity of the first Witt index of a quadratic form

Let X be a smooth projective quadric over a field F, and \overline{F} /F a splitting field extension for X, i.e. an extension such that the quadric \overline{X} = X\overline{F} contains a projective space of the maximal possible dimension d (d is the greatest integer such that 2d ≤ dim X). The variety X and the field extension \overline{F} /F satisfy the assumption of the previous section. Indeed a basis for the free abelian group CH(\overline{X}) is given by elements h^{i}, l_{i}, 0 ≤ i ≤ d, see for example [EKM08]. The cycle h ∈ CH^{1}(\overline{X}) is the hyperplane class, and l_{i} ∈ CH_{i}(\overline{X}) is the class of a projective subspace of dimension i.

A cycle in Ch(\overline{X}) is said to be rational if it is the restriction of a cycle in Ch(X).

Lemma 6.1. Let X be a smooth projective quadric, and \overline{X} the restriction of X to a splitting field. Then we have in Ch_{i−1}(\overline{X})

Sq\overline{X} ; 1(l_{i}) = (i + 1) · l_{i−1}.

Proof. Let j: ℙ^{i} ↪ \overline{X} be a closed embedding representing the cycle l_{i}. Then since j_{∗} ∘ Sq_{1}^{ℙ^{i}} = Sq\overline{X} ; 1 ∘j_{∗}, and by Proposition 5.4 we have

Sq\overline{X} ; 1(l_{i}) = j_{∗} ∘ c_{1}(T_{ℙ^{i}}).

By [EKM08, Example 104.20], this is equal to

j_{∗} ∘ c_{1}((i + 1) · [O(1)] − [O_{ℙ^{i}}]) = j_{∗}((i + 1) · [ℙ^{i−1}]) = (i + 1) · l_{i−1}.  □

Theorem 6.2. Let φ be an anisotropic non-degenerate quadratic form over an arbitrary field. Let 𝔦_{1} be the first Witt index of φ. If dim φ − 𝔦_{1} is odd then 𝔦_{1} = 1.

Proof. Let F be the ground field, X be the smooth projective quadric of φ, and \overline{F} /F a splitting field extension for X. We assume that 𝔦_{1} ≠ 1. We use the notion of a cycle contained in another ([EKM08, p.313]). Let π ∈ Ch(\overline{X}) be the 1-primordial cycle of X ([EKM08, p.323]). It is the “minimal” ([EKM08, Definition 73.5]) rational cycle containing h^{0} × l_{𝔦_{1}−1}. Write

π = h^{0} × l_{𝔦_{1}−1} + l_{𝔦_{1}−1} × h^{0} + v.

Then v ∈ Ch(\overline{X}) does not contain h^{0}×l_{𝔦_{1}−1} nor l_{𝔦_{1}−1}×h^{0} by [EKM08, Lemma 73.15]. Also the rational cycle π, hence v, does not contain h^{1}×l_{𝔦_{1}} nor l_{𝔦_{1}}×h^{1} (these points lie outside of the “shell triangles”, i.e. are forbidden by [EKM08, Lemma 73.12]). Using Proposition 5.3, we see that the cycle Sq\overline{X} ; 1(v) does not contain any of the cycles h^{0} × l_{𝔦_{1}−2}, h^{1} × l_{𝔦_{1}−1}, l_{𝔦_{1}−2} × h^{0}, l_{𝔦_{1}−1} × h^{1}.

Next we compute Sq\overline{X} ; 1(π), which is rational by the considerations of the previous section. We have, by Proposition 5.3, Lemma 6.1 and Proposition 5.4,

Sq\overline{X} ; 1(h^{0} × l_{𝔦_{1}−1}) = h^{0} × Sq\overline{X} ; 1(l_{𝔦_{1}−1}) + c_{1}(T_{X}) × l_{𝔦_{1}−1} = 𝔦_{1} · (h^{0} × l_{𝔦_{1}−2}) + c_{1}(T_{X}) × l_{𝔦_{1}−1}.

By [EKM08], we know that the modulo two total Chern class of the tangent bundle T_{X} is (1 + h)^{dim X+2}, hence c_{1}(T_{X}) = (dim X + 2)h^{1}. This gives

Sq\overline{X} ; 1(π) = 𝔦_{1} · (h^{0} × l_{𝔦_{1}−2} + l_{𝔦_{1}−2} × h^{0}) + (dim φ) · (h^{1} × l_{𝔦_{1}−1} + l_{𝔦_{1}−1} × h^{1}) + Sq\overline{X} ; 1(v).

Now by [EKM08, Corollary 73.21] and the property of Sq\overline{X} ; 1(v) mentioned above, we see that dim φ and 𝔦_{1} must have the same parity. □

References

[Ada61] John F. Adams. On Chern characters and the structure of the unitary group. Proc. Cambridge Philos. Soc., 57:189–199, 1961.

[Bou06] Nicolas Bourbaki. Éléments de mathématique. Algèbre commutative. Chapitres 5 à 7. Springer-Verlag, Berlin, 2006. Reprint of the 1985 original.

[Bou07] Nicolas Bourbaki. Éléments de mathématique. Algèbre commutative. Chapitre 10. Springer-Verlag, Berlin, 2007. Reprint of the 1998 original.

[EKM08] Richard Elman, Nikita Karpenko, and Alexander Merkurjev. The algebraic and geometric theory of quadratic forms, volume 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[Ful98] William Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, 1998.

[Gro61] A. Grothendieck. Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math., (8):222, 1961.

[Kar90] Nikita A. Karpenko. Algebro-geometric invariants of quadratic forms. Algebra i Analiz,

2(1):141–162, 1990.

E-mail address: olivier.haution@gmail.com

Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France.