ON GENERIC QUADRATIC FORMS

NIKITA A. KARPENKO

Abstract. Based on Totaro’s computation of the Chow ring of classifying spaces for orthogonal groups, we compute the Chow rings of all orthogonal Grassmannians associated with a generic quadratic form of any dimension. This closes the gap between the known particular cases of the quadric and the highest orthogonal Grassmannian. We also relate two different notions of generic quadratic forms.

Contents

1. Introduction 1 2. Tautological Chern subring 2 3. The standard generic quadratic form 4 4. Chow rings of classifying spaces 4 5. Chow rings of classifying spaces for orthogonal groups 6 6. Main Theorem and its consequences 7 7. Proof of Main Theorem 7 8. The generic quadratic form in characteristic ≠ 2 9 9. The generic quadratic form in characteristic 2 9 References 10

1. Introduction

Let k be a field of characteristic different from 2 and let F = k(t_{1}, …, t_{n}) be the field of rational functions over k in variables t_{1}, …, t_{n} for some n ≥ 2. We call generic the diagonal quadratic form q := ⟨t_{1}, …, t_{n}⟩ over F.

The Chow ring of the projective quadric given by q has been computed in [7, Corollary 2.2]. The Chow ring of the highest orthogonal Grassmannian of a generic quadratic form has been computed in [13], but this was done for a different notion of generic, which we call here standard generic. In the present paper we determine the Chow ring of all orthogonal Grassmannians associated with the generic and the standard generic quadratic forms. (The characteristic assumption is removed in the latter case.) We use computation of the Chow ring of classifying spaces for orthogonal groups O(n) performed in [12] as well as in [15] over the field of complex numbers and later in [11] over an arbitrary field of characteristic not 2. We actually need only a piece of this computation which is made in [15] over arbitrary field (of arbitrary characteristic), see Section 5.

Date: 10 June 2017. Revised: 20 October 2017.

Key words and phrases. Algebraic groups; projective homogeneous varieties; Chow groups. Mathematical Subject Classification (2010): 20G15; 14C25.

This work has been supported by a Discovery Grant from the National Science and Engineering Board of Canada.

Note that the algebraic group O(n) over a field k is not connected if n is even or char k ≠ 2. In the remaining case (when n is odd and char k = 2) the algebraic group O(n) is not smooth. In contrast, the special orthogonal group O^{+}(n) is always smooth and connected. But since O(n)-torsors correspond to all non-degenerate n-dimensional quadratic forms while O^{+}(n)-torsors correspond to quadratic forms of trivial discriminant, it is more appropriate to work with O(n) for the question raised in this paper. On the other hand, since orthogonal Grassmannians depend only on the similarity class of the quadratic form in question and any odd-dimensional quadratic form is similar to that of trivial discriminant, O(n) can be replaced by O^{+}(n) for odd n.

2. Tautological Chern subring

In this section we consider an arbitrary non-degenerate quadratic form q : V → F of an arbitrary dimension n ≥ 2 over an arbitrary field F. (Characteristic 2 is not excluded; non-degenerate quadratic forms are defined as in [4, §7.A].) In particular, V is an n-dimensional F -vector space. We fix an integer 1 ≤ m ≤ n/2 and write X for the orthogonal Grassmannian of isotropic m-planes (i.e., totally isotropic m-dimensional subspaces) in V. Note that the variety X is smooth projective; it is geometrically connected if and only if m ≠ n/2.

Let T = T_{X} be the tautological (rank-m) vector bundle on X: the fiber of T over a point of X, given by an isotropic m-plane, is this very m-plane itself. We define the tautological Chern subring CT X in the Chow ring CH X as the subring generated by the Chern classes c_{1}(T ), …, c_{m}(T ) . The goal of this section is to determine the ring CT X by providing a list of defining relations on its generators.

The variety X is a closed subvariety of the usual Grassmannian Γ of all m-planes in V. The Chow ring CH Γ is known to be generated by the Chern classes of the tautological (rank-m) vector bundle on Γ. Therefore the pull-back CH Γ → CH X with respect to the closed imbedding X ↪ Γ provides an epimorphism CH Γ →→ CT X. Since a description of the ring CH Γ by generators and relations is available (see [2, Lemma 1.2] or [5, Example 14.6.6]), we fulfill our goal if we describe the kernel of the epimorphism CH Γ →→ CT X in terms of generators of CH Γ. For this, it is more convenient to use the generators c_{1}, …, c_{n−m} ∈ CH Γ given by the Chern classes of −[T ] rather than of T = T_{Γ} itself. By [T ] here we mean the class of T in the Grothendieck ring K(Γ). The Chern classes of −[T ] are the Segre classes of T , i.e. the components of the multiplicative inverse to the total Chern class c(T ). The tautological vector bundle T is a subbundle of the trivial (rank-n) vector bundle V and c_{1}, …, c_{n−m} are the Chern classes of the quotient V/T . We define c_{i} ∈ CH^{i}(Γ) for every integer i by setting c_{i} := c_{i}(−[T ]) = c_{i}(V/T ). Therefore c_{0} = 1 and c_{i} = 0 for i < 0 as well as for i > n − m.

Theorem 2.1. The kernel of the epimorphism CH Γ →→ CT X is generated by the elements

(2.2)  c_{i}^{2} − 2c_{i−1}c_{i+1} + 2c_{i−2}c_{i+2} − ⋯ + (−1)^{i}2c_{0}c_{2i} with i > n/2 − m

and c_{n−m}. The abelian group CT X is free with a basis consisting of the images of the products c_{1}^{α_{1}} … c_{n−m−1}^{α_{n−m−1}} with α_{1} + ⋯ + α_{n−m−1} ≤ m and α_{i} ≤ 1 for i > n/2 − m.

Proof. Let us first check that the elements (2.2) lie in the kernel. The ith element is mapped to the Chern class c_{2i}(−[T ] − [T^{∨}]) ∈ CT X, where T = T_{X} and T^{∨} is the dual vector bundle. The isomorphism V/T^{⊥} = T^{∨}, where T^{⊥} is the vector bundle given by the orthogonal complement, shows that −[T ] − [T^{∨}] = −[T ] + [T^{⊥}] = [T^{⊥}/T ]. Since the rank of the quotient T^{⊥}/T is n − 2m (cf. [4, Proposition 1.5]), its Chern classes vanish in degrees > n − 2m.

In order to show that c_{n−m} is in the kernel, we proceed similarly to [18, Proof of Proposition 2.1]. One notice that the projective bundle ℙ(T ) over X can be identified with the variety of flags of totally isotropic subspaces in V of dimensions 1 and m. In particular, besides of the projection π : ℙ(T ) → X, we have a projection π_{1} : ℙ(T ) → X_{1} to the projective quadric X_{1} (the orthogonal Grassmannian of 1-planes). Moreover, the tautological line bundle on the projective bundle ℙ(T ) is the pull-back π_{1}^{∗}(T_{1}) of the tautological line bundle T_{1} on X_{1}. It follows by [4, §58] or [5, Chapter 3] that c_{i}(−[T ]) = π_{∗}(π_{1})^{∗}c_{i+m−1}(−[T_{1}]) for any i. Since dim X_{1} = n−2, the Chern class c_{n−1}(−[T_{1}]) vanishes implying the vanishing of c_{n−m}(−[T ]).

In order to show that the kernel is generated by the elements (2.2) and c_{n−m}, we construct additive generators of the quotient C of the ring CH Γ by the ideal generated by the elements (2.2) and c_{n−m}. We recall that the group CH Γ is free, a basis is given by the products c_{1}^{α_{1}} … c_{n−m}^{α_{n−m}} with α_{1} + ⋯ + α_{n−m} ≤ m. Using the additional relations in C, we can eliminate squares of c_{i} for i > n/2 − m. Indeed, in the quotient of C by the subgroup generated by the products satisfying the additional condition, any element is divisible by an arbitrary 2-power and therefore is 0 since C is finitely generated.

It follows that the group C is generated by the products c_{1}^{α_{1}} … c_{n−m−1}^{α_{n−m−1}} satisfying the additional condition α_{i} ≤ 1 for i > n/2 − m. It turns out that these are free generators. Moreover, they remain free when we map them to CT X and this finishes the proof of the theorem.

Our products are free in CT X because their images in the ℚ-vector space ℚ ⊗ CH X̄ are free, where X̄ is X over an algebraic closure of F. For odd n this follows from [2, Theorem 2.2(b) and formula (15)] (see Remark 2.3). For even n this follows from [2, Theorem 3.2(b) and formula (40)]. □

Remark 2.3. The paper [2], applied in the above proof, actually deals with the singular cohomology ring instead of the Chow ring. The link is explained by the following two well-known facts: the variety X̄ is cellular and the ring CH X̄ does not depend on the base field. If the base field is ℂ, then the cycle map from CH X̄ to the corresponding singular cohomology ring is an isomorphism, [5, Example 19.1.11(b)].

Remark 2.4. In the case of the highest orthogonal Grassmannian, the ring CH X̄ has been described in [17] (see also [4, Proposition 86.16 and Theorem 86.12]).

Remark 2.5. Theorem 2.1 shows that the ring CT X only depends on the integers n and m.

Remark 2.6. For odd n, the ring CT X can be identified with the full Chow ring CH Y of the variety of isotropic m-planes in an n−1-dimensional vector space endowed with a nondegenerate alternating bilinear form: there is an isomorphism CH Y → CT X mapping the Segre classes of the tautological vector bundle on Y to the Segre classes of T_{X}. (See [2, Theorem 1.2] for a description of the ring CH Y by generators and relations.) This funny observation in the case of the highest orthogonal Grassmannian turned out to be very useful in [16]. We do not use it here.

Our next and ultimate goal is to show that CT X = CH X in the case of generic q. First we need clearness in what is generic. We start with the notion of

3. The standard generic quadratic form

For a field k (of any characteristic) and an integer n ≥ 2, the standard generic ndimensional quadratic form is defined as follows.

We consider the orthogonal group O(n) over k and its tautological imbedding into the general linear group GL(n). The generic fiber of the quotient map

GL(n) → GL(n)/O(n)

is an O(n)-torsor over the function field F := k(GL(n)/O(n)). It determines an ndimensional quadratic form over F (via the identification of [3, Chapitre III, §5, 2.1]; for the case of smooth O(n) see also [10, (29.28)]) which we call the standard generic one.

In order to describe it explicitly, we use the well-known interpretation of the quotient variety GL(n)/O(n) as the variety Q of non-degenerate quadratic forms on the vector space V := k^{n}. The variety of all quadratic forms on V is an affine space (of dimension n(n + 1)/2) and Q is its open subvariety. The group GL(n) acts on Q in the evident way, O(n) is the stabilizer of a split quadratic form q_{0}, and the quotient variety GL(n)/O(n) is identified with Q this way.

For any field extension L/k, an L-point of Q is a non-degenerate quadratic form q on the L-vector space V_{L}; the fiber of the quotient map GL(n) → Q over this point is an O(n)-torsor E over L, and q is the quadratic form corresponding to E. In particular, the quadratic form given by the generic fiber of GL(n) → Q is defined over the field of rational functions F = k(t_{ij})_{1≤i≤j≤n} (t_{ij} are indeterminates, F/k is purely transcendental of the transcendence degree n(n + 1)/2) by the formula

(x_{1}, …, x_{n}) ↦ ∑_{1≤i≤j≤n} t_{ij}x_{i}x_{j}.

4. Chow rings of classifying spaces

Let F be a field (of arbitrary characteristic) and let G be an affine algebraic group over F, not necessarily smooth. The Chow ring CH_{G} of the classifying space of G, introduced in [15], is the G-equivariant Chow ring CH_{G}(Spec F). This is a graded ring, the grading is given by codimension of cycles.

The ring CH_{G} if cofunctorial in G: a homomorphism G′ → G of affine algebraic groups produces a homomorphism of graded rings CH_{G} → CH_{G′} (see [11, §2]).

If G is a split torus, the homomorphism of graded rings S(Ĝ) → CH_{G} is an isomorphism, where Ĝ is the character lattice of G, S(Ĝ) is the symmetric ℤ-algebra, and a character χ ∈ Ĝ = S^{1}(Ĝ), viewed as a G-equivariant line bundle over Spec F, is mapped to its first equivariant Chern class in CH_{G}^{1}.

Proposition 4.1. Let G′ be a closed normal subgroup of G such that the quotient T := G/G′ is a split torus. Then the restriction homomorphism CH_{G} → CH_{G′} is surjective and its kernel is generated by some elements in CH_{G}^{1}. More precisely, the kernel is generated by the image of the (additive) homomorphism

T̂ = S^{1}(T̂) = CH_{T}^{1} → CH_{G}^{1}

induced by the quotient homomorphism G → T.

Proof. For any integer i, let us consider a generically free G-representation V possessing an open G-equivariant subset U ⊂ V such that codim_{V} (V \ U) ≥ i and there are a Gtorsor U → U/G and a G′-torsor U → U/G′. By definition of CH_{G} (and similarly for G′ in place of G), we have a ring homomorphism CH_{G} → CH(U/G) which is bijective in codimensions < i. Moreover, the diagram

CH_{G} ⟶ CH_{G′}
↓  ↓
CH(U/G) ⟶ CH(U/G′)

commutes, where the bottom map is the pull-back homomorphism with respect to the T -torsor U/G′ → U/G. Therefore, in order to prove surjectivity of CH_{G} → CH_{G′} is suffices to prove surjectivity of CH(U/G) → CH(U/G′). Moreover, to get the description of the kernel for CH_{G} → CH_{G′} it suffices to prove the similar description for the kernel of CH(U/G) → CH(U/G′), where the homomorphism T̂ → CH^{1}(U/G) is the composition T̂ → CH_{G}^{1} → CH^{1}(U/G).

Let us first consider the case of T = 𝔾_{m}. Let L be the line bundle ((U/G′) × 𝔸^{1})/T over U/G. Then U/G′ is an open subvariety in L and its complement is the zero section. By the homotopy invariance and the localization property of Chow groups ([4, Theorem 57.13 and Proposition 57.9]) we have an exact sequence

CH(U/G) → CH(U/G) → CH(U/G′) → 0,

where the first map is the multiplication by the first Chern class of L. This finishes the proof for T = 𝔾_{m}.

In the general case, we induct on the rank of T. We decompose T as 𝔾_{m} × T_{1} and define an intermediate subgroup G_{1} with G′ ⊂ G_{1} ⊂ G as the kernel of the composition G → T → T_{1}. The quotient G/G_{1} is then T_{1} and the quotient G_{1}/G′ is 𝔾_{m}. The homomorphism CH_{G} → CH_{G′} decomposes in the composition CH_{G} → CH_{G_{1}} → CH_{G′}. The surjectivity statement follows because both maps in the composition are surjective by induction. It remains to determine the kernel.

Let x ∈ CH_{G} be an element vanishing in CH_{G′}, then the image of x in CH_{G_{1}} is the product yx_{1} for some x_{1} ∈ CH_{G_{1}}, where y ∈ CH_{G_{1}}^{1} is the image of a character of 𝔾_{m}. Extending the character to T, we get an element y′ ∈ CH_{G} lying in the image of T̂ → CH_{G}^{1} and mapped to y. Using the surjectivity of CH_{G} → CH_{G_{1}}, we find an element x′_{1} ∈ CH_{G} mapped to x_{1}. The difference x−y′x′_{1} is then in the kernel of CH_{G} → CH_{G_{1}} and therefore, by induction, lies in the ideal generated by the image of T̂. It follows that x itself lies in the ideal. □

Corollary 4.2. In the situation of Proposition 4.1, if the ring CH_{G′} is generated by Chern classes, then the ring CH_{G} is also generated by Chern classes.

Proof. By [6, Lemma 5.4], for any i ≥ 0 and any x ∈ CH_{G}^{i} there exists an element x′ ∈ CH_{G}^{i}, lying in the Chern subring, such that the difference x − x′ vanishes in CH_{G′}. By Proposition 4.1, x − x′ belongs to the ideal in CH_{G} generated by CH_{G}^{1} so that we can proceed by induction on i. □

Example 4.3. Taking for G a split reductive algebraic group and for G′ ⊂ G the semisimple group given by the commutator subgroup of G, we are in the situation of Proposition 4.1: G/G′ is a split torus. Therefore Proposition 4.1 describes the relation between the Chow ring of the classifying space of a split reductive group G and that of its semisimple part G′. In particular, by Corollary 4.2, if CH_{G′} is generated by Chern classes, then CH_{G} is also generated by Chern classes. This has been proved (by a different method) in [6, Proposition 5.5] in the case of special (split reductive) G, where special means that every G-torsor any field extension of the base field is trivial.

5. Chow rings of classifying spaces for orthogonal groups

The following proposition is a (slightly modified) particular case of [15, Proposition 14.2]. We provide a proof because it is shorter than that of the original statement.

Proposition 5.1. For any algebraic group G (over any field) and any imbedding of G into a special algebraic group H, the homomorphism CH_{H} → CH_{G} is surjective provided that the Chow groups of the quotient H/G over any field extension of the base field are trivial in positive codimensions.

Proof. As usual, we replace the homomorphism in question by the pull-back homomorphism CH(U/H) → CH(U/G) with respect to the morphism U/G → U/H, where U is an open subvariety in an H-representation, an H-torsor over U/H, and a G-torsor over U/G. Since H is special, every H-torsor is Zariski-locally trivial, [1]. It follows that the fiber of U/G → U/H over any point x ∈ U/H is isomorphic to the quotient variety H/G with scalars extended to the residue field of x and therefore has trivial Chow groups in positive codimensions. The statement follows from the spectral sequence of [14, Corollary 8.2] computing the K-cohomology groups of the total space of the fibration U/G → U/H in terms of the K-cohomology groups of the base and of the fibers. □

We get the following statement for arbitrary base field of arbitrary characteristic:

Corollary 5.2. The homomorphism CH_{GL(n)} → CH_{O(n)}, given by the tautological imbedding O(n) ↪ GL(n), is surjective.

Proof. As explained in Section 3, the quotient variety GL(n)/O(n) is identified with the variety Q of n-dimensional non-degenerate quadratic forms. Since Q is an open subvariety in the affine space of all n-dimensional quadratic forms, we have CH^{>0}(Q) = 0 by the homotopy invariance and the localization property of Chow groups. □

6. Main Theorem and its consequences

In this section, k is a field (of any characteristic), n is an integer ≥ 2, F is the function field k(GL(n)/O(n)), E is the standard generic O(n)-torsor given by the generic fiber of GL(n) → GL(n)/O(n), and q is the corresponding standard generic quadratic form.

For m with 1 ≤ m ≤ n/2, let X be the mth orthogonal Grassmannian of q. We would like to determine the ring CH X. The main result is expressed in terms of the tautological (rank-m) vector bundle on X. Its proof will be given in the next section.

Theorem 6.1. The ring CH X is generated by the Chern classes of the tautological vector bundle.

Theorem 6.1 claims that CH X = CT X and the ring CT X has been computed in Section 2.

Before proving Theorem 6.1, let us list some consequences. Let Y be any (partial) flag variety of totally isotropic subspaces in q. Let us consider the standard graded epimorphism CH Y → GK(Y) onto the graded ring associated with the topological filtration (i.e., the filtration by codimension of support) on the Grothendieck ring K(Y).

Corollary 6.2. The abelian group CH Y is free and, in particular, torsion-free. The ring epimorphism CH Y → GK(Y) is an isomorphism. The topological filtration on K(Y) coincides with the gamma filtration.

Proof. The variety Y is the variety of flags of totally isotropic subspaces in q of some dimensions m_{1} < ⋯ < m_{d}. Let X be the orthogonal Grassmannian of m-planes with m = m_{d}. The projection Y → X is a partial flag variety of subspaces in the tautological vector bundle on X. Therefore, it suffices to prove Corollary 6.2 for X instead of Y.

We have: CH X = CT X (Theorem 6.1) and CT X is torsion-free (Theorem 2.1).

The kernel of the epimorphism is contained in the torsion subgroup. Since CH X is torsion-free, the epimorphism is an isomorphism.

Since the Chow ring CH X is generated by Chern classes, the topological filtration on K(X) coincides with the gamma filtration. □

7. Proof of Main Theorem

We continue to work over a field k of arbitrary characteristic. We realize the orthogonal group O(n) as the automorphism group of the following split quadratic form q_{0} on the k-vector space V := k^{n}:

k^{n} ∋ (x_{1}, …, x_{n/2}, y_{n/2}, …, y_{1}) ↦ x_{1}y_{1} + x_{2}y_{2} + ⋯ + x_{n/2}y_{n/2}

if n is even and

k^{n} ∋ (x_{1}, …, x_{(n−1)/2}, z, y_{(n−1)/2}, …, y_{1}) ↦ x_{1}y_{1} + x_{2}y_{2} + ⋯ + x_{(n−1)/2}y_{(n−1)/2} + z^{2}

if n is odd.

Instead of the mth orthogonal Grassmannian X (for some m with 1 ≤ m ≤ n/2), we consider the variety Y of flags of totally isotropic subspaces in q_{0} of dimensions 1, …, m. The group O(n) acts on Y and the variety Y is the quotient O(n)/P, where P is the stabilizer of the rational point of Y given by the standard flag V_{1} ⊂ ⋯ ⊂ V_{m} with V_{i} being the span of the first i vectors in the standard basis of V.

Note that any orthogonal transformation stabilizing this flag also stabilizes the orthogonal complements

V_{m}^{⊥} = V_{n−m} ⊂ ⋯ ⊂ V_{1}^{⊥} = V_{n−1}.

Let F be the variety of flags of all subspaces in V of dimensions 1, …, m, n−m, …, n−1. The group GL(n) acts on F and F = GL(n)/S, where S is the stabilizer of the standard flag V_{1} ⊂ ⋯ ⊂ V_{m} ⊂ V_{n−m} ⊂ ⋯ ⊂ V_{n−1}.

Let E be the standard generic O(n)-torsor given by the generic fiber of GL(n) → GL(n)/O(n). Let E be the corresponding GL(n)-torsor obtained vie the imbedding O(n) ↪ GL(n). We have a commutative square

CH_{S} ⟶ CH(E/S)
↓  ↓
CH_{P} ⟶ CH(E/P)

with surjective horizontal mappings (cf. [8, Lemma 2.1]). The subgroup S′ := 𝔾_{m}^{m} × GL(n − 2m) × 𝔾_{m}^{m} ⊂ S is a Levi subgroup of S, its intersection with P ⊂ S is P′ := 𝔾_{m}^{m} × O(n − 2m). The imbedding P′ ↪ S′ is the product of the map 𝔾_{m}^{m} ↪ 𝔾_{m}^{m} × 𝔾_{m}^{m}, x ↦ (x, x^{−1}) and the tautological imbedding O(n − 2m) ↪ GL(n − 2m).

We claim that the homomorphism CH_{S} → CH_{P} is surjective. Having the claim, we conclude that the pull-back homomorphism CH(E/S) → CH(E/P) = CH Y from the above commutative square is surjective too. Since the group GL(n) is special, the GL(n)- torsor E is trivial implying that E/S = F. We get a surjection CH F →→ CH Y implying that the ring CH Y is generated by the Chern classes of the m tautological vector bundles on Y (given by the components of the flags). It follows (see [9, Lemma 4.3]) that CH X = CT X.

We finish by proving the claim. In the commutative square

CH_{S} ⟶ CH_{GL(n−2m)}
↓  ↓
CH_{P} ⟶ CH_{O(n−2m)}

the horizontal maps are epimorphisms by Proposition 4.1. The map on the right is an epimorphism by Corollary 5.2. We can now prove the surjectivity of the map on the left in every codimension i ≥ 0 using induction on i.

For i = 0 there is nothing to prove. For i = 1, we have a commutative diagram

{ \atop 𝔾_{m}^{m} × 𝔾_{m}^{m}} ⟶ CH_{S}^{1} ⟶^{onto} CH_{GL(n−2m)}^{1}
↓^{onto}  ↓  ↓^{onto}
𝔾̂_{m}^{m} ⟶ CH_{P}^{1} ⟶^{onto} CH_{O(n−2m)}^{1}

with a surjection on the left. Since the lower row is exact (by Proposition 4.1),^{1} the statement for i = 1 follows.

1The upper row is also exact but we don’t care.

For i ≥ 2, it suffices to show that any element x ∈ CH_{P}^{i}, vanishing in the group CH_{O(n−2m)}, is in the image of CH_{S}. Since x = y_{1}x_{1} + ⋯ + y_{r}x_{r} for some r ≥ 0, some y_{1}, …, y_{r} ∈ CH_{P}^{1} , and some x_{1}, …, x_{r} ∈ CH_{P}^{i−1} by Proposition 4.1, we are done.

8. The generic quadratic form in characteristic ≠ 2

For a field k of characteristic not 2 and an integer n ≥ 2, we already defined in the introduction the generic n-dimensional quadratic form as the form q_{g} := ⟨t_{1}, …, t_{n}⟩ over the field of rational functions F_{g} := k(t_{1}, …, t_{n}). Now we are going to compare q_{g} with the standard generic quadratic form q (of Section 3) defined over the field F := k(t_{ij})_{1≤i≤j≤n}.

Proposition 8.1. The field F_{g} can be k-identified with a subfield in F the way that the field extension F/F_{g} is purely transcendental and the generic quadratic form q_{g} with the scalars extended to the field F becomes isomorphic to the standard generic form q.

Corollary 8.2. Theorem 6.1 as well as Corollary 6.2 hold for the generic quadratic form in place of the standard generic one.

Proof. In case of a purely transcendental field extension, the change of field homomorphism for Chow rings is an isomorphism. □

Proof of Proposition 8.1. Let us apply the standard orthogonalization procedure to the standard basis e_{1}, …, e_{n} of F^{n}, where the orthogonality refers to the symmetric bilinear form associated with q. This means that we construct an orthogonal basis e′_{1}, …, e′_{n} by taking for e′_{i} the sum of e_{i} and a linear combination of e_{1}, …, e_{i−1}, where the coefficients of the linear combination are determined by the condition that e′_{i} is orthogonal to e_{1}, …, e_{i−1}. The procedure works for q because its restriction to the span of e_{1}, …, e_{i} is non-degenerate for every i.

Then t_{i} := q(e′_{i}) equals t_{ii}+ a rational function in t_{11}, …, t_{i−1i−1} and t_{rs} with 1 ≤ r < s ≤ n. It follows that the elements t_{rs} (1 ≤ r < s ≤ n) and t_{1}, …, t_{n} all together generate the field F over k and therefore – since their number is the transcendence degree – are algebraically independent over k. In particular, t_{1}, …, t_{n} are algebraically independent so that the field F_{g} is identified with the subfield k(t_{1}, …, t_{n}) ⊂ F. This identification has the required properties. □

9. The generic quadratic form in characteristic 2

In characteristic 2 (actually, in arbitrary characteristic), any non-degenerate quadratic form, depending on the parity of n, is isomorphic to the form [a_{1}, a_{2}]⊥ … [a_{n−1}, a_{n}] or to the form [a_{1}, a_{2}]⊥ … [a_{n−2}, a_{n−1}⊥ ⟨a_{n}⟩, where a_{1}, …, a_{n} are constants from the base field and a_{n} ≠ 0 in the case of odd n. The notation [a_{1}, a_{2}] stands for the 2-dimensional form (x_{1}, x_{2}) ↦ a_{1}x_{1}^{2} + x_{1}x_{2} + a_{2}x_{2}^{2}. So, the generic n-dimensional quadratic form q_{g} will be defined as the form [t_{1}, t_{2}]⊥ … [t_{n−1}, t_{n}] or [t_{1}, t_{2}]⊥ … [t_{n−2}, t_{n−1}]⊥ ⟨t_{n}⟩ over the rational functions field F_{g} := k(t_{1}, …, t_{n}).

Proposition 9.1. Proposition 8.1 and Corollary 8.2 hold in characteristic 2 as well.

Proof. We only need to identify the field F_{g} with a subfield in F = k(t_{ij})_{1≤i≤j≤n} the way that the field extension F/F_{g} is purely transcendental and the generic quadratic form q_{g} with the scalars extended to the field F becomes isomorphic to the standard generic q.

Starting with the standard basis e_{1}, …, e_{n} of the vector space F^{n}, we construct a new basis e′_{1}, …, e′_{n} as follows. For every odd i, the vector e′_{i} is e_{i}+ a linear combination of e_{1}, …, e_{i−1} and if i < n then the vector e′_{i+1} is e_{i+1}+ a linear combination of e_{1}, …, e_{i−1}, where the coefficients of the linear combinations are determined by the condition that the new vectors are orthogonal to each of e_{1}, …, e_{i−2}. Additionally, for every even i, we divide the vector e′_{i} by the non-zero scalar (e′_{i−1}, e′_{i}).

With respect to the new basis, the standard generic quadratic form q has the shape [t_{1}, t_{2}]⊥ … [t_{n−1}, t_{n}] or [t_{1}, t_{2}]⊥ … [t_{n−2}, t_{n−1}]⊥ ⟨t_{n}⟩, where t_{i} := q(e′_{i}). For odd i, t_{i} equals t_{ii}+ a rational function in t_{11}, …, t_{i−1i−1} and t_{rs} with 1 ≤ r < s ≤ n. For even i, t_{i} equals t_{ii}/f_{i}+ a rational function in t_{11}, …, t_{i−2i−2} and t_{rs} with 1 ≤ r < s ≤ n, where f_{i} is also a rational function in t_{11}, …, t_{i−1i−1} and t_{rs} with 1 ≤ r < s ≤ n.

It follows that the elements t_{rs} (1 ≤ r < s ≤ n) and t_{1}, …, t_{n} all together generate the field F over k and therefore are algebraically independent over k. In particular, t_{1}, …, t_{n} are algebraically independent so that the field F_{g} is identified with the subfield k(t_{1}, …, t_{n}) ⊂ F. This identification has the required properties. □

Acknowledgements. I thank Alexander Merkurjev for useful consultations, comments, and suggestions. I thank Raphaël Fino for noticing that the generator c_{n−m} was missing in Theorem 2.1 in a previous version of the paper.

References

[1] Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications. Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958.

[2] Buch, A. S., Kresch, A., and Tamvakis, H. Quantum Pieri rules for isotropic Grassmannians. Invent. Math. 178, 2 (2009), 345–405.

[3] Demazure, M., and Gabriel, P. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice ıt Corps de classes local par Michiel Hazewinkel.

[4] Elman, R., Karpenko, N., and Merkurjev, A. The algebraic and geometric theory of quadratic forms, vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[5] Fulton, W. Intersection theory, second ed., vol. 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, 1998.

[6] Karpenko, N. A. On generically split generic flag varieties. Linear Algebraic Groups and Related Structures (preprint server) 583 (29 April 2017), 17 pages.

[7] Karpenko, N. A. Chow groups of quadrics and the stabilization conjecture. In Algebraic K-theory, vol. 4 of Adv. Soviet Math. Amer. Math. Soc., Providence, RI, 1991, pp. 3–8.

[8] Karpenko, N. A. Chow groups of some generically twisted flag varieties. Annals of K-Theory 2, 2 (2017).

[9] Karpenko, N. A. Chow ring of generically twisted varieties of complete flags. Adv. Math. 306 (2017), 789–806.

[10] Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P. The book of involutions, vol. 44 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits.

[11] Molina Rojas, L. A., and Vistoli, A. On the Chow rings of classifying spaces for classical groups. Rend. Sem. Mat. Univ. Padova 116 (2006), 271–298.

[12] Pandharipande, R. Equivariant Chow rings of O(k), SO(2k + 1), and SO(4). J. Reine Angew. Math. 496 (1998), 131–148.

[13] Petrov, V. A. Chow ring of generic maximal orthogonal Grassmannians. Zap. Nauchn. Sem. S.- Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 443 (2016), 147–150.

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

[15] Totaro, B. The Chow ring of a classifying space. In Algebraic K-theory (Seattle, WA, 1997), vol. 67 of Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1999, pp. 249–281.

[16] Totaro, B. The torsion index of the spin groups. Duke Math. J. 129, 2 (2005), 249–290.

[17] Vishik, A. On the Chow groups of quadratic Grassmannians. Doc. Math. 10 (2005), 111–130 (electronic).

[18] Vishik, A. Fields of u-invariant 2^{r} + 1. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, vol. 270 of Progr. Math. Birkhäuser Boston Inc., Boston, MA, 2009, pp. 661–685.

Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

  E-mail address: karpenko at ualberta.ca, web page:  www.ualberta.ca/~karpenko