ON THE FIRST WITT INDEX OF QUADRATIC FORMS

NIKITA A. KARPENKO

Abstract. We prove Hoffmann’s conjecture determining the possible values of the first Witt index of anisotropic quadratic forms of any given dimension. The proof makes use of the Steenrod type operations on the modulo 2 Chow groups constructed by P. Brosnan.

Let F be a field of characteristic ≠ 2. For an anisotropic quadratic form ϕ over F with dim(ϕ) ≥ 2, the first Witt index i_{1}(ϕ) is the Witt index (i.e., the dimension of a maximal totally isotropic subspace) of the form ϕ_{F (ϕ)} over the function field F (ϕ) = F (X_{ϕ}) of the projective quadric given by ϕ. Clearly, this i_{1}(ϕ) is the minimal positive Witt index of ϕ_{E}, when E runs over all field extension of F.

We are going to prove the following

Conjecture 0.1 (Hoffmann). Let us write the integer dim ϕ − 1 as a sum of powers of 2:

dim(ϕ) − 1 = 2^{n_{1}} + 2^{n_{2}} + ⋯ + 2^{n_{r}}

with 0 ≤ n_{1} < n_{2} < ⋯ < n_{r}. Then the integer i_{1}(ϕ) − 1 is a partial sum of this sum (including the empty one and not including the whole sum):

i_{1}(ϕ) − 1 = 2^{n_{1}} + 2^{n_{2}} + ⋯ + 2^{n_{s}}

for some 0 ≤ s ≤ r − 1.

We remark that for any given n ≥ 2, all the values of i_{1}(ϕ), prescribed by Conjecture 0.1 for forms ϕ with dim(ϕ) = n, are really possible, that is, do occur for suitable ϕ over suitable F. To see it, we take a field k with an anisotropic r-fold Pfister form ⟨ ; ⟨a_{1}, …, a_{r}⟩ ; ⟩. Then we take indeterminates t_{0}, …, t_{m}, set F = k(t_{0}, …, t_{m}), and notice that the first index of the quadratic F -form q = ⟨ ; ⟨a_{1}, …, a_{r}⟩ ; ⟩ ⊗ ⟨t_{0}, …, t_{m}⟩ is equal to 2^{r}. Therefore, by [4, lemma 7.3], for every j = 0, 1, …, 2^{r} − 1, the first Witt index of certain j-codimensional subform ϕ ⊂ qF̃ over certain purely transcendental field extension F̃/F is 2^{r} − j (while dim(ϕ) − i_{1}(ϕ) is still 2^{r}m). ^{1}

We prove Conjecture 0.1 in §3. In §1 and §2 we introduce two different tools needed in the proof.

Date: 14 June 2002.

Key words and phrases. Quadratic forms, first Witt index, Chow groups, Steenrod operations, correspondences. 2000 Mathematical Subject Classifications: 11E04; 14C25.

1By a theorem of A. Vishik, one may even take F̃ = F with any j-codimensional subform ϕ ⊂ q.

All the time we are working with Chow group CH^{∗}(Y) of various equidimensional F -varieties (in fact, Y is always a projective quadric or a product of projective quadrics here). We underline that this Chow group CH^{∗}(Y) is the modulo 2 Chow group. It is graded in the usual way: CH^{∗}(Y) = ⨁_{i=0}^{d} CH^{i}(Y) (d = dim Y), where CH^{i}(Y) is the group of classes of cycles of codimension i. Sometimes, the lower indices CH_{i}(Y) = CH^{d−i}(Y), indicating the dimension of cycles, are used. If we consider CH^{∗}(Y) as a usual group (or ring), not a graded one, in a place, we write simply CH(Y), omitting ∗ in the notation.

Acknowledgements: I am grateful to A. S. Merkurjev who explained me Brosnan’s construction.

1. Vishik’s principle

In this section we formulate and prove an extremely useful observation, due to A. Vishik, concerning correspondences on projective quadrics. We refer to this as to Vishik’s principle. Since originally neither the formulation nor the proof are not in the elementary terms used here, we give a different formulation and a complete proof. We restrict ourself to the case of odddimensional quadrics: this is enough for our purposes and slightly simplifies the consideration.

Let X be the projective quadric over F given by an anisotropic quadratic form ϕ. We set d = dim(X) (that is, d = dim(ϕ) − 2) and assume that this integer is odd. Therefore, over an algebraic closure F̄ of F, the variety X̄ = XF̄ is cellular with one cell in each dimension. Moreover, writing e^{i} ∈ CH^{i}(X̄) (i = 0, 1, …, d) for the classes of the closures of the cells, we have: for i < d/2, the element e^{i} is represented by an i-codimensional linear section of X̄ (and therefore is defined over F), while the element e_{i} = e^{d−i} is represented by an i-dimensional linear subspace of X̄. In particular, the product e^{1} · e^{i} is e^{i+1} for i ∉ {(d − 1)/2, d} and is 0 otherwise.

It follows that the exterior products e^{i} × e^{j} form a basis of the Chow group CH^{∗}(X̄ × X̄) (considered as a vector space over the field ℤ/2ℤ). For any α ∈ CH^{∗}(X ×X) we define α_{ij} ∈ {0, 1} as the coefficients in the representation of ᾱ = αF̄ ∈ CH^{∗}(X̄ × X̄) as a linear combination of the basis elements, so that we have

ᾱ = ∑_{i,j} α_{ij}(e^{i} × e^{j}) .

Clearly, for a homogeneous α ∈ CH^{n}(X × X) the coefficient α_{ij} can be non-zero only if j = n − i. For such α, we set α_{i} = α_{i n−i}.

Proposition 1.1 (Vishik’s principle). Setting i_{1} = i_{1}(ϕ), for every element α ∈ CH^{d}(X × X) and every i = 0, 1, …, i_{1} − 1, we have:

α_{i} = α_{d−i_{1}+1+i} .

Proof. Let us first consider the case of i_{1} = 1. Here we have only one equality to prove, namely, α_{0} = α_{d}. Since α_{0} coincides with the index (sometimes also called degree, [3, def. of example 16.1.4]) of the correspondence α, while α_{d} is the index of the transposition of α, the equality needed is given by [4, th. 6.4].

In the general case (i_{1} is arbitrary), let us take a subquadric X′ ⊂ X of codimension i_{1} − 1 such that i_{1}(X′) = 1. We can certainly find such a subquadric, at least if we extend the scalars up to a purely transcendental extension of the base field, according to [4, lemma 7.3] (in fact, by a stronger [4, cor. 8.3], which we do not really need here, any subquadric X′ ⊂ X of codimension i_{1} − 1 has this property).

Since the Witt index of the quadratic form ϕ_{F (X)} is greater that i, the quadric X_{F (X′)} contains a linear subspace of dimension i. Its closure in X′ × X gives an element β ∈ CH^{d−i}(X′ × X) (this β is a preimage of the class of the linear subspace under the surjective flat pull-back CH^{d−i}(X′×X) →→ CH^{d−i}(X_{F (X′)}) with respect to the evident injective morphism of schemes X_{F (X′)} → X′ × X). We may consider β as a correspondence from X′ to X (see [3, §16.1]). Note that the dimension of the cycle representing β is not the same as the dimension of the graphs of morphisms X′ → X (which is equal to dim X′), that is, the degree ([3, example 16.1.1]) of β is not 0. More precisely, deg β = −i.

Let now Y ⊂ X be a subquadric of codimension i. Then the quadric X′_{F (Y)} has a rational point, and its closure in Y × X′ gives an element γ ∈ CH^{d+i−i_{1}+1}(X × X′) (the push-forward of the element in CH^{d−i_{1}+1}(Y × X′)). This γ, viewed as a correspondence from X back to X′, is of degree i. The composition ([3, def. 16.1.1]) γ ∘ α ∘ β of the three correspondences (note that the degree of the correspondence α is 0) is a correspondence from X′ to X′ of degree −i + 0 + i = 0. It is then an exercise on the calculus of correspondences (note that all the computations needed can be done over F̄) to check that the index of this correspondence is α_{i}, while the index of its transposition is a multiple of α_{d−i_{1}+1+i}. Thus we are done by [4, th. 6.4]. □

2. Brosnan’s construction

Here we recall the construction and basic properties of the cohomological Steenrod type operations on the modulo 2 Chow groups given in [1]. Certainly, the idea that it could be possible and useful to construct in an algebraic context something similar to the well-known topological Steenrod operations, is due to Voevodsky. Brosnan’s construction described bellow is completely independent of and much simpler than those of Voevodsky, however.

Let X be a smooth quasi-projective scheme over a field F. Let G be the group of 2-roots of unity viewed as an algebraic group over F. Since char F ≠ 2 as everywhere in the paper, G is identified with the finite group ℤ/2ℤ and therefore acts on the product X × X by the factors exchange. For any ncodimensional cycle α on X, the product α × α is a 2n-codimensional cycle on X × X invariant with respect to the action of G. Therefore, α × α determines an element of the equivariant Chow group CH_{G}^{2n}(X × X) as defined in [2]. The map to CH_{G}^{2n}(X × X) one obtains this way factors through the rational equivalence ([1, prop. 4.2]) and therefore gives a map CH^{n}(X) → CH_{G}^{2n}(X×X).

Furthermore, the diagonal imbedding δ : X → X × X is equivariant with respect to the trivial action of G on X, giving a pull-back of the equivariant Chow groups:

δ^{∗}: CH_{G}^{2n}(X × X) → CH_{G}^{2n}(X) .

Finally, the equivariant graded Chow ring CH_{G}^{∗}(X) is identified with the graded ring CH^{∗}(X)[t] of polynomials in one variable t over the usual modulo 2 Chow ring CH^{∗}(X), where t is of degree 1 ([1, th. 6.1]). In particular,

δ^{∗}(α × α) = ∑_{i≥0} S^{n−i}(α) · t^{i}

with some S^{i}(α) ∈ CH^{n+i}(X).

The maps S^{i}: CH^{∗}(X) → CH^{∗+i}(X) with i ≥ 0 obtained this way, one calls them Steenrod operations; their sum (which is in fact finite because S^{i} = 0 for i > dim X)

S = S_{X} = S^{0} + S^{1} + ⋯: CH(X) → CH(X)

is the total Steenrod operation (we omit the ∗ in the notation of the Chow group to notify that S is not homogeneous). They have the following basic properties (see [1] for the proofs): for any smooth quasi-projective F -scheme X, the total operation S: CH(X) → CH(X) is a (non-homogeneous) ring homomorphism such that for every morphism f: Y → X of smooth quasi-projective F -schemes and for every field extension E/F, the squares

CH(Y) ⟶^{S_{Y}} CH(Y)
↑^{f^{∗}}  ^{f^{∗}}↑
CH(X) ⟶^{S_{X}} CH(X)

and

CH(X_{E}) ⟶^{S_{X_{E}}} CH(X_{E})
↑^{res_{E/F}}  ^{res_{E/F}} ↑
CH(X) ⟶^{S_{X}} CH(X)

are commutative. Moreover, the restriction S^{i}|_{CH^{n}(X)} is 0 for n < i and the squaring for n = i; finally S^{0} is the identity.

Since S is a ring homomorphism commuting with the pull-backs, one has the Cartan formula: S_{X_{1}×X_{2}}(α_{1} × α_{2}) = S_{X_{1}}(α_{1}) × S_{X_{2}}(α_{2}) for any α_{i} ∈ CH(X_{i}) with X_{i} smooth quasi-projective.

Also, the total Steenrod operation satisfies the following Riemann-Roch type formula:

f_{∗}(S_{Y} (α) · c(−T_{Y})) = S_{X}(f_{∗}(α)) · c(−T_{X})

(in other words, S corrected by c(−T) this way, commutes with the pushforwards) for any proper f: Y → X and any α ∈ CH(Y), where f_{∗}: CH(Y) → CH(X) is the push-forward, c is the total Chern class, T_{X} is the tangent bundle of X, and c(−T_{X}) = c^{−1}(T_{X}) (the expression −T_{X} has a sense if one considers T_{X} as the element of K_{0}(X)). This formula is proved in [1]. Also it follows from the previous formulated properties of S by the general Riemann-Roch theorem of Panin-Smirnov.

A particular case of the Riemann-Roch formula (f is a closed imbedding and α is the class of Y itself) is the Wu formula: for every closed imbedding i: Y ↪ X with smooth quasi-projective X and Y, one has S([Y]) = i_{∗}(c(N)), where N is the normal bundle of the imbedding i.

3. Hoffmann’s conjecture

Conjecture 0.1 is proved after some preliminary computations.

Lemma 3.1. Let X be a smooth projective quadric of dimension d and let i: X ↪ P its imbedding in the (d + 1)-dimensional projective space P. We write e ∈ CH^{1}(X) for the class of a hyperplane section of X, that is, e = i^{∗}(h) with h ∈ CH^{1}(P) being the class of a hyperplane in P. Finally, we write T_{X} for the tangent bundle of X. Then the total Chern class c(T_{X}) in the modulo 2 Chow group CH(X) is equal to (1 + e)^{d+2}.

Proof. The exact sequence of vector X-bundles

0 → T_{X} → i^{∗}(T_{P}) → i^{∗}(O_{P} (2)) → 0

gives the equality c(T_{X})·i^{∗}(c(O_{P} (2))) = i^{∗}(c(T_{P})). Since c(O_{P} (2)) = 1+2h = 1 (we are working with the modulo 2 Chow groups) and c(T_{P}) = (1 + h)^{d+2}, we get c(T_{X}) = (1 + e)^{d+2}. □

Corollary 3.2. Let X be a smooth isotropic projective quadric of dimension d with the Witt index i_{W} (X) > n for some n ≥ 0. Let i: P ↪ X be the class of an n-dimensional projective space lying on X (P is given by an (n + 1)- dimensional totally isotropic subspace of the quadratic space defining X). Then the total Chern class c(N) of the normal bundle N of the imbedding P ⊂ X is equal to (1 + h)^{d+1−n}, where h ∈ CH^{1}(P) is the class of a hyperplane.

Proof. The exact sequence of vector P -bundles ([3, B.7.2])

0 → T_{P} → i^{∗}(T_{X}) → N → 0

gives the equality c(N) = i^{∗}(c(T_{X})) · c(T_{P})^{−1}. Since i^{∗}(e) = h, we get by Lemma 3.1 that i^{∗}(c(T_{X})) is equal to (1 + h)^{d+2} . Since c(T_{P}) = (1 + h)^{n+1}, we get the equality required. □

Corollary 3.3. In the situation of Corollary 3.2, one has

S([P]) = [P] · (1 + e)^{d+1−n} ,

where [P] ∈ CH_{n}(X) is the class of P, while S: CH(X) → CH(X) is the total Steenrod operation.

Proof. By the Wu formula (see §2), we have S([P]) = i_{∗}(c(N)). Using Corollary 3.2 we get

i_{∗}(c(N)) = i_{∗}((1 + h)^{d+1−n}) = (i_{∗} ∘ i^{∗})(1 + e)^{d+1−n} .

Since the composition i_{∗} ∘ i^{∗} coincides with the multiplication by [P], the desired formula follows. □

Proof of Conjecture 0.1. Let ϕ be an anisotropic quadratic form giving a counterexample to Conjecture 0.1. Then for the integer r such that the difference dim(ϕ)−i_{1}(ϕ) is divisible by 2^{r} and not divisible by 2^{r+1}, one has i_{1}(ϕ) ≥ 2^{r}+1. Extending the scalars to certain purely transcendental extension and replacing ϕ by its certain subform of codimension i_{1}(ϕ) − (2^{r} + 1), we can come to the situation where i_{1}(ϕ) is precisely 2^{r} + 1 ([4, lemma 7.3]). Note that the difference dim(ϕ) − i_{1}(ϕ) is still the same.

Let X be the projective quadric given by ϕ. We set d = dim(X) (that is, d = dim(ϕ) − 2) and notice that this integer is odd. Therefore, we are in the situation of §1 and for every homogeneous α ∈ CH^{∗}(X × X) the coefficients α_{i} ∈ ℤ/2ℤ are defined.

Since the Witt index of the quadratic form ϕ_{F (X)} is 2^{r} +1, the quadric X_{F (X)} contains a linear subspace of dimension 2^{r}. Its closure (or the transposition of the closure) in X × X gives an element α ∈ CH^{d−2^{r}}(X × X) with α_{d−2^{r}} = 1 (using Proposition 1.1, it can be easily seen that α_{0} = 1 as well; we do not need this statement however). Let us check that α_{1} = ⋯ = α_{2^{r}} = 0 for this α (in fact, the “symmetric” coefficients, namely, α_{d−2^{r}−1}, …, α_{d−2^{r+1}} are 0 as well, but we do not need this statement).

The product β = α · (e^{0} × e^{2^{r}}) (where we consider e^{0} and e^{2^{r}} as elements of CH^{2^{r}}(X) what is possible because 2^{r} < i_{1}(ϕ) ≤ dim(ϕ)/2 whereby 2^{r} < d/2) is an element of CH^{d}(X × X) with β_{i} = α_{i} for i = 0, 1, …, d − 2^{r} and β_{i} = 0 for i = d − 2^{r} + 1, …, d. Since β_{i} = 0 for i = d − 2^{r} + 1, …, d and i_{1}(ϕ) = 2^{r} + 1, Proposition 1.1 says that β_{1} = ⋯ = β_{2^{r}} = 0. Since α_{i} = β_{i} for i = 0, 1, …, d − 2^{r}, we get that α_{1} = ⋯ = α_{2^{r}} = 0.

Let us keep in mind the last relation and consider the element

γ = S^{2^{r}}(α) ∈ CH^{d}(X × X) ,

where S^{2^{r}} is the Steenrod operation. We are going to show that γ_{2^{r}} = 0 (in fact we will even see that γ_{1} = ⋯ = γ_{2^{r}} = 0). By the Cartan formula (see §2), one has S(e^{a} × e^{b}) = S(e^{a}) × S(e^{b}) for any a, b. Since the non-zero graded components of the elements S(e^{a}), S(e^{b}) ∈ CH^{∗}(X̄) are in codimensions ≥ a and ≥ b respectively, the element S(e^{a} × e^{b}) is a linear combination of the elements e^{i} × e^{j} only with i ≥ a and j ≥ b. In particular, for every i = 2^{r} + 1, …, d − 2^{r}, the element S^{2^{r}}(e^{i} × e^{d−2^{r}−i}) ∈ CH^{d}(X̄ × X̄) is a linear combination of the elements e^{j} × e^{d−j} with j > 2^{r}. Besides, since S is the identity on CH^{0} and in particular on e^{0} ∈ CH^{0}(X̄), the element S^{2^{r}}(e^{0} × e^{d−2^{r}}) is a multiple of e^{0} × e^{d}. Now, the relation α_{1} = ⋯ = α_{2^{r}} = 0 we dispose means that ᾱ is a linear combination of e^{0} × e^{d−2^{r}} and e^{i} × e^{d−2^{r}−i} with i = 2^{r} + 1, …, d − 2^{r}. It follows that the element γ = S^{2^{r}}(α) is a linear combination of e^{0} × e^{d} and e^{j} × e^{d−j} with j > 2^{r}. In particular, γ_{2^{r}} = 0.

Applying Proposition 1.1, we get that γ_{d} = 0. We are going to get a contradiction showing by a direct computation that γ_{d} = 1. Note that γ_{d} satisfies the relation S^{2^{r}}(e^{d−2^{r}} × e^{0}) = γ_{d} · (e^{d} × e^{0}) (here we recall that α_{d−2^{r}} = 1) and, therefore, the simplified relation S^{2^{r}}(e_{2^{r}}) = γ_{d}e_{0}. We recall that the element e_{2^{r}} ∈ CH_{2^{r}}(X̄) is the class of a 2^{r}-dimensional projective space P ⊂ X̄.

Therefore, by Corollary 3.3, we have:

S(e_{2^{r}}) = e_{2^{r}} · (1 + e^{1})^{d−2^{r}+1} .

In particular, S^{2^{r}}(e_{2^{r}}) = \binom{d−2^{r}+1}{2^{r}} · e_{0} and we get that γ_{d} = \binom{d−2^{r}+1}{2^{r}} mod 2. Since the integer d − 2^{r} + 1 = dim(ϕ) − i_{1}(ϕ) is divisible by 2^{r} and is not divisible by 2^{r+1}, the binomial coefficient \binom{d−2^{r}+1}{2^{r}} is odd. □

References

[1] P. Brosnan. Steenrod Operations in Chow Theory. K-theory Preprint Archives 0370 (1999), 1–19.

[2] D. Eddidin and W. Graham. Equivariant intersection theory. Inventiones Math. 131 (1998), no. 3, 595–634.

[3] W. Fulton. Intersection theory. Springer-Verlag, Berlin, 1984.

[4] N. A. Karpenko. On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15 (2000), no. 1, 1–22.

Laboratoire des Mathématiques, Faculté des Sciences Jean Perrin, Université d’Artois, Rue Jean Souvraz SP 18, 62307 Lens Cedex, France

  E-mail address: karpenko@euler.univ-artois.fr