THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS

DANIEL DUGGER AND DANIEL C. ISAKSEN

Abstract. We settle an old question about the existence of certain ‘sumsof-squares’ formulas over a field F, related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of characteristic 0, then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic p > 2.

1. Introduction

Fix a field F. A classical problem asks for what values of r, s, and n do there exist identities of the form

( ∑_{i=1}^{r} x_{i}^{2})·( ∑_{i=1}^{s} y_{i}^{2})= ∑_{i=1}^{n} z_{i}^{2} (1.1)

where the z_{i}’s are bilinear expressions in the x’s and y’s. Equation (1.1) is to be interpreted as a formula in the polynomial ring F [x_{1}, …, x_{r}, y_{1}, …, y_{s}]; we call it a sum-of-squares formula of type [r, s, n].

The question of when such formulas exist has been extensively studied: [La] and [S1] are excellent survey articles, and [S2] is a detailed sourcebook. In this paper we prove the following result, solving Problem C of [La]:

Theorem 1.2. If F is a field of characteristic not equal to 2, and a sum-of-squares formula of type [r, s, n] exists over F, then \binom{n}{i} must be even for n − r < i < s.

We now give a little history. It is common to let r ∗_{F} s denote the smallest n for which a sum-of-squares formula of type [r, s, n] exists. Many papers have studied lower bounds on r∗_{F} s, but for a long time such results were known only for fields of characteristic 0: one reduces to a geometric problem over ℝ, and then topological methods are used to obtain the bounds (see [La] for a summary). In this paper we begin the process of extending such results to characteristic p, replacing the topological methods by those of motivic homotopy theory.

The most classical result along these lines is Theorem 1.2 for the particular case F = ℝ, which leads to lower bounds for r ∗_{ℝ} s. It seems to have been proven in three places, namely [B], [Ho], and [St]; but in modern times the given condition on binomial coefficients is usually called the ‘Hopf condition’. The paper [S1] gives some history, and explains how K. Y. Lam and T. Y. Lam deduced the condition for arbitrary fields of characteristic 0. Problem C of [La, p. 188] explicitly asked whether the same condition holds over fields of characteristic p > 2. Work on this question had previously been done by Adem [A1, A2] and Yuzvinsky [Y] for special values of r, s, and n. In [SS] a weaker version of the condition was proved for arbitrary fields and arbitrary values of r, s, and n.

Stiefel’s proof of the condition for F = ℝ used Stiefel-Whitney classes; Behrend’s (which worked over any formally real field) used some basic intersection theory; and Hopf deduced it using singular cohomology. Our proof of the general theorem uses a variation of Hopf’s method and motivic cohomology. It can be regarded as purely algebraic—at least, as ‘algebraic’ as things like group cohomology and algebraic K-theory. These days it is perhaps not so clear that there exists a point where topology ends and algebra begins.

We now explain Hopf’s proof, and our generalization, in more detail. Given a sum-of-squares formula of type [r, s, n], one has in particular a bilinear map ϕ: F^{r} × F^{s} → F^{n} given by (z_{1}, …, z_{n}). If we let q be the quadratic form on F^{k} given by q(w_{1}, …, w_{k}) = w_{1}^{2} + ⋯ + w_{k}^{2}, then we have q(ϕ(x, y)) = q(x)q(y). When F = ℝ one has that q(w) = 0 only when w = 0, and so ϕ restricts to a map (ℝ^{r} − 0) × (ℝ^{s} − 0) → (ℝ^{n} − 0). The bilinearity of ϕ tells us in particular that we can quotient by scalar-multiplication to get ℝP^{r−1} × ℝP^{s−1} → ℝP^{n−1}.

On mod 2 cohomology this gives ℤ/2[x]/x^{n} → ℤ/2[a]/a^{r} ⊗ ℤ/2[b]/b^{s}, and the bilinearity of ϕ shows that x ↦ a+b. Since x^{n} = 0 and we have a ring map, it follows that (a + b)^{n} = 0 in the target ring. The Hopf condition falls out immediately.

This proof used, in a seemingly crucial way, the fact that over ℝ a sum of squares is 0 only when all the numbers were zero to begin with. This of course does not work over fields of characteristic p (or over ℂ, for that matter). Our bilinear form gives us a map of schemes ϕ: 𝔸^{r} × 𝔸^{s} → 𝔸^{n}, but we cannot say that it restricts to (𝔸^{r} − 0) × (𝔸^{s} − 0) → (𝔸^{n} − 0) as we did above.

To remedy the situation, let Q_{k} denote the projective quadric in ℙ^{k+1} defined by the equation w_{1}^{2} + ⋯ + w_{k+2}^{2} = 0. The bilinear map ϕ induces

(ℙ^{r−1} − Q_{r−2}) × (ℙ^{s−1} − Q_{s−2}) → (ℙ^{n−1} − Q_{n−2}).

In effect, we have removed all possible numbers whose sum-of-squares would give us zero. Let DQ_{k} denote the deleted quadric ℙ^{k} − Q_{k−1} (our convention is that the subscript on a scheme always denotes its dimension). We will compute the mod 2 motivic cohomology of DQ_{k} (Theorem 2.3), find that it is close to being a truncated polynomial algebra, and repeat Hopf’s argument in this new context. As an amusing exercise (cf. [Ln, 6.3]) one can show that over the field ℂ the space DQ_{k}—with the complex topology—has the same homotopy type as ℝP^{k}; so our argument is in some sense ‘the same’ as Hopf’s in this case.

The idea of using deleted quadrics to deduce the Hopf condition first appeared in [SS]. In that paper the Chow groups of the deleted quadrics were computed, but these are only enough to deduce a weaker version of the Hopf condition (one that is approximately half as powerful). This is explained further in Remark 2.7. On the other hand, we should point out that the full power of motivic cohomology is not completely necessary in this paper: one can also derive the Hopf condition using étale cohomology, by the same arguments (see Remark 2.8). Since in this case computing étale cohomology involves exactly the same steps as computing motivic cohomology, we have gone ahead and computed the stronger invariant (to be used in future work).

1.3. Organization. Section 2 shows how to deduce the Hopf condition from a few easily stated facts about motivic cohomology. Section 3 outlines in more detail the basic properties of motivic cohomology needed in the rest of the paper. This list is somewhat extensive, but our hope is that it will be accessible to readers not yet acquainted with the motivic theory—most of the properties are analogs of familiar things about singular cohomology. Finally, Section 4 carries out the necessary calculations. We also include an appendix on the Chow groups of quadrics, as several facts about these play a large role in the paper.

2. The basic argument

Because of the nature of the computations that we will make, we use slightly different definitions for the varieties Q_{n} and DQ_{n} than those in Section 1. These definitions will remain in effect for the entire paper. Unfortunately, the usefulness of these choices will not become clear until Section 4.

From now on the field F is always assumed not to have characteristic 2.

Definition 2.1. When n = 2k, let Q_{n} be the projective quadric in ℙ^{n+1} defined by the equation a_{1}b_{1} + a_{2}b_{2} + ⋯ + a_{k+1}b_{k+1} = 0. When n = 2k + 1, let Q_{n} be the projective quadric in ℙ^{n+1} defined by the equation a_{1}b_{1}+a_{2}b_{2}+⋯ a_{k+1}b_{k+1}+c^{2} = 0. In either case, let DQ_{n+1} be ℙ^{n+1} − Q_{n}.

Note that Q_{0} is isomorphic to Spec F ⨿ Spec F, and Q_{1} ≅ ℙ^{1}. One possible isomorphism ℙ^{1} → Q_{1} sends [x, y] to [−x^{2}, y^{2}, xy].

Occasionally we will need to equip DQ_{n+1} with a basepoint, in which case we’ll always choose [1, 1, 0, 0, …, 0] (although the choice turns out not to matter).

Lemma 2.2. Suppose that the ground field F has a square root of −1 (call it i). Then Q_{n} is isomorphic to the projective quadric in ℙ^{n+1} defined by the equation w_{1}^{2} + ⋯ + w_{n+2}^{2} = 0.

Proof. When n = 2k, use the change of coordinates a_{j} = w_{2j−1} + iw_{2j}, b_{j} = w_{2j−1} − iw_{2j}. When n = 2k + 1, use the same formulas as above for 1 ≤ j ≤ k + 1 and also let c = w_{n+2}. □

We regard ℙ^{2k} ↪ ℙ^{2k+1} as the subscheme defined by a_{k+1} = b_{k+1}, and we regard ℙ^{2k−1} ↪ ℙ^{2k} as the subscheme defined by c = 0. These choices have the advantage that they give us inclusions Q_{n−2} ↪ Q_{n−1} and DQ_{n−1} ↪ DQ_{n}.

The following theorem states the computation of the motivic cohomology ring H^{∗,∗}(DQ_{n}; ℤ/2). In order to understand the statement, the reader needs to know just a few basic facts about motivic cohomology; a more complete account of these facts appears in Section 3. First, H^{∗,∗}(−; ℤ/2) is a contravariant functor defined on smooth F -schemes, taking its values in bi-graded commutative rings of characteristic 2. If we set 𝕄_{2} = H^{∗,∗}(Spec F; ℤ/2), the map induced by X → Spec F makes H^{∗,∗}(X; ℤ/2) into an 𝕄_{2}-algebra. It is known that 𝕄_{2}^{0,0} ≅ ℤ/2, 𝕄_{2}^{0,1} ≅ ℤ/2, and the generator τ ∈ 𝕄_{2}^{0,1} is not nilpotent.

Theorem 2.3. Assume that every element of F is a square and that char(F) ≠ 2.

(a) If n = 2k + 1 then H^{∗,∗}(DQ_{n}; ℤ/2) ≅ 𝕄_{2}[a, b]/(a^{2} = τb, b^{k+1}), where a has degree (1, 1) and b has degree (2, 1).

(b) If n = 2k then H^{∗,∗}(DQ_{n}; ℤ/2) ≅ 𝕄_{2}[a, b]/(a^{2} = τb, b^{k+1}, ab^{k}) where a and b are as in part (a).

(c) The map H^{∗,∗}(DQ_{n+1}; ℤ/2) → H^{∗,∗}(DQ_{n}; ℤ/2) sends a to a, and b to b.

In fact, b is the unique nonzero class in H^{2,1}, and a is the unique nonzero class in H^{1,1} which becomes zero when restricted to the basepoint Spec F → DQ_{n}. These facts are needed below in the proof of Proposition 2.5. See the comments before Proposition 4.6 for more details.

Note that if τ were equal to 1 then the above rings would be truncated polynomial algebras (in analogy with the singular cohomology of ℝP^{n}).

A more general version of this theorem, without any assumptions on F, appears as Theorem 4.9. The proof is slightly involved, and so deferred until Section 4. However, let us at least record how the above statements follow from the more general version:

Proof. If every element of F is a square, then 𝕄_{2}^{1,1} = 0 (see Section 3.2). Therefore, in Theorem 4.9 both ρ and ϵ are zero. This gives us the formulas in part (a) and (b). Part (c) is Proposition 4.6. □

For us, the most important consequence of the theorem is the following:

Corollary 2.4. In H^{∗,∗}(DQ_{n}; ℤ/2) we have a^{n+1} = 0 and a^{i} ≠ 0 for i ≤ n.

Proof. The claims are immediate from the calculation since all the powers of τ are nonzero. □

Proof of Theorem 1.2. Suppose we have a sum-of-squares formula of type [r, s, n] over F. This remains true if we extend F, so we may as well assume that every element of F is a square. Therefore, Theorem 2.3 applies.

As explained in Section 1, the sum-of-squares formula gives a map p: DQ^{r−1} × DQ^{s−1} → DQ^{n−1} (this uses Lemma 2.2) and we will consider the induced map on motivic cohomology. There is a Künneth formula for computing motivic cohomology of products of certain ‘cellular’ varieties (see Proposition 3.9), and the deleted quadrics belong to this class by Proposition 4.2. In order to apply Proposition 3.9, we also have to observe that H^{∗,∗}(DQ_{r−1}; ℤ/2) is free over 𝕄_{2}, which is apparent from Theorem 2.3.

Therefore p^{∗} is a map

H^{∗,∗}(DQ_{n−1}; ℤ/2) → H^{∗,∗}(DQ_{r−1}; ℤ/2) ⊗_{𝕄_{2}} H^{∗,∗}(DQ_{s−1}; ℤ/2).

We will use the letters a and b to denote the generators of H^{∗,∗}(DQ_{n−1}; ℤ/2), a_{1} and b_{1} for the generators of H^{∗,∗}(DQ_{r−1}; ℤ/2), and a_{2} and b_{2} for the generators of H^{∗,∗}(DQ_{s−1}; ℤ/2).

We show in the following proposition that p^{∗}(a) = a_{1} + a_{2}. Since the above corollary says that a^{n} = 0, it will follow that (a_{1} + a_{2})^{n} = 0. Using the corollary again, this can only happen if \binom{n}{i} is even for n − r < i < s. □

Proposition 2.5. Suppose that F is a field of characteristic not 2 in which every element is a square. If p^{∗}, a, a_{1}, and a_{2} are as in the above proof, then p^{∗}(a) = a_{1} + a_{2}.

Before we can give the proof, we need to state a few more properties of motivic cohomology. Once again, more details are given in Section 3. First, 𝕄_{2}^{p,q} is nonzero only in the range q ≥ 0. Second, when every element of F is a square one has 𝕄_{2}^{1,1} = 0. Finally, motivic cohomology is 𝔸^{1}-homotopy invariant in the following sense. Let i_{0} and i_{1} denote the inclusions {0} ↪ 𝔸^{1} and {1} ↪ 𝔸^{1}, respectively. If H: X × 𝔸^{1} → Y is a map of smooth schemes, then the composites H(Id × i_{0}) and H(Id × i_{1}) induce the same map H^{∗,∗}(Y; ℤ/2) → H^{∗,∗}(X; ℤ/2). Such a map H is called an ‘𝔸^{1}-homotopy’ from H(Id × i_{0}) to H(Id × i_{1}).

Proof. Because p^{∗}(a) has degree (1, 1), it must be of the form ϵ_{1}a_{1} + ϵ_{2}a_{2} + m · 1, where m belongs to 𝕄_{2}^{1,1} and ϵ_{1} and ϵ_{2} belong to 𝕄_{2}^{0,0} ≅ ℤ/2. Since 𝕄_{2}^{1,1} = 0 under our assumptions on F, we can ignore m. To show that ϵ_{1} = 1, in light of Theorem 2.3(c) it would suffice to verify that the map

DQ_{1} × {∗} → DQ_{r−1} × DQ_{s−1} → DQ_{n−1}

is 𝔸^{1}-homotopic to the standard inclusion DQ_{1} ↪ DQ_{n−1}. (A similar argument will show that ϵ_{2} = 1.) Actually we will not quite do this, but instead verify that the composition

j: DQ_{1} × {∗} → DQ_{r−1} × DQ_{s−1} → DQ_{n−1} ↪ DQ_{n+1}

is 𝔸^{1}-homotopic to the standard inclusion. By Theorem 2.3(c) again, this is enough.

For the rest of this section we will use the coordinates w_{1}, …, w_{n+2} on ℙ^{n+1} given in Lemma 2.2. Recall that ϕ is our bilinear map F^{r}×F^{s} → F^{n}. Let e_{1}, …, e_{k} be the standard basis for F^{k}, and let ϕ(e_{1}, e_{1}) = (u_{1}, …, u_{n}) and ϕ(e_{2}, e_{1}) = (v_{1}, …, v_{n}). Then the map j: DQ_{1} → DQ_{n+1} has the form

[a, b] ↦ [u_{1}a + v_{1}b, u_{2}a + v_{2}b, …, u_{n}a + v_{n}b, 0, 0],

and the sum-of-squares formula satisfied by ϕ tells us that

u_{1}^{2} + ⋯ + u_{n}^{2} = 1, v_{1}^{2} + ⋯ + v_{n}^{2} = 1, and u_{1}v_{1} + ⋯ + u_{n}v_{n} = 0.

Note that the standard inclusion DQ_{1} ↪ DQ_{n+1} has the same description but where (u_{1}, …, u_{n}) = (1, 0, …, 0) and (v_{1}, …, v_{n}) = (0, 1, 0, …, 0). The following lemma gives the desired 𝔸^{1}-homotopy, since both the map j and the standard inclusion are homotopic to the map [a, b] ↦ [0, 0, …, 0, a, b]. □

For the following statement, recall that we are still using the coordinates on ℙ^{n+1} given by Lemma 2.2.

Lemma 2.6. Suppose that F contains a square root of −1. Let u and v be vectors in F^{n} such that Σ_{j}u_{j}^{2} = 1 = Σ_{j}v_{j}^{2} and Σ_{j}u_{j}v_{j} = 0. Then the map f : DQ_{1} → DQ_{n+1} given by

[a, b] ↦ [u_{1}a + v_{1}b, u_{2}a + v_{2}b, …, u_{n}a + v_{n}b, 0, 0]

is 𝔸^{1}-homotopic to the map [a, b] ↦ [0, 0, …, 0, a, b].

Proof. Let i be a square root of −1. First define a homotopy DQ_{1} × 𝔸^{1} → DQ_{n+1} by the formula

([a, b], t) ↦ [u_{1}a + v_{1}b, u_{2}a + v_{2}b, …, u_{n}a + v_{n}b, ta − tib, tia + tb].

This shows that f is homotopic to g, where g is the map

[a, b] ↦ [u_{1}a + v_{1}b, u_{2}a + v_{2}b, …, u_{n}a + v_{n}b, a − ib, ia + b].

Now define another homotopy DQ_{1} × 𝔸^{1} → ℙ^{n+1} by the formula

([a, b], t) ↦ [tu_{1}a + tv_{1}b, tu_{2}a + tv_{2}b, …, tu_{n}a + tv_{n}b, a − tib, tia + b].

The assumptions on the u’s and v’s imply that the sum of the squares in the image is exactly equal to a^{2} + b^{2}, which is nonzero because [a, b] lies in DQ_{1}. So this is actually a homotopy DQ_{1} × 𝔸^{1} → DQ_{n+1}, showing that g is homotopic to the desired map. □

Remark 2.7. In [SS] a weaker version of the Hopf condition was obtained by computing the Chow ring CH^{∗}(DQ_{n}), which essentially corresponds to the subring of H^{∗,∗}(DQ_{n}; ℤ/2) generated by b (see Property (A) in Section 3). This amounts to seeing about half of what motivic cohomology sees.

Remark 2.8. When F has a square root of −1, a theorem of [Lv] says that the étale cohomology ring H_{et}^{∗}(DQ_{n}; μ_{2}^{⊗∗}) is isomorphic to H^{∗,∗}(DQ_{n}; ℤ/2)[τ^{−1}] ≅ H^{∗,∗}(DQ_{n}; ℤ/2) ⊗_{𝕄_{2}} 𝕄_{2}[τ^{−1}] (see Property (I) below). Since H^{∗,∗}(DQ_{n}; ℤ/2) is free over 𝕄_{2}, this localization is particularly simple: it is precisely a truncated polynomial algebra 𝕄_{2}[τ^{−1}][a]/a^{n+1}. So the Hopf condition could have been proven using étale cohomology.

Remark 2.9. When every element of F is a square, it follows from the proof of the Milnor conjecture [V3] that 𝕄_{2} ≅ ℤ/2[τ]. We never needed this, but it’s useful to keep in mind.

3. Review of motivic cohomology

The theory now called motivic cohomology was first developed in two main places, namely [Bl1] and [VSF] (together with many associated papers). The paper [V4] proved that the two approaches give isomorphic theories. Below we recall the basic properties of motivic cohomology needed in the paper. For various reasons it is difficult to give simple references to [VSF] so most of our citations will be to [SV, Sec. 3] and the lecture notes [MVW].

3.1. Basic properties. For every field F, motivic cohomology is a contravariant functor H^{∗,∗}(−) from the category of smooth schemes of finite type over F to the category of bi-graded commutative rings. Commutativity means that if a ∈ H^{p,q}(X) and b ∈ H^{s,t}(X) then ab = (−1)^{ps}ba. For the basic construction we refer the reader to [SV, Sec. 3] or [MVW, Sec. 3]. The list of properties below is far from complete, and in some cases we only give crude versions of more interesting properties—but this is all we will need in the present paper.

The scheme Spec F will often be denoted by “pt”, and we denote H^{∗,∗}(pt) by 𝕄. The ring 𝕄 can be very complicated (and is, in general, unknown). The motivic cohomology of a scheme is naturally a graded-commutative algebra over 𝕄.

Property A. The graded subring ⊕_{n}H^{2n,n}(X) is naturally isomorphic to the Chow ring CH^{∗}(X). [Bl1, p. 268], [MVW, p. 4; Lect. 17].

In particular, 𝕄^{0,0} = ℤ. In general, H^{∗,∗}(X) is isomorphic to the higher Chow groups of X [V4, Cor. 1.2].

Property B. For a closed inclusion j: Z ↪ X of smooth schemes of codimension c, there is a long exact sequence of the form

⋯ → H^{∗−2c,∗−c}(Z) ⟶^{j_{!}} H^{∗,∗}(X) → H^{∗,∗}(X − Z) → H^{∗−2c+1,∗−c}(Z) → ⋯

The map j_{!} is called the ‘Gysin map’ or the ‘pushforward’, and it is a map of 𝕄-modules. The long exact sequence is called the Gysin, localization, or purity sequence. [Bl1, Sec. 3], [Bl2].

Property C. Let i_{0} and i_{1} denote the inclusions {0} ↪ 𝔸^{1} and {1} ↪ 𝔸^{1}, respectively. If H: X × 𝔸^{1} → Y is a map of smooth schemes, then the composites H(Id × i_{0}) and H(Id × i_{1}) induce the same map H^{∗,∗}(Y) → H^{∗,∗}(X). Such a map H is called an 𝔸^{1}-homotopy from H(Id × i_{0}) to H(Id × i_{1}). [Bl1, Sec. 2], [SV, Prop. 4.2].

Property D. H^{∗,∗}(ℙ^{n}) = 𝕄[t]/(t^{n+1}), where t has degree (2, 1). [SV, Prop. 4.4].

Property E. If E → B is an algebraic fiber bundle (i.e., a map which is locally a product in the Zariksi topology) whose fiber is an affine space 𝔸^{n}, then H^{∗,∗}(B) → H^{∗,∗}(E) is an isomorphism.

Property (E) is easy to prove by inducting on the size of a trivializing cover, and using the Mayer-Vietoris sequence [SV, Prop. 4.1] together with Property (C).

Property F. 𝕄^{p,q} = 0 if q < 0, if p > q ≥ 0, or if q = 0 and p < 0. [MVW, p. 4; Th. 3.5]

Property G. 𝕄^{1,1} = F^{∗} and 𝕄^{0,1} = 0. [Bl1, Th. 6.1], [MVW, p. 4,(2)].

3.2. Finite coefficients. For every n ∈ ℤ there is also a theory H^{∗,∗}(−; ℤ/n) which is related to H^{∗,∗}(−) by a natural long exact sequence of the form

⋯ → H^{∗,∗}(X) ⟶^{×n} H^{∗,∗}(X) → H^{∗,∗}(X; ℤ/n) → H^{∗+1,∗}(X) ⟶^{×n} ⋯ (3.3)

For the definition see [MVW, Def. 3.4]. The theory satisfies the analogs of Properties (B) through (F) above.

Let 𝕄_{2} denote H^{∗,∗}(pt; ℤ/2). Since 𝕄 may contain 2-torsion, 𝕄_{2} is not necessarily the same as 𝕄/(2)—rather, there is a long exact sequence of the form

⋯ → 𝕄^{p,q} ⟶^{×2} 𝕄^{p,q} → 𝕄_{2}^{p,q} → 𝕄^{p+1,q} → ⋯

This sequence, together with Property (F) and the fact that 𝕄^{0,0} = ℤ, tells us that 𝕄_{2}^{0,0} = ℤ/2. Note that H^{∗,∗}(X; ℤ/2) is naturally a commutative algebra over 𝕄_{2}.

Since 𝕄^{1,1} = F^{∗} and 𝕄^{0,1} = 𝕄^{2,1} = 0, we get the exact sequence

0 → 𝕄_{2}^{0,1} → F^{∗} ⟶^{×2} F^{∗} → 𝕄_{2}^{1,1} → 0 (3.4)

where the map F^{∗} → F^{∗} sends x to x^{2}. The usual notation is to let τ ∈ 𝕄_{2}^{0,1} denote the class which maps to −1, and to let ρ ∈ 𝕄_{2}^{1,1} denote the image of −1. If F has a square root of −1 then ρ = 0. Moreover, if every element of F is a square then 𝕄_{2}^{1,1} = 0.

3.5. The Bockstein. The Bockstein map β: H^{∗,∗}(−; ℤ/2) → H^{∗+1,∗}(−; ℤ/2) is defined in the usual manner from the maps in the sequence (3.3). A direct consequence of the definition (as in topology) is that β^{2} = 0. Note that β(τ) = ρ.

Property H. For all a, b ∈ H^{∗,∗}(X; ℤ/2), β(ab) = β(a)b + aβ(b). [Lv, Lem. 6.1].

3.6. Relation with étale cohomology. There is a natural map of bi-graded rings η: H^{∗,∗}(X; ℤ/n) → H_{et}^{∗}(X; μ_{n}^{⊗∗}) (cf. [MVW, Th. 10.2], for example). In the case n = 2, the element τ maps to the class of −1 in H_{et}^{0}(pt; μ_{2}) ≅ {1, −1}, and multiplication by this class is an isomorphism on étale cohomology. Note in particular that this implies that the powers of τ are all nonzero in H^{0,∗}(pt; ℤ/2).

Property I. The induced map H^{∗,∗}(X; ℤ/2)[τ^{−1}] → H_{et}^{∗}(X; μ_{2}^{⊗∗}) is an isomorphism for any smooth scheme X, provided that F has a square root of −1. [Lv].

The construction of the map η from [MVW] makes it clear that the Bockstein on H^{∗,∗}(−; ℤ/2) (which can be regarded as induced by the extension 0 → ℤ/2 → ℤ/4 → ℤ/2 → 0) is compatible with the Bockstein on étale cohomology induced by 0 → μ_{2} → μ_{4} → μ_{2} → 0. If the field contains a square root of −1 then we can identify μ_{4} with ℤ/4, and of course μ_{2} with ℤ/2. These observations will be used in the proof of Theorem 4.9.

3.7. Reduced cohomology. Given any basepoint of a scheme X (i.e., a map pt → X), the kernel of the induced map H^{∗,∗}(X) → H^{∗,∗}(pt) is the reduced cohomology of X and is denoted by H̃^{∗,∗}(X). A similar definition applies with ℤ/n-coefficients. The above map has a splitting (induced by X → pt), and thus H^{∗,∗}(X) ≅ 𝕄 ⊕ H̃^{∗,∗}(X). Similarly, H^{∗,∗}(X; ℤ/2) ≅ 𝕄_{2} ⊕ H̃^{∗,∗}(X; ℤ/2).

3.8. A Künneth theorem. Let 𝒞 denote the smallest class of smooth schemes satisfying the following properties:

(1) 𝒞 contains the affine spaces 𝔸^{k}.

(2) If Z ↪ X is a closed inclusion of smooth schemes and 𝒞 contains two of X, Z, and X − Z, then it also contains the third.

(3) If E → B is an algebraic fiber bundle whose fiber is an affine space, then E ∈ 𝒞 if and only if B ∈ 𝒞. The following result is a modest generalization of [J, Th. 4.5], and can be proven

using the same techniques. A complete proof, for a more general class of schemes than 𝒞, is given in [DI2].

Proposition 3.9. Suppose X and Y are smooth schemes, with at least one of them belonging to 𝒞. If either H^{∗,∗}(X) or H^{∗,∗}(Y) is free as an 𝕄-module, then one has a Künneth isomorphism of bi-graded rings

H^{∗,∗}(X) ⊗_{𝕄} H^{∗,∗}(Y) ≅ H^{∗,∗}(X × Y).

Similarly, if either H^{∗,∗}(X; ℤ/n) or H^{∗,∗}(Y; ℤ/n) is free as an H^{∗,∗}(pt; ℤ/n)- module, then one has a Künneth isomorphism of bi-graded rings

H^{∗,∗}(X; ℤ/n) ⊗_{H^{∗,∗}(pt;ℤ/n)} H^{∗,∗}(Y; ℤ/n) ≅ H^{∗,∗}(X × Y; ℤ/n).

4. Computations

In this section F is an arbitrary ground field not of characteristic 2. We will study the quadrics Q_{n} and DQ_{n}. Note that in this generality Lemma 2.2 does not apply; therefore, Q_{n} and DQ_{n} cannot necessarily be redefined in terms of sums of squares. We assume char(F) ≠ 2 so that Q_{n} is smooth for all n, not just even n.

Proposition 4.1. If n is odd, H^{∗,∗}(Q_{n}) is a free module over 𝕄 with generators in degrees (0, 0), (2, 1), (4, 2), …, (2n, n). If n is even, H^{∗,∗}(Q_{n}) is a free 𝕄-module with generators in degrees (0, 0), (2, 1), …, (2n, n) plus an extra generator in degree (n, \frac{n}{2}).

Proof. The proof is by induction. The result for Q_{0} is obvious, and the result for Q_{1} ≅ ℙ^{1} is Property (D).

Except for the base cases in the previous paragraph, the argument for the odd and even cases is identical. We give details only for the even case, so let n = 2k. Let Z be the (n−1)-dimensional subscheme defined by a_{1} = 0, and let U = Q_{n} −Z. Note that Z is singular (it’s the projective cone on Q_{n−2}), and that U ≅ 𝔸^{n}. Let Q′ = Q_{n} − {[0, 1, 0, 0, …, 0]}, and let Z′ = Z − {[0, 1, 0, 0, …, 0]}. Then Z′ ↪ Q′ is a smooth pair, with complement 𝔸^{n}. So the localization sequence for Z′ ↪ Q′ gives us an isomorphism H̃^{∗,∗}(Q′) ≅ H^{∗−2,∗−1}(Z′). The projection map Z′ → Q_{n−2} which forgets the first two homogeneous coordinates is a fiber bundle with fiber 𝔸^{1}, hence H^{∗,∗}(Z′) ≅ H^{∗,∗}(Q_{n−2}) by Property (E).

Taking the computations of the previous paragraph together, we conclude that H^{∗,∗}(Q′) ≅ H^{∗−2,∗−1}(Q_{n−2}) ⊕ 𝕄. By induction, this is free over 𝕄 with one generator in each degree (0, 0), (2, 1), …, (2n − 2, n − 1) plus an extra generator in degree (n, \frac{n}{2}).

Finally, we consider the localization sequence for {[0, 1, 0, …, 0]} ↪ Q_{n}. This has the form

⋯ ← H^{∗−2n+1,∗−n}(pt) ⟵^{δ} H^{∗,∗}(Q′) ← H^{∗,∗}(Q_{n}) ← H^{∗−2n,∗−n}(pt) ← ⋯

The generators for H^{∗,∗}(Q′) (as an 𝕄-module) must map to zero under δ for dimension reasons. It follows that

0 ← H^{∗,∗}(Q′) ← H^{∗,∗}(Q_{n}) ← H^{∗−2n,∗−n}(pt) ← 0

a short exact sequence of 𝕄-modules, in which the outer terms are known to be free. So the middle term is a direct sum of the outer terms. The right term provides a generator of degree (2n, n), and the left term provides the rest of the generators. □

The above proof also shows the following:

Proposition 4.2. The schemes Q_{n} and DQ_{n} belong to the class 𝒞 from Section 3.8.

Proof. If one knows by induction that Q_{n−2} belongs to 𝒞 then so do Z′, Q′, and Q_{n}, in that order.

The fact that projective spaces belong to the class 𝒞 is trivial: one uses the standard algebraic cell decomposition (cf. [F, 1.9.1]). Then since Q_{n−1} and ℙ^{n} are both in 𝒞, so is DQ_{n}. □

By the above result, in order to understand the ring structure on H^{∗,∗}(Q_{n}) it suffices just to understand the subring H^{2∗,∗}(Q_{n}) ≅ CH^{∗}(Q_{n}), because the 𝕄algebra generators lie in degrees (2∗, ∗). The computation of this Chow ring is well-known; the additive computation can be found in [Sw, 13.3], for instance, and the ring structure is stated in [KM]. For the reader’s convenience, and because we need several of the auxiliary facts, we give a complete account in Appendix A. These ideas lead to the following result, whose proof is essentially the content of Theorem A.4 and Theorem A.10.

Proposition 4.3.

(a) If n = 2k + 1 then as a ring H^{∗,∗}(Q_{n}) = 𝕄[x, y]/(x^{k+1} − 2y, y^{2}) where x has degree (2, 1) and y has degree (2k + 2, k + 1).

(b) If n = 2k and k is odd, then H^{∗,∗}(Q_{n}) = 𝕄[x, y]/(x^{k+1} − 2xy, y^{2}) where x has degree (2, 1) and y has degree (2k, k).

(c) If n = 2k and k is even, then H^{∗,∗}(Q_{n}) = 𝕄[x, y]/(x^{k+1} −2xy, x^{k+1}y, y^{2} −x^{k}y) where x has degree (2, 1) and y has degree (2k, k).

We will now consider the motivic cohomology of the deleted quadrics DQ_{n}. The idea is to use the localization sequence (4.4)

\xymatrix{⋯ & H^{∗−1,∗−1}(Q_{n−1}) \ar[l] & H^{∗,∗}(DQ_{n}) \ar[l] & H^{∗,∗}(ℙ^{n}) \ar[l]_{i^{∗}} & H^{∗−2,∗−1}(Q_{n−1}) \ar[l]_{j_{!}} & ⋯ \ar[l]}

By Proposition 4.1 the cohomology of Q_{n−1} has generators as an 𝕄-module in degrees (2∗, ∗), so we can completely determine the 𝕄-module map j_{!} just by understanding the pushforward map CH^{∗−1}(Q_{n−1}) → CH^{∗}(ℙ^{n}) of Chow groups. For the quadrics, this is discussed in detail in the appendix: all maps are either the identity or multiplication by 2. However, a problem now occurs. Because the ground ring 𝕄 might have 2-torsion, the kernel and cokernel of j_{!} will not necessarily be free over 𝕄—so we run into complicated extension problems. As a result, we haven’t been able to compute the integral motivic cohomology of DQ_{n}. The problem goes away if we work with ℤ/2 coefficients.

Proposition 4.5. If F is a field with char(F) ≠ 2, then H^{∗,∗}(DQ_{n}; ℤ/2) is a free 𝕄_{2}-module with one generator in degree (i, ⌈\frac{i}{2}⌉) for each 0 ≤ i ≤ n, where ⌈\frac{i}{2}⌉ is the smallest integer that is at least \frac{i}{2}.

Proof. The argument from Proposition 4.1 shows that H^{∗,∗}(Q_{n−1}; ℤ/2) is free over 𝕄_{2} on the same set of generators as before, and the map of subrings H^{2∗,∗}(Q_{n−1}) → H^{2∗,∗}(Q_{n−1}; ℤ/2) is just quotienting by the ideal (2).

By Lemma A.6, we know that the Gysin map j_{!} : H^{2i,i}(Q_{n−1}) → H^{2i+2,i+1}(ℙ^{n}) is multiplication by 2 for 0 ≤ i < \frac{n−1}{2}, and is an isomorphism for \frac{n−1}{2} < i ≤ n − 1. If n is odd, then it is the fold map ℤ ⊕ ℤ → ℤ for i = \frac{n−1}{2}.

The goal is to use the ℤ/2-analog of (4.4), so we first have to understand the Gysin map j_{!} with ℤ/2-coefficients. Since H^{2∗,∗}(Q_{n−1}; ℤ/2) and H^{2∗,∗}(ℙ^{n}; ℤ/2) are both obtained from integral cohomology simply by quotienting by the ideal (2), it follows that the Gysin map with ℤ/2-coefficients is an isomorphism, zero, or the fold map in all degrees (2∗, ∗). Since the generators (as 𝕄_{2}-modules) live in these degrees, we find that the kernel and cokernel of j_{!} : H^{∗,∗}(Q_{n−1}; ℤ/2) → H^{∗,∗}(ℙ^{n}; ℤ/2) are both free over 𝕄_{2}.

If n = 2k, then the generators for coker j_{!} are in degrees (0, 0), (2, 1), (4, 2), …, (2k, k), and the generators for ker j_{!} are in degrees (0, 0), (2, 1), …, (2k − 2, k − 1). If n = 2k + 1, then the generators are the same, except that ker j_{!} has another generator in degree (2k, k).

From the ℤ/2-analog of (4.4), we have the short exact sequence

0 ← ker j_{!} ← H^{∗,∗}(DQ_{n}; ℤ/2) ← coker j_{!} ← 0.

It follows that the middle group is also free over 𝕄_{2}. Be aware that the left map shifts degrees by (−1, −1). □

We know 𝕄_{2}^{0,0} ≅ ℤ/2. From Property (F) it follows that 𝕄_{2}^{p,q} = 0 if q < 0, if q = 0 and p < 0, or if p > q ≥ 0. So the above calculation shows that H^{1,1}(DQ_{n}; ℤ/2) ≅ 𝕄_{2}^{1,1}⊕𝕄_{2}^{0,0}, where the first summand comes from the motivic cohomology of Spec F. Hence, there is a unique nonzero element a ∈ H̃^{1,1}(DQ_{n}; ℤ/2). When n > 1 the calculation gives H^{2,1}(DQ_{n}; ℤ/2) ≅ ℤ/2, and we let b denote the unique nonzero element. For n = 1 we have DQ_{1} ≅ 𝔸^{1}−0, and it is known that H^{2,1}(𝔸^{1}−0; ℤ/2) = 0 (see, for instance, [V2, Lem. 6.8]). In this case we define b = 0 by convention.

Proposition 4.6. The map H^{∗,∗}(DQ_{n+1}; ℤ/2) → H^{∗,∗}(DQ_{n}; ℤ/2) induced by the inclusion takes a to a and b to b.

Proof. In light of the definitions of a and b in the previous paragraph, we just need to show that H^{i,1}(DQ_{n+1}; ℤ/2) → H^{i,1}(DQ_{n}; ℤ/2) is surjective for i = 1 or i = 2. Consider the diagram

\xymatrix{H^{i+1,1}(ℙ^{n+1}; ℤ/2) \ar[d] & H^{i−1,0}(Q_{n}; ℤ/2) \ar[d] \ar[l] & H^{i,1}(DQ_{n+1}; ℤ/2) \ar[d] \ar[l] & H^{i,1}(ℙ^{n+1}; ℤ/2) \ar[d] \ar[l] \\ H^{i+1,1}(ℙ^{n}; ℤ/2) & H^{i−1,0}(Q_{n−1}; ℤ/2) \ar[l] & H^{i,1}(DQ_{n}; ℤ/2) \ar[l] & H^{i,1}(ℙ^{n}; ℤ/2) \ar[l]}

in which the rows are localization sequences. The map between the groups on the left is the identity. A similar remark applies to the groups on the right. Finally, the cohomology groups of the quadrics are both isomorphic to 𝕄_{2}^{i−1,0}, and the map between them is also the identity. It follows from a diagram chase that the desired map is surjective. □

Lemma 4.7. β(a) = b in H^{∗,∗}(DQ_{n}; ℤ/2).

Proof. For brevity write DQ = DQ_{n}. We look at the long exact sequence

⋯ → H^{1,1}(DQ; ℤ) ⟶^{×2} H^{1,1}(DQ; ℤ) → H^{1,1}(DQ; ℤ/2) ⟶^{δ} H^{2,1}(DQ; ℤ) → · · · .

The localization sequence (4.4) for integral cohomology, together with the identification of j_{!} in Lemma A.6, shows that DQ ↪ ℙ^{n} induces an isomorphism on H^{1,1}(−; ℤ). It follows that if a were the mod 2 reduction of an integral class, it would also be the image of a class in H^{1,1}(ℙ^{n}; ℤ/2). But ∗ ↪ ℙ^{n} induces an isomorphism on H^{1,1}(−; ℤ/2), whereas the class a in H^{1,1}(DQ; ℤ/2) restricts to zero on the basepoint. We conclude that a can’t be the mod 2 reduction of an integral class, and therefore δ(a) is nonzero.

The sequence (4.4) (again with our knowledge of j_{!}) also shows that H^{2,1}(DQ; ℤ) is isomorphic to ℤ/2, with the generator being in the image of H^{2,1}(ℙ^{n}; ℤ) → H^{2,1}(DQ; ℤ). It follows that δ(a) is the unique nonzero element of H^{2,1}(DQ; ℤ), and the mod 2 reduction of δ(a) is b. □

We need one more lemma before stating the final result.

Lemma 4.8. H^{2k+1,k+1}(DQ_{2k+2}; ℤ/2) → H^{2k+1,k+1}(DQ_{2k+1}; ℤ/2) is injective.

Proof. Consider the diagram

[figure: H^{2k,k}(Q_{2k+1}; ℤ/2); H^{2k+1,k+1}(DQ_{2k+2}; ℤ/2); H^{2k+1,k+1}(ℙ^{2k+2}; ℤ/2); H^{2k,k}(Q_{2k}; ℤ/2); H^{2k+1,k+1}(DQ_{2k+1}; ℤ/2); H^{2k+1,k+1}(ℙ^{2k+1}; ℤ/2); f]

in which the rows are portions of localization sequences. From Proposition 4.1 and Properties (D) and (F), f must be isomorphic to a map 𝕄_{2}^{1,1} → 𝕄_{2}^{1,1}, but Lemma A.6 tells us that this map is the zero map. Also, the right-most vertical map is an isomorphism. A diagram chase would give us the desired result, if we knew that the left vertical map was injective. This map is equal to the map CH^{k}(Q_{2k+1}) → CH^{k}(Q_{2k}) after reducing modulo 2. We look at the diagram

[figure: ℤ; CH^{k}(Q_{2k+1}); CH^{k}(Q_{2k}); ℤ ⊕ ℤ; ℤ; CH^{k}(ℙ^{2k+2}); ≅; ≅; CH^{k}(ℙ^{2k+1}); ℤ]

Lemma A.6 shows that the left vertical map is an isomorphism. Lemma A.9 identifies the right vertical map as the diagonal, and from that information the result follows at once. □

Theorem 4.9. Let F be a field with char(F) ≠ 2.

(a) If n = 2k + 1 then H^{∗,∗}(DQ_{n}; ℤ/2) ≅ 𝕄_{2}[a, b]/(a^{2} = ρa + τb, b^{k+1}) where a has degree (1, 1) and b has degree (2, 1).

(b) If n = 2k, there exists an element ϵ in 𝕄_{2}^{1,1} such that H^{∗,∗}(DQ_{n}; ℤ/2) ≅ 𝕄_{2}[a, b]/(a^{2} = ρa + τb, b^{k+1}, ab^{k} = ϵb^{k}) where a and b are as in (a).

Remark 4.10. We haven’t been able to identify the class ϵ in any nontrivial case. This is not important for proving the Hopf condition, but it would be satisfying to resolve the issue of whether ϵ is equal to 0, or ρ, or some other element.

Proof. For convenience we will drop subscripts and superscripts: Q = Q_{n−1}, ℙ = ℙ^{n}, and DQ = DQ_{n}. We know H^{∗,∗}(DQ; ℤ/2) additively, we just need to determine the ring structure.

Note that the map H^{2i,i}(ℙ) → H^{2i,i}(DQ) is surjective because it is the map CH^{i}(ℙ) → CH^{i}(DQ). Therefore, the nonzero element t of H^{2,1}(ℙ; ℤ/2) goes to the nonzero element b of H^{2,1}(DQ; ℤ/2). Then t^{i} maps to b^{i}, and surjectivity implies that b^{i} must be the unique nonzero element in H^{2i,i}(DQ; ℤ/2) for 1 ≤ i ≤ \frac{n}{2}.

Lemma 4.7 showed that β(a) = b. Since β^{2} = 0 one has β(b) = 0, so Property (H) implies that β(ab^{i}) = b^{i+1}. In particular ab^{i} is nonzero for 0 ≤ i ≤ \frac{n}{2} − 1.

Now H̃^{1,1}(DQ; ℤ/2) ≅ 𝕄_{2}^{0,0}a and H^{2i−1,i}(DQ; ℤ/2) ≅ 𝕄_{2}^{0,0} ⊕ 𝕄_{2}^{1,1}b^{i−1} for 1 ≤ i ≤ \frac{n+1}{2}, where the first factor arises from the generator in degree (2i − 1, i). Property (H) and the fact that 𝕄_{2}^{2,1} = 0 implies that β(x) = 0 for any x ∈ 𝕄_{2}^{1,1}b^{i−1}. So we cannot have ab^{i} ∈ 𝕄_{2}^{1,1}b^{i−1}.

Based on our knowledge of H^{∗,∗}(DQ; ℤ/2) as an 𝕄_{2}-module, we can now conclude that when n = 2k the classes 1, b, b^{2}, …, b^{k} and a, ab, ab^{2}, …, ab^{k−1} are a free basis for H^{∗,∗}(DQ; ℤ/2) over 𝕄_{2}.

The argument is slightly harder when n = 2k + 1, because we must show that ab^{k} is nonzero (even though its Bockstein is zero). However, we already know that ab^{k} is nonzero in H^{∗,∗}(DQ_{n+1}; ℤ/2). The map H^{2k+1,k+1}(DQ_{n+1}; ℤ/2) → H^{2k+1,k+1}(DQ_{n}; ℤ/2) is an injection by Lemma 4.8 and takes ab^{k} to ab^{k} by Proposition 4.6. It follows that 1, b, …, b^{k}, a, ab, …, ab^{k} is a free basis for H^{∗,∗}(DQ; ℤ/2) when n = 2k + 1.

We next identify a^{2}. This part of the argument exactly parallels [V2, pp. 22-23]. The class a^{2} ∈ H̃^{2,2}(DQ; ℤ/2) must be a linear combination over 𝕄_{2} of the elements a and b: a^{2} = Aa + Bb where A ∈ 𝕄_{2}^{1,1} and B ∈ 𝕄_{2}^{0,1} ≅ ℤ/2. To identify A it’s sufficient to look at the image of a^{2} under H^{∗,∗}(DQ; ℤ/2) → H^{∗,∗}(DQ_{1}; ℤ/2), since Aa + Bb goes to Aa under this map by Proposition 4.6 and the fact that b = 0 in H^{∗,∗}(DQ_{1}; ℤ/2). Note that DQ_{1} is isomorphic to 𝔸^{1} − 0, and one knows that H^{∗,∗}(𝔸^{1} − 0; ℤ/2) ≅ 𝕄_{2}[a]/(a^{2} = ρa) by [V2, Lem. 6.8]. So A = ρ.

To identify B, let K be the field consisting of F with a square root of −1 adjoined (unless F already has a square root of −1, in which case K = F). Let DQ_{K} be the base change of DQ along the map Spec K → Spec F. Under the induced map H^{∗,∗}(DQ; ℤ/2) → H^{∗,∗}(DQ_{K}; ℤ/2), ρ maps to zero; so ρa+Bb maps to Bb. Hence it suffices to assume that F contains a square root of −1 and show that a^{2} = τb.

Under the map H^{∗,∗}(DQ; ℤ/2) → H_{et}^{∗}(DQ; μ_{2}^{⊗∗}) the element τ becomes invertible (cf. Property (I)), and so we can write a = τa′ (in H_{et}^{∗}(DQ; μ_{2}^{⊗∗})), for some a′ ∈ H_{et}^{1}(pt; μ_{2}^{0}). This group is sheaf cohomology with coefficients in the constant sheaf ℤ/2; if β_{et} is the Bockstein on étale cohomology induced by 0 → ℤ/2 → ℤ/4 → ℤ/2 → 0 one has that β_{et}(a′) = (a′)^{2} by a standard property of the Bockstein on sheaf cohomology (the proof is the same as the one in topology). Our remarks in Section 3.6 show that the Bocksteins in motivic and étale cohomology are compatible, because F has a square root of −1. So we now compute that

a^{2} = τ^{2}(a′)^{2} = τ^{2}β_{et}(a′) = τ · β(τa′) = τ · β(a) = τ · b (4.11)

in H_{et}^{∗}(DQ; μ_{2}^{⊗∗}). Note that the third equality uses the analog of Property (H) for étale cohomology, together with the fact that β(τ) = ρ = 0 (by our assumption on F).

As a consequence of (4.11), we have in particular that a^{2} is nonzero in H^{∗,∗}(DQ; ℤ/2)[τ^{−1}]. But a^{2} = Bb, so B must be nonzero. From the sequence (3.4) we recall that 𝕄_{2}^{0,1} = {0, τ}, so B = τ. We have therefore shown that a^{2} = ρa + τb ∈ H^{2,2}(DQ; ℤ/2).

This finishes part (a) of the theorem. For part (b) we just observe that ab^{k} ∈ H^{2k+1,k+1}(DQ; ℤ/2), and H^{2k+1,k+1}(DQ; ℤ/2) ≅ 𝕄_{2}^{1,1}b^{k}. So for some ϵ ∈ 𝕄_{2}^{1,1} we have ϵb^{k} = ab^{k}. This finishes part (b). □

Remark 4.12. When n is odd, the cohomology of DQ_{n} is the same as the cohomology of the scheme (𝔸^{n} − 0)/ ± 1, which was essentially computed by Voevodsky in [V2, Th. 6.10]. It seems likely that these two schemes are 𝔸^{1}-homotopy equivalent, but we haven’t proven this.

Appendix A. Chow groups of quadrics

This appendix contains a calculation of the Chow rings of the quadrics Q_{n}, as well as various pushforward and pullback maps. This is classical, but the details are useful and we don’t know a suitable reference. We assume a basic familiarity with the Chow ring; see [F] or [H, App. A].

Let CH_{i}(X) be the Chow group of dimension i cycles on X. If Z ↪ X is a closed subscheme there is an exact sequence

CH_{i}(Z) → CH_{i}(X) → CH_{i}(X − Z) → 0

where the first map is pushforward and the second map is restriction.

If X ⊆ ℙ^{n} is a closed subscheme, we let / ΣX ⊆ ℙ^{n+1} denote the projective cone on X. Let / Σ: CH_{i}(X) → CH_{i+1}(/ ΣX) be the map sending a cycle to the projective cone on the cycle, and recall that this is an isomorphism for i ≥ 0. Also note that CH_{0}(/ ΣX) = ℤ no matter what X is. Finally, recall that CH_{i}(𝔸^{n}) = 0 if i ≠ n, whereas CH_{n}(𝔸^{n}) = ℤ.

When X is nonsingular one defines CH^{i}(X) = CH_{dim X−i}(X).

The following discussion is modeled on [Sw, 13.3].

A.1. The odd-dimensional case. Consider the quadric Q_{2k+1} ↪ ℙ^{2k+2} defined by a_{1}b_{1} + ⋯ + a_{k+1}b_{k+1} + c^{2} = 0. We let j be the inclusion.

Lemma A.2. For all 0 ≤ i ≤ 2k + 1, the Chow group CH_{i}(Q_{2k+1}) is isomorphic to ℤ. The pushforward map j_{∗} : CH_{i}(Q_{2k+1}) → CH_{i}(ℙ^{2k+2}) is an isomorphism if 0 ≤ i ≤ k, and is multiplication by 2 (as a map ℤ → ℤ) if k + 1 ≤ i ≤ 2k + 1.

Proof. The first claim follows immediately from Proposition 4.1 and Property (A).

The proof of the second statement is by induction. The base case Q_{1} is isomorphic to ℙ^{1}, and Q_{1} is imbedded in ℙ^{2} as a degree two hypersurface. So j_{∗} is an isomorphism for i = 0 and is multiplication by 2 for i = 1.

If Z is the closed subscheme defined by a_{1} = 0, we know Q_{2k+1} − Z ≅ 𝔸^{2k+1} and Z ≅ / ΣQ_{2k−1}. The resulting localization sequence gives us a diagram

\xymatrix{CH_{i}(/ ΣQ_{2k−1}) \ar[r] \ar[d] & CH_{i}(Q_{2k+1}) \ar[r] \ar[d]^{j_{∗}} & CH_{i}(𝔸^{2k+1}) \\ CH_{i}(/ Σℙ^{2k}) \ar[r] & CH_{i}(ℙ^{2k+2})}

in which the top row is exact. Since / Σℙ^{2k} is isomorphic to ℙ^{2k+1}, the bottom horizontal arrow is an isomorphism for all 0 ≤ i ≤ 2k + 1, and both groups in the bottom row are isomorphic to ℤ.

For 0 ≤ i ≤ 2k, the first two groups in the top row are also isomorphic to ℤ. For 0 ≤ i ≤ k, the left vertical arrow is known by induction to be an isomorphism. The only possibility is that the map j_{∗} is an isomorphism in this range.

Now for k + 1 ≤ i ≤ 2k, the left vertical arrow is known by induction to be multiplication by 2. Since the upper left horizontal arrow is a surjection, the only possibility is that the map j_{∗} is multiplication by 2.

Finally, for the case i = 2k + 1 note that Q_{2k+1} is a degree 2 hypersurface in ℙ^{2k+2}. Thus, the fundamental class [Q_{2k+1}] maps to twice the generator of CH_{2k+1}(ℙ^{2k+2}). □

By analyzing the above proof, one can give explicit generators for CH_{i}(Q_{2k+1}). If 0 ≤ i ≤ k, the generator is the class of the cycle determined by setting all coordinates equal to zero except for b_{1}, …, b_{i+1}. Note that this cycle is isomorphic to ℙ^{i}. On the other hand, if k + 1 ≤ i ≤ 2k + 1, then the generator is the class of the cycle determined by setting a_{1}, …, a_{2k+1−i} equal to zero. Note that this cycle is the iterated projective cone on Q_{2i−2k+1}, and also the intersection of Q_{2k+1} with a copy of ℙ^{i+1}.

We next want to compute the ring structure on CH^{∗}(Q_{2k+1}) as well as the pullback map j^{∗} : CH_{i}(ℙ^{2k+2}) → CH_{i−1}(Q_{2k+1}). It is easier to do the latter first.

Proposition A.3. The map j^{∗}: CH_{i}(ℙ^{2k+2}) → CH_{i−1}(Q_{2k+1}) is an isomorphism if k + 2 ≤ i ≤ 2k + 2 and is multiplication by 2 if 1 ≤ i ≤ k + 1.

Proof. The projection formula j_{∗}(a · j^{∗}b) = (j_{∗}a) · b gives us

j_{∗}(j^{∗}[ℙ^{i}]) = j_{∗}([Q_{2k+1}] · j^{∗}[ℙ^{i}]) = j_{∗}([Q_{2k+1}]) · [ℙ^{i}] = 2[ℙ^{2k+1}] · [ℙ^{i}] = 2[ℙ^{i−1}].

In other words the composition j_{∗}j^{∗} : CH_{i}(ℙ^{2k+2}) → CH_{i−1}(ℙ^{2k+2}) is multiplication by 2. When 1 ≤ i ≤ k + 1, the map j_{∗} is an isomorphism, so j^{∗} must be multiplication by 2. When k + 2 ≤ i ≤ 2k + 2, the map j_{∗} is multiplication by 2, so j^{∗} must be an isomorphism. □

It is now easy to deduce the ring structure on CH^{∗}(Q_{2k+1}), using the map from CH^{∗}(ℙ^{2k+2}). Note that when k = 0 we are looking at Q_{1} ≅ ℙ^{1}, and so CH^{∗}(Q_{1}) is isomorphic to ℤ[x]/x^{2}, where x has degree 1.

Theorem A.4. If k ≥ 0, then CH^{∗}(Q_{2k+1}) ≅ ℤ[x, y]/(x^{k+1} − 2y, y^{2}), where x has degree 1 and y has degree k + 1.

Proof. The map j^{∗}: CH^{i}(ℙ^{2k+2}) → CH^{i}(Q_{2k+1}) (which now preserves the grading because we are grading by codimension) is an isomorphism if 0 ≤ i ≤ k and is multiplication by 2 if k + 1 ≤ i ≤ 2k + 1. This follows immediately from the previous proposition simply by regrading.

Let t be the generator of CH^{1}(ℙ^{2k+2}), and let x = j^{∗}(t). Then x^{k+1} = j^{∗}(t^{k+1}) is twice a generator of CH^{k+1}(Q_{2k+1}), and we let y be this generator. The desired isomorphism of rings follows immediately from our knowledge of the groups CH^{∗}(Q_{2k+1}) and the description of j^{∗} in the previous paragraph. □

A.5. The even-dimensional case. This case is a little harder. The quadric Q_{2k} is defined by a_{1}b_{1} +⋯+a_{k+1}b_{k+1} = 0. As before, let j be the inclusion Q_{2k} → ℙ^{2k+1}. Many of the results from the previous section carry over to this section with identical proofs.

The base case is Q_{0}, which is pt ⨿ pt. Note that j_{∗} : CH_{0}(Q_{0}) → CH_{0}(ℙ^{1}) is the fold map ℤ ⊕ ℤ → ℤ.

We already know from Proposition 4.1 that the Chow group CH_{i}(Q_{2k}) is isomorphic to ℤ for all 0 ≤ i ≤ 2k, except that CH_{n}(Q_{2k}) is isomorphic to ℤ ⊕ ℤ. The same arguments as in the proof of Lemma A.2 allow us to conclude that the pushforward map j_{∗} : CH_{i}(Q_{2k}) → CH_{i}(ℙ^{2k+1}) is an isomorphism if 0 ≤ i ≤ k − 1, is multiplication by 2 if k + 1 ≤ i ≤ 2k, and is the fold map if i = k. We summarize these facts (with cohomological grading) in the following lemma, which we state because it is critical for the computations in Section 4.

Lemma A.6. For any n, the map j_{∗}: CH^{i}(Q_{n−1}) → CH^{i+1}(ℙ^{n}) is multiplication by 2 for 0 ≤ i < \frac{n−1}{2}, and is an isomorphism for \frac{n−1}{2} < i ≤ n − 1. If n is odd, then it is the fold map ℤ ⊕ ℤ → ℤ for i = \frac{n−1}{2}.

Once again, one can give explicit generators for CH_{i}(Q_{2k}). For i ≠ k, the description of these generators is the same as in the odd case. For i = k, one generator is determined by b_{1} = b_{2} = ⋯ = b_{k+1} = 0, and the other generator is determined by a_{1} = b_{2} = ⋯ = b_{k+1} = 0. We let α and β represent these two codimension k cycles.

Lemma A.7. Let α′ be the cycle determined by a_{1} = a_{2} = ⋯ = a_{k+1} = 0, and let β′ be the cycle determined by b_{1} = a_{2} = ⋯ = a_{k+1} = 0. If k is odd, then α = α′ and β = β′ in CH_{n}(Q_{2k}). If k is even, then α = β′ and β = α′ in CH_{n}(Q_{2k}).

Proof. If k is odd, then α and α′ do not meet. Then [HP, Th. XII.4.III] says that α and α′ are rationally equivalent. Similarly, β = β′.

The same argument applies to the even case. □

Lemma A.8. Let [∗] be the fundamental class of a point in CH^{2k}(Q_{2k}). If k is odd, then α · α = 0 = β · β and α · β = [∗] in the Chow ring CH^{∗}(Q_{2k}). If k is even, then α · α = [∗] = β · β and α · β = 0.

Proof. When k is odd, α · α = α · α′. However, α and α′ do not intersect, so α · α′ = 0. Similarly, β · β = 0. Now α and β′ intersect transversely at a point, so α · β = α · β′ = [∗]. Similar arguments apply to the even case. □

As in Proposition A.3, the map j^{∗}: CH_{i}(ℙ^{2k+1}) → CH_{i−1}(Q_{2k}) is an isomorphism if k + 2 ≤ i ≤ 2k + 1 and is multiplication by 2 if 1 ≤ i ≤ k. After regrading by codimension, this says j^{∗}: CH^{i}(ℙ^{2k+1}) → CH^{i}(Q_{2k}) is an isomorphism for 0 ≤ i < k and multiplication by 2 for k < i ≤ 2k. The same argument with the projection formula also shows that when i = k, j^{∗} takes the generator to uα + (2 − u)β for some u ∈ ℤ.

Lemma A.9. The map j^{∗}: CH^{k}(ℙ^{2k+1}) → CH^{k}(Q_{2k}) sends the generator t^{k} to α + β.

Proof. We already know that j^{∗}(t^{2k}) = 2[∗], where [∗] is the fundamental class of a point in Q_{2k} and is also the generator of CH^{2k}(Q_{2k}). Therefore,

2[∗] = j^{∗}(t^{2k}) = (j^{∗}(t^{k}))^{2} = (uα + (2 − u)β)^{2} = u^{2}α^{2} + 2u(2 − u)αβ + (2 − u)^{2}β^{2}.

If k is odd, Lemma A.8 lets us rewrite this equation as 2[∗] = 2u(2−u)[∗], so u = 1. If k is even Lemma A.8 gives 2[∗] = (u^{2} + (2 − u)^{2})[∗], so again u = 1. □

Theorem A.10. If k is odd, then there is an isomorphism of rings CH^{∗}(Q_{2k}) ≅ ℤ[x, y]/(x^{k+1} −2xy, y^{2}), where x has degree 1 and y has degree k. If k is even, then CH^{∗}(Q_{2k}) ≅ ℤ[x, y]/(x^{k+1} − 2xy, x^{k+1}y, y^{2} − x^{k}y), where x has degree 1 and y has degree k.

Proof. Let t be the generator [ℙ^{2k}] of CH^{1}(ℙ^{2k+1}), and let x = j^{∗}(t). Then j^{∗}(t^{i}) = x^{i}. As we know that j^{∗} takes generators to generators for 0 ≤ i ≤ k − 1, it follows that x^{i} is a generator for CH^{i}(Q_{2k}) in these dimensions.

Now x^{k} = j^{∗}(t^{k}) = α+β by the previous lemma. If we let y equal α, then x^{k} and y are two generators for CH^{k}(Q_{2k}). Note that Lemma A.8 implies that x^{k}y = [∗] since α(α + β) = [∗] in both the even and odd cases.

Next we can compute that

j_{∗}(x^{i} · y) = j_{∗}(j^{∗}(t^{i}) · y) = t^{i} · j_{∗}y = t^{i} · t^{k} = t^{k+i}

for 1 ≤ t ≤ k. Since j_{∗} is an isomorphism in codimension k + i, it follows that x^{i}y is a generator in CH^{k+i}(Q_{2k}).

Now x^{k+1} = j^{∗}(t^{k+1}) = j^{∗}(j_{∗}(xy)) = 2xy. Also, for dimension reasons x^{k+1}y = 0. Finally, Lemma A.8 shows that y^{2} = 0 if k is odd and y^{2} = [∗] = x^{k}y if k is even.

Thus, we have shown that the additive generators for CH^{∗}(Q_{2k}) are 1, x, x^{2}, …, x^{k}, y, xy, …, x^{k}y, where the elements are listed in order of increasing degree. Moreover, we have constructed a ring map from the desired ring to CH^{∗}(Q_{2k+1}), which is an additive isomorphism. □

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Department of Mathematics, University of Oregon, Eugene, OR 97403

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

E-mail address: ddugger@math.uoregon.edu
E-mail address: isaksen@math.wayne.edu