Algebraic K-Theory and Sums-of-Squares Formulas

Daniel Dugger, Daniel C. Isaksen

Received: January 20, 2005 Revised: July 25, 2005

Communicated by Stefan Schwede

Abstract. We prove a result about the existence of certain ‘sums-of-squares’ formulas over a field F. A classical theorem uses topological K-theory to show that if such a formula exists over ℝ, then certain powers of 2 must divide certain binomial coefficients. In this paper we use algebraic K-theory to extend the result to all fields not of characteristic 2.

1. Introduction

Let F be a field. A classical problem asks for which values of r, s, and n does there exist an identity of the form

(x_{1}^{2} + ⋯ + x_{r}^{2}) (y_{1}^{2} + ⋯ + y_{s}^{2}) = z_{1}^{2} + ⋯ + z_{n}^{2}

in the polynomial ring F [x_{1}, …, x_{r}, y_{1}, …, y_{s}], where the z_{i}’s are bilinear expressions in the x’s and y’s. Such an identity is called a sums-of-squares formula of type [r, s, n]. For the history of this problem, see the expository papers [L, Sh]. The main theorem of this paper is the following:

Theorem 1.1. Assume F is not of characteristic 2. If a sums-of-squares formula of type [r, s, n] exists over F, then 2^{⌊ \frac{s−1}{2} ⌋−i+1} divides \binom{n}{i} for n − r < i ≤ ⌊\frac{s−1}{2}⌋.

As one specific application, the theorem shows that a formula of type [13, 13, 16] cannot exist over any field of characteristic not equal to 2. Previously this had only been known in characteristic zero. (Note that the case char(F) = 2, which is not covered by the theorem, is trivial: formulas of type [r, s, 1] always exist). In the case F = ℝ, the above theorem was essentially proven by Atiyah [At] as an early application of complex K-theory; the relevance of Atiyah’s paper to the sums-of-squares problem was only later pointed out by Yuzvinsky [Y]. The result for characteristic zero fields can be deduced from the case F = ℝ by an algebraic argument due to K. Y. Lam and T. Y. Lam (see [Sh]). Thus, our contribution is the extension to fields of non-zero characteristic. In this sense the present paper is a natural sequel to [DI], which extended another classical condition about sums-of-squares. We note that sums-of-squares formulas in characteristic p were first seriously investigated in [Ad1, Ad2]. Our proof of Theorem 1.1, given in Section 2, is a modification of Atiyah’s original argument. The existence of a sums-of-squares formula allows one to make conclusions about the geometric dimension of certain algebraic vector bundles. A computation of algebraic K-theory (in fact just algebraic K^{0}), given in Section 3, determines restrictions on what that geometric dimension can be—and this yields the theorem. Atiyah’s result for F = ℝ is actually slightly better than our Theorem 1.1. The use of topological KO-theory rather than complex K-theory yields an extra power of 2 dividing some of the binomial coefficients. It seems likely that this stronger result holds in non-zero characteristic as well and that it could be proved with Hermitian algebraic K-theory.

1.2. Restatement of the main theorem. The condition on binomial coefficients from Theorem 1.1 can be reformulated in a slightly different way. This second formulation surfaces often, and it’s what arises naturally in our proof. We record it here for the reader’s convenience. Each of the following observations is a consequence of the previous one:

• By repeated use of Pascal’s identity \binom{c}{d} = \binom{c−1}{d−1} + \binom{c−1}{d}, the number \binom{n+i−1}{k+i} is a ℤ-linear combination of the numbers \binom{n}{k+1}, \binom{n}{k+2}, …, \binom{n}{k+i}. Similarly, \binom{n}{k+i} is a ℤ-linear combination of \binom{n}{k+1}, \binom{n+1}{k+2}, …, \binom{n+i−1}{k+i}.

• An integer b is a common divisor of \binom{n}{k+1}, \binom{n}{k+2}, …, \binom{n}{k+i} if and only if it is a common divisor of \binom{n}{k+1}, \binom{n+1}{k+2}, …, \binom{n+i−1}{k+i}.

• The series of statements

2^{N} | \binom{n}{k+1} , 2^{N−1} | \binom{n}{k+2} , …, 2^{N−i+1} | \binom{n}{k+i}

is equivalent to the series of statements

2^{N} | \binom{n}{k+1} , 2^{N−1} | \binom{n+1}{k+2} , …, 2^{N−i+1} | \binom{n+i−1}{k+i} .

• If N is a fixed integer, then 2^{N−i+1} divides \binom{n}{i} for n − r < i ≤ N if

and only if 2^{N−i+1} divides \binom{r+i−1}{i} for n − r < i ≤ N. The last observation shows that Theorem 1.1 is equivalent to the theorem below. This is the form in which we’ll actually prove the result.

Theorem 1.3. Suppose that F is not of characteristic 2. If a sums-of-squares formula of type [r, s, n] exists over F, then 2^{⌊ \frac{s−1}{2} ⌋−i+1} divides the binomial coefficient \binom{r+i−1}{i} for n − r < i ≤ ⌊\frac{s−1}{2}⌋.

1.4. Notation. Throughout this paper K^{0}(X) denotes the Grothendieck group of locally free coherent sheaves on the scheme X. This group is usually denoted K_{0}(X) in the literature.

Algebraic K-Theory and Sums-of-Squares Formulas 359

2. The main proof

In this section we fix a field F not of characteristic 2. Let q_{k} be the quadratic form on 𝔸^{k} defined by q_{k}(x) = ∑_{i=1}^{k} x_{i}^{2}. A sums-of-squares formula of type [r, s, n] gives a bilinear map ϕ: 𝔸^{r} × 𝔸^{s} → 𝔸^{n} such that q_{r}(x)q_{s}(y) = q_{n}(ϕ(x, y)). We begin with a simple lemma:

Lemma 2.1. Let F ↪ E be a field extension, and let y ∈ E^{s} be such that q_{s}(y) ≠ 0. Then for x ∈ E^{r} one has ϕ(x, y) = 0 if and only if x = 0.

Proof. Let ⟨−, −⟩ denote the inner product on E^{k} corresponding to the quadratic form q_{k}. Note that the sums-of-squares identity implies that

⟨ϕ(x, y), ϕ(x′, y)⟩ = q_{s}(y)⟨x, x′⟩

for any x and x′ in E^{r}. If one had ϕ(x, y) = 0 then the above formula would imply that q_{s}(y)⟨x, x′⟩ = 0 for every x′; but since q_{s}(y) ≠ 0, this can only happen when x = 0. □

Let V_{q} be the subvariety of ℙ^{s−1} defined by q_{s}(y) = 0. Let ξ denote the restriction to V_{q} of the tautological line bundle 𝒪(−1) of ℙ^{s−1}.

Proposition 2.2. If a sums-of-squares formula of type [r, s, n] exists over F, then there is an algebraic vector bundle ζ on ℙ^{s−1} − V_{q} of rank n − r such that

r[ξ] + [ζ] = n

as elements of the Grothendieck group K^{0}(ℙ^{s−1} − V_{q}) of locally free coherent sheaves on ℙ^{s−1} − V_{q}.

Proof. We’ll write q = q_{s} in this proof, for simplicity. Let S = F [y_{1}, …, y_{s}] be the homogeneous coordinate ring of ℙ^{s−1}. By [H, Prop. II.2.5(b)] one has ℙ^{s−1} − V_{q} = Spec R, where R is the subring of the localization S_{q} that consists of degree 0 homogeneous elements. The group K^{0}(ℙ^{s−1} − V_{q}) is naturally isomorphic to the Grothendieck group of finitely-generated projective R-modules. Let P denote the subset of S_{q} consisting of homogeneous elements of degree −1, regarded as a module over R. Then P is projective and is the module of sections of the vector bundle ξ. To see explicitly that P is projective of rank 1, observe that there is a split-exact sequence 0 → R^{s−1} → R^{s} ⟶^{π} P → 0 where π(p_{1}, …, p_{s}) = ∑ p_{i} · \frac{y_{i}}{q} and the splitting χ: P → R^{s} is χ(f) = (y_{1}f, y_{2}f, …, y_{s}f). From our bilinear map ϕ: 𝔸^{r} × 𝔸^{s} → 𝔸^{n} we get linear forms ϕ(e_{i}, y) ∈ S^{n} for 1 ≤ i ≤ r. Here e_{i} denotes the standard basis for F^{r}, and y = (y_{1}, …, y_{s}) is the vector of indeterminates from S. If f belongs to P, then each component of f · ϕ(e_{i}, y) is homogeneous of degree 0—hence lies in R. Define a map α: P^{r} → R^{n} by

(f_{1}, …, f_{r}) ↦ f_{1}ϕ(e_{1}, y) + f_{2}ϕ(e_{2}, y) + ⋯ + f_{r}ϕ(e_{r}, y).

We can write α(f_{1}, …, f_{r}) = ϕ((f_{1}, …, f_{r}), y), where the expression on the right means to formally substitute each f_{i} for x_{i} in the defining formula for ϕ. If R → E is any map of rings where E is a field, we claim that α ⊗_{R} E is an injective map E^{r} → E^{n}. To see this, note that R → E may be extended to a map u: S_{q} → E (any map Spec E → ℙ^{s−1} − V_{q} lifts to the affine variety q ≠ 0, as the projection map from the latter to the former is a Zariski locally trivial bundle). One obtains an isomorphism P ⊗_{R} E → E by sending f ⊗ 1 to u(f). Using this, α ⊗_{R} E may be readily identified with the map x ↦ ϕ(x, u(y)). Now apply Lemma 2.1. Since R is a domain, we may take E to be the quotient field of R. It follows that α is an inclusion. Let M denote its cokernel. The module M will play the role of ζ in the statement of the proposition, so to conclude the proof we only need show that M is projective. An inclusion of finitely-generated projectives P_{1} ↪ P_{2} has projective cokernel if and only if P_{1} ⊗_{R} E → P_{2} ⊗_{R} E is injective for every map R → E where E is a field (that is to say, the map has constant rank on the fibers)—this follows at once using [E, Ex. 6.2(iii),(v)]. As we have already verified this property for α, we are done. □

Remark 2.3. The above algebraic proof hides some of the geometric intuition behind Proposition 2.2. We outline a different approach more in the spirit of [At]. Let Gr_{r}(𝔸^{n}) denote the Grassmannian variety of r-planes in affine space 𝔸^{n}. We claim that ϕ induces a map f: ℙ^{s−1} − V_{q} → Gr_{r}(𝔸^{n}) with the following behavior on points. Let [y] be a point of ℙ^{s−1} represented by a point y of 𝔸^{s} such that q_{s}(y) ≠ 0. Then the map ϕ_{y} : x ↦ ϕ(x, y) is a linear inclusion by Lemma 2.1. Let f([y]) be the r-plane that is the image of ϕ_{y}. Since ϕ is bilinear, we get that ϕ_{λy} = λ · ϕ_{y} for any scalar y. This shows that f([y]) is well-defined. We leave it as an exercise for the reader to carefully construct f as a map of schemes. The map f has a special property related to bundles. If η_{r} denotes the tautological r-plane bundle over the Grassmannian, we claim that ϕ induces a map of bundles f̃: rξ → η_{r} covering the map f. To see this, note that the points of rξ (defined over some field E) correspond to equivalence classes of pairs (y, a) ∈ 𝔸^{s} × 𝔸^{r} with q(y) ≠ 0, where (λy, a) ∼ (y, λa) for any λ in the field. The pair (y, a) gives us a line ⟨y⟩ ⊆ 𝔸^{s} together with r points a_{1}y, a_{2}y, …, a_{r}y on the line. One defines f̃ so that it sends (y, a) to the element of η_{r} represented by the vector ϕ(a, y) lying on the r-plane spanned by ϕ(e_{1}, y), …, ϕ(e_{r}, y). This respects the equivalence relation, as ϕ(λa, y) = ϕ(a, λy). So we have described our map f̃: rξ → η_{r}. We again leave it to the reader to construct f as a map of schemes. One readily checks that f̃ is a linear isomorphism on geometric fibers, using Lemma 2.1. So f̃ gives an isomorphism rξ ≅ f^{∗}η_{r} of bundles on ℙ^{s−1} − V_{q}. The bundle η_{r} is a subbundle of the rank n trivial bundle, which we denote by n. Consider the quotient n/η_{r}, and set ζ = f^{∗}(n/η_{r}). Since n = [η_{r}]+[n/η_{r}] in K^{0}(Gr_{r}(𝔸^{n})), application of f^{∗} gives n = [f^{∗}η_{r}] + [ζ] in K^{0}(ℙ^{s−1} − V_{q}). Now recall that f^{∗}η_{r} ≅ rξ. This gives the desired formula in Proposition 2.2.

The next task is to compute the Grothendieck group K^{0}(ℙ^{s−1} − V_{q}). This becomes significantly easier if we assume that F contains a square root of −1. The reason for this is made clear in the next section.

Proposition 2.4. Suppose that F contains a square root of −1 and is not of characteristic 2. Let c = ⌊\frac{s−1}{2}⌋. Then K^{0}(ℙ^{s−1} − V_{q}) is isomorphic to ℤ[ν]/(2^{c}ν, ν^{2} = −2ν), where ν = [ξ] − 1 generates the reduced Grothendieck group K̃^{0}(ℙ^{s−1} − V_{q}) ≅ ℤ/2^{c}.

The proof of the above result will be deferred until the next section. Note that K^{0}(ℙ^{s−1} − V_{q}) has the same form as the complex K-theory of real projective space ℝP^{s−1} [A, Thm. 7.3]. To complete the analogy, we point out that when F = ℂ the space ℂP^{s−1} −V_{q}(ℂ) is actually homotopy equivalent to ℝP^{s−1} [Lw, 6.3]. We also mention that for the special case where F is contained in ℂ, the above proposition was proved in [GR, Theorem, p. 303]. By accepting the above proposition for the moment, we can finish the

Proof of Theorem 1.3. Recall that one has operations γ^{i} on K̃^{0}(X) for any scheme X [SGA6, Exp. V] (see also [AT] for a very clear explanation). If γ_{t} = 1 + γ^{1}t + γ^{2}t^{2} + ⋯ denotes the generating function, then their basic properties are:

(i) γ_{t}(a + b) = γ_{t}(a)γ_{t}(b). (ii) For a line bundle L on X one has γ_{t}([L] − 1) = 1 + t([L] − 1). (iii) If E is an algebraic vector bundle on X of rank k then γ^{i}([E] − k) = 0

for i > k. The third property follows from the preceding two via the splitting principle. If a sums-of-squares identity of type [r, s, n] exists over a field F, then it also exists over any field containing F. So we may assume F contains a square root of −1. If we write X = ℙ^{s−1} − V_{q}, then by Proposition 2.2 there is a rank n − r bundle ζ on X such that r[ξ] + [ζ] = n in K^{0}(X). This may also be written as r([ξ] − 1) + ([ζ] − (n − r)) = 0 in K̃^{0}(X). Setting ν = [ξ] − 1 and applying the operation γ_{t} we have

γ_{t}(ν)^{r} · γ_{t}([ζ] − (n − r)) = 1

or

γ_{t}([ζ] − (n − r)) = γ_{t}(ν)^{−r} = (1 + tν)^{−r}.

The coefficient of t^{i} on the right-hand-side is (−1)^{i}\binom{r+i−1}{i}ν^{i}, which is the same as −2^{i−1}\binom{r+i−1}{i}ν using the relation ν^{2} = −2ν. Finally, since ζ has rank n − r we know that γ^{i}([ζ] − (n − r)) = 0 for i > n − r. In light of Proposition 2.4, this means that 2^{c} divides 2^{i−1}\binom{r+i−1}{i} for i > n − r, where c = ⌊\frac{s−1}{2}⌋. When i − 1 < c, we can rearrange the powers of 2 to conclude that 2^{c−i+1} divides \binom{r+i−1}{i} for n − r < i ≤ c. □

3. K-theory of deleted quadrics

The rest of the paper deals with the K-theoretic computation stated in Proposition 2.4. This computation is entirely straightforward, and could have been done in the 1970’s. We do not know of a reference, however.

Let Q_{n−1} ↪ ℙ^{n} be the split quadric defined by one of the equations

a_{1}b_{1} +⋯+a_{k}b_{k} = 0 (n = 2k −1)  or  a_{1}b_{1} +⋯+a_{k}b_{k} +c^{2} = 0 (n = 2k).

Beware that in general Q_{n−1} is not the same as the variety V_{q} of the previous section. However, if F contains a square root i of −1 then one can write x^{2} + y^{2} = (x + iy)(x − iy). After a change of variables the quadric V_{q} becomes isomorphic to Q_{n−1}. These ‘split’ quadrics Q_{n−1} are simpler to compute with, and we can analyze the K-theory of these varieties even if F does not contain a square root of −1. Write DQ_{n} = ℙ^{n}−Q_{n−1}, and let ξ be the restriction to DQ_{n} of the tautological line bundle 𝒪(−1) of ℙ^{n}. In this section we calculate K^{0}(DQ_{n}) over any ground field F not of characteristic 2. Proposition 2.4 is an immediate corollary of this more general result:

Theorem 3.1. Let F be a field of characteristic not 2. The ring K^{0}(DQ_{n}) is isomorphic to ℤ[ν]/(2^{c}ν, ν^{2} = −2ν), where ν = [ξ] − 1 generates the reduced group K̃^{0}(DQ_{n}) ≅ ℤ/2^{c} and c = ⌊\frac{n}{2}⌋.

Remark 3.2. We remark again that we are writing K^{0}(X) for what is usually denoted K_{0}(X) in the algebraic K-theory literature. We prefer this notation partly because it helps accentuate the relationship with topological K-theory.

3.3. Basic facts about K-theory. Let X be a scheme. As usual K^{0}(X) denotes the Grothendieck group of locally free coherent sheaves, and G_{0}(X) (also called K′_{0}(X)) is the Grothendieck group of coherent sheaves [Q, Section 7]. Topologically speaking, K^{0}(−) is the analog of the usual complex K-theory functor KU^{0}(−) whereas G_{0} is something like a Borel-Moore version of KU-homology. Note that there is an obvious map α: K^{0}(X) → G_{0}(X) coming from the inclusion of locally free coherent sheaves into all coherent sheaves. When X is nonsingular, α is an isomorphism whose inverse β: G_{0}(X) → K^{0}(X) is constructed in the following way [H, Exercise III.6.9]. If ℱ is a coherent sheaf on X, there exists a resolution

0 → ℰ_{n} → ⋯ → ℰ_{0} → ℱ → 0

in which the ℰ_{i}’s are locally free and coherent. One defines β(ℱ) = ∑_{i}(−1)^{i}[ℰ_{i}]. This does not depend on the choice of resolution, and now αβ and βα are obviously the identities. This is ‘Poincare duality’ for K-theory. Since we will only be dealing with smooth schemes, we are now going to blur the distinction between G_{0} and K^{0}. If ℱ is a coherent sheaf on X, we will write [ℱ] for the class that it represents in K^{0}(X), although we more literally mean β([ℱ]). As an easy exercise, check that if i: U ↪ X is an open immersion then the image of [ℱ] under i^{∗}: K^{0}(X) → K^{0}(U) is the same as [ℱ |_{U}]. We will use this fact often. If j: Z ↪ X is a smooth embedding and i: X − Z ↪ X is the complement, there is a Gysin sequence [Q, Prop. 7.3.2]

⋯ → K^{−1}(X − Z) ⟶ K^{0}(Z) ⟶^{j_{!}} K^{0}(X) ⟶^{i^{∗}} K^{0}(X − Z) ⟶ 0.

(Here K^{−1}(X − Z) denotes the group usually called K_{1}(X − Z), and i^{∗} is surjective because X is regular). The map j_{!} is known as the Gysin map. If ℱ is a coherent sheaf, then j_{!}([ℱ]) equals the class of its pushforward j_{∗}(ℱ) (also known as extension by zero). Note that the pushforward of coherent sheaves is exact for closed immersions.

3.4. Basic facts about ℙ^{n}. If Z is a degree d hypersurface in ℙ^{n}, then the structure sheaf 𝒪_{Z} can be pushed forward to ℙ^{n} along the inclusion Z → ℙ^{n}; we will still write this pushforward as 𝒪_{Z}. It has a very simple resolution of the form 0 → 𝒪(−d) → 𝒪 → 𝒪_{Z} → 0, where 𝒪 is the trivial rank 1 bundle on ℙ^{n} and 𝒪(−d) is the d-fold tensor power of the tautological line bundle 𝒪(−1) on ℙ^{n}. So [𝒪_{Z}] equals [𝒪] − [𝒪(−d)] in K^{0}(ℙ^{n}). From now on we’ll write [𝒪] = 1. Now suppose that Z ↪ ℙ^{n} is a complete intersection, defined by the regular sequence of homogeneous equations f_{1}, …, f_{r} ∈ k[x_{0}, …, x_{n}]. Let f_{i} have degree d_{i}. The module k[x_{0}, …, x_{n}]/(f_{1}, …, f_{r}) is resolved by the Koszul complex, which gives a locally free resolution of 𝒪_{Z}. It follows that

(3.4)  [𝒪_{Z}] = (1 − [𝒪(−d_{1})])(1 − [𝒪(−d_{2})]) ⋯ (1 − [𝒪(−d_{r})])

in K^{0}(ℙ^{n}). In particular, note that for a linear subspace ℙ^{i} ↪ ℙ^{n} one has

[𝒪_{ℙ^{i}}] = (1 − [𝒪(−1)])^{n−i}

because ℙ^{i} is defined by n − i linear equations. One can compute that K^{0}(ℙ^{n}) ≅ ℤ^{n+1}, with generators [𝒪_{ℙ^{0}}], [𝒪_{ℙ^{1}}], …, [𝒪_{ℙ^{n}}] (see [Q, Th. 8.2.1], as one source). If t = 1 − [𝒪(−1)], then the previous paragraph tells us that K^{0}(ℙ^{n}) ≅ ℤ[t]/(t^{n}) as rings. Here t^{k} corresponds to [𝒪_{ℙ^{n−k}}].

3.5. Computations. Let n = 2k. Recall that Q_{2k−1} denotes the quadric in ℙ^{2k} defined by a_{1}b_{1} + ⋯ + a_{k}b_{k} + c^{2} = 0. The Chow ring CH^{∗}(Q_{2k−1}) consists of a copy of ℤ in every dimension (see [DI, Appendix A] or [HP, XIII.4–5], for example). The generators in dimensions k through 2k − 1 are represented by subvarieties of Q_{2k−1} which correspond to linear subvarieties ℙ^{k−1}, ℙ^{k−2}, …, ℙ^{0} under the embedding Q_{2k−1} ↪ ℙ^{2k}. In terms of equations, ℙ^{k−i} is defined by c = b_{1} = ⋯ = b_{k} = 0 together with 0 = a_{k} = a_{k−1} = ⋯ = a_{k−i+2}. The generators of the Chow ring in degrees 0 through k − 1 are represented by subvarieties Z_{i} ↪ ℙ^{2k} (k ≤ i ≤ 2k − 1), where Z_{i} is defined by the equations

0 = b_{1} = b_{2} = ⋯ = b_{2k−1−i}, a_{1}b_{1} + ⋯ + a_{k}b_{k} + c^{2} = 0.

Note that Z_{2k−1} = Q_{2k−1}.

The following result is proven in [R, pp. 128-129] (see especially the first paragraph on page 129):

Proposition 3.6. The group K^{0}(Q_{2k−1}) is isomorphic to ℤ^{2k}, with generators [𝒪_{ℙ^{0}}], …, [𝒪_{ℙ^{k−1}}] and [𝒪_{Z_{k}}], …, [𝒪_{Z_{2k−1}}].

It is worth noting that to prove Theorem 3.1 we don’t actually need to know that K^{0}(Q_{2k−1}) is free—all we need is the list of generators.

Proof of Theorem 3.1 when n is even. Set n = 2k. To calculate K^{0}(DQ_{2k}) we must analyze the localization sequence

⋯ → K^{0}(Q_{2k−1}) ⟶^{j_{!}} K^{0}(ℙ^{2k}) → K^{0}(DQ_{2k}) → 0.

The image of j_{!} : K^{0}(Q_{2k−1}) → K^{0}(ℙ^{2k}) is precisely the subgroup generated by [𝒪_{ℙ^{0}}], …, [𝒪_{ℙ^{k−1}}] and [𝒪_{Z_{k}}], …, [𝒪_{Z_{2k−1}}]. Since ℙ^{i} is a complete intersection defined by 2k − i linear equations, formula (3.4) tells us that [𝒪_{ℙ^{i}}] = t^{2k−i} for 0 ≤ i ≤ k − 1. Now, Z_{2k−1} is a degree 2 hypersurface in ℙ^{2k}, and so [𝒪_{Z_{2k−1}}] equals 1−[𝒪(−2)]. Note that

1 − [𝒪(−2)] = 2(1 − [𝒪(−1)]) − (1 − [𝒪(−1)])^{2} = 2t − t^{2}.

In a similar way one notes that Z_{i} is a complete intersection defined by 2k−1−i linear equations and one degree 2 equation, so formula (3.4) tells us that

[𝒪_{Z_{i}}] = (1 − [𝒪(−1)])^{2k−1−i} · (1 − [𝒪(−2)]) = t^{2k−1−i}(2t − t^{2}).

The calculations in the previous two paragraphs imply that the kernel of the map K^{0}(ℙ^{2k}) → K^{0}(DQ_{2k}) is the ideal generated by 2t−t^{2} and t^{k+1}. This ideal is equal to the ideal generated by 2t − t^{2} and 2^{k}t, so K^{0}(DQ_{2k}) is isomorphic to ℤ[t]/(2^{k}t, 2t − t^{2}). If we substitute ν = [ξ] − 1 = −t, we find ν^{2} = −2ν. To find K̃^{0}(DQ_{2k}), we just have to take the additive quotient of K^{0}(DQ_{2k}) by the subgroup generated by 1. This quotient is isomorphic to ℤ/2^{k} and is generated by ν. □

This completes the proof of Theorem 3.1 in the case where n is even. The computation when n is odd is very similar:

Proof of Theorem 3.1 when n is odd. In this case Q_{n−1} is defined by the equation a_{1}b_{1} + ⋯ + a_{k}b_{k} = 0 with k = \frac{n+1}{2}. The Chow ring CH^{∗}(Q_{n−1}) consists of ℤ in every dimension except for k − 1, which is ℤ ⊕ ℤ. The generators are the Z_{i}’s (k − 1 ≤ i ≤ 2k − 2) defined analogously to before, together with the linear subvarieties ℙ^{0}, ℙ^{1}, …, ℙ^{k−1}. By [R, pp. 128–129], the group K^{0}(Q_{n−1}) is again free of rank 2k on the generators [𝒪_{Z_{i}}] and [𝒪_{ℙ^{i}}]. One finds that K^{0}(DQ_{n}) is isomorphic to ℤ[t]/(2t − t^{2}, t^{k}) = ℤ[t]/(2t − t^{2}, 2^{k−1}t). Everything else is as before. □

References

[A] J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632.

[Ad1] J. Adem, On the Hurwitz problem over an arbitrary field I , Bol. Soc. Mat. Mexicana (2) 25 (1980), no. 1, 29–51.

[Ad2] J. Adem, On the Hurwitz problem over an arbitrary field II , Bol. Soc. Mat. Mexicana (2) 26 (1981), no. 1, 29–41.

[At] M. F. Atiyah, Immersions and embeddings of manifolds, Topology 1 (1962), 125–132.

[AT] M. F. Atiyah and D. O. Tall, Group representations, λ-rings and the J-homomorphism, Topology 8 (1969), 253–297.

[DI] D. Dugger and D. C. Isaksen, The Hopf condition for bilinear forms over arbitrary fields, preprint, 2003.

[E] D. Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995.

[GR] A. V. Geramita and L. G. Roberts, Algebraic vector bundles on projective space, Invent. Math. 10 (1970), 298–304.

[H] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, 1977.

[HP] W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, Vol. II, Cambridge University Press, 1952.

[L] K. Y. Lam, Topological methods for studying the composition of quadratic forms, Quadratic and Hermitian Forms, (Hamilton, Ont., 1983), pp. 173–192, Canadian Mathematical Society Conference Proceedings 4, Amer. Math. Soc., 1984.

[Lw] P. S. Landweber, Fixed point free conjugations on complex manifolds, Ann. Math. (2) 86 (1967), 491–502.

[Q] D. Quillen, Higher algebraic K-theory I, Algebraic K-theory, I: Higher K-theories (Proc. Conf. Battelle Memorial Inst., Seattle, 1972), pp. 85– 147, Lecture Notes in Mathematics 341, Springer, 1973.

[R] L. Roberts, Base change for K_{0} of algebraic varieties, Algebraic Ktheory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, 1972), pp. 122–134, Lecture Notes in Mathematics 342, Springer, 1973.

[SGA6] Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), by P. Berthelot, A. Grothendieck, and L. Illusie, Lecture Notes in Mathematics 225, Springer, 1971.

[Sh] D. B. Shapiro, Products of sums of squares, Expo. Math. 2 (1984), 235– 261.

[Y] S. Yuzvinsky, Orthogonal pairings of Euclidean spaces, Michigan Math. J. 28 (1981), 131–145.

Daniel Dugger Department of Mathematics University of Oregon Eugene, OR 97403, USA ddugger@math.uoregon.edu Daniel C. Isaksen Department of Mathematics Wayne State University Detroit, MI 48202, USA isaksen@math.wayne.edu