An Application of Hermitian K-Theory : Sums-of-Squares Formulas

By using Hermitian K-theory, we improve D. Dugger and D. Isaksen’s condition (some powers of 2 dividing some binomial coefficients) for the existence of sums-of-squares formulas. 2010 Mathematics Subject Classification: 19G38; 11E25; 15A63


Introduction
A sums-of-squares formula of type [r, s, n] over a field F of characteristic = 2 (with strictly positive integers r, s and n) is a formula r i=1 where z i = z i (X, Y ) for each i ∈ {1, . . ., n} is a bilinear form in X and Y (with coefficients in F ), i.e. z i ∈ F [x 1 , . . ., x r , y 1 , . . ., y s ] is homogeneous of degree 2 and F -linear in X and Y .Here, X = (x 1 , . . ., x r ) and Y = (y 1 , . . ., y s ) are coordinate systems.To be specific, z i = k,j c (i) kj x k y j for c (i) kj ∈ F .An old problem of Adolf Hurwitz concerns the existence of sums-of-squares formulas.Historical remarks can be found in [18] and [20].For any m ∈ Z >0 , we let ϕ(m) denote the cardinality of the set {l ∈ Z : 0 < l ≤ m and l ≡ 0, 1, 2 or 4 (mod 8)}.The aim of this paper is to introduce the following result.

Heng Xie
The proof of Theorem 1.1 over R was provided by [2] and [21].It involves computations of topological KO-theory of real projective spaces and γ i -operations.The statement of Theorem 1.1 over R can be extended to any field of characteristic 0 by an algebraic remark of T.Y.Lam and K.Y.Lam, cf.Theorem 3.3 [18].By using algebraic K-theory, D. Dugger and D. Isaksen prove a similar result over an arbitrary field of characteristic = 2, where ϕ(s − 1) in the above theorem is replaced by ⌊ s−1 2 ⌋, cf.Theorem 1.1 [7].They actually conjectured the above statement.Since ϕ(s − 1) ≥ ⌊ s−1 2 ⌋, our main theorem generalizes theirs.One may wish to look at the following table.[15,10,16] which does not exist over F by the above theorem.Neither Hopf's condition [8] nor the weaker condition in [7] can give the non-existence of [15,10,16].
Remark 1.1.The necessary condition of our main theorem does not imply the existence of [r, s, n].To illustrate, [3,5,5] does not exist over the field F by the Hurwitz-Radon theorem.However, it satisfies the necessary condition.
Remark 1.2.The algebraic K-theory analog (cf.Theorem 1.1 [7]) of our main theorem works even if the assumption 'if a sums-of-squares formula of type [r, s, n] exists over F ' is replaced by 'if a nonsingular bilinear map of size [r, s, n] exists over F '.The statement with the latter assumption is 'stronger'.However, this is not the case under our proof, since we will use the sums-of-squares identity (1).
Remark 1.3.The triplet [r, s, n] is independent of the base fields whenever r ≤ 4 and whenever s ≥ n − 2 (cf.Corollary 14.21 [20]), so that the main theorem is true.There is a bold conjecture which states that the existence of [r, s, n] is independent of the base field F (of characteristic = 2), cf.Conjecture 3.8 [18] or Conjecture 14.22 [20].Our main theorem and Dugger-Isaksen's Hopf condition (cf.[8]) suggest this conjecture to some extent.However, as Shapiro points out in Chapter 14 [20], there is indeed very little evidence to support this conjecture.
In [22], it is shown that the Grothendieck-Witt group of a complex cellular variety is isomorphic to the KO-theory of its set of C-rational points with analytic topology.The set of C-rational points of a deleted quadric is homotopy equivalent to the real projective space of the same dimension, cf.Lemma 6.3 [15].Moreover, the computation of topological KO-theory of a real projective space is well-known, cf.Theorem 7.4 [1].We therefore have motivations to work on the Grothendieck-Witt group of a deleted quadric and on the γ i -operations.The proof of our main theorem requires the computation of Grothendieck-Witt group of a deleted quadric which will be explored in Section 3.

Terminology, notation and remark
Let (E, * , η) be a Z[ 1  2 ]-linear exact category with duality.For i ∈ Z, Walter's Grothendieck-Witt groups GW i (E, * , η) are defined in Section 4.3 [16].The triplet (Vect(X), Hom( , L), can) (notation in Example 2.3 [16]) is an exact category with duality.If X is any Z[ 1  2 ]-scheme, then we define By the symbols GW i (X), we mean the groups GW i (X, O).Note that GW 0 (X) is just Knebusch's L(X) which is defined in [14].The notation in [3] is used for the Witt theory.For KO-theory and comparison maps, we refer to [22].
Definition 2.1.Let T be a scheme.For us, a smooth T -variety X is called T -cellular if it has a filtration by closed subvarieties T for each k.In this paper, the following notations are introduced for convenience:

F
-a field of characteristic = 2; K -an algebraically closed field of characteristic = 2; V -the ring of Witt vectors over K; L -the field of fractions of V ; X F -the base-change scheme X × Z[ 1  2 ] F for any Z[ -the quadratic polynomial q s (y) = y 2 1 + . . .+ y 2 s ; V + (q s ) -the closed subscheme of P s−1 defined by q s ; D + (q s ) -the open subscheme P s−1 − V + (q s ) of P s−1 ; ξ -the line bundle O(−1) of P s−1 F restricted to D + (q s ) F ; R -the ring of elements of total degrees 0 in S qs ; P -the R-module of elements of total degrees −1 in S qs ; Remark 2.1.(i) Let E be a field containing √ −1 and of characteristic = 2.Note that (Q s−2 ) E is isomorphic to the projective variety V + (q s ) E , cf.Lemma 2.2 [8].This map induces an isomorphism i E : (DQ s−1 ) E → D + (q s ) E .(ii) Observe that V is a complete DVR with the quotient field K, cf.Chapter II [17].Also, note that the fraction field L of V has characteristic 0, cf.loc.cit.. (iii) The scheme D + (q s ) F is affine over the base field F , since D + (q s ) F and Spec R are isomorphic, cf. the proof of Proposition 2.2 [7].
where n is the trivial bilinear space of the rank n.
Proof.The K-theory analog has been proved, cf.Proposition 2.2 [7].It is clear that the group GW 0 (D + (q s ) F ) is isomorphic to GW 0 (R) by Remark 2.1 (iii).If the equation ( 1) exits, we are able to construct a graded S-module where Y = (y 1 , . . ., y s ) is the coordinate system introduced in Section 1.This map induces a homomorphism α : P r → R n of R-modules by localizing it at q s .The isomorphism It follows that α is injective and (P r , r i=1 σ) can be viewed as a non-degenerate subspace of (R n , −, − R n ) via α.Define ζ to be its orthogonal complement (P r ) ⊥ with the unit form −, − Rn restricting to (P r ) ⊥ .By a basic fact of quadratic form theory, ζ is non-degenerate and ζ⊥(P r , where ϕ(s − 1) is the number defined in Section 1.Therefore, for any rational point ς : Spec K → D + (q s ) K , the reduced Grothendieck-Witt ring Theorem 3.1 will be proved in the next section.
Proof of Theorem 1.1.It is enough to show this theorem over the algebraic closure F of F .Indeed, if [r, s, n] exists over F , then it also exists over F .In order to apply the standard trick (cf. the proof of Theorem 1.3 [7]), we have to take care of γ i -operations on GW 0 (D + (q s ) F ).To be specific, this standard trick can not be applied without the list of three properties (cf.Properties (i)-(iii) in loc.cit.) of γ i -operations and their generating power series γ t = 1+ i>0 γ i t i on GW 0 (D + (q s ) F ). Due to the lack of reference, we will develop γ i -operations on K(Bil(X)) and prove these three properties (see Appendix A).It is enough for our purpose because GW 0 (X) is just K(Bil(X)) if X is affine (see Remark A.1), and the scheme D + (q s ) F is affine by Remark 2.1 (iii).Hence, together with Lemma 3.1, we are allowed to apply the standard trick.One checks that details are the same as in the proof of Theorem 1.3 [7] by replacing K-theory analogs with GW -theory and ⌊ s−1 2 ⌋ with ϕ(s − 1).Combining with a reformulation of powers of 2 dividing correspondent binomial coefficients (cf.Section 1.2 [7]), we are done.
4 Proof of Theorem 3.1

Rigidity and Hermitian K-theory of cellular varieties
On the right-hand side of the diagram (2), the maps of Witt groups are all induced by the correspondent ring maps of the left-hand side for a fixed i ∈ Z.All these Witt groups are trivial if i ≡ 0 (mod 4), cf.Theorem 5.6 [5].Note that β 0 is an isomorphism by Satz 3.3 [13].It is also clear that W 0 (K) is isomorphic to Z/2 and that all the maps on the right-hand side of the diagram (2) preserve multiplicative identities for i = 0. Since Witt groups are four periodicity in shifting, we obtain Moreover, by Karoubi induction (cf.Section 3 [6]), the lefthand side of the digram (2) gives an isomorphism GW i (K) → GW i (C) of Grothendieck-Witt groups.
Lemma 4.2.Let X be a smooth Z[ 1  2 ]-cellular variety.Let f : A → B be a map of regular local rings of finite Krull dimensions with 1/2.Suppose that the map W i (A) → W i (B) induced by f is an isomorphism for each i, then f gives an isomorphism of Witt groups (resp.Grothendieck-Witt groups) for each i and any line bundle L over X.

Heng Xie
Proof.We may use W i (X, L) * to simplify the notation W i (X * , L * ).We wish to prove the Witt theory case by induction on cells.Firstly, note that the pullback maps ) are isomorphisms by homotopy invariance, cf.Theorem 3.1 [4].It follows that In general, the closed subvarieties Z k may not be smooth.However, let U k be the open subvariety Consider the following commutative diagram of localization sequences.
Here, W i C k (U k , L) means the L-twisted ith-Witt group of U k with supports on C k .Note that any line bundle over (C k ) A is trivial, since Pic(A n A ) ∼ = Pic(A) = 0 (A is regular local and so it is a UFD).
By the dévissage theorem (cf.[10]), we deduce that Moreover, by induction hypothesis, Applying the 5-lemma, one sees that the middle vertical map is an isomorphism.Since the K-theory analog of this theorem is also true by induction on cells, the GW -theory cases follow by Karoubi induction, cf.Section 3 [6].
Corollary 4.1.The Witt group (resp.the Grothendieck-Witt group) for each i and any line bundle L over X.

Comparison maps and rank one bilinear spaces
If X is a smooth variety over C, we let X(C) be the set of C-rational points of X with analytic topology.One can define comparison maps (cf.Section 2 [22]) where KO K 0 (X(C)) means the cokernel of the realification map from K 0 (X) to KO 0 (X(C)).Let GW 0 top (X(C)) be the Grothendieck-Witt group of complex bilinear spaces over X(C).The map gw 0 consists of the composition of the following two maps where the map f takes a class [M, φ] on X to the class [M (C), φ(C)] on X(C).
The map g sends a class [N, ǫ] on X(C) to the class represented by the underlying real vector bundle R(N, ǫ) such that R(N, ǫ) ⊗ R C = N and that ǫ| R(N,ǫ) is real and positive definite, cf.Lemma 1.3 [22].Let Q(X) (resp.Q top (X)) denote the group of isometry (resp.isomorphism) classes of rank one bilinear spaces (resp.rank one complex bilinear spaces) over X (resp.X(C)) with the group law defined by the tensor product.There are maps of sets Let Pic R (X(C)) be the group of isomorphism classes of rank one real vector bundles over X(C).
Lemma 4.3.The following diagram is commutative Proof.The square on the left-hand side is obviously commutative.It remains to show that the right-hand side square is commutative.Check that the map g is well-defined.Note that, for each couple of complex bilinear spaces (L ′ , ǫ ′ ) and This contradicts the assumption that the bundle L has rank one.Then, it is clear that g Set S p,q = (S 1 s ) ∧(p−q) ∧ (S 1 t ) ∧q with p ≥ q ≥ 0.Then, S 2,1 and T are A 1weakly equivalent.See Section 3.2 [9] for details and Section 1.4 [22] for discussion.One may take these objects to SH(C).The category SH(C) is triangulated with translation functor S 1,0 ∧ −.Set KO p,q (X ) := [Σ ∞ X , S p,q ∧ KO] and KO p,q (X) := [Σ ∞ X + , S p,q ∧ KO] where X ∈ H • (C) and X ∈ H(C).The object KO ∈ SH(C) is the geometric model of Hermitian K-theory in the A 1 -homotopy theory defined by Schlichting and Tripathi (See Section 1.5 [22]).Moreover, there are isomorphisms GW q (X) ∼ = KO 2q,q (X) and W q (X) ∼ = KO 2q−1,q−1 (X).One defines comparison maps (cf.Section 2 [22]) kp,q h (X ) : KO p,q (X ) → KO p (X (C)) k p,q h (X) : KO p,q (X) → KO p (X(C)).
In particular, when X is a complex smooth variety, we have Theorem 4.1.Let X be a complex smooth cellular variety.Assume further that Z is cellular and closed in X, and let U := X − Z.Then, the map k 2q,q h (U ) is an isomorphism and the map k 2q+1,q h (U ) is injective.
Proof.When Z = ∅, this theorem is a special case of Theorem 2.6 [22].We slightly modify the proof of Theorem 2.6 [22] to show this theorem by induction on cells.
and S 2d,d are A 1 -weakly equivalent, where d is the codimension of C k in U k , cf.Proposition 2.17 (page 112) [9].We can therefore deduce the commutative ladder diagram in Figure 1 (page 486) [22].Assume by induction, the theorem is true for U k , and we want to prove it for U k+1 .It is known that k2q,q h (S 2d,d ) and k2q+1,q h (S 2d,d ) are isomorphisms and that k2q+2,q h (S 2d,d ) is injective, cf. the proof of Theorem 2.6 [22].The results follow by the 5-lemma.

Grothendieck-Witt group of a deleted quadric
In this subsection, we simply write Theorem 4.2.The comparison map gw q : GW q (DQ C ) → KO 2q (DQ(C)) is an isomorphism for each q ∈ Z.
Proof.This theorem is a consequence of Theorem 4.1.
Proof.Applying Corollary 4.1 and the dévissage theorem, we observe that the vertical maps of W and GW -groups in the following commutative diagram are all isomorphisms where all vertical maps are induced from the left-hand side of the diagram (2) (use the 5-lemma to see the middle map Ω is an isomorphism).
Recall the isomorphism of varieties i K : DQ K → X K in Remark 2.1 (i).Note that i C : DQ C → X C gives a homeomorphism i (C) : DQ(C) → X(C) by taking C-rational points.Besides, let υ : RP s−1 → X(C) be the natural embedding.The space RP s−1 is a deformation retract of the space X(C) in the category of real spaces, cf.Lemma 6.3 [15].These maps that induce isomorphisms in KO-theory or GW -theory are described in the diagram (4).
Hermitian K-theory Topological KO-theory Proof of Theorem 3.1.Let ξ top denote the tautological line bundle over RP s−1 .
Recall that there is an isomorphism of rings where ν top represents the class [ξ top ]−1, cf.Section 7 [1] or Chapter IV [12].Note that Pic R (RP s−1 ) is isomorphic to Z/2.Let ϑ : GW 0 (X K ) → KO 0 (RP s−1 ) be the composition of maps in the diagram (4).We have known ϑ is an isomorphism.Therefore, to prove Theorem 3.1, we only need to show ϑ(ν) = ν top .To achieve this, we give the following lemma.
Proof.There is an exact sequence (cf.Chapter IV.1 (page 229) [14]) where 2 Pic(X K ) means the subgroup of elements of order ≤ 2 in Pic(X K ) and where F is the forgetful map.Note that 2 Pic(X K ) ∼ = Z/2, cf.[19].In addition, observe that O(X K ) * ∼ = R * = K * and that the group K * /K (obtained in an obvious way) such that the following diagram is commutative The map i is injective (Note that [ξ] and 1 are distinct elements in K 0 (X K ) by its computation in Proposition 2.4 [7]).The map j is injective by the computation of KO 0 (RP s−1 ).Then, we see that θ is bijective and must send [ξ, σ] to [ξ top ].Therefore, ϑ([ξ, σ]) = [ξ top ], so that ϑ(ν) = ν top .

A Operations on the Grothendieck-Witt group
The γ i -operations on GW 0 of an affine scheme are analogous to those on the topological KO-theory which have been explained in Section 1 and 2 in [2].For readers' convenience, details have been added.
Let Bil(X) be the set of isometry classes of bilinear spaces over a scheme X.The orthogonal sum and the tensor product of bilinear spaces over the scheme X make Bil(X) a semi-ring with a zero and a multiplicative identity.Then, by taking the associated Grothendieck ring K(Bil(X)), we have a homomorphism of the underlying semi-rings ι : Bil(X) → K(Bil(X)) satisfying the universal property (see Chapter I.4 (page 137) [14] for details).
Definition A.1 (Chapter IV.3 (page 235) [14]).Let (F , φ) be a bilinear space over a scheme X.Let i be a strictly positive integer.The i-th exterior power of (F , φ), denoted by Λ i (F , φ), is the symmetric bilinear space (Λ i F , Λ i φ) over X, where Λ i F is the i-th exterior power of the locally free sheaf F and where is a morphism of sheaves consisting of a symmetric bilinear form for each open subscheme U of X.The exterior power Λ 0 (F , φ) for every bilinear space (F , φ) (over X) is defined as 1 = (O, id).
If I : (F , φ) → (G, ψ) is an isometry of bilinear spaces, so is the natural map Then, the map Λ t is well-defined.Furthermore, Lemma A.1 (b) implies that Λ t is a homomorphism of the underlying monoids.By the universal property of K-theory, we can lift Λ t to a homomorphism of groups λ t : K(Bil(X)) → A(X) such that λ t • ι = Λ t .Taking coefficients of λ t , we get operators (not homomorphisms in general) λ i : K(Bil(X)) → K(Bil(X)).
Explicitly, we deduce i≥0 Hence, the γ i are certain Z-linear combinations of the λ s .By definition, the map γ t is a homomorphism of groups.Hence, for all x, y ∈ K(Bil(X)), we have

Lemma 3 . 1 .
If a sums-of-squares formula of type [r, s, n] exists over F , then there exist a non-degenerate bilinear form σ

Heng Xie 4 . 3 1 s− 1 t−
Comparison maps and cellular varieties Let H(C) (resp.SH(C)) be the unstable A 1 -homotopy category (resp.the stable A 1 -homotopy category) over C. Let H • (C) be the pointed version of H(C).There are objects in H • (C): S the constant sheaf represented by ∆ 1 /∂∆ 1 pointed canonically; S the sheaf represented by A 1 − {0} pointed by 1; T − the sheaf represented by the projective line P 1 pointed by ∞.