ON THE MOTIVIC COMMUTATIVE RING SPECTRUM BO

IVAN PANIN AND CHARLES WALTER

Abstract. We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on 𝒮m𝒪p/S the cohomology theory (X, U) ↦ BO^{p,q}(X_{+}/U_{+}) and Schlichting’s hermitian K-theory functor (X, U) ↦ KO_{2q−p}^{[q]}(X, U) are canonically isomorphic. We use the motivic weak equivalence ℤ × HGr ⟶^{∼} KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is Spec ℤ[ \frac{1}{2} ], this monoid structure and the induced ring structure on the cohomology theory BO^{∗,∗} are the unique structures compatible with the products

KO_{0}^{[2m]}(X) × KO_{0}^{[2n]}(Y) → KO_{0}^{[2m+2n]}(X × Y). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO^{∗,∗}(T ∧T) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space ⟨−1⟩.

1. Introduction

In a recent paper [16] we defined motivic versions of symplectically oriented cohomology theories A and of quaternionic Grassmannians HGr(r, n). This HGr(r, n) is the open subscheme of the ordinary Grassmannian Gr(2r, 2n) parametrizing subspaces on which the standard symplectic form on 𝒪^{⊕2n} is nondegenerate. We defined Pontryagin classes of symplectic bundles in such theories and calculated

A(HGr(r, n)) = A(pt)[p_{1}, …, p_{r}]/(h_{n−r+1}, …, h_{n})  (1.1)

where the p_{i} are the Pontryagin classes of the tautological bundle on HGr(r, n), and the h_{i} are the polynomials in the p_{i} corresponding to the complete symmetric polynomials. This is the same formula as the one which describes the cohomology of an ordinary Grassmannian in terms of the Chern classes of an oriented cohomology theory.

In this paper we begin to apply those results to the hermitian K-theory of regular noetherian separated schemes X of finite Krull dimension with \frac{1}{2} ∈ Γ(X, 𝒪_{X}). We write KO^{[n]}(X, U) for Schlichting’s hermitian K-theory space for bounded complexes of vector bundles on X which are acyclic on the open subscheme U ⊂ X and which are symmetric with respect to the shift by n of the usual duality. We write KO_{i}^{[n]}(X, U) for its homotopy groups (for i ≥ 0) or for Balmer’s Witt groups W^{n−i}(X, U) (for i < 0).

One of our main results is the following.

Date: September 20, 2010.

The first author thanks for the support and excellent working conditions Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice Sophia Antipolis, the DFG–RFFI project 09-01-91333-NNIO-a the Presidium of RAS Program “Fundamental Research in modern mathematics” and the RFFI-grant 10-01-00551.

Theorem 1.1. For a regular separated noetherian scheme S of finite Krull dimension with \frac{1}{2} ∈ Γ(S, 𝒪_{S}) Schlichting’s hermitian K-theory is a ring cohomology theory with an SL^{c} Thom classes theory.

Here ring cohomology theory is used in the sense of [14, Definitions 2.1 and 2.13]. An SL^{c} Thom classes theory specifies a Thom class th(E, L, λ) ∈ KO_{0}^{[n]}(E, E − X) for every SL_{n}^{c}-bundle, by which we mean a triple (E, L, λ) with E a vector bundle of rank n over X, L a line bundle and λ: L ⊗ L → det E an isomorphism. These classes are functorial, multiplicative, and induces isomorphisms ∪ th(E, L, λ): KO_{i}^{[m]}(X) → KO_{i}^{[m+n]}(E, E−X) for all i and m. The Thom classes restrict to Euler classes e(E, L, λ) ∈ KO_{0}^{[n]}(X). An SL^{c} Thom classes theory gives Thom classes for all special linear, special orthogonal and symplectic bundles. So by the theory of [16] there are Pontryagin classes p_{i}(E, ϕ) ∈ KO_{0}^{[2i]}(X) for symplectic bundles. The p_{1}(E, ϕ) of a symplectic bundle of rank 2r is the class corresponding to [E, ϕ] − r[H] ∈ KSp_{0}(X, U) = GW^{−}(X) under the natural isomorphism KSp ≅ KO^{[2]}. Here H is the trivial symplectic bundle of rank 2. The higher Pontryagin classes will be calculated elsewhere.

We also construct several motivic spectra representing hermitian K-theory. The first construction is a T -spectrum whose spaces (KO^{[0]}, KO^{[1]}, KO^{[2]}, …) are fibrant replacements of presheaves composed of Schlichting’s Waldhausen-like hermitian K-theory spaces for bounded complexes of vector bundles with shifted dualities [19]. The structure maps KO^{[n]} ∧ T → KO^{[n+1]} are adjoint to the maps KO^{[n]}(−) → KO^{[n+1]}(− ∧ T) which are essentially multiplication by the Thom class th ∈ KO_{0}^{[1]}(T) of the trivial line bundle. Note the use of the appearance of the Thom classes in the very structure of the spectrum. For (X, U) in 𝒮m𝒪p/S there are functorial isomorphisms

KO_{i}^{[n]}(X, U) ≅ KO_{i}^{[n]}(X_{+}/U_{+}) ≅ BO^{2n−i,n}(Σ_{T}^{∞}(X_{+}/U_{+})),  (1.2)

and the boundary maps ∂: KO_{i}^{[n]}(U) → KO_{i−1}^{[n]}(X, U) and

∂: BO^{2n−i,n}(Σ_{T}^{∞}(U_{+})) → BO^{2n−i+1,n}(Σ_{T}^{∞}(X_{+}/U_{+}))

correspond. Because it is based on the hermitian K-theory of chain complexes, this spectrum has advantages in certain situations over the one constructed several years ago by Hornbostel [8]. It treats all shifts/weights uniformly instead of dealing in one way with the K^{h}-theory of the even weights and in another way with the U-theory and V -theory of the odd weights. It naturally handles non-affine schemes X and even pairs (X, U) with U ⊂ X open. Finally we can easily identify the Thom, Euler and Pontryagin classes in the Grothendieck-Witt groups of chain complexes GW^{[n]}(X on X − U).

We show that the Morel and Voevodsky’s theorem on Grassmannians and algebraic Ktheory extends to the symplectic context. There are in truth only a few things to verify for symplectic groups beyond what is in Morel and Voevodsky’s paper. (Orthogonal groups are much more problematic. However it has been done recently by M.Schlichting and Shanker Tripathi.)

Theorem 1.2. Let HGr = colim HGr(n, 2n) be the infinite quaternionic Grassmannian. Then ℤ × HGr and KSp are isomorphic in the motivic unstable homotopy category H_{•}(S).

Next we define the × product. The final group of theorems in the paper concerns the product structure. The groups KO_{0}^{[n]}(X, U), which are the Grothendieck-Witt groups of bounded chain complexes of vector bundles which are symmetric with respect to shifted but untwisted dualities, have a naive product induced by the tensor product of chain complexes

KO_{0}^{[m]}(X, U) × KO_{0}^{[n]}(Y, V) → KO_{0}^{[m+n]}(X × Y, X × V ∪ U × Y).  (1.3)

Let ⟨1⟩ and ⟨−1⟩ in KO_{0}^{[0]}(pt) denote the Grothendieck-Witt classes of the rank one symmetric bilinear forms. The product is ⟨−1⟩-commutative meaning that for α ∈ BO^{p,q}(A) and β ∈ BO^{p′,q′}(B) we have α × β = (−1)^{pp′}⟨−1⟩^{qq′}σ^{∗}(β × α) where σ: A × B → B × A switches the factors. Recall that a motivic space A is called small if Hom_{SH(S)}(Σ_{T}^{∞}A, −) commutes with arbitrary coproducts.

Theorem 1.3. The cohomology theory (BO^{∗,∗}, ∂) on the category M_{•}^{small}(S) of small motivic spaces over S has a product × which is associative and ⟨−1⟩-commutative with the unit 1 = ⟨1⟩ ∈ BO^{0,0}(pt_{+}), which has α × Σ_{𝔾_{m}}1 = Σ_{𝔾_{m}}α and α × Σ_{S_{s}^{1}}1 = Σ_{S_{s}^{1}}α for all α, and which

restricts via the isomorphism (1.2) to the naive ring structure (1.3) on the groups KO_{0}^{[2n]}(X) for X ∈ 𝒮m/S. It is the unique product with these properties.

This is Theorems 11.5 and 11.6. Restricting to pairs (X, U) with U ⊂ X an open subscheme of a scheme smooth over X, we get the following result.

Theorem 1.4. There is a canonical ring structure on the cohomology theory (KO_{∗}^{[∗]}, ∂) on 𝒮m𝒪p/S which is associative and ⟨−1⟩-commutative with unit ⟨1⟩ ∈ KO_{0}^{[0]}(pt) and which restricts to the naive product on the Grothendieck-Witt groups of chain complexes KO_{0}^{[2n]}(X). This product and the Thom classes of SL^{c}-bundles make (KO_{∗}^{[∗]}, ∂) ring cohomology theory with an SL^{c} Thom classes theory.

Our strongest result on the product is the following theorem.

Theorem 1.5. There exist morphisms m: BO ∧ BO → BO and e: Σ_{T}^{∞}1 → BO in SH(S) which make (BO, m, e) a commutative monoid in SH(S) and which are compatible with the naive product in the following sense.

(1) For all X and Y in 𝒮m/S and all even integers 2p and 2q the naive products

KO_{0}^{[2p]}(X) × KO_{0}^{[2q]}(Y) → KO_{0}^{[2p+2q]}(X × Y)

and the product

BO^{4p,2p}(X_{+}) × BO^{4q,2q}(Y_{+}) → BO^{4p+4q,2p+2q}(X_{+} ∧ Y_{+})

induced by m correspond under the isomorphisms (1.2).

(2) The elements ⟨1⟩ ∈ KO_{0}^{[0]}(pt) and e ∈ BO^{0,0}(S^{0,0}) correspond under the isomorphisms (1.2).

Moreover, if for the base scheme S the groups KO_{1}(S) and KSp_{1}(S) are finite (for example S = Spec ℤ[\frac{1}{2}]), then the monoid structure (m, e) with these properties is unique.

For the proof of the theorem see Theorem 13.5.

We explain the basic ideas in the proofs of these three theorems. Gille and Nenashev’s method [7] for constructing pairings in Witt groups of triangulated categories can be used in hermitian K-theory to construct pairings between hermitian K-theory groups and Grothendieck-Witt groups

KO_{r}^{[i]}(X, U) × KO_{0}^{[j]}(Y, V) → KO_{r}^{[i+j]}(X × Y, X × V ∪ U × Y)  (1.4)
KO_{0}^{[i]}(X, U) × KO_{r}^{[j]}(Y, V) → KO_{r}^{[i+j]}(X × Y, X × V ∪ U × Y)  (1.5)

(see (5.4c)–(5.4d)). They respect the boundary maps of the cohomology theorem KO_{∗}^{[∗]}. This gives the hermitian K-theory groups the structure of a cohomology theory with a partial multiplicative structure with Thom classes for all SL^{c} bundles including symplectic bundles. Although this is less structure than we assumed while writing [16], it is enough to prove the quaternionic projective bundle theorem and the symplectic splitting principle and to calculate the cohomology of quaternionic Grassmannians. Thus formula (1.1) holds for A = KO_{∗}^{[∗]} with the p_{i} ∈ KO_{0}^{[2i]}(HGr(r, n)). Because of the isomorphism (1.2) it also holds for A = BO^{∗,∗} with the p_{i} ∈ BO^{4i,2i}(HGr(r, n)).

The isomorphism (1.2) also transplants these pairings to BO, giving in particular pairings

⊠: BO^{2i−r,i}(X_{+}) × BO^{2j,j}(Y_{+}) → BO^{2i+2j−r,i+j}(X_{+} ∧ Y_{+})  (1.6)

Theorem 1.2 and the isomorphisms KSp ≅ KO^{[4k+2]} gives us a canonical elements τ_{4k+2} ∈ BO^{8k+4,4k+2}(ℤ × HGr). Write [−n, n] = {m ∈ ℤ | −n ≤ m ≤ n} and set

HGr_{n} = [−n, n] × HGr(n, 2n).

We have ℤ × HGr = colim HGr_{n}. A standard formula for homotopy colimits in triangulated categories gives us an exact sequence

0 → \underleftarrow{lim}^{1} BO^{8k+3,4k+2}(HGr_{n}) → BO^{8k+4,4k+2}(ℤ × HGr) → \underleftarrow{lim} BO^{8k+4,4k+2}(HGr_{n}) → 0

The inclusions HGr_{n} → HGr_{n+1} induce surjections on cohomology for any symplectically oriented theory. So the \underleftarrow{lim}^{1} vanish, and we have

BO^{8k+4,4k+2}(ℤ × HGr) ≅ \underleftarrow{lim} BO^{8k+4,4k+2}(HGr_{n}).  (1.7)

For essentially the same reasons we have isomorphisms

BO^{16k+8,8k+4}((ℤ × HGr) ∧ (ℤ × HGr)) ≅ \underleftarrow{lim} BO^{16k+8,8k+4}(HGr_{n} ∧ HGr_{n}).  (1.8)

Our class τ_{4k+2} ∈ BO^{8k+4,4k+2}(ℤ × HGr) and the pairing (1.6) gives us a system of classes

τ_{4k+2}|_{HGr_{n}} ⊠ τ_{4k+2}|_{HGr_{n}} ∈ BO^{16k+8,8k+4}(HGr_{n} ∧ HGr_{n}).  (1.9)

and therefore a class

τ_{4k+2} ⊠ τ_{4k+2} ∈ BO^{16k+8,8k+4}(KO^{[4k+2]} ∧ KO^{[4k+2]}).

There is also an exact sequence of the form

0 → \underleftarrow{lim}^{1} BO^{4i−1,2i}(KO^{[i]} ∧ KO^{[i]}) → BO^{0,0}(BO ∧ BO) → \underleftarrow{lim} BO^{4i,2i}(KO^{[i]} ∧ KO^{[i]}) → 0.

The elements τ_{4k+2} ⊠ τ_{4k+2} define an element m̄ ∈ \underleftarrow{lim} BO^{4i,2i}(KO^{[i]} ∧ KO^{[i]}). This m̄ is the unique element compatible with the naive product (1.9) and the isomorphisms (1.7) and (1.8). Lifting this element to m ∈ Hom_{SH(S)}(BO ∧ BO, BO) gives an element we can use to define the product ×. On small motivic spaces × depends only on m̄ and not on the choice of m. We deduce the associativity and bigraded commutativity of × on small motivic spaces from the associativity and commutativity of the naive product on the KO_{0}^{[2r]}(HGr_{n}). This gives us Theorem 1.3.

Theorem 1.5 is more subtle. The obstructions to the uniqueness of m and to the associativity, commutativity and unit property of the monoid it defines all live in certain \underleftarrow{lim}^{1} groups. We show in §13 that when S = Spec R with \frac{1}{2} ∈ R and with KO_{1}(R) and KSp_{1}(R) finite groups, the \underleftarrow{lim}^{1} vanish. This uses the construction in §12 of three new spectra using a new motivic sphere.

The geometry used to prove the quaternionic projective bundle theorem in [16] also shows that the pointed quaternionic projective line (HP^{1}, x_{0}) is isomorphic to T^{∧2} in the motivic homotopy category H_{•}(S). The pointed scheme HP^{1+} which is the A^{1} mapping cone of the pointing x_{0} : pt → HP^{1} is therefore also homotopy equivalent to T^{∧2}. It is the union of HP^{1} and A^{1} with x_{0} ∈ HP^{1} identified with 0 ∈ A^{1}, pointed at 1 ∈ A^{1}. Therefore the motivic stable homotopy categories of T -spectra and of HP^{1+}-spectra are equivalent.

We construct three HP^{1+}-spectra BO_{HP^{1+}}, BO^{geom} and BO^{fin} representing hermitian Ktheory. The spaces of BO_{HP^{1+}} are the even-indexed spaces (KO^{[0]}, KO^{[2]}, KO^{[4]}, …) of the T -spectrum. The spaces of BO^{geom} are alternately ℤ×RGr and ℤ×HGr. The spaces of BO^{fin} are finite unions of finite-dimensional real and quaternionic Grassmannians. Here a real Grassmannian RGr(r, 2n) is the open subscheme of the ordinary Grassmannian Gr(r, 2n) where the hyperbolic quadratic form on 𝒪^{⊕2n} is nondegenerate, while RGr = colim RGr(n, 2n). For the details of BO^{fin} see Theorem 12.3. The structure maps of the spectra are all essentially multiplication with the Euler class −p_{1}(𝒰) of the tautological rank 2 symplectic subbundle on HP^{1}. The bonding maps BO_{2i}^{∗} ∧ HP^{1+} → BO_{2i+2}^{∗} of the two geometric spectra are morphisms of schemes or ind-schemes which are constant on the wedge BO_{2i}^{∗} ∨ HP^{1+}. The inclusion BO^{fin} → BO^{geom} is a motivic stable weak equivalence, while the isomorphism BO^{geom} ≅ BO_{HP^{1}} in SH(S) is constructed from classifying maps

τ_{4k}: ℤ × RGr → KO^{[4k]},  τ_{4k+2} : ℤ × HGr ⟶^{∼} KO^{[4k+2]},

in H_{•}(S). The τ_{4k+2} are the isomorphisms of Theorem 1.2, while the τ_{4k} are constructed from the τ_{4k+2}. We do not know if the τ_{4k} are isomorphisms, although stably they have a right inverse (Proposition 12.7).

The \underleftarrow{lim}^{1} calculated with BO and with BO^{fin} are the same. Calculations based on the quaternionic projective bundle theorem show that for BO^{fin} the \underleftarrow{lim}^{1} are over inverse systems of groups which are finite direct sums of copies of KO_{1}(S) and KSp_{1}(S). When those groups are finite, the \underleftarrow{lim}^{1} vanishes. This gives Theorem 1.5 for S = Spec ℤ[\frac{1}{2}]. For other S one pulls the structure back from Spec ℤ[\frac{1}{2}] using the closed motivic model structure of [15].

Theorem 1.6. The products of Theorems 1.3, 1.4 and 1.5 are compatible with all the naive products of (1.3) and with the partial multiplicative structure of (1.4) and (1.5).

We do not know how to prove this theorem using our construction of the hermitian Ktheory product. But Marco Schlichting has described to us (oral communication) how to put a pairing on his hermitian K-theory spaces when one has a pairing of complicial exact categories with weak equivalences and duality. When applied to our situation his product is isomorphic to ours on small motivic spaces by Theorem 1.3. Since Schlichting’s product is compatible with all the naive products of (1.3) and with the partial multiplicative structure of (1.4) and (1.5), therefore ours is as well.

We finish the paper in §14 by giving the analogue for algebraic K-theory of the spectra BO^{fin} and BO^{geom} of hermitian K-theory of §12. The BGL^{fin} seems to be completely new.

We write CGr(r, n) for the affine Grassmannian, CP^{1} = CGr(1, 2) for the affine version of P^{1}, and CP^{1+} for the A^{1} mapping cone of the pointing map of CP^{1}.

Theorem 1.7. There are CP^{1+}-spectra BGL^{fin} and BGL^{geom} isomorphic to BGL_{CP^{1+}} in SH_{CP^{1+}}(S) with spaces

BGL_{n}^{fin} = [−4^{n}, 4^{n}] × CGr(4^{n}, 2 · 4^{n}),  BGL_{n}^{geom} = ℤ × CGr,

which are unions of affine Grassmannians. The bonding maps BGL_{n}^{∗} ∧ CP^{1+} → BGL_{n+1}^{∗} of the two spectra are morphisms of schemes or ind-schemes which are constant on the wedge BGL_{n}^{∗} ∨ CP^{1+}.

The spectum BGL^{fin} can be used to give alternate proofs of the uniqueness results of [15] concerning the P^{1}-spectrum representing algebraic K-theory and the commutative monoid structure on that spectrum. These proofs avoid the use of topological realization and apply to any noetherian base scheme S of finite Krull dimension with finite K_{1}(S).

2. Cohomology theories

We fix a base scheme S which is regular noetherian separated of finite Krull dimension and with \frac{1}{2} ∈ Γ(S, 𝒪_{S}). The hermitian K-theory of such schemes is simpler than for other schemes, and we wish to avoid the complications of negative hermitian K-theory and of characteristic 2.

Let 𝒮m/S be the category of smooth S-schemes of finite type. Let 𝒮m𝒪p/S be the category whose objects are pairs (X, U) with X in 𝒮m/S and U ⊂ X an open subscheme and whose morphisms f: (X, U) → (Y, V) are morphisms f: X → Y of S-schemes having f(U) ⊂ V. We write X for (X, ∅). The base scheme itself will often be written as S = pt.

A cohomology theory on 𝒮m𝒪p/S [14, Definition 2.1] is a pair (A, ∂) with A a contravariant functor from 𝒮m/S to the category of abelian groups having localization exact sequences and satisfying étale excision and homotopy invariance. The ∂ is a morphism of functors with components ∂_{X,U}: A(U) → A(X, U) which are the boundary maps of the localization exact sequences. A ring cohomology theory in the sense of [14, Definition 2.13] has products

×: A(X, U) × A(Y, V) → A(X × Y, (X × V) ∪ (U × Y))

which are functorial, bilinear and associative and and which have a two-sided unit 1_{A} ∈ A(pt) and satisfy ∂(α × β) = ∂α × β.

A cohomology theory also defines groups A(X, x) for pointed smooth schemes and their smash products such as

A((X_{1}, x_{1}) ∧ (X_{2}, x_{2})) = ker(A(X_{1} × X_{2}) ⟶^{(x_{1}^{∗}×1,1×x_{2}^{∗})} A(pt × X_{2}) ⊕ A(X_{1} × pt)). (2.1)

A bigraded cohomology theory (A^{∗,∗}, ∂) is one in which the groups are bigraded, that is A^{∗,∗}(X, U) = ⨁_{p,q∈ℤ} A^{p,q}(X, U), the pullback maps are homogeneous of bidegree (0, 0) and the boundary maps ∂_{X,U} are homogeneous of bidegree (1, 0). In a bigraded ring cohomology theory (A, ∂, ×, 1) the × products respects the bigrading and we have 1 ∈ A^{0,0}(pt).

Definition 2.1. Let (A, ∂, ×, 1) bigraded ring cohomology theory, and suppose ε ∈ A^{0,0}(pt) = A^{0,0}(pt) satisfies ε^{2} = 1. Then (A, ∂, ×, 1) is ε-commutative if for α ∈ A^{p,q}(X, U) and β ∈ A^{r,s}(Y, V) one has σ^{∗}(α × β) = β × α × (−1)^{pr}ε^{qs} where σ: Y × X → X × Y switches the factors.

Equivalently a bigraded ring cohomology is ε-commutative if the associated cup product satisfies α∪β = (−1)^{pr}ε^{qs} β∪α for α ∈ A^{p,q}(X, U) and β ∈ A^{r,s}(X, V). For such a cohomology theory the A^{∗,∗}(X) are bigraded-commutative rings, and for any (X, U) the A^{∗,∗}(X, U) and A^{∗,∗}(U) are right and left bigraded A^{∗,∗}(X)-modules. The ∂_{X,U} are morphisms of right A^{∗,∗}(X)-modules.

Sometimes it is easier to define certain products than others. For a bigraded cohomology theory (A^{∗,∗}, ∂) set A^{0}(X, U) = ⨁_{p∈ℤ} A^{2p,p}(X, U). We need the following notion.

Definition 2.2. Let (A^{∗,∗}, ∂) be a bigraded cohomology theory as above. An ε-commutative partial multiplication on (A^{∗,∗}, ∂) is given by

(1) pairings ×: A^{p,q}(X, U) × A^{2r,r}(Y, V) → A^{p+2r,q+r}((X, U) ∧ (Y, V)) which are bilinear and functorial, and

(2) elements 1 and ε in A^{0,0}(pt) satisfying

(a) α × (b × c) = (α × b) × c for α ∈ A^{p,q}(X, U), b ∈ A^{2r,r}(Y, V), c ∈ A^{2s,s}(Z, W);

(b) α × 1 = α for α ∈ A^{p,q}(Y, V),

(c) ε × ε = 1,

(d) a × b = σ^{∗}(b × a) × ε^{rs} for a ∈ A^{2r,r}(X, U), b ∈ A^{2s,s}(Y, V) where σ: X × Y → Y × X switches the factors;

(e) ∂_{Y ×X,V ×X}(α × b) = ∂_{Y,V} (α) × b for α ∈ A^{p,q}(V), b ∈ A^{2r,r}(X).

If (A^{∗,∗}, ∂) has such a partial multiplication, then for α ∈ A^{p,q}(X, V) and b ∈ A^{2r,r}(X, U) one has a cup product

α ∪ b = Δ^{∗}(α × b) ∈ A^{p+2r,q+r}(X, U ∪ V).

If (A, ∂) is equipped with a partial multiplicative structure (×, 1, ε), then the functor (X, U) ↦ A^{0}(X, U) is an ε-commutative graded ring functor in the sense that the properties (a), (b), (c) and (d) hold for α ∈ A^{0}(Y, V).

Moreover, A is a bigraded right A^{0}-module in the same sense with ∂ a morphism of bigraded right A^{0}-modules which is homogeneous of bidegree (1, 0).

The switch σ: X × Y → Y × X allows us to define pairings ×: A^{2r,r}(X, U) × A^{p,q}(Y, V) → A^{p+2r,q+r}((X, U) × (Y, V)) by b × α = σ^{∗}(α × b) × ε^{qr}. There are also cup products b ∪ α = Δ^{∗}(b × α). The two pairings are compatible by (d). Thus A is a bigraded left and right A^{0}-module, with ∂ a morphism of right A^{0}-modules.

3. SL and SL^{c} orientations

We discuss SL oriented cohomology theories. Hermitian K-theory will turn out to be one. We also include a discussion of Thom classes for vector bundles whose structural group is the double cover SL_{n}^{c} of GL_{n}. It contains SL_{n}. We believe this is the true level at which Witt groups and hermitian K-theory are oriented.

An SL bundle on X is a pair (E, λ) with E a vector bundle over X and λ: 𝒪_{X} ≅ det E an isomorphism. An isomorphism of SL bundles f: (E, λ) ≅ (E_{1}, λ_{1}) is an isomorphism f: E ≅ E_{1} such that λ_{1} = det f ∘ λ.

Definition 3.1. An SL orientation on a bigraded cohomology theory A^{∗,∗} with an ε-commu-tative partial multiplication or ring structure is an assignment to every SL bundle (E, λ) over every X in 𝒮m/S of a class th(E, λ) ∈ A^{2n,n}(E, E − X) for n = rk E satisfying the following conditions:

(1) For an isomorphism f: (E, λ) ≅ (E_{1}, λ_{1}) we have th(E, λ) = f^{∗} th(E_{1}, λ_{1}).

(2) For u: Y → X we have u^{∗} th(E, λ) = th(u^{∗}E, u^{∗}λ) in A^{2n,n}(u^{∗}E, u^{∗}E − Y).

(3) The maps − ∪ th(E, λ): A^{∗,∗}(X) → A^{∗+2n,∗+n}(E, E − X) are isomorphisms.

(4) We have

th(E_{1} ⊕ E_{2}, λ_{1} ⊗ λ_{2}) = q_{1}^{∗} th(E_{1}, λ_{1}) ∪ q_{2}^{∗} th(E_{2}, λ_{2}),

where q_{1}, q_{2} are the projections from E_{1} ⊕ E_{2} onto its factors. The class th(E, λ) is the Thom class of the SL bundle, and e(E, λ) = z^{∗} th(E, λ) ∈ A^{2n,n}(X) is its Euler class.

This definition is analogous to the Thom classes theory version of the definition of an orientation [14, Definition 3.32] or of a symplectic orientation [16, Definition 14.2].

The Thom and Euler classes of SL bundles are not necessarily central in contrast with the classes in the oriented and symplectically oriented theories of [14] and [16]. But for an SL bundle of rank n the Thom and Euler classes are in bidegree (2n, n), and such classes need not be central when n is odd and ε ≠ 1. Centrality occcurs for oriented theories because they have ε = 1 and for symplectically oriented theories because the Thom and Pontryagin classes of symplectic bundles are in bieven bidegrees (4r, 2r).

Twisted versions of cohomology groups with coefficients in a line bundle can be defined for any SL oriented theory by

A^{p,q}(X; L) = A^{p+2,q+1}(L, L − X)

and more generally by A^{p,q}(X, X − Z; L) = A^{p+2,q+1}(L, L − Z) for closed subsets Z ⊂ X.

Theorem 3.2. Let E be a vector of rank n over X. Suppose that A^{∗,∗} is an SL oriented bigraded cohomology theory. Then there are canonical isomorphisms of bigraded right A^{0}(X) or A^{∗,∗}(X)-modules A^{∗+2n,∗+n}(E, E − X) ≅ A^{∗,∗}(X; det E).

Proof. Write L_{E} = det E. There are canonical isomorphisms

λ_{1} : 𝒪_{X} ≅ det(E ⊕ L_{E}^{∨}),  λ_{2} : 𝒪_{X} ≅ det(L_{E} ⊕ L_{E}^{∨}).

This gives us SL bundles (E ⊕ L_{E}^{∨}, λ_{1}) and (L_{E} ⊕ L_{E}^{∨}, λ_{2}) over X. The pullback of the first bundle along q: L_{E} → X gives an SL bundle whose structural map is the first projection L_{E} ⊕ E ⊕ L_{E}^{∨} ↠ L_{E}. The pullback of the second bundle along p: E → X and permutation of the summands gives an SL bundle whose structural map is the second projection L_{E} ⊕ E ⊕ L_{E}^{∨} ↠ E. We now have canonical isomorphisms

\xymatrix{A^{∗+2n,∗+n}(E, E − X) \ar[r]_{≅}^{p^{∗} th(L_{E}⊕L_{E}^{∨},λ_{2})∪} & A^{∗+2n+4,∗+n+2}(E ⊕ L_{E} ⊕ L_{E}^{∨}, E ⊕ L_{E} ⊕ L_{E}^{∨} − X) \ar[d]_{≅}^{ε^{n}} \\ A^{∗+2,∗+1}(L_{E}, L_{E} − X), \ar[r]^{≅}_{q^{∗} th(E⊕L_{E}^{∨},λ_{1})∪} & A^{∗+2n+4,∗+n+2}(L_{E} ⊕ E ⊕ L_{E}^{∨}, L_{E} ⊕ E ⊕ L_{E}^{∨} − X)}

and the bottom left module is A^{∗,∗}(X; det E) by definition. The sign ε^{n} is appropriate when one permutes the rank 1 bundles and the rank n bundle. □

Hermitian K-theory and Witt groups have more Thom classes than just those for SL bundles because of what are often called periodicity isomorphisms such as W^{∗}(X; L) ≅ W^{∗}(X; L ⊗ L_{1}^{⊗2}). However, these periodicity isomorphisms depend on choices. A good way to structure these choices is to talk about SL^{c} bundles, using extra structure analogous to the Spin^{c} structures frequently used in differential geometry.

An SL^{c} vector bundle on X is a triple (E, L, λ) with E a vector bundle, L a line bundle, and λ: L ⊗ L ≅ det E an isomorphism. The structural group of an SL^{c} bundle of rank n is SL_{n}^{c} which is the kernel of

GL_{n} × 𝔾_{m} ⟶^{(det^{−1}, t ↦ t^{2})} 𝔾_{m}.

There is a natural exact sequence 1 → μ_{2} → SL_{n}^{c} → GL_{n} → 1. The notation SL^{c} is in imitation of Spin^{c}. The role of this double cover of GL_{n} in direct images in real topological K-theory merited a mention by Atiyah [1, p. 55] nearly forty years ago.

Definition 3.3. An SL^{c} orientation on a cohomology theory A^{∗,∗} with a ε-commutative ring structure or partial multiplication is an assignment to every SL^{c} bundle (E, L, λ) over every scheme X in 𝒮m/S of a class th(E, L, λ) ∈ A^{2n,n}(E, E − X) where n = rk E satisfying the conditions (1)–(4) of Definition 3.1.

4. Schlichting’s hermitian K-theory and the Gille-Nenashev pairing

In [19, §2.7] Schlichting defines the hermitian K-theory space of a complicial exact category with weak equivalences and duality in the style of Waldhausen’s K-theory. We will denote his space by KO(C, w, ♯, η). More generally we write

KO^{[n]}(C, w, ♯, η) = KO((C, w, ♯, η)[n])

for the hermitian K-theory space for the n^{th} shifted duality, and KO_{i}^{[n]}(C, w, ♯, η) for its homotopy groups. A symmetric object of degree n in (C, w, ♯, η) is a pair (X, ϕ) with ϕ: X → X^{♯}[n] a weak equivalence which is symmetric ϕ = ϕ^{t} for the shifted duality. There is a natural definition of a Grothendieck-Witt group of symmetric objects of degree n, and The π_{0} of the hermitian K-theory space is the Grothendieck-Witt group of degree n symmetric objects

KO_{0}^{[n]}(C, w, ♯, η) = GW^{n}(C, w, ♯, η).

When C is ℤ[\frac{1}{2}]-linear, that is the same as the triangulated Grothendieck-Witt group of the homotopy category Ho(C, w) = C[w^{−1}] for the duality (♯, η), defined à la Balmer.

For a duality-preserving exact functor (F, f): (C, w, ♯, η) → (D, v, ♮, ϖ) there are induced maps of spaces KO^{[n]}(C, w, ♯, η) → KO^{[n]}(D, v, ♮, ϖ). A weak equivalence between dualitypreserving exact functors (F, f) ≃ (G, g) produces a homotopy between the maps.

There are natural periodicity isomorphisms KO^{[n]}(C, w, ♯, η) ≃ KO^{[n+4k]}(C, w, ♯, η). Moreover, we may write

KSp^{[n]}(C, w, ♯, η) = KO^{[n]}(C, w, ♯, −η)

because the effect of changing the sign is to interchange symmetric and skew-symmetric forms. Then there are isomorphisms KSp^{[n]}(C, w, ♯, η) ≃ KO^{[n+4k+2]}(C, w, ♯, η) induced by the duality preserving functor X ↦ X[2k+1]. However, it is more useful to use the identifications

\begin{matrix}KSp_{i}^{[n]}(C, w, ♯, η) & ⟶ & KO_{i}^{[n+4k+2]}(C, w, ♯, η) \\ ξ & ⟼ & − ξ[2k+1] \end{matrix}  (4.1)

because these commute with the forgetful maps to Waldhausen’s K-theory K_{i}(C, w).

Among the many important results Schlichting proves is localization. Suppose that \overline{C}_{1} ⊂ Ho(C, w) is a thick triangulated subcategory which is stable under the duality. Let C_{1} ⊂ C be the full exact subcategory with the same objects as \overline{C}_{1}. Let w_{1} be the set of all morphisms in C whose mapping cone is in C_{1}.

Theorem 4.1 ([19, Theorem 6]). If (C, w, ♯, η) is a complicial exact category with weak equivalences and duality, and C_{1} is as above, then

KO(C_{1}, w, ♯, η) → KO(C, w, ♯, η) → KO(C, w_{1}, ♯, η)

is a fibration sequence up to homotopy.

Gille and Nenashev [7] have defined pairings for Witt groups of triangulated categories. We explain how their construction can be applied to hermitian K-theory to give pairings in the spirit of the partial multiplicative structure of Definition 2.2. A pairing

(⊠, t_{1}, t_{2}, λ): (C, w, ♯, η) × (D, v, ♭, θ) → (E, u, ♮, ϖ)

of complicial exact categories with weak equivalences and duality is an additive bifunctor ⊠: C ×D → E which commutes with the translations up to specified functorial isomorphisms t_{1,X,Y} : X[1] ⊠ Y ≅ (X ⊠ Y)[1] and t_{2,X,Y} : X ⊠ Y [1] ≅ (X ⊠ Y)[1] plus functorial weak equivalences λ_{X,Y} : X^{♯} ⊠ Y^{♭} → (X ⊠ Y)^{♮} such that for any X in C and any Y in D the functors X ⊠ − and − ⊠ Y are exact and preserve weak equivalences and such that all the conditions of [7, Definitions 1.2 and 1.11] hold.

Suppose given a symmetric object (M, ϕ) of degree r in (C, w, ♯, η) and a symmetric object (N, ψ) of degree s in (D, v, ♭, θ). Gille and Nenashev show how to define duality-preserving exact functors [7, Lemma 1.14]

(− ⊠ (N, ψ), ℜ(N, ψ)): (C, w, ♯, η) → (E, u, ♮, ϖ)[s]
((M, ϕ) ⊠ −, 𝔏(M, ϕ)): (D, v, ♭, θ) → (E, u, ♮, ϖ)[r]

It follows that these induce maps of KO spaces, which we will write as

(− ⊠ (N, ψ))_{∗}: KO^{[n]}(C, w, ♯, η) → KO^{[n]}(E, u, ♮, ϖ)[s]  (4.3a)
((M, ϕ) ⊠ −)_{∗}: KO^{[n]}(D, v, ♭, θ) → KO^{[n]}(E, u, ♮, ϖ)[r]  (4.3b)

The duality-preserving functor (1_{E}, −1), which acts on symmetric objects by (Z, ξ) ↦ (Z, −ξ), induces a map

ε: KO^{[n]}(E, u, ♮, ϖ) → KO^{[n]}(E, u, ♮, ϖ).  (4.4)

These sign involutions exist for the hermitian K-theory of any complicial exact category with weak equivalences and duality, and they satisfy ε^{2} = 1 exactly. (In general ε is not the same as the −1 which is the inverse map for the H-space structure induced by the orthogonal direct sum.) The methods of Gille and Nenashev show [7, Lemma 1.15] that the effect of the two functors on the Grothendieck-Witt classes [M, ϕ] ∈ GW^{r}(C, w, ♯, η) and [N, ψ] ∈ GW^{s}(D, v, ♭, θ) is

(− ⊠ (N, ψ))_{∗}[M, ϕ] = ε^{rs}((M, ϕ) ⊠ −)_{∗}[N, ψ].  (4.5)

Proposition 4.2. The homotopy classes of the maps (4.3a) and (4.3b) on hermitian Ktheory spaces depend only on the classes [N, ψ] ∈ GW^{s}(D, v, ♭, θ) and [M, ϕ] ∈ GW^{r}(C, w, ♯, η) respectively.

Proof. There are three relations in the definition of the Grothendieck-Witt groups [19, Definition 1]. The maps on homotopy groups are compatible with the relations [N, ψ] + [N_{1}, ψ_{1}] = [N ⊕ N_{1}, ψ ⊕ ψ_{1}] because the orthogonal direct sum of symmetric objects induces a monoidal structure on the KO^{[n]}(E, u, ♮, ϖ) giving a naive additivity for orthogonal direct sums of duality-preserving functors.

The maps are compatible with the relations [N, ψ] = [N_{2}, σ^{♭}ψσ] for a weak equivalence σ: N_{2} → N because σ induces a natural weak equivalence of duality-preserving functors.

The maps are compatible with the third relation related to lagrangians because of Schlichting’s Additivity Theorem [19, Theorem 5]. □

We thus get pairings

KO_{j}^{[m]}(C, w, ♯, η) × KO_{0}^{[s]}(D, v, ♭, θ) → KO_{j}^{[m+s]}(E, u, ♮, ϖ)  (4.6a)
KO_{0}^{[r]}(C, w, ♯, η) × KO_{i}^{[n]}(D, v, ♭, θ) → KO_{i}^{[n+r]}(E, u, ♮, ϖ)  (4.6b)

which we call the right pairing and the corrected left pairing. They coincide on KO_{0}^{[m]}×KO_{0}^{[n]}.

The Gille-Nenashev pairings have a number of other properties such as functoriality, associativity, compatibility with localization sequences. Using the right pairing means that the boundary map on the homotopy groups in the localization sequences satisfies ∂(α∪ξ) = ∂α∪ξ for α ∈ KO_{i}^{[n]}(C, w_{1}, ♯, η) and ξ ∈ KO_{0}^{[r]}(D, v, ♭, θ).

In [19] Schlichting constructs a hermitian K-theory space for (C, w, ♯, η) but not a spectrum. So in principle what we have discussed so far does not yield any negative homotopy groups. In practice it is known that if the categories are ℤ[\frac{1}{2}]-linear, then Balmer’s triangulated Witt groups for the homotopy category Ho(C, w) = C[w^{−1}] can function as negative homotopy groups

KO_{i}^{[n]}(C, w, ♯, η) = W^{n−i}(C[w^{−1}], ♯, η) for i < 0.

This is explained in [18]. The localization sequences for the homotopy groups of the hermitian K-theory spaces and Balmer’s localization sequence for triangulated Witt groups [2, Theorem 6.2] attach to each other because the π_{0} are triangulated Grothendieck-Witt groups.

Two other important theorems are the following.

Theorem 4.3 (Fundamental Theorem [18]). Let (C, w, ♯, η) be a ℤ[\frac{1}{2}]-linear complicial exact category with weak equivalences and duality. Then for all n

KO^{[n−1]}(C, w, ♯, η) ⟶^{F} K(C, w) ⟶^{H} KO^{[n]}(C, w, ♯, η)

is a homotopy fiber sequence, where F is the forgetful map and H the hyperbolic map.

Theorem 4.4 ([18]). Let (F, f): (C, w, ♯, η) → (E, u, ♮, ϖ) be a duality-preserving exact functor between ℤ[\frac{1}{2}]-linear complicial exact categories with weak equivalences and duality. If (F, f) induces a homotopy equivalence K(C, w) ≃ K(E, u) of Waldhausen K-theory spaces and isomorphisms W^{i}(C[w^{−1}], ♯, η) ≅ W^{i}(E[u^{−1}], ♮, ϖ) of Balmer’s triangulated Witt groups, then (F, f) induces homotopy equivalences KO^{[n]}(C, w, ♯, η) ≃ KO^{[n]}(E, u, ♮, ϖ) for all n.

In particular if (F, f) induces an equivalence of ℤ[\frac{1}{2}]-linear triangulated categories with duality (C[w^{−1}], ♯, η) ≃ (E[u^{−1}], ♮, ϖ), then it induces a homotopy equivalence KO(C, w, ♯, η) ≃ KO(E, u, ♮, ϖ) by Thomason’s theorem and by the fact that Balmer’s Witt groups are a functorial over the category with objects ℤ[\frac{1}{2}]-linear triangulated categories with duality and arrows isomorphism classes of duality-preserving triangulated functors [3, Lemma 4.1].

5. The cohomology theory KO_{∗}^{[∗]} on the category 𝒮m𝒪p/k

Let S be a regular noetherian separated scheme of finite Krull dimension with \frac{1}{2} ∈ Γ(S, 𝒪_{S}). For every S-scheme X consider the category V BX of big vector bundles over X in the sense of [5, Appendix C.4]. The assignments X ↦ V BX and (f: Y → X) ↦ f^{∗}: V BX → V BY then form a strict functor (𝒮m/S)^{op} → 𝒞at because one has equalities (f ∘ g)^{∗} = g^{∗} ∘ f^{∗} instead of simply isomorphisms.

For any X ∈ 𝒮m/S, let Ch^{b}(V BX) denote the additive category of bounded complexes of big vector bundles on X. We will consider Ch^{b}(V BX) as a complicial exact category with weak equivalences, the conflations being the degreewise-split short exact sequences, and the weak equivalences w_{X} being the quasi-isomorphisms. When we further endow Ch^{b}(V BX) with the duality consisting of the functor ^{∨} = ℋom_{𝒪_{X}} (−, 𝒪_{X}) and the natural biduality maps η_{X}: 1 ≅ ^{∨∨}, we will write it as Ch^{b}(V BX).

Ch^{b}(V BX) = (Ch^{b}(V BX), w_{X},^{∨}, η_{X})

Now suppose U ⊂ X is an open subscheme and Z = X − U. Let w_{U} be the set of chain maps whose restriction to U is a quasi-isomorphism. Let Ch^{b}(V BX)^{w_{U}} be the full additive subcategory of complexes which are acyclic on U. We have two new families of complicial exact categories with weak equivalences and duality

Ch^{b}(V BX on Z) = (Ch^{b}(V BX)^{w_{U}} , w_{X},^{∨}, η_{X}),
Ch^{b}(V BX on U) = (Ch^{b}(V BX), w_{U},^{∨}, η_{X}).

We then have hermitian K-theory spaces

KO^{[n]}(X) = KO^{[n]}(Ch^{b}(V BX))
KO^{[n]}(X, U) = KO^{[n]}(Ch^{b}(V BX on Z)).

with KO^{[n]}(X) = KO^{[n]}(X, ∅). Let D^{b}(V BX on Z) be the homotopy category equipped with the triangulated duality (^{∨}, η_{X}). We define the hermitian K-theory groups as

KO_{i}^{[n]}(X, U) = \begin{cases}π_{i}KO^{[n]}(X, U) & for i ≥ 0, \\ W^{n−i}(D^{b}V BX on Z) & for i < 0. \end{cases}  (5.1)

For f: (X, U) → (Y, V) a morphism in 𝒮m𝒪p/S write W = Y − V. Then the functor f^{∗}: Ch^{b}(V BX on Z) → Ch^{b}(V BY on W) can by made duality-preserving by equipping it with the natural isomorphism f^{∗}ℋom_{𝒪_{X}} (−, 𝒪_{X}) ≅ ℋom_{𝒪_{Y}} (f^{∗}−, 𝒪_{Y}). This gives us maps

f^{∗}: KO^{[n]}(X, U) → KO^{[n]}(Y, V),  (5.2a)
f^{∗}: KO_{i}^{[n]}(X, U) → KO_{i}^{[n]}(Y, V).  (5.2b)

By Schlichting’s localization theorem (Theorem 4.1) the sequences

KO^{[n]}(X, U) → KO^{[n]}(X) → KO^{[n]}(Ch^{b}(V BX on U))

are fibration sequences up to homotopy. The restriction map Ch^{b}(V BX on U) → Ch^{b}(V BU) is a duality-preserving functor which induces an equivalence on the homotopy categories D^{b}(V BX on U) ≃ D^{b}(V BU). (Schlichting actually gives a different argument in [19, §9] which is valid with fewer restrictions on the schemes.) So we get fibration sequences up to homotopy

KO^{[n]}(X, U) → KO^{[n]}(X) → KO^{[n]}(U)  (5.3a)

and therefore long exact sequences

⋯ → KO_{i}^{[n]}(X, U) → KO_{i}^{[n]}(X) → KO_{i}^{[n]}(U) ⟶^{∂} KO_{i−1}^{[n]}(X, U) → ⋯  (5.3b)

Now suppose (Y, V) is in 𝒮m𝒪p/S and write W = Y − V. There is then a pairing of complicial exact categories with weak equivalences and duality

⊠: Ch^{b}(V BX on Z) × Ch^{b}(V BY on W) → Ch^{b}(V B(X × Y) on Z × W).

For a degree r symmetric complex (N, ψ) on Y which is acyclic on V and for a degree s symmetric complex (M, ϕ) on X which is acyclic on U we have maps of spaces

(− ⊠ (N, ψ))_{∗}: KO^{[n]}(X, U) → KO^{[n+r]}(X × Y, (X × V) ∪ (U × Y))  (5.4a)
ε^{ns}((M, ϕ) ⊠ −)_{∗}: KO^{[n]}(Y, V) → KO^{[n+s]}(X × Y, (X × V) ∪ (U × Y))  (5.4b)

This leads to a right pairing and a corrected left pairing

KO_{i}^{[n]}(X, U) × KO_{0}^{[r]}(Y, V) → KO_{i}^{[n+r]}(X × Y, (X × V) ∪ (U × Y)),  (5.4c)
KO_{0}^{[n]}(X, U) × KO_{i}^{[r]}(Y, V) → KO_{i}^{[n+r]}(X × Y, (X × V) ∪ (U × Y)).  (5.4d)

Now suppose (E, L, λ) is an SL^{c} bundle of rank n over X. Let p: E → X be the structural map. We may construct Thom isomorphisms for hermitian K-theory using the same method that Nenashev used for Witt groups [13, §2]. Namely, the pullback p^{∗}E = E ⊕ E → E has a canonical section s, the diagonal. There is a Koszul complex

K(E) = (0 → Λ^{n}p^{∗}E^{∨} → Λ^{n−1}p^{∗}E^{∨} → ⋯ → Λ^{2}p^{∗}E^{∨} → E^{∨} → 𝒪_{E} → 0)

(considered as a chain complex in homological degrees n to 0) in which each boundary map the contraction with s. It is a locally free resolution of the coherent sheaf z_{∗}𝒪_{X}. There is a canonical isomorphism Θ(E): K(E) → K(E)^{∨} ⊗ det p^{∗}E^{∨}[n] which is symmetric for the (det p^{∗}E^{∨})-twisted shifted duality. The composition

Θ(E, L, λ): K(E) ⊗ p^{∗}L ⟶^{Θ(E)} K(E)^{∨} ⊗ det p^{∗}E^{∨} ⊗ p^{∗}L[n] ⟶^{1⊗p^{∗}(λ^{∨}⊗L)} K(E)^{∨} ⊗ p^{∗}L^{∨}[n]

is symmetric for the untwisted shifted duality. We consider the Grothendieck-Witt class

th(E, L, λ) = [K(E) ⊗ p^{∗}L, Θ(E, L, λ)] ∈ KO_{0}^{[n]}(E, E − X).  (5.5)

When L is trivial this is an SL Thom class th(E, λ) = [K(E), Θ(E, λ)]. Finally for g a nowhere vanishing function on X we let ⟨g⟩ = [𝒪_{X}, g] ∈ KO_{0}^{[0]}(X) be the Grothendieck-Witt class of the rank one symmetric bilinear bundle.

Theorem 5.1. Let S be a regular noetherian separated scheme of finite Krull dimension with \frac{1}{2} ∈ Γ(S, 𝒪_{S}). Then the groups KO_{i}^{[n]}(X, U) of (5.1), the maps f^{∗} of (5.2b) and ∂ of (5.3b), the pairings of (5.4c) and (5.4d), the classes 1 = ⟨1⟩ and ε = ⟨−1⟩ in KO_{0}^{[0]}(pt) and the classes th(E, L, λ) of (5.5) form an SL^{c} oriented cohomology theory with an ε-commutative partial multiplication.

In particular the products with the Thom classes are isomorphisms

− ∪ th(E, L, λ): KO_{i}^{[m]}(X) ⟶^{≅} KO_{i}^{[m+n]}(E, E − X)  (5.6)

Sketch of the proof. The verifications are all straightforward. For instance étale excision and homotopy invariance amount to having certain pullback maps be isomorphisms, and pullback maps are induced by duality-preserving functors. Since these duality-preserving functors give isomorphisms for Quillen’s K-theory and Balmer’s Witt groups, they give isomorphisms for hermitian K-theory as well under our hypotheses.

The Thom maps come from duality-preserving functors. The functor part Ch^{b}(V BX) → Ch^{b}(V BE) is given by ℱ ↦ π^{∗}ℱ ⊗_{𝒪_{E}} K(E) with the target quasi-isomorphic to the coherent sheaf z_{∗}ℱ. These produce dévissage isomorphisms in both Quillen-Waldhausen K-theory and Balmer’s Witt groups [6].

For the ε-commutativity of the partial multiplicative structure, σ^{∗}(b × a) is calculated by applying the right pairing for a with respect to b and then switching. That is equivalent to applying the uncorrected left pairing for a with respect to b. But that satisfies (4.5). □

We will discuss later the Pontryagin classes associated to the Thom classes.

It is sometimes inconvenient that the KO_{i}^{[n]}(X, U) for i < 0 are not defined as homotopy groups. But actually they are naturally isomorphic to direct summands of homotopy groups. For we have the S-scheme 𝔾_{m} = A^{1} − 0 (pointed by 1) and groups KO_{i}^{[n]}(𝔾_{m}^{∧r} × X, 𝔾_{m}^{∧r} × U) defined as in (2.1).

Lemma 5.2. For all i and n and all r ≥ 1 and all (X, U) in 𝒮m𝒪p/S there are natural isomorphisms KO_{i+r}^{[n+r]}(𝔾_{m}^{∧r} × X, 𝔾_{m}^{∧r} × U) ≅ KO_{i}^{[n]}(X, U).

Proof. For r = 1 we have a localization sequence which splits and a Thom isomorphism

[figure: KO_{i+1}^{[n+1]}(A^{1} × X, A^{1} × U); KO_{i+1}^{[n+1]}(𝔾_{m} × X, 𝔾_{m} × U); ∂; (in_{1}×1_{X})^{∗}; KO_{i+1}^{[n+1]}(X, U); ≅; KO_{i}^{[n+1]}(A^{1} × X, A^{1} × U ∪ 𝔾_{m} × X); KO_{i}^{[n]}(X, U); ≅; th ×]

with th ∈ KO_{0}^{[1]}(A^{1}, A^{1} − 0) the Thom class of the trivial rank one SL bundle. Hence we have a natural isomorphism

KO_{i+1}^{[n+1]}(𝔾_{m} × X, 𝔾_{m} × U) ≅ KO_{i+1}^{[n+1]}(X, U) ⊕ KO_{i}^{[n]}(X, U).

By induction KO_{i+r}^{[n+r]}(𝔾_{m}^{×r} × X, 𝔾_{m}^{×r} × U) is a direct sum of 2^{r} terms of which exactly one is KO_{i}^{[n]}(X, U). □

Definition 5.3. The periodicity element β_{8} ∈ KO_{0}^{[4]}(pt) is the element corresponding to 1 ∈ KO_{0}^{[0]}(pt) under the periodicity isomorphisms KO_{i}^{[n]} ≅ KO_{i}^{[n+4]} of the hermitian Ktheory of chain complexes.

Then for all X and n the periodicity isomorphisms KO_{i}^{[n]}(X) ≅ KO_{i}^{[n+4]}(X) coincides with − × β_{8} up to homotopy.

6. KO_{∗}^{[∗]} of motivic spaces

In this section we recall what the category of pointed motivic spaces is and extend the functor KO_{∗}^{[∗]} to a functor KO_{∗}^{[∗]} on that category.

The basic definitions, constructions and model structures we use are given in [21]. A motivic space over S is a simplicial presheaf on the site 𝒮m/S of smooth S-schemes of finite type. A pointed motivic space over S is a pointed simplicial presheaf on the site 𝒮m/S. We write M_{•}(S) for the category of pointed motivic spaces over S.

We equip the category M_{•}(S) with the local injective model structure [21, p. 181] and with the motivic model structure [21, p. 194]. In both model structures the cofibrations are the monomorphisms. The weak equivalences and fibrations of the local injective model structure are called local weak equivalences and global fibrations. Those of the motivic model structure are called motivic weak equivalences and motivic fibrations.

We write H_{•}(S) for the pointed motivic unstable homotopy category obtained by inverting the motivic weak equivalences. The homotopy category H_{•}(S) is equivalent to the motivic homotopy category of [12]. For a morphism f: A → B of pointed motivic spaces we will write [f] for the class of f in H_{•}(S).

Notation 6.1. There is a global fibrant model functor G: Id_{M_{•}(S)} → (−)^{f} functor in M_{•}(S). The natural transformation G is a local weak equivalence, but we do not require it to be injective.

Lemma 6.2. Let KO^{[i]} = (KO^{[i]})^{f}. Then the map G: KO^{[i]} → KO^{[i]} is a schemewise weak equivalence, and the space KO^{[i]} is motivically fibrant.

Proof. The space KO^{[i]} satisfies Nisnevich descent because the homotopy groups KO_{r}^{[i]} satisfy étale excision. Therefore [21, Theorem 5.21] the morphism G: KO^{[i]} → KO^{[i]} is a schemewise weak equivalence. For every X ∈ 𝒮m/S the projection A_{X}^{1} → X induces a weak equivalence of simplicial sets KO^{[i]}(X) → KO^{[i]}(A_{X}^{1}), since this the case for the space KO^{[i]} and G : KO^{[i]} → KO^{[i]} is a schemewise weak equivalence. This proves [21, p. 195] that the globally fibrant space KO^{[i]} is motivically fibrant. □

We write S_{s}^{r} for the r-sphere Δ[r]/∂Δ[r] in sSet and in the homotopy category H_{•} of pointed simplicial sets and for the corresponding constant simplicial presheaf in M_{•}(S). We write 𝔾_{m} for the pointed scheme (A^{1} − 0, 1).

Definition 6.3. For any pointed motivic space A and any i define

KO_{r}^{[i]}(A) = \begin{cases}Hom_{H_{•}(S)}(A ∧ S_{s}^{r}, KO^{[i]}) & for r ≥ 0, \\ Hom_{H_{•}(S)}(A ∧ 𝔾_{m}^{∧(−r)}, KO^{[i−r]}) & for r < 0. \end{cases}

Lemma 6.4. For any r and X ∈ 𝒮m/S one has functorial isomorphisms

α_{X} : KO_{r}^{[i]}(X) ⟶^{≅} KO_{r}^{[i]}(X_{+}).

Proof. In fact, the following chain of isomorphisms give the desired one

Hom_{H_{•}}(S_{s}^{r}, KO^{[i]}(X)) ⟶^{≅} Hom_{H_{•}}(S_{s}^{r}, KO^{[i]}(X)) ⟶^{adj} Hom_{H_{•}(S)}(X_{+} ∧ S_{s}^{r}, KO^{[i]})

π_{0}(KO^{[i−r]}(X × 𝔾_{m}^{×−r})) ⟶^{≅} π_{0}(KO^{[i−r]}(X × 𝔾_{m}^{×−r})) ⟶^{adj} Hom_{H_{•}(S)}((X × 𝔾_{m}^{×−r})_{+}, KO^{[i−r]})

All the arrows are indeed isomorphisms by Lemma 6.2. □

Now suppose (X, U) ∈ 𝒮m𝒪p/S with j: U ↪ X the inclusion and Z = X−U. Schlichting’s localization sequence (5.3a)

KO^{[i]}(X, U) ⟶^{e^{∗}} KO^{[i]}(X) ⟶^{j^{∗}} KO^{[i]}(U)

can be described more precisely than we have done so far. The composite map j^{∗}e^{∗} is induced by the functor Ch^{b}(V BX on Z) → Ch^{b}(V BU) such that the morphism of functors j^{∗}e^{∗} → 0 is a natural weak equivalence. This natural weak equivalence gives a homotopy from the map of spaces j^{∗}e^{∗} to the trivial map, and that gives a factorization of e^{∗} as

KO^{[i]}(X, U) ⟶^{e_{X,U}} hofib(j^{∗}) ⟶^{can_{X,U}} KO^{[i]}(X).  (6.1)

Schlichting’s theorem is that e_{X,U} a homotopy equivalence. This factorization is functorial in (X, U).

For a pointed motivic space A write KO^{[i]}(A) = hom_{•}(A, KO^{[i]}) for the pointed mapping space, which is the pointed simplicial set [n] ↦ Hom_{M_{•}(S)}(A ∧ Δ[n]_{+}, KO^{[i]}). For A = X_{+} with X a smooth scheme we have KO^{[i]}(A) = KO^{[i]}(X). For j: U ↪ X as above we can fill out a commutative diagram of pointed simplicial sets

\xymatrix{KO^{[i]}(X, U) \ar[d]_{e_{X,U}} \\ hofib(j^{∗}) \ar[r]^{can_{X,U}} \ar[d]_{G_{1}} & KO^{[i]}(X) \ar[r]^{j^{∗}} \ar[d]_{G(X)} & KO^{[i]}(U) \ar[d]^{G(U)} \\ KO^{[i]}(Cone(j_{+})) = hofib(j_{+}^{∗}) \ar[r]^{nat_{X,U}} & KO^{[i]}(X_{+}) \ar[r]^{j_{+}^{∗}} & KO^{[i]}(U_{+}) \\ KO^{[i]}(X_{+}/U_{+}) \ar[u]^{q_{X,U}}}

We claim that all the vertical arrows are homotopy equivalences. This is true for G(X) and G(U) by Lemma 6.2 and therefore for G_{1} because it is the map between the homotopy fibers. The map e_{X,U} is a homotopy equivalence by Schlichting’s theorem. The map q_{X,U} is a homotopy equivalence because it is obtained by applying hom_{•}(−, KO^{[i]}) with KO^{[i]} fibrant to the schemewise weak equivalence between cofibrant objects Cone(j_{+}) → X_{+}/U_{+}.

The diagram is functorial in (X, U). We conclude:

Theorem 6.5. For (X, U) in 𝒮m𝒪p/S there is a functorial zigzag of homotopy equivalences KO^{[i]}(X, U) → KO^{[i]}(Cone(U_{+} → X_{+})) ← KO^{[i]}(X_{+}/U_{+}) which for (X, ∅) reduces to the G(X): KO^{[i]}(X) → KO^{[i]}(X) of Lemma 6.2. These induce functorial isomorphisms of groups KO_{r}^{[i]}(X, U) ≅ KO_{r}^{[i]}(X_{+}/U_{+}) for all integers i and r.

Notation 6.6. Denote by α : KO_{∗}^{[i]}(−, −) → KO_{∗}^{[i]}(−_{+}/−_{+}) the functor isomorphism described in Theorem 6.5.

7. The T -spectrum BO and the cohomology theory BO^{∗,∗}

Let T = A^{1}/(A^{1} − 0) be the Morel-Voevodsky object. A T -spectrum E over S consists of a sequence (E_{0}, E_{1}, …) of pointed motivic spaces over S plus structure maps σ_{n}: E_{n} ∧ T → E_{n+1}. Let SH(S) denote the stable homotopy category of T -spectra as described in [9]. It is canonically equivalent to the motivic stable homotopy category constructed in [20].

Here we define a T -spectrum BO. Its spaces are (KO^{[0]}, KO^{[1]}, KO^{[2]}, KO^{[3]}, …). We now define the structure maps.

Let A^{1} → pt be the trivial rank one SL bundle, and let th ∈ KO_{0}^{[1]}(A^{1}, A^{1} − 0) be its Thom class as defined by (5.5). Because KO_{∗}^{[∗]} is SL^{c} oriented, the maps

− × th: KO_{r}^{[n]}(X) → KO_{r}^{[n+1]}(X × A^{1}, X × (A^{1} − 0))  (7.1)

are isomorphisms. Recall that th is defined by (5.5) as the class of the symmetric complex

\xymatrix{K(𝒪) \ar[d]^{≅}_{Θ(𝒪)} & 0 \ar[r] & 𝒪_{A^{1}} \ar[r]^{x} \ar[d]_{−1} & 𝒪_{A^{1}} \ar[r] \ar[d]^{1} & 0 \\ K(𝒪)^{∨}[1] & 0 \ar[r] & 𝒪_{A^{1}} \ar[r]^{−x} & 𝒪_{A^{1}} \ar[r] & 0}  (7.2)

of degree 1 in Ch^{b}(V BA^{1} on 0). The maps − × th are induced by the maps of spaces

(− ⊠ (K(𝒪), Θ(𝒪)))_{∗}: KO^{[n]}(X) → KO^{[n+1]}(X × A^{1}, X × (A^{1} − 0))

These maps are thus homotopy equivalences, and KO^{[n]} → KO^{[n+1]}(− × A^{1}, − × (A^{1} − 0)) is a schemewise weak equivalence.

From Lemma 6.2 and Theorem 6.5 we now have a zigzag

[figure: KO^{[n]}; G; ∼; KO^{[n]}; × th; ∼; KO^{[n+1]}(− × A^{1}, − × (A^{1} − 0)); ∼; KO^{[n+1]}(Cone(− ∧ (A^{1} − 0)_{+} → − ∧ A_{+}^{1})); ∼; KO^{[n+1]}(− ∧ T)]

of schemewise weak equivalences in M_{•}(S). Their composition is an isomorphism KO^{[n]} ≅ KO^{[n+1]}(− ∧ T) in the homotopy category.

There is a Quillen adjunction with left adjoint − ∧ T and right adjoint F (−) ↦ F (− ∧ T). It follows that KO^{[n+1]}(−∧T) is fibrant, while KO^{[n]} is cofibrant, so there exists a morphism σ_{n}^{∗}: KO^{[n]} → KO^{[n+1]}(− ∧ T) in M_{•}(S) representing the same isomorphism in the homotopy category as the zigzag. Let σ_{n}: KO^{[n]} ∧ T → KO^{[n+1]} be the adjoint morphism. Since the KO^{[n]} and KO^{[n]} are periodic modulo 4, we may choose the σ_{n}^{∗} and σ_{n} so they are also periodic.

Definition 7.1. The T -spectrum BO consists of the sequence of pointed motivic spaces

(KO^{[0]}, KO^{[1]}, KO^{[2]}, KO^{[3]}, …)  (7.3)

together with the structure maps σ_{n} : KO^{[n]} ∧ T → KO^{[n+1]} just described.

The spaces KO^{[n]} are motivically fibrant and the adjoints σ_{n}^{∗} of the structure maps are schemewise weak equivalences. So we have [9, Lemma 2.7]:

Theorem 7.2. The T -spectrum BO is stably motivically fibrant.

As explained in [20] any T -spectrum E defines a bigraded cohomology theory (E^{∗,∗}, ∂) on the category M_{•}(S) with

E^{p,q}(A) = Hom_{SH_{•}(S)}(A, E ∧ S_{s}^{p−q} ∧ 𝔾_{m}^{∧q}).

The differential ∂ increases the bidegree by (1, 0).

For any A ∈ M_{•}(S) the adjunction map induces isomorphisms

KO_{i}^{[n]}(A) = Hom_{H_{•}(S)}(A ∧ S_{s}^{i}, KO^{[n]}) ≅ Hom_{SH(S)}(A ∧ S_{s}^{i}, BO ∧ T^{∧n}) = BO^{2n−i,n}(A),

KO_{i}^{[n]}(A) = Hom_{H_{•}(S)}(A ∧ 𝔾_{m}^{∧−i}, KO^{[n−i]}) ≅ Hom_{SH(S)}(A ∧ 𝔾_{m}^{∧−i}, BO ∧ T^{∧n−i})

= BO^{2n−i,n}(A),

for i ≥ 0 and i < 0 respectively. This gives us an isomorphism of functors on M_{•}(S)

β_{A} : KO_{i}^{[n]} ⟶^{≅} BO^{2n−i,n}.

Corollary 7.3. The composition isomorphism β|_{𝒮m𝒪p/k} ∘ α : KO_{∗}^{[∗]} → BO^{∗,∗}|_{𝒮m𝒪p/S} respects the boundary homomorphisms in both cohomology theories on 𝒮m𝒪p/S. So it is an isomorphism

γ : KO_{∗}^{[∗]} → BO^{∗,∗}|_{𝒮m𝒪p/S}

of cohomology theories in the sense of [14].

Definition 7.4. Using the cohomology isomorphism γ we transplant to BO|_{𝒮m𝒪p/S} the partial multiplicative structure of (KO_{∗}^{[∗]}, ∂) and the Thom and Pontryagin classes of its SL^{c} orientation described in Theorem 5.1. The unit of this partial multiplicative structure is the element e = γ(⟨1⟩) ∈ BO^{0,0}(S^{0}).

Theorem 7.5 (Bott periodicity). The adjoints of the structure maps and the categorical periodicity isomorphisms give levelwise weak equivalences

BO ⟶^{∼ level} Ω_{T}^{4} BO(4) ⟶^{≅} Ω_{T}^{4} BO.

8. A symplectic version of the Morel-Voevodsky theorem

In [12, Theorem 4.3.13] Morel and Voevodsky showed that ℤ × Gr represents algebraic Ktheory in the motivic unstable homotopy category. If one replaces the ordinary Grassmannians by quaternionic Grassmannians, the same holds for symplectic K-theory.

We write H for the trivial rank 2 symplectic bundle (𝒪^{⊕2}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )). The orthogonal direct sum H^{⊕n} is the trivial symplectic bundle of rank 2n. We will sometimes write H_{X}^{⊕n} to designate the trivial symplectic bundle over the scheme X.

The quaternionic Grassmannian HGr(r, n) = HGr(r, H^{⊕n}) is defined as the open subscheme of Gr(2r, 2n) = Gr(2r, H^{⊕n}) parametrizing subspaces of dimension 2r of the fibers of H^{⊕n} on which the symplectic form of H^{⊕n} is nondegenerate. We write 𝒰_{r,n} for the restriction to HGr(r, n) of the tautological subbundle of Gr(2r, 2n). The symplectic form of H^{⊕n} restricts to a symplectic form on 𝒰_{r,n} which we denote by ϕ_{r,n}. The pair (𝒰_{r,n}, ϕ_{r,n}) is the tautological symplectic subbundle of rank 2r on HGr(r, n). Morphisms X → HGr(r, n) are classified by subbundles E ⊂ H_{X}^{⊕n} of rank 2r such that the symplectic form of H_{X}^{⊕n} is nondegenerate on every fiber.

More generally, given a symplectic bundle (E, ϕ) of rank 2n over X, the quaternionic Grassmannian bundle HGr(r, E, ϕ) is the open subscheme of the Grassmannian bundle Gr(2r, E) parametrizing subspaces of dimension 2r of the fibers of E on which ϕ is nondegenerate.

For r = 1 we have quaternionic projective spaces and bundles HP^{n} = HGr(1, n + 1) and HP (E, ϕ) = HGr(1, E, ϕ).

There are commuting morphisms

\xymatrix{HGr(r, n) \ar[r]^{α_{r,n}} \ar[d]_{β_{r,n}} & HGr(r, n + 1) \ar[d]^{β_{r,n+1}} \\ HGr(r + 1, n + 1) \ar[r]^{α_{r+1,n+1}} & HGr(r + 1, n + 2)}  (8.1)

with α_{r,n} classified by the rank 2r subbundle 𝒰_{r,n} ⊕ 0 ⊂ H^{⊕n} ⊕ H and β_{r+1,n+1} classified by the rank 2r + 2 subbundle H ⊕ 𝒰_{r,n} ⊂ H ⊕ H^{⊕n}. Composition gives us maps HGr(n, 2n) → HGr(n + 1, 2n + 2). We define HGr = colim HGr(n, 2n). We consider ℤ × HGr pointed by (0, HGr(0, 0)). It has a universal property.

Theorem 8.1. Suppose X ∈ 𝒮m/S is affine. Then for every ξ ∈ GW^{−}(X) there is a morphism of ind-schemes f: X → ℤ × HGr such that ξ = f^{∗}τ. Moreover f is unique up to naive A^{1}-homotopy.

This is the equivalence π_{0}ℒ_{6} ≅ π_{0}GW^{−} of [4, Proposition 6.2.1.5] plus the isomorphism π_{0}GW^{−} = GW^{−} which happens for our schemes which are regular with \frac{1}{2}.

For a smooth S-scheme X let KSp(X) = KO(Ch^{b}(V BX), w_{X},^{∨}, −η) be its symplectic K-theory space. There are natural isomorphisms KSp ≅ KO^{[4n+2]}.

Theorem 8.2. The objects ℤ × HGr and KSp are isomorphic in the motivic unstable homotopy category H_{•}(S).

The proof of this theorem is identical to Morel and Voevodsky’s proof for ordinary K-theory except for one point. Let HU(r, n) → HGr(r, n) be the principal Sp_{2r}-bundle associated to the tautological rank 2r symplectic subbundle on HGr(r, n). There is an isomorphism HU(r, n) ≅ Sp_{2n}/Sp_{2n−2r}. To establish Theorem 8.2 by following the proof of Morel and Voevodsky exactly we would need the HU(r, n) to form an admissible gadget in the sense of [12, Definition 4.2.1]. It does not seem as if they do. Nor can we use the admissible gadget used by Morel and Voevodsky, for that would substitute for HU(r, n) the principal GL(2r)-bundle associated to the tautological subbundle on Gr(2r, 2n), called U_{2r,2n} in [12]. (This U_{2r,2n} is the space of 2r × 2n matrices of maximal rank.) But the quotient U_{2r,2n}/Sp_{2r} is not the quaternionic Grassmannian HGr(r, n), and its cohomology risks being much less tractable.

So we give a new definition based on the property actually used in the proof of [12, Proposition 4.2.3]. For a commutative ring B let

Δ_{B}^{n} = Spec B[t_{1}, …, t_{n}],  ∂Δ_{B}^{n} = Spec B[t_{1}, …, t_{n}]/(t_{1}t_{2} ⋯ t_{n}(1 − ∑t_{i})) ⊂ Δ_{B}^{n}

Definition 8.3. An acceptable gadget over an S-scheme X is a sequence of smooth quasiprojective X-schemes (U_{i})_{i∈ℕ} and closed embeddings U_{i} → U_{i+1} of X-schemes such that for any henselian regular local ring B and any commutative square

\xymatrix{∂Δ_{B}^{n} \ar[r] \ar[d]_{inclusion} & U_{i} \ar[d]^{projection} \\ Δ_{B}^{n} \ar[r] & X}

there exists a j ≥ i and a map Δ_{B}^{n} → U_{j} making the following diagram commute.

\xymatrix{∂Δ_{B}^{n} \ar[r] \ar[d]_{inclusion} & U_{i} \ar[r]^{gadget map} & U_{j} \ar[d]^{projection} \\ Δ_{B}^{n} \ar[rr] \ar[urr] &  & X}

Sections of the principal bundle HU(r, n) → HGr(r, n) are given by symplectic frames (i.e. symplectic bases) of the tautological symplectic subbundle. Therefore giving a morphism V → HU(r, n) is equivalent to giving an embedding H_{V}^{⊕r} ⊂ H_{V}^{⊕n} such that the symplectic form on H_{V}^{⊕r} is the restriction of the symplectic form on H_{V}^{⊕n}. This is also equivalent by duality to giving 2n sections s_{1}, …, s_{2n} of the bundle of linear forms H_{V}^{⊕r∨} such that ∑_{i=1}^{n} s_{2i−1} ∧ s_{2i} is equal to the symplectic form of H_{V}^{⊕r}.

We need the HU(r, n) to form an acceptable gadget in the relative case as well (cf. [12, Lemma 4.2.8]). Given a symplectic bundle (E, ϕ) of rank 2r over X, the relative HU(E, ϕ; n) is constructed by taking the Sp_{2r}-torsor P → X associated to (E, ϕ) and forming the quotient (P × HU(r, n))/Sp_{2r} by the diagonal action of Sp_{2r} on the two torsors. This gives us a fiber bundle HU(E, ϕ; n) → X × HGr(r, n) with fibers Sp_{2r} and structural group Sp_{2r} × Sp_{2r} acting on the fibers by left and right translation. Giving a morphism V → HU(E, ϕ; n) is equivalent to giving a triple (f, g, ι) with f: V → X and g: V → HGr(r, n) morphisms of schemes and ι: f^{∗}(E, ϕ) ≅ g^{∗}(𝒰_{r,n}, ϕ_{r,n}) an isometry of symplectic bundles. This is equivalent to giving (f, u_{1}, …, u_{2n}) with f: V → X a map and the u_{i} sections of f^{∗}E^{∨} such that ∑_{i=1}^{n} u_{2i−1} ∧ u_{2i} = f^{∗}ϕ.

Lemma 8.4. Let R be a commutative ring, let g ∈ R and let \overline{R} = R/(g). Let (E, ϕ) a symplectic R-module, and let (\overline{E}, \overline{ϕ}) be the associated symplectic \overline{R}-module. Let u_{1}, …, u_{2n} ∈ E^{∨} be linear forms such that ∑_{i=1}^{n} \overline{u}_{2i−1} ∧ \overline{u}_{2i} = \overline{ϕ} in Λ^{2}\overline{E}^{∨}. Then there exist linear forms v_{1}, …, v_{2n} and w_{1}, …, w_{2m} in E^{∨} such that

∑_{i=1}^{n}(u_{2i−1} + gv_{2i−1}) ∧ (u_{2i} + gv_{2i}) + ∑_{j=1}^{m} gw_{2j−1} ∧ gw_{2j} = ϕ.

Proof. The \overline{u}_{1}, …, \overline{u}_{2n} generate \overline{E}^{∨}. So there exist v_{1}, …, v_{2n} ∈ E^{∨} such that

ϕ − ∑_{i=1}^{n} u_{2i−1} ∧ u_{2i} ≡ g ∑_{j=1}^{2n} u_{j} ∧ (−1)^{j}v_{j} (mod g^{2}).  □

Proposition 8.5. For any symplectic bundle (E, ϕ) of rank 2r over a scheme X, the schemes (HU(E, ϕ; n))_{n≥r} together with the closed embeddings HU(E, ϕ; n) → HU(E, ϕ; n+1) induced by the inclusions H^{⊕n} ⊂ H^{⊕n} ⊕ H form an acceptable gadget.

Proof. Let R = B[t_{1}, …, t_{n}] and g = t_{1}t_{2} ⋯ t_{n}(1 − ∑t_{i}) ∈ R. Giving the first diagram of Definition 8.3 is then equivalent to giving the (E, ϕ) and (\overline{u}_{1}, …, \overline{u}_{2n}) of Lemma 8.4. The map Δ_{B}^{n} → U_{j} is then given by (u_{1} + gv_{1}, …, u_{2n} + gv_{2n}, gw_{1}, …, gw_{2m}). The top triangle of the second diagram commutes because modulo g this last vector is (\overline{u}_{1}, …, \overline{u}_{2n}, 0, …, 0). The lower triangle commutes because of the equation involving ϕ. □

Substituting the acceptable gadgets HU(r, n) → HU(r, n + 1) → ⋯ and the quaternionic Grassmannians HGr(i, n) for the admissible gadgets U_{n,i} → U_{n,i+1} → ⋯ and Grassmannians Gr(i, n) of Morel and Voevodsky, the proof of [12, Theorem 4.3.13] can be used to prove Theorem 8.2. Proposition 8.5 substitutes for the last paragraph of the proof of Proposition 4.2.3 and for Lemma 4.2.8. We get hocolim_{n} HU(r, n) ≅ pt and

BSp_{2r} ≅ B_{et}Sp_{2r} ≅ HGr(r, ∞)

in H(S), and we get the theorem. At the end one needs the equivalence of hermitian Ktheories based on group completion and of Schlichting’s Waldhausen-like construction. But Schlichting has shown that each is equivalent to the hermitian Q-construction: see [17, Theorem 4.2] and [19, Proposition 6].

Similar but less definitive results can be proven for the orthogonal group. Let H_{+}^{⊕n} denote the trivial orthogonal bundle (𝒪_{S}^{⊕2n}, q_{2n}) with the split quadratic form q_{2n} = ∑_{i=1}^{n} x_{2i−1}x_{2i}. Let RGr(r, 2n) = RGr(r, H_{+}^{⊕n}) be the open subscheme of Gr(r, 2n) parametrizing subspaces of rank r on which q_{2n} is nondegenerate. Then over RGr(r, 2n) we have a tautological rank r orthogonal subbundle (U|_{RGr}, q_{2n}|_{RGr}) whose structural group scheme is the orthogonal group scheme O(r, q_{2n}|_{RGr}) → RGr(r, 2n). The associated principal bundle is RU(r, 2n) → RGr(r, 2n). To give a morphism V → RU(r, 2n) one gives a quadratic bundle (E, q) of rank r over V and 2n sections s_{1}, …, s_{2n} of E^{∨} such that q = ∑_{i=1}^{n} s_{2i−1}s_{2i}. The data (E, q, s_{1}, …, s_{2n}) and (E_{1}, q_{1}, t_{1}, …, t_{2n}) define the same morphism if and only if there is an isomorphism ϕ: E ≅ E_{1} such that q = ϕ^{∗}q_{1} and s_{i} = ϕ^{∗}t_{i} for all i. The relative case is like the relative symplectic case described earlier.

Proposition 8.6. (a) For any quadratic bundle (E, q) the RU(E, q, 2n) and the inclusions RU(E, q, 2n) → RU(E, q, 2n + 2) corresponding to (E, q, s_{1}, …, s_{2n}) ↦ (E, q, s_{1}, …, s_{2n}, 0, 0) form an acceptable gadget, as do their relative versions.

(b) For any quadratic bundle (E, q) of rank r over X the fiber bundles RU(E, q, 2n) → X have sections for n ≥ r over any open subscheme over which the vector bundle E trivializes.

Part (b) of the proposition is of concern [12, Proposition 4.1.20 and Definition 4.2.4.3]. It is not an issue for symplectic bundles because symplectic bundles are locally trivial in the Zariski topology. It holds because to give a section of RU(E, q, 2n) → X is to give sections s_{1}, …, s_{2n} of E^{∨} such that q = ∑ s_{2i−1}s_{2i}. If E trivializes over U with coordinates x_{1}, …, x_{r} and q = ∑_{i≤j} a_{ij}x_{i}x_{j}, then one can give a local section over U by s_{2i−1} = x_{i} and s_{2i} = ∑_{j=i}^{n} a_{ij}x_{j} for 1 ≤ i ≤ r and s_{i} = 0 for i > 2r.

The results of Proposition 8.6 and of the arguments of Morel and Voevodsky are isomorphisms B_{et}O_{r} ≅ RGr(r, ∞) and B_{et}O ≅ RGr in H_{•}(S). However, neither of the natural maps ℤ × BO → ℤ × B_{et}O or ℤ × BO → KO^{[0]} are isomorphisms in H_{•}(S) because orthogonal bundles are not always locally trivial in the Nisnevich topology.

We also do not know how to calculate KO_{∗}^{[∗]}(RGr).

9. The cohomology of quaternionic Grassmannians

We review the calculation of the cohomology of quaternionic Grassmannians of [16]. We reformulate the definitions and some of the theorems for a bigraded ε-commutative partial multiplication. In [16] we assumed we had a full ring structure. We do not redo the proofs because no change is needed: all the needed products are with the Thom and Pontryagin classes of symplectic bundles or with pullbacks of such classes, and those lie in the A^{4i,2i}.

Definition 9.1 ([16, Definition 7.1]). A symplectic Thom structure on a bigraded cohomology theory (A^{∗,∗}, ∂) with an ε-commutative partial multiplication or ring structure is a rule assigning to every rank 2 symplectic bundle (E, ϕ) over an X in 𝒮m/S an element th(E, ϕ) ∈ A^{4,2}(E, E − X) with the following properties:

(1) For an isomorphism u : (E, ϕ) ≅ (E_{1}, ϕ_{1}) one has th(E, ϕ) = u^{∗} th(E_{1}, ϕ_{1}).

(2) For a morphism f : Y → X with pullback map f_{E} : f^{∗}E → E one has f_{E}^{∗} th(E, ϕ) = th(f^{∗}E, f^{∗}ϕ).

(3) For the trivial rank 2 bundle H over pt the map

− × th(H): A^{∗,∗}(X) → A^{∗,∗}(X × A^{2}, X × (A^{2} − {0}))

is an isomorphism for all X. The Pontryagin class of (E, ϕ) is p(E, ϕ) = −z^{∗} th(E, ϕ) ∈ A^{4,2}(X) where z: X → E is the zero section.

From Mayer-Vietoris one sees that for any rank 2 symplectic bundle

∪ th(E, ϕ): A^{∗,∗}(X) ⟶^{≅} A^{∗,∗}(E, E − X)

is an isomorphism. The sign in the Pontryagin class is simply conventional. It is chosen so that if A^{∗,∗} is an oriented cohomology theory with an additive formal group law, then the Chern and Pontryagin classes satisfy the traditional formula p_{i}(E, ϕ) = (−1)^{i}c_{2i}(E). The following is [16, Theorem 8.2].

Theorem 9.2 (Quaternionic projective bundle theorem). Let (A^{∗,∗}, ∂) be a bigraded cohomology theory with an ε-commutative partial multiplication or ring structure and a symplectic Thom structure. Let (𝒰, ϕ|_{𝒰}) be the tautological rank 2 symplectic subbundle over the quaternionic projective bundle HP (E, ϕ), and let t = p(𝒰, ϕ|_{𝒰}) be its Pontryagin class. Then we have an isomorphism of graded A^{0}(X)-modules

(1, t, …, t^{n−1}): A^{∗,∗}(X) ⊕ A^{∗,∗}(X) ⊕ ⋯ ⊕ A^{∗,∗}(X) → A^{∗,∗}(HP_{X}(E, ϕ))

and an isomorphism of graded modules over A^{0}(X) = ⨁_{p} A^{2p,p}(X):

(1, t, …, t^{n−1}): A^{0}(X) ⊕ A^{0}(X) ⊕ ⋯ ⊕ A^{0}(X) → A^{0}(HP_{X}(E, ϕ)).

Definition 9.3. Under the hypotheses of Theorem 9.2 there are unique elements p_{i}(E, ϕ) ∈ A^{4i,2i}(X) for i = 1, 2, …, n such that

t^{n} − p_{1}(E, ϕ) ∪ t^{n−1} + p_{2}(E, ϕ) ∪ t^{n−2} − ⋯ + (−1)^{n}p_{n}(E, ϕ) = 0.

The classes p_{i}(E, ϕ) are called the Pontryagin classes of (E, ϕ) with respect to the symplectic Thom structure of the cohomology theory (A, ∂). For i > n and i < 0 one sets p_{i}(E, ϕ) = 0, and one sets p_{0}(E, ϕ) = 1.

The quaternionic projective bundle theorem has consequences the symplectic splitting principle and the Cartan sum formula for Pontryagin classes [16, Theorems 10.2, 10.5]. With them one can compute the cohomology of quaternionic Grassmannians. Let e_{i} denote the i^{th} elementary symmetric polynomial, and let h_{i} be the i^{th} complete symmetric polynomial, the sum of all monomials of degree i. There are formulas relating them, including the recurrence relation h_{k} + ∑_{i≥1}(−1)^{i}e_{i}h_{k−i} = 0. Let Π_{r,n−r} be the set of all partitions whose Young diagrams fit inside an r × (n − r) box. (More formally these are partitions λ with l(λ) = λ′_{1} ≤ r and λ_{1} ≤ n−r.) Associated to any such partition is a Schur polynomial, which can be written in terms of the e_{i}.

Theorem 9.4. For any bigraded ring cohomology theory A^{∗,∗} with an ε-commutative partial multiplication or ring structure and a symplectic Thom structure and any X the map

A^{∗,∗}(X)[e_{1}, …, e_{r}]/(h_{n−r+1}, …, h_{n}) ⟶^{≅} A^{∗,∗}(HGr(r, n) × X)  (9.1)

sending e_{i} ↦ p_{i}(𝒰_{r,n}, ϕ_{r,n}) for all i is an isomorphism, as is the map

A^{∗,∗}(X)^{⊕\binom{n}{r}} ⟶^{(s_{λ}(𝒰_{r,n},ϕ_{r,n}))_{λ∈Π_{r,n−r}}} A^{∗,∗}(HGr(r, n) × X).  (9.2)

Theorem 9.5 ([16, Theorem 11.4]). For any bigraded ring cohomology theory A^{∗,∗} with an ε-commutative partial multiplication or ring structure and a symplectic Thom structure and any X the α_{r,n} and β_{r,n} of (8.1) induces split surjections

(α_{r,n} × 1_{X})^{∗}: A(HGr(r, n + 1) × X) ↠ A(HGr(r, n) × X)
(β_{r,n} × 1_{X})^{∗}: A(HGr(r + 1, n + 1) × X) ↠ A(HGr(r, n) × X)

which the isomorphisms (9.2) identify with the surjections A(X)^{⊕\binom{n+1}{r}} ↠ A(X)^{⊕\binom{n}{r}} and A(X)^{⊕\binom{n+1}{r+1}} ↠ A(X)^{⊕\binom{n}{r}} which are the identity on the summands corresponding to λ ∈ Π_{n,r} and which vanish on the summands corresponding to λ ∉ Π_{n,r}. We have isomorphisms

A^{∗,∗}(X)[[p_{1}, …, p_{r}]]^{homog} ⟶^{≅} \underleftarrow{lim}_{n→∞} A^{∗,∗}(HGr(r, n) × X)  (9.3)
A^{∗,∗}(X)[[p_{1}, p_{2}, p_{3}, …]]^{homog} ⟶^{≅} \underleftarrow{lim}_{n→∞} A^{∗,∗}(HGr(n, 2n) × X)  (9.4)

with each variable p_{i} sent to the inverse system of i^{th} Pontryagin classes (p_{i}(𝒰_{r,n}))_{n≥r} or (p_{i}(𝒰_{n,2n}))_{n∈ℕ}.

The notation on the left in (9.3)–(9.4) is the bigraded ring of homogeneous power series. We have a simple lemma.

Lemma 9.6. If A is a T -spectrum, then for any pointed motivic spaces X and Y the canonical map X × Y → X ∧ Y induces a split injection A^{r,s}(X ∧ Y) ↪ A^{r,s}(X × Y). The image of the injection coincides with the kernel of the map

[(id_{X} × y)^{∗}, (x × id_{Y}^{∗})]: A^{r,s}(X × Y) → A^{r,s}(X × y) ⊕ A^{r,s}(x × Y).

We write [−n, n] = {i ∈ ℤ | −n ≤ i ≤ n}. We have a sequential colimit of pointed spaces

(ℤ × HGr, (0, x_{0})) = colim(([−n, n] × HGr(n, 2n), (0, x_{0}))

to which Theorem 10.1 applies. Theorem 9.5 and Lemma 9.6 now give the following result.

Theorem 9.7. Let A be a T -spectrum whose associated cohomology theory (A^{∗,∗}, ∂) has an ε-commutative partial multiplication or ring structure and a symplectic Thom structure. Then the natural map

A^{∗,∗}((ℤ × HGr), (0, x_{0})) → \underleftarrow{lim} A^{∗,∗}([−n, n] × HGr(n, 2n), (0, x_{0}))

is an isomorphism. More generally for any r and s the natural map

A^{∗,∗}((ℤ×HGr, (0, x_{0}))^{∧r}∧(HP^{1}, x_{0})^{∧s}) → \underleftarrow{lim} A^{∗,∗}(([−n, n]×HGr(n, 2n), (0, x_{0}))^{∧r}∧(HP^{1}, x_{0})^{∧s})

is an isomorphism.

We complete our review of parts of [16] by looking at the geometry of HP^{1} = HGr(1, H^{⊕2}).

Theorem 9.8. In H_{•}(S) we have a canonical isomorphism η: (HP_{+}^{1}, pt) ≅ T^{∧2}.

Proof. By definition HP^{1} is by the open subscheme of Gr(2, 4) parametrizing 2-dimensional symplectic subspaces of the 4-dimensional trivial symplectic bundle. It contains two distinguished points x_{0} = [H ⊕ 0] and x_{∞} = [0 ⊕ H]. The x_{∞} is the origin of an open cell A^{4} ⊂ Gr(2, 4) of the usual Grassmannian consisting of subspaces with basis of the form (y_{1}, y_{2}, 1, 0) and (y_{3}, y_{4}, 0, 1). Within the A^{4} there are two transversal loci N^{+} ≅ A^{2} defined by y_{2} = y_{4} = 0 and N^{−} ≅ A^{2} defined by y_{1} = y_{3} = 0. They are closed in HP^{1}.

The complement HP^{1} − N^{+} is the quotient of A^{5} by a free action of 𝔾_{a} [16, Theorem 3.4]. Consequently the structural map HP^{1} − N^{+} → pt and its section x_{0} : pt → HP^{1} − N^{+} are motivic weak equivalences. This gives us motivic weak equivalences

[figure: T^{∧2} ≅ N^{−}/(N^{−} − 0); A; 2 out of 3; 2 out of 3; HP^{1}/(HP^{1} − N^{+}); (HP^{1}, x_{0}); /A; A^{4}/(A^{4} − N^{+}); excision; (A^{4} ∩ HP^{1})/((A^{4} ∩ HP^{1}) − N^{+}); excision]  (9.5)

The zigzag on the top line is the canonical isomorphism in H_{•}(S). □

A symplectic bundle (E, ϕ) is naturally a special linear bundle (E, λ_{ϕ}) with λ_{ϕ} the inverse of the Pfaffian of ϕ. Hence special linear Thom classes of hermitian K-theory (Theorem 5.1) give BO^{∗,∗} a symplectic Thom structure. So there are Pontryagin classes for symplectic bundles in hermitian K-theory, and all the formulas of this section are valid for them.

We may compute the Pontryagin classes induced by the Thom classes of this particular symplectic Thom structure. We need the isomorphism of (4.1), which becomes the isomorphism

\begin{matrix}GW^{−}(X) & ⟶^{≅} & KO_{0}^{[2]}(X) \\ [X, ϕ] & ⟼ & −[(X, ϕ)[1]]. \end{matrix}  (9.6)

The sign makes the isomorphism commute with the forgetful maps to K_{0}(X). For a rank 2 symplectic bundle (E, ϕ) has Pontryagin class p_{1}(E, ϕ) = −z^{∗} th(E, 𝒪_{X}, λ_{ϕ}), which is the image under the isomorphism of [E, ϕ] − [H] ∈ GW^{−}(X). The symplectic splitting principle [16, Theorem 10.2] now gives the following.

Proposition 9.9. Let (F, ψ) be a symplectic bundle of rank 2r on X. Its first Pontryagin class p_{1}(F, ψ) ∈ KO_{0}^{[2]}(X) is the image under the isomorphism (9.6) of [F, ψ] − r[H] ∈ GW^{−}(X).

Formulas for higher Pontryagin classes in terms of exterior powers of (F, ψ) will be given in [22].

10. The strategy for putting a ring structure on A^{∗,∗}

We explain our strategy for turning our partial multiplicative structure on BO^{∗,∗} into a full ring structure.

We will need the following standard facts about spectra. Recall that a motivic space X is small if Hom_{SH(S)}(Σ_{T}^{∞}X, −) commutes with arbitrary coproducts. We write M_{•}^{small}(S) for the full subcategory of small motivic spaces.

Theorem 10.1 ([15, Lemma A.34]). Let D^{(0)} → D^{(1)} → D^{(2)} → ⋯ be a sequence of morphisms in SH(S) with hocolim D^{(i)} = D, let X be a small motivic space, and let A be a T -spectrum. Then there is a canonical isomorphism

Hom_{SH(S)}(Σ_{T}^{∞}X, D) = colim Hom_{H_{•}(S)}(X, D^{(n)}).

and a canonical short exact sequence:

0 → \underleftarrow{lim}^{1} A^{p−1,q}(D^{(i)}) → A^{p,q}(D) → \underleftarrow{lim} A^{p,q}(D^{(i)}) → 0.

Particular cases of this are the following.

Theorem 10.2 ([16, Theorem 5.2]). Let X be a small motivic space and A a T -spectrum. Then we have

Hom_{SH(S)}(Σ_{T}^{∞}X, A) = colim Hom_{H_{•}(S)}(X ∧ T^{∧n}, A_{n}).

Theorem 10.3 ([15, Corollaries 3.4, 3.5]). Let A and E be T -spectra. Then for any r there is a canonical short exact sequences

0 → \underleftarrow{lim}^{1} A^{∗−2rn−1,∗−rn}(E_{n}^{∧r}) → A^{∗,∗}(E^{∧r}) → \underleftarrow{lim} A^{∗−2rn,∗−rn}(E_{n}^{∧r}) → 0.

Definition 10.4. An almost commutative monoid in SH(S) is a triple (A, μ, e) with A a T -spectrum and μ: A ∧ A → A and e: Σ_{T}^{∞}1 → A morphisms in SH(S) such that

(1) the morphism μ ∘ (μ ∧ 1) − μ ∘ (1 ∧ μ) ∈ Hom_{SH(S)}(A ∧ A ∧ A, A) lies in the subgroup \underleftarrow{lim}^{1} A^{∗−6n−1,∗−3n}(A_{n} ∧ A_{n} ∧ A_{n}),

(2) for σ: A∧A → A∧A the morphism switching the two factors, the morphism μ−μ∘σ ∈ Hom_{SH(S)}(A ∧ A, A) lies in the subgroup \underleftarrow{lim}^{1} A^{∗−4n−1,∗−2n}(A_{n} ∧ A_{n}),

(3) the map 1 − μ ∘ (1 ∧ e) ∈ Hom_{SH(S)}(A, A) lies in the subgroup \underleftarrow{lim}^{1} A^{∗−2n−1,∗−n}(A_{n}).

An almost commutative monoid (A, μ, e) defines pairings

×: A^{p,q}(X) × A^{r,s}(Y) → A^{p+r,q+s}(X ∧ Y)  (10.1)

for X and Y in M_{•}(S) as follows. For α : Σ_{T}^{∞}X → A ∧ S^{p,q} and β : Σ_{T}^{∞}Y → A ∧ S^{r,s} define α × β ∈ A^{p+r,q+s}(X ∧ Y) as the composition

Σ_{T}^{∞}(X ∧ Y) ≅ Σ_{T}^{∞}X ∧ Σ_{T}^{∞}Y ⟶^{α∧β} A ∧ S^{p,q} ∧ A ∧ S^{r,s} ≅ A ∧ A ∧ S^{p+r,q+s} ⟶^{m∧1} A ∧ S^{p+r,q+s}.

The unit e ∈ Hom_{SH(S)}(Σ_{T}^{∞}pt_{+}, A) defines an element 1 ∈ A^{0,0}(pt_{+}). There is then a unique element ε ∈ A^{0,0}(pt_{+}) such that Σ_{T} ε ∈ Hom_{SH(S)}(T, A ∧ T) is the composition of the endomorphism of T induced by the endomorphism x ↦ −x of A^{1} with e ∧ 1_{T}.

Theorem 10.5. For an almost commutative monoid (A, μ, e) in SH(S) the cohomology theory (A^{∗,∗}, ∂) with the pairing × of (10.1) and the element 1 ∈ A^{0,0}(pt_{+}) form an εcommutative ring cohomology theory on M_{•}(S)^{small} and on 𝒮m𝒪p/S.

Proof. There are canonical elements a_{n}: Σ_{T}^{∞}A_{n}(−n) → A. The definition of an almost commutative monoid is equivalent to having (A, μ, e) such that the induced pairing satisfies (a_{n} × a_{n}) × a_{n} = a_{n} × (a_{n} × a_{n}) and σ^{∗}(a_{n} × a_{n}) = ε^{n}a_{n} × a_{n} and a_{n} × e = a_{n} for all n. It then also satisfies

(Σ^{p,q}a_{n} × Σ^{p′,q′}a_{n}) × Σ^{p′′,q′′}a_{n} = Σ^{p,q}a_{n} × (Σ^{p′,q′}a_{n} × Σ^{p′′,q′′}a_{n})

for all (p, q), (p′, q′) and (p′′, q′′). By Theorem 10.2 for a small motivic space X any morphism Σ_{T}^{∞}X → A ∧ S^{p,q} factors as

Σ_{T}^{∞}X → Σ_{T}^{∞}A_{n}(−n) ∧ S^{p,q} ⟶^{Σ^{p,q}a_{n}} A ∧ S^{p,q}

for some n. So on small motivic spaces the multiplication is associative. The ε-commutativity and the unit property are treated similarly. The signs in the commutativity come from permuting the spheres. □

11. The universal elements

The isomorphism τ: (ℤ × HGr, (0, x_{0})) ⟶^{≅} KSp of Theorem 8.2 is classified by an element which has restrictions

τ|_{{i}×HGr(n,2n)} = [𝒰_{n,2n}, ϕ_{n,2n}] + (i − n)[H] ∈ KSp_{0}(HGr(n, 2n)).  (11.1)

Its image under the isomorphism (9.6) is a class τ_{2} ∈ KO_{0}^{[2]}(ℤ × HGr, (0, x_{0})) with

τ_{2}|_{{i}×HGr(n,2n)} = p_{1}(𝒰_{n,2n}, ϕ_{n,2n}) + ih ∈ KO_{0}^{[2]}(HGr(n, 2n))

according to Proposition 9.9. Here h ∈ KO_{0}^{[2]}(pt) = BO^{4,2}(pt) is the class corresponding to [H] ∈ GW^{−}(pt) under the isomorphism. By Theorem 9.5 we have an isomorphism

BO^{∗,∗}(ℤ × HGr) ≅ ∏_{i∈ℤ} BO^{∗,∗}(pt)[[p_{1}, p_{2}, p_{3}, …]]^{homog}

with the product taken in the category of graded rings. Setting

p_{1} = (p_{1})_{i∈ℤ} ∈ BO^{∗,∗}(ℤ × HGr),  \frac{1}{2} rk = (i1_{BO})_{i∈ℤ} ∈ BO^{∗,∗}(ℤ × HGr)

we see we have τ_{2} = p_{1} + (\frac{1}{2} rk)h.

For any k we have a composition of isomorphisms in the homotopy category H_{•}(S)

(ℤ × HGr, (0, x_{0})) ⟶^{τ} KSp ⟶^{trans_{2k+1}} KO^{[4k+2]} ⟶^{−1} KO^{[4k+2]}

where the trans_{2k+1} comes from translation and the −1 is the inverse operation of the H-space structure, which we add as in (4.1) so that the forgetful maps to K-theory should commute up to homotopy. (The inverse in the H-space structure comes from the Ω_{T} -spectrum structure and to the authors’ knowledge not from a duality-preserving functor.)

Definition 11.1 (Universal element). We denote by

τ_{4k+2} ∈ KO_{0}^{[4k+2]}(ℤ × HGr, (0, x_{0})) ≅ BO^{8k+4,4k+2}(ℤ × HGr, (0, x_{0}))

the element corresponding to the composition. It is given by τ_{4k+2} = (p_{1} + (\frac{1}{2} rk)h) ∪ β_{8}^{k}.

We write [−n, n] = {i ∈ ℤ | −n ≤ i ≤ n}. The class τ ∈ GW^{−}(ℤ × HGr) giving the isomorphism ℤ × HGr ≅ KSp has a universal property.

Lemma 11.2. There are unique μ and μ_{8k+4} in H_{•}(S) making the diagram commute

\xymatrix{(ℤ × HGr, (0, x_{0})) ∧ (ℤ × HGr, (0, x_{0})) \ar[r]^{μ} \ar[d]_{τ_{4k+2}∧τ_{4k+2}}^{≅} & KO^{[0]} \ar[d]^{translation}_{≅} \\ KO^{[4k+2]} ∧ KO^{[4k+2]} \ar[r]_{μ_{8k+4}} & _{KO_{[8k+4]}}}

and such that for each i, j and n the restriction of μ

({i} × HGr(n, 2n)) × ({j} × HGr(n, 2n) → KO^{[0]}

is the class in H_{•}(S) of the morphisms of ind-schemes represing the orthogonal Grothendieck-Witt class which is

([𝒰_{n,2n}, ϕ_{n,2n}] + (i − n)[H]) ⊠ ([𝒰_{n,2n}, ϕ_{n,2n}] + (j − n)[H]).

The μ with the asserted restrictions exists and is unique because of Theorem 9.7. It factors through the wedge space because of Lemma 9.6. Then μ_{8k+4} is the map making the diagram commute.

Lemma 11.3. The following diagram commutes in H_{•}(S).

\xymatrix{KO^{[4k−2]} ∧ KO^{[4k−2]} ∧ T^{∧8} \ar[r]^{μ_{8k−4}∧1} \ar[d]_{τ_{4k+2}∧τ_{4k+2}}^{≅} & KO^{[8k−4]} ∧ T^{∧8} \ar[dd]^{structure maps} \\ KO^{[4k−2]} ∧ T^{∧4} ∧ KO^{[4k−2]} ∧ T^{∧4} \ar[d]^{structure maps} \\ KO^{[4k+2]} ∧ KO^{[4k+2]} \ar[r]_{μ_{8k+4}} & _{KO_{[8k+4]}}}

Proof. Because of Theorem 9.7 it is enough to observe that the compositions of the two paths of arrows with the composition

\xymatrix{(\{i\} × HGr(n, 2n)) × (\{j\} × HGr(n, 2n)) × (HP^{1}, x_{0})^{×4} \ar@{^{(}->}[d] \\ (ℤ × HGr) × (ℤ × HGr) × (HP^{1}, x_{0})^{×4} \ar[r]^{τ_{4k−2}∧τ_{4k−2}∧η^{∧4}}_{≅} & KO^{[4k−2]} ∧ KO^{[4k−2]} ∧ T^{∧8}}

are the same for all i, j and n. □

By Theorem 10.3 we have a surjection

Hom_{SH(S)}(BO ∧ BO, BO) → \underleftarrow{lim} BO^{16k+8,8k+4}(KO^{[4k+2]} ∧ KO^{[4k+2]}) → 0

the compositions

Σ_{T}^{∞}(KO^{[4k+2]} ∧ KO^{[4k+2]})(−8k − 4) ⟶^{μ_{4k+2}} Σ_{T}^{∞}KO^{[8k+4]}(−8k − 4) ⟶^{canonical} BO

form a system of elements of the groups in the inverse limit. They are compatible with the connecting maps of the inverse limit because the diagrams of Lemma 11.3 commute. Let

m̄ = (μ_{8k+4})_{k∈ℕ} ∈ \underleftarrow{lim} BO^{16k+8,8k+4}(KO^{[4k+2]} ∧ KO^{[4k+2]})  (11.2)

and let

m ∈ Hom_{SH(S)}(BO ∧ BO)  (11.3)

be an element lifting it. As in (10.1) m defines a pairing

×: BO^{p,q}(A) × BO^{r,s}(B) → BO^{p+r,q+s}(A ∧ B).  (11.4)

Theorem 11.4. Suppose S satisfies the hypotheses of Theorem 13.1. For X and Y in 𝒮m/S and all p and q the pairing

BO^{4p,2p}(X_{+}) × BO^{4q,2q}(Y_{+}) → BO^{4p+4q,2p+2q}(X_{+} ∧ Y_{+})

induced by m is identified via the isomorphism γ of Corollary 7.3 with the naive pairing

KO_{0}^{[2p]}(X) × KO_{0}^{[2q]}(Y) → KO_{0}^{[2p+2q]}(X × Y).

Proof. Because of Jouanolou’s trick it is enough to consider affine X and Y.

Let α ∈ KO_{0}^{[2p]}(X) and β ∈ KO_{0}^{[2q]}(Y).

When we have 2p = 2q = 4k + 2 and X = Y = [−n, n] × HGr(n, 2n) and α = β is the restriction of the universal class τ_{4k+2}, then we have the identification by the construction of m.

When we have 2p = 2q = 4k + 2 but X, Y, α and β are general then by Theorem 8.1 there exists an n and morphisms f_{α}: X → [−n, n]×HGr(n, 2n) and f_{β}: Y → [−n, n]×HGr(n, 2n) in 𝒮m/S such that f_{α}^{∗}(τ_{4k+2}|_{[−n,n]×HGr(n,2n)}) = α and f_{β}^{∗}(τ_{4k+2}|_{[−n,n]×HGr(n,2n)}) = β. The identification of the two pairings for α and β now follows from that for the restrictions of the universal classes because both pairings are functorial for morphisms in 𝒮m/S.

For general p and q pick 4k + 2 ≥ max(2p, 2q). The identification of the two pairings on α and β follows from the identification of the two pairings on

α × p_{1}(U, ϕ)^{×2k+1−p} ∈ KO_{0}^{[4k+2]}(X × (HP^{1})^{×2k+1−p}),
β × p_{1}(U, ϕ)^{×2k+1−q} ∈ KO_{0}^{[4k+2]}(Y × (HP^{1})^{×2k+1−q}).  □

Theorem 11.5. Let m be as in (11.3), and let e ∈ BO^{0,0}(pt_{+}) be the unit of the partial multiplicative structure on (BO^{∗,∗}, ∂) of Definition 7.4. Then (BO, m, e) is an almost commutative monoid in SH(S), and the × and 1 induced by m and e make (BO^{∗,∗}, ∂, ×, 1) a ⟨−1⟩-commutative bigraded ring cohomology theory on M_{•}^{small}(S) and on 𝒮m𝒪p/S.

Proof. We prove associativity. The morphisms

\xymatrix{BO ∧ BO ∧ BO \ar[r]^{m∘(m∧id_{BO})} \ar[r]_{m∘(id_{BO}∧m)} & BO}

define two elements of BO^{0,0}(BO ∧ BO ∧ BO) with images in

\underleftarrow{lim} BO^{24k+12,12k+6}(KO^{[4k+2]} ∧ KO^{[4k+2]} ∧ KO^{[4k+2]}).

This last group is isomorphic to

\underleftarrow{lim} BO^{24k+12,12k+6}((ℤ × HGr, (0, x_{0}))^{∧3})

So it is enough to show that the corresponding elements (τ_{4k+2} ⊠ τ_{4k+2}) ⊠ τ_{4k+2} and τ_{4k+2} ⊠ (τ_{4k+2} ⊠ τ_{4k+2}) in each group of the inverse system are equal. However, since we have

BO^{24k+12,12k+6}((ℤ×HGr, (0, x_{0}))^{∧3}) ≅ \underleftarrow{lim} BO^{24k+12,12k+6}(([−n, n]×HGr(n, 2n), (0, x_{0}))^{∧3})

by Theorem 9.7 it is enough to show that the restrictions to the smooth affine schemes [−n, n] × HGr(n, 2n) are equal. But then by Theorem 11.4 the pairings coincide with the naive products, and those are associative.

The proofs of commutativity and of the unit property are similar.

Thus (BO, m, e) is an almost commutative monoid in SH(S). The rest of the statement of the theorem follows from that fact by Theorem 10.5 except for the value of ε. For that note that Σ_{T} 1 ∈ BO^{2,1}(T) is the class in GW^{[1]}(A^{1}, A^{1} − 0) of the symmetric complex (K(𝒪), Θ(𝒪)) of (7.2). The pullback of the complex along the endomorphism x ↦ −x of A^{1} is isometric to (K(𝒪), −Θ(𝒪)). So we have ε = ⟨−1⟩. □

Theorem 11.6. Suppose × and ×′ are two products on (BO^{∗,∗}, ∂) on M_{•}^{small}(S) which associative and ⟨−1⟩-commutative with unit 1 = ⟨1⟩, and such that α × Σ_{𝔾_{m}}1 = Σ_{𝔾_{m}}α and α × Σ_{S_{s}^{1}}1 = Σ_{S_{s}^{1}}α for all α. If × and ×′ are both compatible with the products KO_{0}^{[2m]}(X) ×

KO_{0}^{[2n]}(Y) → KO_{0}^{[m+n]}(X × Y) induced on Grothendieck-Witt groups of smooth schemes by the tensor product, then we have × = ×′.

Proof. Suppose first that we have α ∈ BO^{2i,i}(A) and β ∈ BO^{2j,j}(B) with A and B small pointed motivic spaces. By Theorems 10.1 and 10.2 there exist m and n such that Σ_{T}^{4m+2−i}α ∈ BO^{8m+4,4m+2}(A ∧ T^{∧2m}) has a factorisation

A ∧ T^{∧4m+2} → ([−n, n] × HGr(n, 2n), (0, x_{0})) ↪ ℤ × HGr ≅ KO^{[4m+2]} → BO(4m + 2)

and such that Σ_{T}^{4m+2−j}β has a similar factorization. Since Σ_{T}^{4m+2−i}α and Σ_{T}^{4m+2−j}β are thus pullbacks of classes on which × and ×′ agree, the products agree on Σ_{T}^{4m+2−i}α and Σ_{T}^{4m+2−j}β. The compatibility of the products with the suspension and their ε-commutativity imply that we also have Σ_{T}^{8m+4−i−j}(α × β) = Σ_{T}^{8m+4−i−j}(α ×′ β). Since the suspension operation is a bijection on cohomology groups, we have α × β = α ×′ β.

For α ∈ BO^{p,q}(A) and β ∈ BO^{p′,q′}(B) with for instance p < 2q and p′ > 2q′ the products agree on Σ_{S_{s}^{1}}^{2q−p}α ∈ BO^{2q,q}(A) and Σ_{𝔾_{m}}^{p′−2q′}β ∈ BO^{2p′−2q′,p′−q′}(B) by the previous case. By the same sort of argument they also agree on α and β. The other cases are similar. □

Theorem 11.7. The assertions of Theorem 1.3 and 1.4 hold.

This follows from Theorems 11.5 and 11.6.

12. Spectra of finite and infinite real and quaternionic Grassmannians

We now wish to switch our sphere from T to the pointed HP^{1}. This is possible because the pointed HP^{1} is isomorphic to T^{∧2} in H_{•}(S). The result is a spectrum BO_{HP^{1}} very similar to BO except that the structural maps are induced by a product with the Pontryagin class of a symplectic bundle on a pointed scheme rather than a Thom class on a Thom space. This is important because the universal property of ℤ × HGr deals with Grothendieck-Witt classes of bundles on schemes not of chain complexes on an X acyclic over U.

The zigzag (9.5) also gives us an equivalence between the stable homotopy categories SH_{T^{∧2}}(S) and SH_{(HP^{1},x_{∞})}(S) [9, Proposition 2.13]. There is also a Quillen equivalences between Spt_{T} (S) and Spt_{T^{∧2}}(S) given by the forgetful functor and its adjoints.

Theorem 12.1. The stable homotopy categories of T -spectra and of (HP^{1}, x_{0})-spectra are equivalent.

The class −p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}) ∈ KO_{0}^{[2]}(HP^{1}, x_{∞}) corresponds to th^{×2} ∈ KO_{0}^{[2]}(A^{2}/A^{2} − 0) under the identifications. So we may define our new spectrum.

Definition 12.2. The HP^{1}-spectrum BO_{HP^{1}} corresponding to the T -spectrum BO has spaces (KO^{[0]}, KO^{[2]}, KO^{[4]}, …) and bonding maps adjoint to the maps

− × −p_{1}(𝒰_{HP^{1}}, ϕ_{HP^{1}}): KO^{[2n]}(−) → KO^{[2n+2]}(− ∧ (HP^{1}, x_{∞}))

The purpose of this section is to prove the following result. For parallelism with the labelling of real and complex Grassmannians, we write HGr′(2r, 2n) = HGr(r, n) so that RGr(2r, 2n) and HGr′(2r, 2n) are both open subschemes of the ordinary Grassmannian Gr(2r, 2n) where a certain bilinear form is nondegenerate. We also write

[−n, n]′ = {i ∈ ℤ | −n ≤ i ≤ n and i ≡ n mod 2}.

For even n the scheme [−n, n]′ × RGr(n, 2n) is pointed in the component corresponding to 0 ∈ [−n, n]′ by the point corresponding to H_{+}^{⊕n/2} ⊕ 0 ⊂ H_{+}^{⊕n}. For odd n we either do not use a base point or we use a disjoint base point. To compactify our notations we write

HGr′_{2n} = [−n, n] × HGr′(2n, 4n),  (12.1)
RGr_{2n} = ([−2n, 2n]′ × RGr(2n, 4n)) ∪ ([−2n + 1, 2n − 1]′ × RGr(2n − 1, 4n − 2)). (12.2)

The first step in the proof of this theorem is the following general construction. For (X, x) a pointed scheme over S, let (X, x)^{+} be the pushout of

\xymatrix{A^{1} & pt \ar[r]^{x} \ar[l]^{∼}_{0} & X}  (12.3)

pointed by 1 ∈ A^{1}(pt). (This is essentially the A^{1} mapping cone of the inclusion x: pt → X.) The natural projection (X, x)^{+} → (X, x) which is the identity on X and sends A^{1} → x is a motivic weak equivalence.

We abbreviate HP^{1+} = (HP^{1}, x_{0})^{+}. We will actually consider HP^{1+}-spectra. The natural functor SH_{(HP^{1},x_{0})}(S) → SH_{(HP^{1+}}(S) is an equivalence, and let BO_{HP^{1+}} be the (HP^{1}, x_{0})^{+}- spectrum corresponding to BO.

Theorem 12.3. There are HP^{1+}-spectra BO^{fin} and BO^{geom} which are isomorphic to BO_{HP^{1+}} in SH_{HP^{1+}}(S) with spaces

BO_{2i}^{fin} = \begin{cases}RGr_{2·8^{i}} & for even i, \\ HGr′_{2·8^{i}} & for odd i, \end{cases}  (12.4)

BO_{2i}^{geom} = \begin{cases}ℤ × RGr & for even i, \\ ℤ × HGr & for odd i, \end{cases}  (12.5)

which are unions of real and quaternionic Grassmannians. The bonding maps BO_{2i}^{∗}∧HP^{1+} → BO_{2i+2}^{∗} with ∗ = fin or geom are morphisms of schemes and ind-schemes, respectively, which are constant on BO_{2i}^{∗} ∨ HP^{1+}.

Proposition 12.4. Let (X, x), (Y, y) and (Z, z) be pointed schemes. There is a natural bijection between

(1) the set of morphisms of pointed schemes g: (X, x) × (Y, y)^{+} → (Z, z) which restrict to the constant map (X × 1) ∪ (x × (Y, y)^{+}) → z, and

(2) the set of pairs (f, h) where

(a) f: (X × Y, x × y) → (Z, z) is a morphism of schemes which restricts to the constant map x × Y → z and (b) h: (X, x)×A^{1} → (Z, z) is a pointed A^{1}-homotopy between f|_{X×y} and the constant

map X → z.

The idea is that (X, x) × (Y, y)^{+} is the union of two subschemes, and the restrictions of g to these subschemes are g|_{X×Y} = f and g|_{X×A^{1}} = h.

Let U_{2n} and U be the tautological symplectic subbundles on HGr′(2n, 4n) and HP^{1} = HGr′(2, 4) resp., and let V_{16n} be the tautological orthogonal subbundle on RGr(16n, 24n). Let H be the trivial rank 2 symplectic bundle, and let H_{+} be the trivial rank 2 orthogonal bundle for the split quadratic form q(x_{1}, x_{2}) = x_{1}x_{2}.

Lemma 12.5. There exist morphisms of pointed schemes

f_{2n}: ([−n, n] × HGr′(2n, 4n)) × HP^{1} → RGr(16n, 32n)

such that the Grothendieck-Witt classes satisfy

f_{2n}^{∗}([V_{16n}] − 8n[H_{+}]) = ([U_{2n}] − (n − i)[H]) ⊠ ([U] − [H])  (12.6)

(where i ∈ [−n, n] ⊂ ℤ is the index of the component) and such that f_{2n}|_{pt×HP^{1}} is constant, and f_{2n}|_{([−n,n]×HGr′(2n,4n))×pt} is pointed A^{1}-homotopic to a constant map. These maps and homotopies are compatible with the inclusions HGr′(2n, 4n) ↪ HGr′(2(n + 1), 4(n + 1)) and RGr(16n, 32n) ↪ RGr(16(n + 1), 32(n + 1)).

Proof. There are orthogonal direct sums U_{2n} ⊕U_{2n}^{⊥} ≅ H^{⊕2n} and U ⊕U^{⊥} ≅ H^{⊕2}. Consequently we have

([U_{2n}] − (n − i)[H]) ⊠ ([U] − [H]) = [U_{2n} ⊠ U] + [H^{⊕n−i} ⊠ U^{⊥}] + [U_{2n}^{⊥} ⊠ H] − (6n − 2i)[H_{+}].

We have inclusions of orthogonal bundles

U_{2n} ⊠ U ⊂ H^{⊕2n} ⊠ U,  U_{2n}^{⊥} ⊠ H ⊂ H^{⊕2n} ⊠ H,
H^{⊕n−i} ⊠ U^{⊥} ⊂ H^{⊕2n} ⊠ U^{⊥},  H_{+}^{⊕2n+2i} ⊂ H_{+}^{⊕4n}

Consequently the orthogonal subbundle

(U_{2n} ⊠ U) ⊕ (H^{⊕n−i} ⊠ U^{⊥}) ⊕ (U_{2n}^{⊥} ⊠ H) ⊕ H_{+}^{⊕2n+2i}  (12.7)

of

(H^{⊕2n} ⊠ (U ⊕ U^{⊥} ⊕ H)) ⊕ H_{+}^{⊕4n} = H_{+}^{⊕16n}

is classified by a map f_{2n}: ([−n, n] × HGr′(2n, 4n)) × HP^{1} → RGr(16n, 32n).

When we restrict to pt × HP^{1}, we have i = 0, and the direct sum U_{2n} ⊕ U_{2n}^{⊥} becomes H^{⊕n} ⊕ H^{⊕n}. The orthogonal direct sum (12.7) becomes

((H^{⊕n} ⊕ 0) ⊠ (U ⊕ U^{⊥} ⊕ 0)) ⊕ ((0 ⊕ H^{⊕n}) ⊠ (0 ⊕ 0 ⊕ H)) ⊕ (H_{+}^{⊕2n} ⊕ 0).

Since U ⊕ U^{⊥} ⊕ 0 = H ⊕ H ⊕ 0, this is the same as the subbundle

((H^{⊕n} ⊕ 0) ⊠ (H ⊕ H ⊕ 0)) ⊕ ((0 ⊕ H^{⊕n}) ⊠ (0 ⊕ 0 ⊕ H)) ⊕ (H_{+}^{⊕2n} ⊕ 0).  (12.8)

corresponding to the pointing of RGr(16n, 32n). So f_{2n}(pt × HP^{1}) = pt.

A similar argument shows that f_{2n} is compatible with the inclusions of the Grassmannians in higher-dimensional Grassmannians.

When we restrict f_{2n} to ([−n, n] × HGr′(2n, 4n)) × pt, the direct sum over pt ⊂ HP^{1} becomes U ⊕ U^{⊥} = H ⊕ H. The orthogonal direct sum (12.7) becomes

(U_{2n} ⊠ (H ⊕ 0 ⊕ 0)) ⊕ (H^{⊕n−i} ⊠ (0 ⊕ H ⊕ 0)) ⊕ (U_{2n}^{⊥} ⊠ (0 ⊕ 0 ⊕ H)) ⊕ H_{+}^{⊕2n+2i} (12.9)

This direct sum decomposition of the subbundle of rank 16n is compatible with the orthogonal direct sum decomposition of the bundle of rank 32n into two summands

H^{⊕2n} ⊠ (H ⊕ 0 ⊕ H) = (U_{2n} ⊕ U_{2n}^{⊥}) ⊠ (H ⊕ 0 ⊕ H) = (H^{⊕n} ⊕ H^{⊕n}) ⊠ (H ⊕ 0 ⊕ H) (12.10)

and

(H^{⊕2n} ⊠ (0 ⊕ H ⊕ 0)) ⊕ H_{+}^{4n} = H_{+}^{8n}.  (12.11)

Because

M_{1} = (\begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

is a product of matrices which are elementary symplectic and elementary orthogonal, there is a morphism M: A^{1} → Sp_{4} ∩ O_{4} with M(0) = I and M(1) = M_{1}, namely

M(t) = (\begin{matrix}1 − t^{2} & 0 & −t & 0 \\ 0 & 1 − t^{2} & 0 & −2t + t^{3} \\ 2t − t^{3} & 0 & 1 − t^{2} & 0 \\ 0 & t & 0 & 1 − t^{2} \end{matrix})

When we restrict to pt × pt the direct sum U ⊕ U^{⊥} becomes H ⊕ H. We may see that

((1_{H^{⊕n}⊕0} ⊠ 1_{H^{⊕2}}) ⊕ (1_{0⊕H^{⊕n}} ⊠ M))^{−1}((1_{U_{2n}} ⊠ 1_{H^{⊕2}}) ⊕ (1_{U_{2n}^{⊥}} ⊠ M))

is a pointed A^{1}-homotopy between the subbundle

(U_{2n} ⊠ (H ⊕ 0 ⊕ 0)) ⊕ (U_{2n}^{⊥} ⊠ (0 ⊕ 0 ⊕ H))

and the subbundle

((H^{⊕n} ⊕ 0) ⊠ (H ⊕ 0 ⊕ 0)) ⊕ ((0 ⊕ H^{⊕n}) ⊠ (0 ⊕ 0 ⊕ H))

One may construct a similar pointed A^{1}-homotopy between the subbundles

((H^{⊕n−i} ⊕ 0) ⊠ (0 ⊕ H ⊕ 0)) ⊕ (H_{+}^{⊕2n+2i} ⊕ 0)

and

((H^{⊕n} ⊕ 0) ⊠ (0 ⊕ H ⊕ 0)) ⊕ (H_{+}^{⊕2n} ⊕ 0)

Combining the two gives a pointed A^{1}-homotopy between (12.9) and (12.8) and thus between f_{2n}|_{([−n,n]×HGr′(2n,4n))×pt} and the constant map.

The compatibility of the homotopies with the inclusions of Grassmannians is relatively straightforward. □

The next lemma is proven in the same way as the last one.

Lemma 12.6. There exist morphisms of pointed schemes

g_{n}: ([−n, n]′ × RGr(n, 2n)) × HP^{1} → HGr′(8n, 16n)

such that the Grothendieck-Witt classes satisfy

g_{n}^{∗}([U_{8n}] − 4n[H]) = ([V_{n}] − \frac{1}{2}(n − i)[H_{+}]) ⊠ ([U] − [H])  (12.12)

(where i ∈ [−n, n]′ ⊂ ℤ is the index of the component) and such that g_{n}|_{pt×HP^{1}} is constant, and g_{n}|_{([−n,n]′×RGr(n,2n))×pt} is pointed A^{1}-homotopic to a constant map. These maps and homotopies are compatible with the inclusions RGr(n, 2n) ↪ RGr(n + 2, 2n + 4) and HGr′(8n, 16n) ↪ HGr′(8(n + 2), 16(n + 2)).

Proof of Theorem 12.3. By Proposition 12.4 the maps and homotopies of Lemmas 12.5 and 12.6 give us maps

F_{n}: HGr′_{2n} ∧ HP^{1+} → RGr_{16n},
G_{n}: RGr_{2n} ∧ HP^{1+} → HGr′_{16n}.

We now define BO^{fin} to be the HP^{1}-spectrum with spaces as in the statement of Theorem 12.3 and with bonding maps the compositions

RGr_{2·8^{2i}} ∧ HP^{1+} → HGr′_{2·8^{2i+1}} → HGr′_{2·8^{2i+2}}
HGr′_{2·8^{2i}} ∧ HP^{1+} → RGr_{2·8^{2i+1}} → RGr_{2·8^{2i+2}}

of the appropriate F_{n} or G_{n} with the maps induced by the inclusions of Grassmannians. We define BO^{geom} to be the HP^{1}-spectrum with spaces

BO_{2i}^{geom} = \begin{cases}colim_{n} RGr_{2n} = ℤ × RGr & for even i, \\ colim_{n} HGr′_{2n} = ℤ × HGr & for odd i, \end{cases}

and with bonding maps induced by the F_{n} and G_{n}.

We claim that the inclusion map BO^{fin} → BO^{geom} is a stable weak equivalence. To show this we need to show that the maps colim_{i} Ω_{HP^{1}}^{i}(BO_{2i+2j}^{fin})^{f} → colim_{i} Ω_{HP^{1}}^{i}(BO_{2i+2j}^{geom})^{f} are weak equivalences for all j. The (−)^{f} denotes fibrant replacement. This is because we have two ℕ^{2}-indexed families of spaces

E_{n,i} = \begin{cases}Ω_{HP^{1}}^{i}(RGr_{2·8^{i}})^{f} & for even i, \\ Ω_{HP^{1}}^{i}(HGr′_{2·8^{i}})^{f} & for odd i, \end{cases}

E′_{n,i} = \begin{cases}Ω_{HP^{1}}^{i}(HGr′_{2·8^{i}})^{f} & for even i, \\ Ω_{HP^{1}}^{i}(RGr_{2·8^{i}})^{f} & for odd i, \end{cases}

The inclusions of Grassmannians the F_{n} and G_{n} give us maps E_{n,i} → E_{n+1,i} and E_{n,i} → E_{n+1,i+1} which commute, and similarly for the E′_{n,i}. Thus the E_{n,i} and E′_{n,i} are filtered systems of spaces indexed by a category with set of objects ℕ^{2} such that there is a unique arrow (n, i) → (n′, i′) if and only if n ≤ n′ and i − n ≤ i′ − n′. By cofinality we have isomorphisms colim_{i} E_{i,2i+2j} = colim_{n,i} E_{n,i} = colim_{i}(colim_{n} E_{n,i}) for all j and similarly for E′_{n,i}. These are the required weak equivalences for even and odd j respectively.

We now construct an isomorphism BO^{geom} ≅ BO_{HP^{1+}} in SH_{HP^{1+}}(S). By Theorem 10.3 we have an exact sequence

0 → \underleftarrow{lim}^{1} BO_{HP^{1+}}^{4n−1,2n}(BO_{2n}^{geom}) → BO_{HP^{1+}}^{0,0}(BO^{geom}) → \underleftarrow{lim} BO_{HP^{1+}}^{4n,2n}(BO_{2n}^{geom}) → 0.

For every odd n = 2k + 1 the universal element of Definition 11.1 gives an isomorphism τ_{4k+2} : ℤ × HGr ≅ KO^{[4k+2]} in H_{•}(S) and by adjunction a τ′_{4k+2} ∈ BO^{8k+4,4k+2}(BO_{4k+2}^{geom}).

The inverse system sends τ′_{4k+2} ↦ τ′_{4k−2} because in the diagram

\xymatrix{(ℤ × HGr) ∧ HP^{1+} ∧ HP^{1+} \ar[r] \ar[d]_{τ_{4k−2}∧1∧1}^{≅} & ℤ × HGr \ar[d]_{τ_{4k+2}}^{≅} \\ KO^{[4k−2]} ∧ HP^{1+} ∧ HP^{1+} \ar[r] & _{KO_{[4k+2]}}}  (12.13)

both horizontal maps are a × product with p_{1}(U)^{×2} = ([U] − [H])^{×2}. So the τ′_{4k+2} define an inverse system τ′ ∈ \underleftarrow{lim} BO_{HP^{1}}^{8k+4,4k+2}(BO_{4k+2}^{geom}). Since the τ′_{4k+2} are all isomorphisms, any element of BO_{HP^{1}}^{0,0}(BO^{geom}) lifting τ′ is an isomorphism by general facts about homotopy colimits of sequential direct systems in triangulated categories. □

The inverse system τ′ also lies in \underleftarrow{lim} BO_{HP^{1}}^{4i,2i}(BO_{2i}^{geom}). So it also gives us maps τ_{4k}: ℤ × RGr → KO^{[4k]} in H_{•}(S). Essentially these are the compositions of τ_{4k+2} and the maps in the motivic unstable homotopy category induced by the adjoint bonding maps of the two spectra

ℤ × RGr → Ω_{HP^{1+}}(ℤ × HGr) ⟶^{∼} Ω_{HP^{1+}}KO^{[4k+2]} ⟵^{∼} KO^{[4k]}.

We do not know whether these are isomorphisms in H_{•}(S). The best we know how to do is:

Proposition 12.7. The morphism Ω_{HP^{1}}(τ_{4k}) has a right inverse in H_{•}(S).

This is because in the commutative diagram

\xymatrix{ℤ × HGr \ar[r] \ar[d]^{∼} & Ω_{HP^{1+}}(ℤ × RGr) \ar[d] \\ _{KO_{[4k−2]}} \ar[r]^{∼} & Ω_{HP^{1+}}KO^{[4k]}}

the arrows on the left side and bottom of the square are weak equivalences by Theorem 8.2 and 7.2 respectively.

13. The commutative monoid structure in SH(S)

The main technical result of this section is the following theorem. We use it to show that the almost commutative monoid structure on the T -spectrum BO we constructed in Theorem 11.5 is actually a commutative monoid for S = Spec ℤ[\frac{1}{2}].

Theorem 13.1. Let S be a regular noetherian separated ℤ[\frac{1}{2}]-scheme of finite Krull dimension. Suppose that KO_{1}(S) and KSp_{1}(S) are finite. Then for all m the natural map

BO^{0,0}(BO^{∧m}) → \underleftarrow{lim} BO^{2mi,mi}((KO^{[i]})^{∧m})

is an isomorphism.

Proof. We prove the theorem for the HP^{1}-spectrum BO_{HP^{1}}. The theorem then follows for the T -spectrum BO.

By Theorem 10.3 and Theorem 12.3 there is a commutative diagram with exact rows

[figure: \underleftarrow{lim}^{1}_{i} BO_{HP^{1}}^{4mi−1,2mi}((KO^{[2i]})^{∧m}); BO_{HP^{1}}^{0,0}(BO_{HP^{1}}^{∧m}); ≅; \underleftarrow{lim}_{i} BO_{HP^{1}}^{4mi,2mi}((KO^{[2i]})^{∧m}); ≅ by cofinality; \underleftarrow{lim}^{1}_{i} BO_{HP^{1}}^{4mi−1,2mi}((BO_{2i}^{fin})^{∧m}); BO_{HP^{1}}^{0,0}((BO^{fin})^{∧m}); \underleftarrow{lim}_{i} BO_{HP^{1}}^{4mi,2mi}((BO_{2i}^{fin})^{∧m})]

The middle vertical arrow is an isomorphism because the morphism (BO_{2i}^{fin})^{∧m} → BO_{HP^{1}}^{∧m} is a stable weak equivalence. The righthand vertical arrow is an isomorphism by a cofinality argument as in the proof of Theorem 12.3. It follows that the lefthand vertical arrow is also an isomorphism.

The \underleftarrow{lim}^{1} in the lower row is isomorphic to

\underleftarrow{lim}^{1}_{i odd} KO_{1}(HGr′_{2N_{i}}^{∧m})  or \underleftarrow{lim}^{1}_{i odd} KSp_{1}(HGr′_{2N_{i}}^{∧m})

for even m and odd m, respectively, where N_{i} = 8^{2i}. By Theorem 9.4 each group in the system is a direct sum of a finite number of copies of KO_{1}(S) and of KSp_{1}(S). By the hypothesis it follows that each group in this system is finite, and so the \underleftarrow{lim}^{1} in the lower row of the diagram vanishes. Therefore the \underleftarrow{lim}^{1} in the upper row also vanishes, proving the isomorphism. □

Theorem 13.2. Let R be a Euclidean domain. Then we have KSp_{1}(R) = 0.

This is classical. It is proven essentially by showing that the action of the group ESp_{2n}(R) on unimodular vectors is transitive.

Theorem 13.3. Let R be a Euclidean domain with \frac{1}{2} ∈ R. Then we have KO_{1}(R) ≅ ℤ/2ℤ × R^{×}/R^{×2}.

This must be very well known to the experts. We include the proof for completeness’ sake.

Proof of Theorem 13.3. We use the long exact sequences of Karoubi’s fundamental theorem [11, 18]

⋯ → KO_{i}^{[n]}(R) ⟶^{F} K_{i}(R) ⟶^{H} KO_{i}^{[n+1]}(R) ⟶^{η} KO_{i−1}^{[n]}(R) → · · · .

with F the forgetful map and H the hyperbolic map. This amounts to four exact sequences including

⋯ → _{−1}V → K_{1} → KO_{1} → _{1}U ⟶^{0} K_{0} ↪ GW^{+} → W^{0} → 0  (13.1)
⋯ → KSp_{1} → K_{1} ↠ _{−1}V ⟶^{0} GW^{−} ↪ K_{0} → _{1}U → W^{−1} → 0  (13.2)

with the GW^{+} and GW^{−} the Grothendieck-Witt groups of symmetric and skew-symmetric bilinear forms respectively, the _{−1}V and _{1}U the groups of [10, Appendices 2 et 3], and the W^{i} the Witt groups with cohomological indexing à la Balmer.

For a principal ideal domain containing \frac{1}{2} we have W^{i} = 0 for i ≡ 2 or 3 (mod 4), while the the forgetful map GW^{−} → K_{0} of (13.2) is the inclusion 2ℤ ⊂ ℤ, and the hyperbolic map K_{0} → GW^{+} of (13.1) is injective. It then follows from the two sequences that we have a short exact sequence

0 → coker(K_{1} ↠ _{−1}V → K_{1}) → KO_{1} → ℤ/2ℤ → 0.

By [11, §4.1] or [4, §4.5] the composition K_{1} ↠ _{−1}V → K_{1} is [x] ↦ [x] − [^{t}x̄^{−1}]. For a euclidean domain with trivial involution, this is [x] ↦ [x^{2}]. So we have an exact sequence 1 → R^{×}/R^{×2} → KO_{1} → ℤ/2ℤ → 0. The image of [b] ∈ R^{×}/R^{×2} in KO_{1} is the class of the auto-isometry [ \begin{matrix}b & 0 \\ 0 & b^{−1} \end{matrix} ] of the hyperbolic quadratic form q(x_{1}, x_{2}) = x_{1}x_{2}. The class of the auto-isometry [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] is not in that image because its determinant is −1. It provides a splitting of the surjection in the exact sequence. □

For a nice discussion of the group_{1}V and a bit of the other two exact sequences of Karoubi’s fundamental theorem see [4, §4.5].

Theorem 13.4. Suppose KO_{1}(S) and KSp_{1}(S) are finite, for instance S = Spec ℤ[\frac{1}{2}]. Let m ∈ Hom_{SH(S)}(BO ∧ BO, BO) be the morphism of (11.3). Let e ∈ Hom_{SH(S)}(pt_{+}, BO) = BO^{0,0}(pt_{+}) be the element corresponding to ⟨1⟩ ∈ GW^{+}(pt) = KO_{0}^{[0]}(pt).

(a) Then (BO, m, e) is a commutative monoid in SH(S).

(b) The map m is the unique element of Hom_{SH(S)}(BO ∧ BO, BO) defining a pairing which, when restricted to pairing

BO^{4p,2p}(X_{+}) × BO^{4q,2q}(Y_{+}) → BO^{4p+4q,2p+2q}(X_{+} ∧ Y_{+})

with X, Y ∈ 𝒮m/S coincides with the tensor product pairing

KO_{0}^{[2p]}(X) × KO_{0}^{[2q]}(Y) → KO_{0}^{[2p+2q]}(X × Y)

of Grothendieck-Witt groups.

Proof. (a) By Theorem 11.5 (B, m, e) is an almost commutative monoid in SH(S). By Definition 10.4 the obstructions to (B, m, e) being a commutative monoid are three classes in the kernels of the maps of Theorem 13.1 for m = 1, 2, 3. Those classes vanish.

(b) The product on the KO_{0}^{[2p]}(X_{+}) determined uniquely the m̄ of (11.2). By Theorem 13.1 m is the unique element of Hom_{SH(S)}(BO ∧ BO, BO) mapping onto m̄. □

We now wish to use the closed motivic model structure of [15, Appendix A]. Among its properties are:

(1) The closed motivic model structure and the local injective motivic model structure used in §§6–7 have the same weak equivalences, but the closed motivic model structure has fewer cofibrations and more fibrations than the local injective motivic model structure.

(2) A pointed smooth S-scheme (X, x_{0}) is cofibrant in the closed motivic model structure. More generally, a closed embedding Z ↣ X in 𝒮m/S induces a cofibration Z_{+} ↣ X_{+} [15, Lemma A.10].

Hence the pointed scheme HP^{1+} is cofibrant for the closed motivic model structure, so we may define levelwise and stable closed motivic model structures for HP^{1+}- spectra.

(3) For any morphism u: S → S′ of noetherian schemes of finite Krull dimension, the pullback u^{∗}: M_{•}(S′) → M_{•}(S) is a strict symmetric monoidal left Quillen functor for the closed motivic model structure [15, Theorem A.17]. Consequently Lu^{∗}: SH(S′) → SH(S) can be computed by taking levelwise closed cofibrant replacements and then applying u^{∗}.

To extend m to other base schemes S, we need to discuss base change for morphisms u: S → S′. For any X ∈ 𝒮m/S′ there is a duality-preserving pullback functor inducing morphisms of hermitian K-theory spaces (1 × u)^{∗}: KO^{[n]}(X) → KO^{[n]}(X ×_{S′} S). This gives us maps KO_{S′}^{[n]} → u_{∗}KO_{S}^{[n]} and adjoint maps u^{∗}KO_{S′}^{[n]} → KO_{S}^{[n]}. These maps are compatible with Thom isomorphisms, inducing maps of spectra. The maps u^{∗}BO_{S′}^{geom} → BO_{S}^{geom} are isomorphisms in SH(S) because u^{∗} acts as base change on the quaternionic and real Grassmannians and their direct colimits. The maps Lu^{∗}BO_{S′}^{geom} → u^{∗}BO_{S′}^{geom} are isomorphisms SH(S) because for the closed motivic model structure BO_{S′}^{geom} is levelwise cofibrant and u^{∗} is a levelwise left Quillen functor.

Setting S′ = Spec ℤ[\frac{1}{2}] with u: S → Spec ℤ[\frac{1}{2}] the canonical map, we can now define the monoidal structure on BO_{S} in SH(S) as in [15, Definition 3.7] as the composition

m_{S}: BO_{S} ∧BO_{S} ≅ u^{∗}BO_{ℤ[ \frac{1}{2} ]} ∧u^{∗}BO_{ℤ[ \frac{1}{2} ]} ≅ u^{∗}(BO_{ℤ[ \frac{1}{2} ]} ∧BO_{ℤ[ \frac{1}{2} ]}) ⟶^{u^{∗}m_{ℤ[ \frac{1}{2} ]}} u^{∗}BO_{ℤ[ \frac{1}{2} ]} ≅ BO_{S}

Theorem 13.5. The assertions of Theorem 1.5 hold.

For S = Spec ℤ[\frac{1}{2}] this is part of Theorem 13.4. For other S it is deduced by base change from Spec ℤ[\frac{1}{2}].

Theorem 13.6. The assertions of Theorem 1.1 hold.

Theorem 5.1 shows that hermitian K-theory is an SL^{c}-oriented cohomology theory with a partial multiplicative structure. The ring structure is given by Theorem 13.5. The compatibility of the two multiplications is Theorem 11.4.

Schlichting’s multiplicative structure, which we mentioned when discussing Theorem 1.6, could replace our partial multiplicative structure for Theorems 1.1, 1.3, 1.4, etc. However, as we understand it, Schlichting’s product is defined in unstable homotopy theory. To get our main Theorem 1.5 with the monoid structure for T -spectra, we need our argument with the \underleftarrow{lim}^{1}.

14. CP^{1+}-spectra BGL^{fin} and BGL^{geom} for algebraic K-theory

The HP^{1+}-spectra constructed in §12 have an analogue for ordinary algebraic K-theory: the CP^{1}-spectra BGL^{fin} and BGL^{geom}. We sketch their construction. The first can be used to show that the uniqueness results concerning the algebraic K-theory spectrum BGL and its × product of [15, Remark 2.19 and Theorem 3.6] hold for any base scheme S which is noetherian of finite Krull dimension with finite K_{1}(S) and not just for S = Spec ℤ.

We use the affine Grassmannians which can be defined as

CGr(m, n) = GL_{n}/(GL_{m} × GL_{n−m})

or as the open subscheme

CGr(m, n) ⊂ Gr(m, n) × Gr(n − m, n)

where the two tautological subbundles of 𝒪^{⊕n} are supplementary or as the closed subscheme of the space on n × n matrices parametrizing projectors of rank m. Each CGr(m, n) is affine over the base scheme and an A^{m(n−m)}-bundle over the ordinary Grassmannnian Gr(m, n). Morphisms V → CGr(m, n) are in bijection with direct sum decompositions 𝒪_{V}^{⊕n} = U′_{m} ⊕ U′′_{n−m} with U′_{m} and U′′_{n−m} subbundles of ranks m and n − m respectively. We let CGr = colim_{n} CGr(n, 2n).

In particular CP^{1} = CGr(1, 2) ≅ P^{1} × P^{1} − Δ is an A^{1}-bundle over P^{1}. We may point CGr(1, 2) by CGr(0, 0). Let CP^{1+} then be the pointed scheme constructed in (12.3). The motivic stable homotopy categories of P^{1}-spectra, of CP^{1}-spectra and of CP^{1+}-spectra are equivalent. In particular there is a CP^{1+}-spectrum BGL_{CP^{1+}} corresponding to the P^{1}- spectrum BGL of [15]. For any smooth S-scheme X we write n = n[𝒪_{X}] ∈ K_{0}(X).

Lemma 14.1. There exist morphisms of pointed schemes

h_{n}: ([−n, n] × CGr(n, 2n)) × CP^{1} → CGr(4n, 8n)

such that the classes in K_{0} satisfy

h_{n}^{∗}([U′_{4n}] − 4n) = ([U′_{n}] − (n − i)) ⊠ ([U′_{1}] − 1)  (14.1)

(where i ∈ [−n, n] ⊂ ℤ is the index of the component) and such that h_{n}|_{pt×CP^{1}} is constant, and h_{n}|_{([−n,n]×CGr(n,2n))×pt} is pointed A^{1}-homotopic to a constant map. Moreover, these maps and homotopies are compatible with the inclusions CGr(n, 2n) ↪ CGr(n + 1, 2(n + 1)) and CGr(4n, 8n) ↪ CGr(4(n + 1), 8(n + 1)).

This lemma is proven in the same way as Lemma 12.5 using the equality

([U′_{n}] − (n − i)) ⊠ ([U′_{1}] − 1) = [U′_{n} ⊠ U′_{1}] + [𝒪^{⊕n−i} ⊠ U′′_{1}] + [U′′_{n} ⊠ 𝒪] − (3n − i)[𝒪 ⊠ 𝒪]

in K_{0}([−n, n] × CGr(n, 2n)) and the direct sum decompositions of vector bundles

(U′_{n} ⊠ U′_{1}) ⊕ (U′′_{n} ⊠ U′_{1}) = 𝒪^{⊕2n} ⊠ U′_{1},
(𝒪^{⊕n−i} ⊠ U′′_{1}) ⊕ (𝒪^{⊕n+i} ⊠ U′′_{1}) = 𝒪^{⊕2n} ⊠ U′′_{1},
(U′′_{n} ⊠ 𝒪) ⊕ (U′_{n} ⊠ 𝒪) = 𝒪^{⊕2n} ⊠ 𝒪,
(𝒪^{⊕n+i} ⊠ 𝒪) ⊕ (𝒪^{⊕n−i} ⊠ 𝒪) = 𝒪^{⊕2n} ⊠ 𝒪,

yielding a decomposition of the trivial bundle of rank 8n on ([−n, n] × CGr(n, 2n)) × CP^{1} as the direct sum of two subbundles of rank 4n.

Theorem 14.2. There are CP^{1+}-spectra BGL^{fin} and BGL^{geom} isomorphic to BGL_{CP^{1}} in SH_{CP^{1}}(S) with spaces

BGL_{n}^{fin} = [−4^{n}, 4^{n}] × CGr(4^{n}, 2 · 4^{n})  BGL_{n}^{geom} = ℤ × CGr

which are unions of affine Grassmannians. The bonding maps BGL_{n}^{∗} ∧ CP^{1+} → BGL_{n+1}^{∗} of the two spectra are morphisms of schemes or ind-schemes which are constant on the wedge BGL_{n}^{∗} ∨ CP^{1+}.

This theorem is proven in the same way as Theorem 12.3.

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Steklov Institute of Mathematics at St. Petersburg, Russia

Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Université de Nice – Sophia Antipolis, 28 Avenue Valrose, 06108 Nice Cedex 02, France