2.1.1. Given a vector bundle V over a scheme X, we let $\operatorname {\mathrm {Th}}(V)=V/V^\times $ be its Thom space in $\operatorname {\mathrm {SH}}(X)$ . Given a spectrum $\mathbb E$ and integers $(n,i)\in \mathbb Z^2$ , we put as usual:

$$ \begin{align*}\mathbb E^{n,i}\big(\operatorname{\mathrm{Th}}(V)\big)=\operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{SH}}(X)}(\operatorname{\mathrm{Th}}(V),\mathbb E_X(i)[n]). \end{align*} $$

Recall that the Thom space functor is a monoidal functor with respect to direct sums of vector bundles and tensor products of spectra: $\operatorname {\mathrm {Th}}(E \oplus F) \simeq \operatorname {\mathrm {Th}}(E) \otimes \operatorname {\mathrm {Th}}(F)$ . In particular, tensor products of morphisms induce an exterior product

$$ \begin{align*}\mathbb E^{n,i}\big(\operatorname{\mathrm{Th}}(E)\big) \otimes \mathbb E^{m,j}\big(\operatorname{\mathrm{Th}}(F)\big) \rightarrow \mathbb E^{m+n,i+j}\big(\operatorname{\mathrm{Th}}(E \oplus F)\big), x \otimes y \mapsto x \cdot y. \end{align*} $$

2.1.2. We will consider the following $\mathbb Z$ -graded algebraic subgroups of $\mathrm {GL}_*$ :

(2.1.2.a) $$ \begin{align} \mathrm{Sp}_* \rightarrow \mathrm{SL}_* \rightarrow \mathrm{SL}^c_* \rightarrow \mathrm{GL}_*. \end{align} $$

The nth graded part of $\mathrm {Sp}_*$ is $\mathrm {Sp}_{2n}$ so that the first map is homogeneous of degree $2$ . All other maps are homogeneous of degree $1$ . They are all classical algebraic groups except $\mathrm {SL}^c_*$ , which was introduced in [Reference Panin and Walter28, Definition 3.3]. Recall that its nth graded part is definedFootnote ⁸ as the kernel of the map

$$ \begin{align*}\mathrm{GL}_n \times \mathbb{G}_m \rightarrow \mathbb{G}_m, (g,t) \mapsto t^{-2}\det(g). \end{align*} $$

Letting $G=G_*$ be one of these groups, there is a classical correspondence between G-torsors over a given scheme X and vector bundles E over X equipped with extra structures. We summarise the situation in the following table:

Recall that our symplectic forms are always assumed to be nondegenerate. For short, we will say G-bundle for a bundle equipped with the corresponding extra structure. Morphisms of G-bundles over X are morphisms of vector bundles over X that preserve the extra structure in the obvious sense.

One can check that the category ${\mathscr V}(X,G)$ of G-bundles over X is additive. Let $r=2$ (respectively $r=1$ ) when $G=\mathrm {Sp}_*$ (respectively in the other cases). The constant bundle of rank r admits a canonical G-bundle structure and the corresponding object is denoted by

. In particular, given a (ring) spectrum $\mathbb E$ over X, one gets an isomorphism

The following definition is a slight extension of the original one due to Panin and Walter (our reference text will be [Reference Ananyevskiy3]).

2.1.4. Thom isomorphisms. This definition is tailored to generalise the classical notion of orientation in motivic homotopy theory, which corresponds in the above term to a $\mathrm {GL}$ -orientation.

More generally, under the assumptions of the above definition, given any G-bundle $\mathfrak V$ of rank r over X, the properties stated above imply that $\mathbb E^{**}(\operatorname {\mathrm {Th}}(\mathfrak V))$ is a free rank 1 bigraded $\mathbb E^{**}(X)$ -module with base $\mathfrak t(\mathfrak V)$ (see [Reference Ananyevskiy3, Corollary 3.9]).

In other words, the cup product with the Thom class induces a Thom isomorphism:

$$ \begin{align*}\theta_{\mathfrak V}^{\mathfrak t}:{\mathbb E}^{n,i}(X) \xrightarrow \sim \mathbb E^{n+2r,i+r}(\operatorname{\mathrm{Th}}(\mathfrak V)) \end{align*} $$

as in the classical oriented case (see, e.g., [Reference Déglise12, §2.2]).

2.1.10. Gysin morphisms. More generally, it is possible to extend the notions of Gysin maps (e.g., [Reference Déglise12, §3]) to a G-oriented ring spectrum $(\mathbb E,\mathfrak t)$ over B. We plan to come back to this point in a future work. Let us give the example of closed immersions because it is deeply linked with Euler classes.

Let $i:Z \rightarrow X$ be a closed immersion between smooth S-schemes with normal bundle $N_i$ . A G-orientation $\sigma $ on i is a G-bundle $\mathcal N_i$ over Z whose underlying vector bundle is $N_i$ . One defines the Gysin morphism associated to $(i,\sigma )$ :

$$ \begin{align*}i_*^\sigma:\mathbb E^{**}(Z) \xrightarrow{\theta_{\mathcal N_i}^{\mathfrak t}} \mathbb E^{**}(\operatorname{\mathrm{Th}}(N_i)) \xrightarrow{\tau_i} {\mathbb E}^{**}(X/X-Z) \rightarrow \mathbb E^{**}(X), \end{align*} $$

where the first map is the Thom isomorphism (Paragraph 2.1.4), the second one Morel-Voevodsky’s purity isomorphism and the last one is obtained through the canonical map $X \rightarrow (X/X-Z)$ . If $N_i$ is of rank c, the Gysin map is homogeneous of bidegree $(2c,c)$ .

Given a G-bundle $\mathfrak V=(V,\psi )$ , the zero section $s_0:X \rightarrow V$ obviously admits a canonical G-orientation $can$ . Then it follows from the above definition (and the fact that $\tau _{s_0}$ is essentially the identity through the latter identification) that

$$ \begin{align*}(s_0^{can})^*(s_0^{can})_*(1)=e(\mathfrak V). \end{align*} $$

2.2.1. Symplectic bundles. Let us be more specific about symplectic vector bundles, introduced in Paragraph 2.1.2.

First recall that any vector bundle $V/X$ admits a symplectification

$$ \begin{align*}\mathfrak H(V):=\left(V \oplus V^\vee,\begin{pmatrix} 0 & 1 \\ -\mathrm{can} & 0 \end{pmatrix}\right), \end{align*} $$

where $\mathrm {can}:V\to V^{\vee \vee }$ is the usual canonical isomorphism. Using the notation of loc. cit., we get in particular , which we will simply denote by $\mathfrak H$ in the sequel. To comply with the classical notations, the direct sum of symplectic vector bundles will be denoted by $\perp $ . In particular, $\mathfrak H(\mathbb A^n)=\mathfrak H^{\perp ,n}$ . More generally, $\mathfrak H$ sends $\oplus $ to $\perp $ .

2.2.2. As explained by Panin and Walter, $\mathrm {Sp}$ -oriented ring spectra are analogous to ( $\mathrm {GL}$ -)oriented ones. Let us first recall the $\mathrm {Sp}$ -projective bundle theorem. Consider an $\mathrm {Sp}$ -bundle $(V,\phi )$ over a scheme X. In [Reference Panin and Walter26] Panin and Walter introduced the projective $\mathrm {Sp}$ -bundle $\mathrm {H}\mathbb {P}(V,\phi )$ as the open subscheme of the Grassmannian scheme $\mathrm {Gr}(2,V)$ on which the restriction of $\phi $ to the canonical subbundle of rank $2$ is nondegenerate.

We let U be the tautological rank $2$ bundle on $\mathrm {H}\mathbb {P}(V,\psi )$ . By definition, it is equipped with a symplectic structure $\psi $ coming from the restriction of $\phi $ and we set $\mathfrak U=(U,\psi )$ .

The following $\mathrm {Sp}$ -projective bundle theorem is due to Panin and Walter; see [Reference Panin and Walter26, Th. 8.2] for a proof.

2.2.5. Borel classes. It is easy to derive from the previous theorem the theory of Borel classes. Under the above notation, they are the classes $b_i(V,\psi ) \in \mathbb E^{4i,2i}(X)$ for $i \geq 0$ uniquely determined by the following relations:

(2.2.4.a) $$ \begin{align} \sum_{i=0}^n (-1)^i.b_i(V,\psi).b^{n-i}=0, b_0(V,\psi)=1, \forall i>n, b_i(V,\psi)=0. \end{align} $$

In addition, they satisfy the following relations:

1. (1) $\operatorname {\mathrm {rk}}(V)=2n$ , $b_n(V,\psi )=e(V,\psi )$ .

2. (2) Invariance under isomorphisms. $b_i(\mathfrak V)=b_i(\mathfrak W)$ for $\phi :\mathfrak V \xrightarrow \sim \mathfrak W$ .

3. (3) Pullbacks compatibility. $f^*b_i(\mathfrak V)=b_i(f^{-1}\mathfrak V)$ for $f:Y \rightarrow X$ in $\mathscr Sch_B$ .

4. (4) Trivial bundles. $b_i\big [\mathfrak H(\mathbb A^n)\big ]=0$ for $i>0$ , $n>0$ .

5. (5) Whitney sum formula. $b_t(\mathfrak V \perp \mathfrak W)=b_t(\mathfrak V) \cdot b_t(\mathfrak W)$ in $\mathbb E^{**}(X)[[t]]$ , where $b_t$ denotes the total Borel class

   (2.2.4.b) $$ \begin{align} b_t(\mathfrak V)=\sum_i b_i(\mathfrak V).t^i. \end{align} $$

One can reformulate the two last properties by saying that the total Borel class $b_t$ factors through the $0$ th symplectic K-theory group, $\mathbf {KSp}_0(X)$ , and actually induces a morphism of abelian groups, sending sums to products:

(2.2.4.c) $$ \begin{align} b_t:\mathbf{KSp}_0(X)/\mathbb Z\big[\mathfrak H\big] \rightarrow \mathbb E^{**}(X)[[t]]^\times. \end{align} $$

2.2.6. The symplectic splitting principle. As for the classical Chern classes (see, e.g., [Reference Fulton16, Section 3.2]), one derives from the previous theorem a splitting principle. Given any symplectic bundle $\mathfrak V$ over a scheme X, there exists an affine morphism $p:X' \rightarrow X$ inducing a monomorphism $p^*:\mathbb E^{**}(X) \rightarrow \mathbb E^{**}(X')$ with the property that $p^{-1}(\mathfrak V)$ splits as a direct sum of rank $2$ symplectic bundles: $p^{-1}(\mathfrak V)=\mathcal X_1 \perp \ldots \perp \mathcal X_n$ .

One defines the Borel roots of $\mathfrak V$ as the Borel classes $\xi _i=b_1(\mathcal X_i)$ so that by the preceding Whitney sum formula the Borel classes of $\mathfrak V$ are the elementary symmetric polynomials in the variables $\xi _i$ .

As in the classical case, one can compute universal formulas involving Borel classes of $\mathfrak V$ by introducing Borel roots $\xi _i$ , which reduce to rank $2$ symplectic bundles, compute as if $\mathfrak V$ was completely split and then express the resulting formula in terms of the elementary symmetric polynomials in the $\xi _i$ . This principle will be used repeatedly in Section 3.

2.2.8. For the next formula, we need some notation. Let $u \in \mathcal O_X(X)^\times $ be a global unit on a scheme X. Let us consider the isomorphism $\gamma _u:\mathbb A^1_X \rightarrow \mathbb A^1_X$ obtained by multiplication by u. It induces a morphism of Thom spaces: $\gamma _u:\operatorname {\mathrm {Th}}(\mathbb A^1_X) \rightarrow \operatorname {\mathrm {Th}}(\mathbb A^1_X)$ . Following Morel (e.g., [Reference Morel24, Lemma 6.1.3]), we denote by $\langle u \rangle $ the corresponding automorphism of in the stable homotopy category over X. These elements satisfy the following formulas in the group :

• (U1) $\langle u \rangle \cdot \langle v \rangle =\langle uv \rangle $ .

• (U2) $\forall i, \langle u^{2i} \rangle =1$ .

The following proposition is mainly due to [Reference Ananyevskiy3]. At present, we do not know whether it is true for a general $\mathrm {Sp}$ -oriented spectrum.

Proof. Let then $(V,\psi )$ be a symplectic bundle. The symplectic form $\psi $ corresponds to a trivialisation of the determinant $\psi :\det V\simeq \mathbb A^1_X$ , and the symplectic form $u.\psi $ corresponds to the composite

$$ \begin{align*} \det V\stackrel \psi\to \mathbb A^1_X\stackrel {\cdot u}\to \mathbb A^1_X. \end{align*} $$

The claim now follows from [Reference Ananyevskiy3, Lemma 7.3]. For (2), we can simply use the symplectic splitting principle 2.2.6.

2.3.1. It appears clearly from the previous section that Borel classes are to $\mathrm {Sp}$ -oriented spectra what Chern classes are to oriented spectra. It is therefore natural to look for an analogue of the theory of formal group law associated to any oriented ring spectrum.

As observed by Walter,Footnote ¹⁰ the first technical problem that arises is the fact that symplectic bundles are not stable under tensor product. However, if you consider an odd number of symplectic bundles, then their tensor product is equipped with a canonical symplectic form – the tensor product of the symplectic forms of each bundle. The second technical problem is that a triple product of symplectic bundles of rank $2$ is of rank $8$ . This means that we will have to take into account four nontrivial Borel classes $b_i$ for $i=1,\ldots , 4$ . Once taking into account these differences, we can mimic the construction of the associated formal group law in classical orientation theory as follows.

Let us fix an $\mathrm {Sp}$ -oriented ring spectra $(\mathbb E,\mathfrak t)$ over an arbitrary scheme X (Definition 2.1.3). Put $\mathbb E^{**}=\mathbb E^{**}(X)$ .

We consider the following triple product as an ind-smooth X-scheme: $P=\mathrm {H}\mathbb {P}^\infty _X \times _X \mathrm {H}\mathbb {P}^\infty _X \times _X \mathrm {H}\mathbb {P}^\infty _X$ . For $i=1, 2, 3$ , let $\mathfrak U_i$ be the respective canonical rank $2$ symplectic bundle on the ith coordinate, and let $x,y,z=b_1(\mathfrak U_1), b_1(\mathfrak U_2), b_1(\mathfrak U_3)$ . According to the symplectic projective bundle theorem 2.2.3, one gets the following isomorphism of $\mathbb E^{**}$ -bigraded rings:

$$ \begin{align*}\mathbb E^{**}(P) \simeq \mathbb E^{**}[[x,y,z]], \end{align*} $$

the ring of power series in three variables. The next definition follows considerations initiated by Walter.

2.3.3. It is legitimate to look for an analogue of the axioms satisfied by formal group laws in the case of ternary laws. Commutativity is obvious, because the tensor product of three symplectic bundles is commutative. Explicitly, the coefficient $a^l_{ijk}$ above is independent of the order of the indices $i,j,k$ :

(2.3.3.a) $$ \begin{align} a_{ijk}^l=a_{jik}^l=a_{ikj}^l. \end{align} $$

In other words, the power series $F_l(x,y,z)$ is symmetric in the variables $x,y,z$ . To give examples of ternary laws, it is useful to consider a basis for the symmetric polynomials in $x,y,z$ . We choose the monomial basis denoted by

(2.3.3.b) $$ \begin{align} \sigma(x^iy^jz^k)=\sum_{(a,b,c)} x^ay^bz^c, \end{align} $$

where the sum runs over the monomials $x^ay^bz^c$ in the orbit of $x^iy^jz^k$ under the action of the permutations of the variables $x,y,z$ . So taking into account the commutativity constraints, ternary laws can be written as

$$ \begin{align*}F_l(x,y,z)=\sum_{i \geq j \geq k} a_{ijk}^l.\sigma(x^iy^jz^k). \end{align*} $$

Proof. Consider the first assertion. We write $\mathfrak U=(U,\psi )$ for $\mathfrak U_1$ . Given that $b_i(\mathfrak H)=0$ for $i>0$ (Paragraph 2.2.4), we get

$$ \begin{align*}F_l(x,0,0)=b_l\big((U,\psi) \otimes \mathfrak H \otimes \mathfrak H\big). \end{align*} $$

Let $h^\prime $ be the symmetric bundle on $\mathbb Z^2$ given by the matrix $\displaystyle {\begin {pmatrix} 0 & 1 \\ 1 & 0\end {pmatrix}}$ . A direct computation shows that $\mathfrak H \otimes \mathfrak H$ is isometric to $h^\prime \perp h^\prime $ ; that is, $\mathfrak H \otimes \mathfrak H=2h^\prime $ . We have $h^\prime +1= h+1$ as illustrated by the following computation:

$$ \begin{align*} \begin{pmatrix} 1 & 0 & 1 \\ 1 & -1 & 1 \\ 0 & 1 & -1 \end{pmatrix}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix} \begin{pmatrix} 1 & 1 & 0 \\ 0 & -1 & 1\\ 1 & 1 & -1\end{pmatrix}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{pmatrix}. \end{align*} $$

It follows that $\mathfrak H \otimes \mathfrak H+2= 2h^\prime +2=2h+2$ . We then obtain

$$ \begin{align*} b_t\big(((U,\psi) \otimes \mathfrak H \otimes \mathfrak H)\perp(U,\psi)\perp(U,\psi)\big)=b_t\big((2h.(U,\psi))\perp(U,\psi)\perp(U,\psi)\big). \end{align*} $$

We now observe that if $(V_1,\phi _1)$ and $(V_2,\phi _2)$ are symplectic bundles on $\mathrm {H}\mathbb {P}^\infty _X$ such that

$$ \begin{align*} b_t((V_1,\phi_1)\perp (U,\psi))=b_t((V_2,\phi_2)\perp (U,\psi)), \end{align*} $$

then $b_t(V_1,\phi _1)=b_t(V_2,\phi _2)$ . Indeed, we have

$$ \begin{align*} b_t((V_i,\phi_i)\perp (U,\psi))=b_t(V_i,\phi_i)b_t(U,\psi)=b_t(V_i,\phi_i)(1-xt) \end{align*} $$

and the claim follows from the fact that $(1-xt)$ is not a zero divisor in $\mathbb E^{**}(X)[[x]][t]$ . Thus,

$$ \begin{align*} b_t\big((U,\psi) \otimes \mathfrak H \otimes \mathfrak H\big)=b_t\big(2h.(U,\psi)\big)=b_t\big((U,\psi)\perp (U,\psi)\perp (U,-\psi)\perp (U,-\psi)). \end{align*} $$

According to Proposition 2.2.9, one gets

$$ \begin{align*}b_t(U,-\psi)=1-(\epsilon x).t \end{align*} $$

and then

$$ \begin{align*}b_t\big((U_{,}\psi) \otimes \mathfrak H \otimes \mathfrak H\big)=(1+x.t)^2.\big(1-(\epsilon x).t\big)^2 \end{align*} $$

so that expanding the last term gives the desired result.

To get the second point, we have to compute $b_4(\mathfrak U \otimes \mathfrak U \otimes \mathfrak H)$ . We claim that the bundle $\mathfrak U \otimes \mathfrak U \otimes \mathfrak H$ has a nowhere vanishing section. Indeed, consider the short exact sequence

$$ \begin{align*} 0\to \det(\mathfrak U)\to \mathfrak U \otimes \mathfrak U\to \mathrm{Sym}^2(\mathfrak U)\to 0. \end{align*} $$

Because $\mathfrak U$ is symplectic, we have an isomorphism $\det (\mathfrak U)\simeq \mathcal {O}$ and it follows that $\mathfrak U \otimes \mathfrak U$ has a nowhere vanishing section. So does $\mathfrak H$ , and finally we see that $\mathfrak U \otimes \mathfrak U \otimes \mathfrak H$ has a nowhere vanishing section. Consequently, its Euler class vanishes; hence the result (according to point (1) in Paragraph 2.2.4).

2.3.7. The additive oriented case. Suppose that $\mathbb E$ is $\mathrm {GL}$ -oriented and consider the induced $\mathrm {Sp}$ -orientation (Remark 2.1.6). In this case, one can express the total ternary law $F_t(x,y,z)$ in terms of the formal group law associated with the $\mathrm {GL}$ -orientation. We assume for simplicity that the formal group law associated with the $\mathrm {GL}$ -orientation is the additive one. In this case, the Borel classes can be expressed in terms of Chern classes as follows:

(2.3.7.a) $$ \begin{align} b_i(V,\psi)=c_{2i}(V). \end{align} $$

This is easily seen for a symplectic bundle of rank $2$ , using the fact that its first Chern class is trivial.Footnote ¹¹ The result for symplectic bundles of higher rank follows from the splitting principle. So, using the notations of the above definition, one can use the following equality:

(2.3.7.b) $$ \begin{align} c_t(U_1 \otimes U_2 \otimes U_3)=\sum_{i=2l}F_l(x,y,z).t^i, \end{align} $$

where $\mathfrak U_i=(U_i,\psi _i)$ , $x=c_2(U_1)$ , $y=c_2(U_2)$ , $z=c_2(U_3)$ . Using the splitting principle to compute the left-hand side (and the fact the formal group law underlying $\mathbb E$ is additive), one derives from the previous equality the following one:

$$ \begin{align*} &\prod_{e_1,e_2,e_3 \in \{\pm 1\}} (e_1u_1+e_2u_2+e_3u_3)=\sum_{i=2l} F_l(x,y,z).t^i, \\ &\quad x=-u_1^2, y=-u_2^2, z=-u_3^2. \end{align*} $$

The result of this computation yields the following example of FTL.

3.1.1. The aim of this section is to compute the ternary laws (Definition 2.3.2) associated with Chow-Witt groups or, equivalently, Milnor-Witt motivic cohomology. So, in the entire section we will work with the ring of coefficients $A=\mathbb E^{**}$ of some $\mathrm {SL}$ -oriented ring spectrum $\mathbb E$ . The ternary laws live in the ring of formal power series $A[[x,y,z]]$ .

One can already obtain the following simple lemma that follows from the formula for the degree of the coefficients of the ternary laws (2.3.2.a) and Proposition 2.3.4.

3.2.1. We will now determine the ternary laws $F_l^{{\mathbf H} \operatorname {{\underline W}}}$ in the case of Witt unramified cohomology, represented by ${\mathbf H} \operatorname {{\underline W}}$ . The underlying category of schemes $\mathscr Sch$ is that of k-schemes for a fixed base field k of characteristic different from $2$ (see also Notation section).

We can apply Lemma 3.1.2 to this ring spectrum. Recall, moreover, that on ${\mathbf H} \operatorname {{\underline W}}$ one gets $\epsilon =1$ and $h=0$ . Let us restate the lemma in our particular case to clarify the situation:

$$ \begin{align*}F^{{\mathbf H} \operatorname{{\underline W}}}_l(x,y,z)= \begin{cases} 0, & l=1, \\ -2.\sigma(x^2)+a_{110}^2.\sigma(xy), & l=2, \\ a^3_{210}.\sigma(x^2y)+a^3_{111}.xyz, & l=3, \\ \sigma(x^4)+a^4_{310}.\sigma(x^3y)+a^4_{220}.\sigma(x^2y^2)+a^4_{211}.\sigma(x^2yz), & l=4. \end{cases} \end{align*} $$

Proof. Our main tool will be the following computation of symplectic bundles (A.0.5.a)Footnote ¹²:

(3.2.3.a) $$ \begin{align} (U^{\otimes 3},\varphi^{\otimes 3})\simeq (U,\langle 2\rangle\varphi)\perp (U,\langle 6\rangle\varphi) \perp (\mathrm{Sym}^3U,\psi). \end{align} $$

Note that applying point (2) of Proposition 2.2.9, we get

$$ \begin{align*}b^W_1(U,\langle 2\rangle\varphi)=\langle 2\rangle.x, \ b^W_1(U,\langle 6\rangle\varphi)=\langle 6\rangle.x. \end{align*} $$

We start with the proof of (1). We first do the computation of the left-hand column. The symplectic bundle $\operatorname {\mathrm {Sym}}^3\mathfrak U$ is of rank $4$ , so that we already get the required vanishing (by definition, see (2.2.4.a)). We apply the symplectic splitting principle to that bundle (Paragraph 2.2.6): in particular, after pullback along an affine morphism $p:X' \rightarrow \mathrm {H}\mathbb {P}^\infty $ , $\operatorname {\mathrm {Sym}}^3\mathfrak U$ admits a splitting as two rank $2$ symplectic bundles, whose Euler classes we denote respectively $\alpha $ and $\beta $ . Pulling back the decomposition (3.2.3.a) to $X'$ yields a decomposition of $\mathfrak U^{\otimes ,3}$ into a sum of four rank $2$ symplectic bundles and applying the Whitney sum formula of Paragraph 2.2.4, we get

(3.2.3.b) $$ \begin{align} b^W_t(U^{\otimes 3},\varphi^{\otimes 3})=(1+\langle 2\rangle xt)(1+\langle 6\rangle xt)(1+\alpha t)(1+\beta t). \end{align} $$

Because we already know $F_1^W$ and $F_2^W$ (Paragraph 3.2.1 and relation (3.2.2.a)), we obtain the following equation by computing the coefficients of t and $t^2$ :

$$ \begin{align*} \alpha+\beta+(\langle 2\rangle+\langle 6\rangle).x & = 0 \\ \alpha\beta+(\alpha+\beta)(\langle 2\rangle + \langle 6\rangle).x+\langle 3\rangle.x^2 &=-6.x^2. \end{align*} $$

Using the relation satisfied by the Borel roots $\alpha $ and $\beta $ , we deduce that

$$ \begin{align*}b^W_1(\operatorname{\mathrm{Sym}}^3 \mathfrak U)=(\langle -2\rangle+\langle -6\rangle).x, \ b_2^W(\operatorname{\mathrm{Sym}}^3 \mathfrak U)=(-4+\langle 3\rangle).x^2. \end{align*} $$

To conclude the proof of (1), it therefore suffices to show the following equality in $W(k)$ :

$$ \begin{align*} \langle -2\rangle+\langle -6\rangle = -3+ \langle 3\rangle. \end{align*} $$

We need only to prove it either for a finite field (if k is of positive characteristic) or for $\mathbb Q$ in case k is of characteristic zero. The first case is obvious because both forms have the same rank and same discriminant and the second case is obtained via a comparison of residues.

To get (2), it suffices now to finish the computation of (3.2.3.b). But using the computation above, we obtain

$$ \begin{align*}b^W_t(U^{\otimes 3},\varphi^{\otimes 3})=(1+\langle 2\rangle xt)(1+\langle 6\rangle xt)(1+(-3+\langle 3\rangle) xt+(-4+\langle 3\rangle)x^2t^2). \end{align*} $$

Then, an easy computation allows us to conclude.

3.2.4. The case $l=3$ and $l=4$ of point (2) in the preceding computation together with relations (3.2.2.b) and (3.2.2.c) yields

(3.2.4.a) $$ \begin{align} & a^3_{111}=-8, \end{align} $$

(3.2.4.b) $$ \begin{align} & 3-6+a^4_{211}=-3 \Rightarrow a^4_{211}=0. \end{align} $$

Let us write the final result for the FTL associated with ${\mathbf H} \operatorname {{\underline W}}$ :

(3.2.4.c) $$ \begin{align} F_t^{{\mathbf H} \operatorname{{\underline W}}}(x,y,z)=-2\sigma(x^2).t^2-8.xyz.t^3+\lbrack\sigma(x^4)-2\sigma(x^2y^2)\rbrack.t^4. \end{align} $$

Proof. One uses the decomposition

where $\nu :\operatorname {\mathrm {Spec}}{\mathbb Z} \rightarrow \operatorname {\mathrm {Spec}}{\mathbb Q}$ is the canonical open immersion (see [Reference Déglise, Fasel, Khan and Jin15, Cor. 6.2]).

4.1.1. Let S be any $\mathbb Z[{1}/{2}]$ -scheme. Using a method of Morel and Voevodsky, Panin and Walter proved in [Reference Panin and Walter28, Theorem 8.2] that there exists a canonical weak $\mathbb A^1$ -homotopy equivalence

(4.1.1.a) $$ \begin{align} HGr_S \rightarrow B\mathrm{Sp}_S. \end{align} $$

Moreover, when S is regular, they also proved that one gets an isomorphism of abelian groups (we refer the reader to [Reference Schlichting and Tripathi34, Th. 1.3])

(4.1.1.b) $$ \begin{align} KSp_0(S) \rightarrow [X,\mathbb Z \times B\mathrm{Sp}_S], \end{align} $$

where $[-,-]$ denotes the set of morphisms in the unpointed $\mathbb A^1$ -homotopy category over S. Concretely, to a symplectic vector bundle $\mathfrak V$ over X of rank $2r$ the above isomorphism associates the couple $(\gamma _{\mathfrak V},r)$ where $\gamma _{\mathfrak V}:S \rightarrow BSp_S$ is the map classifying $\mathfrak V$ and $2r$ is the rank of $\mathfrak V$ .

In this section, we will use Riou’s method to classify the following ‘operations’.

4.1.3. Let us now consider an $\mathrm {Sp}$ -oriented ring spectrum $\mathbb E$ over a regular scheme S. Let us put $\mathbb E^{**}=\mathbb E^{**}(S)$ as a bigraded ring.

Recall that, according to [Reference Panin and Walter28],Footnote ¹³ one can compute the $\mathbb E$ -cohomology of symplectic Grassmanians as

(4.1.3.a) $$ \begin{align} \mathbb E^{n,i}(HGr_S)=\big(E^{**}[[b_r,r \geq 1]]\big)^{(n,i)}, \end{align} $$

where the exponent on the right-hand side denotes the subgroup of elements of degree $(n,i)$ with the convention that $b_r$ is of degree $(4r,2r)$ . Explicitly, an element of the right-hand side is a formal power series of the form

(4.1.3.b) $$ \begin{align} F=\sum_\alpha \left(a_\alpha.{\textstyle \prod}_{i \in \mathbb N^*} b_i^{\alpha(i)}\right), \end{align} $$

where $\mathbb N^*=(\mathbb N-\{0\})$ and $\alpha $ runs over the applications $\alpha :\mathbb N^* \rightarrow \mathbb N$ with finite support and $a_\alpha $ is an element of $\mathbb E^{**}$ of degree $(n-4.|\alpha |,i-2.|\alpha |)$ .

Using the method of [Reference Riou30], one deduces a complete description of the $\mathrm {Sp}$ -operations with values in $\mathbb E$ .

Proof. This is a consequence of [Reference Riou30, Proposition 1.2.9] applied to the inductive system of smooth schemes $HGr_\bullet =(HGr_{n,d})_{n,d \in \mathbb N}$ and to the H-group $E=\Omega ^\infty (\mathbb E(i)[n])$ . Indeed, formula (4.1.3.a) implies that “ $HGr_\bullet $ does not unveil phantoms in E” in the sense of [Reference Riou30, Definition 1.2.2]. Moreover, “ $\pi _0 HGr$ is generated by $HGr$ ” in the sense of [Reference Riou30, Definition 1.2.5] by application of [Reference Panin and Walter28, Theorem 8.1].

4.1.5. According to [Reference Riou30, Lem. 6.2.2.2], for any integer $n>0$ there exists a unique group morphism, natural in commutative ring A, of the form

$$ \begin{align*}\psi^A_n:(1+t.A[[t]],\times) \rightarrow (A,+) \end{align*} $$

such that $\psi ^A_n(1+a.t)=a^n$ and that vanishes on $1+t^{n+1}.A[[t]]$ . By analogy with the oriented case, one defines an additive $\mathrm {Sp}$ -operation $\tilde {\chi }_{2n}^{\mathbb E}$ of degree $(4n,2n)$ with values in $\mathbb E$ . Given any symplectic bundle $\mathfrak V$ on a smooth S-scheme X, one sets

(4.1.5.a) $$ \begin{align} \tilde{\chi}_{2n}^{\mathbb E}([\mathfrak V])=\psi_n^{\mathbb E^{**}}(b_t(\mathfrak V)), \end{align} $$

where $b_t(\mathfrak V)$ is the total Borel class of $\mathfrak V$ , Formula (2.2.4.b). (The latter formula shows that $\tilde {\chi }_{2n}^{\mathbb E}$ is indeed additive.) We also put $\tilde {\chi }^E_0=1$ . Note in particular that when $\mathfrak V=\mathfrak U$ has rank $2$ , one gets by construction for $n \geq 0$

(4.1.5.b) $$ \begin{align} \tilde{\chi}_{2n}^{\mathbb E}([\mathfrak U])=b_1(\mathfrak U)^n. \end{align} $$

Recall from Panin-Walter’s symplectic projective bundle theorem (Theorem 2.2.3) that one obtains an isomorphism of bigraded $\mathbb E^{**}$ -modules

$$ \begin{align*}\mathbb E^{**}[[b]] \rightarrow \mathbb E^{**}(\mathrm{H}\mathbb{P}^\infty_S), b \mapsto b_1(\mathfrak U), \end{align*} $$

where the indeterminate b has degree $(4,2)$ . As in [Reference Riou30, Prop. 6.2.2.1], one can compute all of the additive $\mathrm {Sp}$ -operations with values in $\mathbb E$ .

Proof. The injectivity of the map follows from the symplectic bundle principle (Paragraph 2.2.6) and the universal property of $\mathrm {H}\mathbb {P}^\infty \simeq B\mathrm {Sp}_2$ . The last assertion follows from relation (4.1.5.b). It remains to prove surjectivity. One considers a power series $\sum _n \lambda _n.b^n$ . Then the infinite sum

$$ \begin{align*}\sum_n \lambda_n.\tilde{\chi}_{2n}^{\mathbb E} \end{align*} $$

gives a well-defined additive operation. Indeed, given any scheme X and a symplectic bundle $\mathfrak V$ over X, it follows from the nilpotency of Borel classes ([Reference Panin and Walter26, Theorem 8.6]) that there exists an integer $N>0$ such that for all $n>N$ , $\chi _{2n}^{\mathbb E}(\mathfrak V)=0$ .

4.1.7. The lemma of Riou used in the above proof is a smart way of constructing $\tilde {\chi }_{2n}$ . A more classical way is to use the symplectic splitting principle 2.2.6. Indeed, $\tilde {\chi }_{2n}(\mathfrak V)$ is uniquely defined in terms of the Borel roots $\xi _i$ of $\mathfrak V$ by the property

$$ \begin{align*}\tilde{\chi}_{2n}(\mathfrak V)=\sum_i \xi_i^n. \end{align*} $$

Thus, one can express $\tilde {\chi }_{2n}(\mathfrak V)$ in terms of the Borel classes of $\mathfrak V$ using the fact that $b_i=b_i(\mathfrak V)$ is the ith elementary symmetric function in the $\xi _i$ and using the classical expression of the symmetric power sum polynomials in terms of the elementary symmetric functions. For example, a classical formula in terms of determinant (see [Reference Macdonald and Zelevinsky22, I.2, p. 28]) is

(4.1.7.a)

A more useful formula is given by Newton’s relations:

(4.1.7.b) $$ \begin{align} \tilde{\chi}_{2n}-b_1\tilde{\chi}_{2n-2}+b_2\tilde{\chi}_{2n-4} + \ldots +(-1)^{n-1}b_{n-1}\tilde{\chi}_2+(-1)^nnb_n=0. \end{align} $$

4.2.1. We now study stable operations over a regular scheme S, still following the method of Riou. A technical difference between symplectic K-theory and usual K-theory is that the former is $(8,4)$ -periodic, whereas the latter is $(2,1)$ periodic. Therefore, the natural sphere that comes into play is .Footnote ¹⁵

Given any spectrum $\mathbb E$ over S, we get a tautological stability isomorphism for any smooth S-scheme X:

$$ \begin{align*}\sigma^{\mathbb E}:\mathbb E^{n-8,i-4}(X) \xrightarrow \sim \tilde{\mathbb E}^{n,i}(H \wedge X_+), \end{align*} $$

where $\tilde {\mathbb E}$ is the associated reduced cohomology theory. When $\mathbb E$ admits a ring structure, this isomorphism can be expressed by the cup-product with a tautological class $\sigma _X \in \tilde {\mathbb E}^{8,4}(H)$ . When $\mathbb E$ is in addition $\mathrm {Sp}$ -oriented, this class is induced by $b_1(\mathfrak U_1) \cdot b_1(\mathfrak U_2) \in \mathbb E^{8,4}(\mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^1)$ , where $\mathfrak U_i$ is the pullback of the tautological symplectic bundle on the ith factor (apply Remark 2.2.5).

Similarly, the multiplication map in symplectic K-theory

$$ \begin{align*}\mathbf{KSp}_0(X) \xrightarrow{(\mathfrak U_i-\mathfrak H) \cdot (\mathfrak U_i-\mathfrak H)} \mathbf{KSp}_0(\mathrm{H}\mathbb{P}^1 \times \mathrm{H}\mathbb{P}^1 \times X) \end{align*} $$

induces, according to the symplectic bundle theorem, an isomorphism

$$ \begin{align*}\sigma^{\mathbf{KSp}}:\mathbf{KSp}_0(X) \rightarrow\tilde{\mathbf{KSp}}_0(H \wedge X_+). \end{align*} $$

Given an $\mathrm {Sp}$ -operation $\theta $ with values in a spectrum $\mathbb E$ of degree $(n,i)$ , we define a new associated $\mathrm {Sp}$ -operation $\omega _H(\theta )$ with values in $\mathbb E$ of degree $(n-8,i-4)$ by the following commutative diagram:

(4.2.1.a)

In particular, we get two projective systems indexed by integers $r \geq 0$ :

$$ \begin{align*} \ldots \operatorname{\mathrm{Hom}}_{\mathrm{Set}}\big(KSp_0,\mathbb E^{n+8r,i+4r}\big) \xleftarrow{\omega_H} \operatorname{\mathrm{Hom}}_{\mathrm{Set}}\big(KSp_0,\mathbb E^{n+8r+8,i+4r+4}\big) \ldots \\ \ldots \operatorname{\mathrm{Hom}}_{\mathrm{Ab}}\big(KSp_0,\mathbb E^{n+8r,i+4r}\big) \xleftarrow{\omega_H} \operatorname{\mathrm{Hom}}_{\mathrm{Ab}}\big(KSp_0,\mathbb E^{n+8r+8,i+4r+4}\big) \ldots \end{align*} $$

Their projective limits agree as stable operations must be additive (see [Reference Vishik38, Proposition 3.5]).

4.2.3. Recall from [Reference Riou29, 6.1] that a naive H-spectrum over a scheme S is the datum of a sequence $(E_n,\sigma _n)_{n \in \mathbb N}$ such that $E_n$ is an object of ${{\mathscr H}_{\bullet }}(S)$ and $\sigma _n:H \wedge E_n \rightarrow E_n$ is a map in ${{\mathscr H}_{\bullet }}(S)$ whose adjoint map $E_n \rightarrow \Omega _H E_n$ is an isomorphism. Every spectrum $\mathbb E$ if $\operatorname {\mathrm {SH}}(S)$ determines a naive H-spectrum whose nth term is $E_n=\Omega ^\infty \Sigma _H^n \mathbb E$ . Reciprocally, any naive H-spectrum $(E_n,\sigma _n)$ admits a lifting to an object of $\operatorname {\mathrm {SH}}(S)$ , and the lifting is unique provided

$$ \begin{align*}\mathbf{R}^1 \lim_{n \in \mathbb N} \operatorname{\mathrm{Hom}}_{{{\mathscr H}_{\bullet}}(S)}(S^1 \wedge E_n,E_n)=0. \end{align*} $$

An important example for us is provided by the naive H-spectrum over any $\mathbb Z[{1}/{2}]$ -scheme S,

$$ \begin{align*}(\mathbb Z \times B\mathrm{Sp},\mathbb Z \times B\mathrm{Sp},\ldots), \end{align*} $$

whose transition maps are all equal to the multiplication by the element $\sigma ^2 \in [H,\mathbb Z\times B\mathrm {Sp}]_*$ corresponding to the element $[\mathfrak U_1-\mathfrak H].[\mathfrak U_2-\mathfrak H]$ in $\mathbf {KSp}_0(\mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^1)$ , where $\mathfrak U_i$ is the tautological symplectic bundle on the ith factor of $\mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^1$ . As explained above, this naive H-spectrum lifts as an object of $\operatorname {\mathrm {SH}}(S)$ . Over $S=\operatorname {\mathrm {Spec}}{\mathbb Z[{1}/{2}]}$ , the ambiguity in this lifting vanishes according to [Reference Panin and Walter28, Theorem 13.2], thus giving a canonical spectrum $\mathbf {KSp}_{\mathbb Z[{1}/{2}]}$ . Over an arbitrary $\mathbb Z[{1}/{2}]$ -scheme S, we define $\mathbf {KSp}_S$ by pullback. According to [Reference Schlichting and Tripathi34], one gets a canonical isomorphism in $\operatorname {\mathrm {SH}}(S)$ :

(4.2.3.a) $$ \begin{align} \mathbf{KSp}_S \simeq \mathbf{GW}_S(2)[4]. \end{align} $$

Let us recall the following proposition from [Reference Riou29, Lem. 6.4].

4.2.5. Let k be a perfect field of characteristic different from $2$ . According to [Reference Déglise and Fasel13, Proposition 4.1.2], we get the following vanishing of the Milnor-Witt motivic cohomology groups of k with integral coefficients:

$$ \begin{align*}H^{n,m}_{MW}(k,\mathbb Z)=0 \text{ if } n>m \text{ or } (m<0 \text{ and } n \neq m). \end{align*} $$

Regarding rational motivic cohomology, we get from [Reference Déglise, Fasel, Khan and Jin15] that

$$ \begin{align*}H^{n,m}_{MW}(\mathbb Z[{1}/{2}],\mathbb Q)=K_{2m-n}^{(m)}(\mathbb Z[{1}/{2}]) \otimes \mathbb Q \oplus H^{n-m}_{\mathrm{Zar}}(\mathbb Q,{\mathbf H} \operatorname{{\underline W}} \otimes \mathbb Q), \end{align*} $$

where ${\mathbf H} \operatorname {{\underline W}}$ is the unramified Witt sheaf over $\mathscr Sm_{\mathbb Q}$ . Given Borel’s computations of the K-theory of integers, these groups vanish in the same range as in the previous case.Footnote ¹⁶

4.2.7. Indeed, in view of the vanishing recalled before the corollary, the projective system appearing on the left-hand side of the short exact sequence in the previous section is just $0$ at each degree. Besides, one can give another expression for the projective system of the right-hand side appearing in Definition 4.2.2. Using Theorem 4.1.6, the above vanishing and the fact that

$$ \begin{align*}H^{0,0}_{MW}(S,R) \simeq GW(S)_R, \end{align*} $$

we get that the two above groups are given by the projective limit of a tower, indexed by $r \geq 0$ , of the form

(4.2.7.a) $$ \begin{align} \ldots \leftarrow \operatorname{\mathrm{GW}}(S)_R.\tilde{\chi}^R_{2n+4r} \xleftarrow{\omega_H} \operatorname{\mathrm{GW}}(S)_R.\tilde{\chi}^R_{2n+4r+4} \leftarrow \ldots \end{align} $$

where we have denoted by $\tilde {\chi }^R_{2n}:KSp_0 \rightarrow H_{MW,R}^{4n,2n}$ the additive $\mathrm {Sp}$ -operations of Example 4.1.8. Note that the desuspension operators $\omega _H$ (Paragraph 4.2.1) are $GW(S)$ -linear (because all morphisms involved in Diagram (4.2.1.a) are $GW(S)$ -linear). The next subsection is devoted to computing explicitly the transition maps in the above projective system.

4.3.1. Consider the notations and assumptions of Corollary 4.2.6 and Paragraph 4.2.7. According to formula (4.2.7.a), we know a priori that for any $n \geq 0$ there exists an isomorphism class of quadratic form $\psi _{2n+4} \in \operatorname {\mathrm {GW}}(S)_R$ such that

(4.3.1.a) $$ \begin{align} \omega_H\left(\tilde{\chi}_{2n+4}^R\right)=\psi_{2n+4}.\tilde{\chi}_{2n}^R. \end{align} $$

For normalisation purposes, we put $\psi _0=\psi _2=1$ .

Proof. To determine an additive $\mathrm {Sp}$ -operation over S, we know from Theorem 4.1.6 that we need only to apply it to the element $[\mathfrak U] \in KSp_0(\mathrm {H}\mathbb {P}^\infty _S)$ . Going back to the defining diagram (4.2.1.a) for $\omega _H$ and using the fact that $\tilde {\chi }_{2n}^R(\mathfrak U)=b_1(\mathfrak U)^n=u^n$ , we get the following relation in the cohomology group $H_{MW}^{4n+8,2n+4}(\mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^\infty _S)$ :

(4.3.2.b) $$ \begin{align} \tilde{\chi}_{2n+4}^R\big((\mathfrak U_1-\mathfrak H)(\mathfrak U_2-\mathfrak H)\mathfrak U\big)=\psi_{2n+4}.u_1u_2u^n, \end{align} $$

where we denoted $u_1$ and $u_2$ for the first Borel class of the tautological symplectic bundle on the first and second coordinates of $\mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^1 \times \mathrm {H}\mathbb {P}^\infty _S$ .

Using the computation of the ternary laws for Milnor-Witt cohomology, it is possible to determine $\psi _{2n}$ . However, it is possible to substantially simplify this computation by remembering that the class $\psi _{2n}$ in $GW(S)_R$ is determined by its rank and its class in the Witt ring $W(S)_R$ . On the other hand, we have two canonical maps

$$ \begin{align*}\mathbf{H}_{\mathrm{MW}} R_S \rightarrow \mathbf{H}_{\mathrm M} R_S,\ \mathbf{H}_{\mathrm{MW}} R_S \rightarrow {\mathbf H} \operatorname{{\underline W}}_{R,S} \end{align*} $$

according to [Reference Déglise and Fasel14, Section 4.3.1] and [Reference Déglise and Fasel13, (proof of) Proposition 4.1.2] under assumption (a) and [Reference Déglise, Fasel, Khan and Jin15, Cor. 6.2] under assumption (b). These maps induce respectively the rank and the projection map on the cohomology groups in degree $(0,0)$ . Thus, we need only to specialise our computations either to motivic cohomology or to unramified Witt cohomology. This will be done in Proposition 4.3.5 and Corollary 4.3.8.

4.3.3. We consider the hypothesis of Theorem 4.3.2. Recall from [Reference Riou30, Proposition 6.2.2.1] that there are canonical operations

$$ \begin{align*}\chi_i:\mathbb Z \times B\mathrm{GL} \rightarrow K\big(R(i),2i\big) \end{align*} $$

where $K(R(i),2i)=\Omega ^\infty \big (\mathbf {H}_{\mathrm M} R_S(i)[2i]\big )$ is the motivic Eilenberg-MacLane space of degree $(2i,i)$ .Footnote ¹⁷

On the other hand, we defined additive $\mathrm {Sp}$ -operations by applying Theorem 4.1.6 to the motivic cohomology ring spectrum $\mathbf {H}_{\mathrm M} R_S$ :

$$ \begin{align*}\tilde{\chi}_{2n}^M:\mathbb Z \times B\mathrm{Sp} \rightarrow K\big(R(2i),4i\big). \end{align*} $$

Proof. Let us denote by $b_i^M$ the Borel classes associated with the canonical $\mathrm {Sp}$ -orientation of $\mathbf {H}_{\mathrm M} R_S$ . We prove the result by induction on $n\geq 1$ .

For $n=1$ and a symplectic bundle $(U,\psi )$ , we get from formulas (4.1.7.a) and (2.3.7.a) ${\tilde {\chi }}^M_2(U,\psi )=b^M_1(U,\psi )=-c_2(U)$ . On the other hand, using [Reference Riou30, Remark 6.2.2.3], we obtain $\chi _{2}=c_1^2(U)-2c_2(U)=-2c_2(U)$ . These two equalities allow us to conclude. Then the induction step is provided by the following computation (where we suppress f for readability):

$$ \begin{align*} \chi_{2n+2} & =-\sum_{i=1}^{n} c_{2i}\chi_{2n-2i+2}-(2n+2)c_{2n+2} \\ & = -2.\sum_{i=1}^{n} (-1)^ib^M_{i}{\tilde{\chi}}^M_{2n-2i+2}-(2n+2)(-1)^{n+1}b^M_{n+1} \\ & = -2\left(\sum_{i=1}^{n} (-1)^ib^M_{i}{\tilde{\chi}}^M_{2n-2i+2}+(-1)^{n+1}(n+1)b^M_{n+1}\right) = 2{\tilde{\chi}}^M_{2n+2}, \end{align*} $$

where the first (respectively second, last) equality follows from [Reference Riou30, Remark 6.2.2.3] (respectively (2.3.7.a) and the induction hypothesis, (4.1.7.b)).

Proof. According to the plus part of formula (4.2.7.a), we have a priori

$$ \begin{align*}\omega_H({\tilde{\chi}}^M_{2n+4})= r_{2n+4}.{\tilde{\chi}}^M_{2n}, \end{align*} $$

where $r_{2n+4}$ is an element of $H^{0,0}_M(S,R)=R$ .Footnote ¹⁸ We know from [Reference Riou30, Lemma 6.2.3.2] that $\Omega _{\mathbb P^1}(\chi _{2n+4})=(2n+4)\chi _{2n+3}$ . Therefore, because $2$ is a nonzero divisor in R, the proposition follows the previous lemma and the obvious fact that $\omega _H({\tilde {\chi }}^M_{2n+4} \circ f) =\Omega _{\mathbb P^1}^4(\tilde {\chi }^M_{2n+4}) \circ f$ . Beware the particular case $n=0$ , as $\tilde {\chi }_0=\chi _0$ .

4.3.6. The main point to prove the above theorem is to determine the Witt part $\bar \psi _{2n+4} \in \operatorname {\mathrm {W}}(S)_R$ of the quadratic form $\psi _{2n+4} \in \operatorname {\mathrm {GW}}(S)_R$ of Paragraph 4.3.1. So we consider the assumptions of this theorem and we let $\tilde {\chi }^W_{2n+4}$ (respectively $b_i^W$ ) be the $\mathrm {Sp}$ -operation (respectively Borel class) associated with the $\mathrm {Sp}$ -oriented ring spectrum ${\mathbf H} \operatorname {{\underline W}}_{R,S}$ . Specialising relation (4.3.2.b), we get for any $n \geq 0$ ,

$$ \begin{align*}{\tilde{\chi}}_{2n+4}^W\big((\mathfrak U_1-\mathfrak H)(\mathfrak U_2-\mathfrak H)\mathfrak U\big)=\bar \psi_{2n+4}.u_1u_2u^n, \end{align*} $$

where $u_1=b_1^W(\mathfrak U_1)$ , $u_2=b_1^W(\mathfrak U_2)$ , $u=b_1^W(\mathfrak U)$ .

Proof. Put $b_i^W=b_i^W(\mathfrak U_1\mathfrak U_2\mathfrak U)$ and ${\tilde {\chi }}^{W}_{2n+4}={\tilde {\chi }}^{W}_{2n+4}(\mathfrak U_1\mathfrak U_2\mathfrak U)$ . Let us first start by computing the Borel classes using the ternary law of unramified Witt theory, Formula (3.2.4.c) and the relation $u_i^2=0$ :

$$ \begin{align*}b_i^W= \begin{cases} 0 & i=1, i>4, \\ -2u^2 & i=2, \\ -8u_1u_2u & i=3, \\ u^4 & i=4. \end{cases} \end{align*} $$

We derive from this computation and Newton’s identity relation (4.1.7.b) the following relation for $n>2$ :

(4.3.7.a) $$ \begin{align} {\tilde{\chi}}^{W}_{2n+4} =2u^2.\tilde{\chi}^{W}_{2n} -8u_1u_2u.\tilde{\chi}^{W}_{2n-2} -u^4.\tilde{\chi}^{W}_{2n-4}. \end{align} $$

On the other hand, one can express the first $\mathrm {Sp}$ -operations using again the first computation and Formula (4.1.7.a):

$$ \begin{align*}\tilde{\chi}^W_{2n+4}= \begin{cases} 4u^2 & n=0, \\ -24u_1u_2u & n=1, \\ 4u^4 & n=2. \end{cases} \end{align*} $$

Finally, one proves the lemma by induction on n using relation (4.3.7.a).

Proof. Using the additivity of ${\tilde {\chi }}_{2n+4}^W$ , we find

$$ \begin{align*} {\tilde{\chi}}_{2n+4}^W\big((\mathfrak U_1-\mathfrak H)(\mathfrak U_2-\mathfrak H)\mathfrak U\big)={\tilde{\chi}}_{2n+4}^W(\mathfrak U_1\mathfrak U_2\mathfrak U)-\sum_{i=1}^2{\tilde{\chi}}_{2n+4}^W(\mathfrak H\mathfrak U_i\mathfrak U)+{\tilde{\chi}}_{2n+4}^W(\mathfrak H\mathfrak H\mathfrak U). \end{align*} $$

To compute the second term, it suffices to replace $u_i$ by $0$ in the expression of Proposition 4.3.7, and the third term is obtained by setting $u_1=u_2=0$ . A direct computation allows to conclude.

4.4.1. Consider again the notation of Paragraph 4.3.1. The next step is to provide conditions under which inverting the quadratic forms $\psi _{2n}$ in the Grothendieck-Witt ring $\operatorname {\mathrm {GW}}(S)_R$ is sensible. For n even, the forms are hyperbolic and inverting them would erase all quadratic information. Thus, we are led to consider the multiplicative system $S_\psi $ of $\operatorname {\mathrm {GW}}(S)_R$ generated by $\{\psi _{2n}\vert n \text { odd}\}$ . Note that

$$ \begin{align*}\operatorname{\mathrm{GW}}(\mathbb Z)_{\mathbb Q} \simeq \mathbb Q.h \oplus \mathbb Q.\epsilon \simeq \operatorname{\mathrm{GW}}(\mathbb Q)_{\mathbb Q}, \end{align*} $$

so we restrict our attention to the case of a perfect field k.

Proof. It suffices to observe that $1$ is a square and then is positive w.r.t. P. It follows that $-1$ is negative and we conclude.

Proof. The above lemma shows that the map

$$ \begin{align*} \mathrm{W}(k)\xrightarrow{\sum s_P} \bigoplus_{P\in\mathcal P} \mathbb Z \end{align*} $$

induces a well-defined map as in the statement; that is, we have a commutative diagram

Besides, the kernel and cokernel of the top homomorphism are $2$ -primary torsion (see Theorem 6.1 and the paragraph preceding this theorem in [Reference Scharlau32]). It follows immediately that the bottom map is surjective. Let now y be in the kernel of

$$ \begin{align*} S_{\psi}^{-1}\mathrm{W}(k)\xrightarrow{\sum s_P} \bigoplus_{P\in\mathcal P}\mathbb Q. \end{align*} $$

We may write $y=\frac xs$ with $s\in S_{\psi }$ , $x\in W(k)$ . One deduces

$$ \begin{align*}\sum s_P(x)\bigg(\sum s_P(s)\bigg)^{-1}=0,\end{align*} $$

thus showing that $\sum s_P(x)=0$ . It follows that $2^rx=0$ for some $r\in \mathbb N$ . Now, we have $8\vert (2n+4)(2n+2)$ in $\mathrm {W}(k)$ and it follows that for any $r\in \mathbb N$ there exists an odd n such that $2^r\vert (\psi _{2n+4}\cdot \psi _{2n}\cdot \ldots \psi _2)$ . The map is thus injective.

Proof. We have an exact sequence of $\operatorname {\mathrm {GW}}(k)$ -modules

$$ \begin{align*} 0\to \operatorname{\mathrm{GW}}(k)\to \mathbb Z\oplus \mathrm{W}(k)\to \mathrm{W}(k)/2\to 0. \end{align*} $$

Localisation being exact, we deduce an exact sequence of $S_{\psi }^{-1}\mathrm {GW}(k)$ -modules. The above proposition shows that

$$ \begin{align*} S_{\psi}^{-1}\mathrm{GW}(k)\simeq S_{\psi}^{-1}\mathbb Z\oplus \bigoplus_{P\in\mathcal P}\mathbb Q \end{align*} $$

and we are left to prove that $S_{\psi }^{-1}\mathbb Z\simeq \mathbb Q$ . Let p be an odd prime. Then, $p-2$ is odd and $(2n+4)(2n+2)=4p(p-1)$ and therefore p is invertible. Because we already know that $2$ is also invertible, the result follows.

4.4.5. Let $S=\operatorname {\mathrm {Spec}}(\mathbb Z[{1}/{2}])$ . We are now in a position to classify all stable symplectic operations with values in rational Milnor-Witt motivic cohomology $\mathbf {H}_{\mathrm {MW}} \mathbb Q_S$ of degree $(4n,2n)$ for $n \in \mathbb Z$ .

We first consider the case $n=0$ . Recall the decomposition in plus and minus parts of Milnor-Witt motivic cohomology:

$$ \begin{align*}\mathbf{H}_{\mathrm{MW}} \mathbb Q_S=\mathbf{H}_{\mathrm{MW}} \mathbb Q_{S+} \oplus \mathbf{H}_{\mathrm{MW}} \mathbb Q_{S-}=\mathbf{H}_{\mathrm M} \mathbb Q_{S} \oplus {\mathbf H} \operatorname{{\underline W}}_{\mathbb Q}. \end{align*} $$

We consider the additive symplectic operation $\tilde {\chi }_{2i}^M:\mathbb Z \times B\mathrm {Sp} \rightarrow K\big (\mathbb Q_S(i),2i\big )$ defined in Paragraph 4.3.3. According to Proposition 4.3.5, we get a stable symplectic operation (Definition 4.2.2) on rational motivic cohomology by considering the following sequence:

$$ \begin{align*}\left(\tilde{\chi}^M_0,\frac 2 {4!} \tilde{\chi}^M_4,\ldots,\frac 2 {4n!} \tilde{\chi}^M_{4n},\ldots\right). \end{align*} $$

Applying Corollary 4.2.6, this uniquely corresponds to a map $\tilde {\operatorname {\mathrm {bo}}}^+_0:\mathbf {KSp}_S\rightarrow \mathbf {H}_{\mathrm M} \mathbb Q_S$ and we deduce a map

$$ \begin{align*}\tilde{\operatorname{\mathrm{bo}}}_0:\mathbf{KSp}_S\xrightarrow{\tilde{\operatorname{\mathrm{bo}}}^+_0} \mathbf{H}_{\mathrm M} \mathbb Q_S \xrightarrow{i_+} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S, \end{align*} $$

where the last map is the canonical inclusion. For any $n \in \mathbb Z$ , we put

$$ \begin{align*}\tilde{\operatorname{\mathrm{bo}}}_{4n}:\mathbf{KSp}_S \simeq \mathbf{KSp}_S(4n)[8n] \xrightarrow{\tilde{\operatorname{\mathrm{bo}}}_0(4n)[8n]} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(4n)[8n]. \end{align*} $$

By definition, when $n>0$ , it is induced as above by the following stable symplectic operation with values in rational motivic cohomology of degree $(4n,8n)$ :

(4.4.5.a) $$ \begin{align} \left(\frac 2 {4n!} \tilde{\chi}^M_{4n},\frac 2 {(4n+4)!} \tilde{\chi}^M_{4n+4},\ldots\right). \end{align} $$

For $n<0$ , just add enough zeroes at the start.

Next we consider the case $n=2$ . Let us consider the following product:

(4.4.5.b) $$ \begin{align} \psi_{2+4n}!=\psi_2 \cdot \psi_6 \cdot \ldots \cdot \psi_{2+4n}. \end{align} $$

Applying the previous corollary, this product of quadratic forms is invertible in $\mathbf {H}_{\mathrm {MW}}^{00}(S,\mathbb Q)=\operatorname {\mathrm {GW}}(S)_{\mathbb Q}=\operatorname {\mathrm {GW}}(\mathbb Q)_{\mathbb Q}$ . Thus, using Theorem 4.3.2, we can introduce the following stable symplectic operation with values in rational Milnor-Witt cohomology over S:

(4.4.5.c) $$ \begin{align} \left(\tilde{\chi}^{\mathbb Q}_2,\frac 1 {\psi_6!} \tilde{\chi}^{\mathbb Q}_6,\ldots,\frac 1 {\psi_{2+4n}!} \tilde{\chi}^{\mathbb Q}_{2+4n},\ldots\right). \end{align} $$

Applying again Corollary 4.2.6, it uniquely corresponds to a morphism

$$ \begin{align*}\tilde{\operatorname{\mathrm{bo}}}_2:\mathbf{KSp}_S \rightarrow \mathbf{H}_{\mathrm{MW}} {\mathbb Q}_S(2)[4]. \end{align*} $$

For $n \in \mathbb Z$ , we put

$$ \begin{align*}\tilde{\operatorname{\mathrm{bo}}}_{2+4n}:\mathbf{KSp}_S \simeq \mathbf{KSp}_S(4n)[8n] \xrightarrow{\tilde{\operatorname{\mathrm{bo}}}_2(4n)[8n]} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(2+4n)[4+8n], \end{align*} $$

which by definition is induced by the following stable symplectic operation for $n>0$ :

(4.4.5.d) $$ \begin{align} \left(\frac 1 {\psi_{2+4n}!} \tilde{\chi}^{\mathbb Q}_{2+4n},\frac 1 {\psi_{6+4n}!} \tilde{\chi}^{\mathbb Q}_{6+4n},\ldots\right). \end{align} $$

Given now any $\mathbb Z[{1}/{2}]$ -scheme S, the operations $\tilde {\operatorname {\mathrm {bo}}}_{2n}$ defined above can be defined over S by taking pullback along the unique morphism $S \rightarrow \operatorname {\mathrm {Spec}}\left (\mathbb Z[{1}/{2}]\right )$ .

Proof. Each statement follows simply from Corollary 4.2.6 and the computation of the projective system (4.2.7.a), whose transition maps are given by Theorem 4.3.2 (note that by $(8,4)$ -periodicity of $\mathbf {KSp}_S$ , one can reduce to the case $n \geq 0$ ). By construction, we have for any symplectic bundle $\mathfrak U$ over S equalities $\tilde {\operatorname {\mathrm {bo}}}_0(\mathfrak U)=\tilde {\chi }_0^{\mathbb Q}(\mathfrak U)=\operatorname {\mathrm {rk}}(\mathfrak U)$ and $\tilde {\operatorname {\mathrm {bo}}}_2(\mathfrak U)=\tilde {\chi }_2^{\mathbb Q}(\mathfrak U)=b_1(\mathfrak U)$ – apply (4.1.7.a). This implies that the operations $\tilde {\operatorname {\mathrm {bo}}}_{2n}$ are nonzero.

4.4.7. Assume $S=\operatorname {\mathrm {Spec}}(\mathbb Z[{1}/{2}])$ (or $\operatorname {\mathrm {Spec}}(k)$ ). Recall from [Reference Riou30, Definition 6.2.3.9] that for any $n \geq 0$ , one has that

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{SH}}(S)}(\mathbf{KGL},\mathbf{H}_{\mathrm M} \mathbb Q_S(n)[2n]) \simeq \mathbb Q \end{align*} $$

is generated by the nth component of the Chern character map $\operatorname {\mathrm {ch}}_n:\mathbf {KGL} \rightarrow \mathbf {H}_{\mathrm M} \mathbb Q_S$ (see footnote 17 for the case $S=\operatorname {\mathrm {Spec}}(\mathbb Z[{1}/{2}])$ ). Using the notations of Paragraph 4.3.3, the map $\operatorname {\mathrm {ch}}_n$ can be viewed as an H-stable operation as

(4.4.7.a) $$ \begin{align} \left(\frac 1 {n!}.\chi_n,\frac 1 {(n+4)!}.\chi_{n+4},\ldots\right). \end{align} $$

Proof. The first point follows directly from comparing formulas (4.4.5.a) and (4.4.7.a) using Lemma 4.3.4. Consider the second point. The case n even is implied by the first point. The case n odd reduces to $n>0$ (in fact, $n=1$ is enough). Then one can compare formulas (4.4.5.d) and (4.4.7.a) using Lemma 4.3.4 and the fact that $\operatorname {\mathrm {rk}}(\psi _{2n}!)=(2n)!/2$ (use Formula (4.3.2.a)).

5.1.1. We rephrase in the next statement the main theorem of the previous section, Theorem 4.4.6, in terms of higher Grothendieck-Witt groups. Let S be a $\mathbb Z[{1}/{2}]$ -scheme S and $n \in \mathbb Z$ be an integer. Using the isomorphism (4.2.3.a) and the notations of Paragraph 4.4.5, we introduce the following maps. When n is even,

$$ \begin{align*} \operatorname{\mathrm{bo}}_{2n}:&\mathbf{GW}_S \simeq \mathbf{KSp}_S(-2)[-4] \\ &\xrightarrow{\tilde{\operatorname{\mathrm{bo}}}_{2+2n}(-2)[-4]} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(2+2n)[4+4n](-2)[-4] \simeq \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(2n)[4n]. \end{align*} $$

and when n is odd.

$$ \begin{align*} \operatorname{\mathrm{bo}}_{2n}:&\mathbf{GW}_S \simeq \mathbf{KSp}_S(-2)[-4] \\ &\xrightarrow{p_+ \circ \operatorname{\mathrm{bo}}_{2+2n}(-2)[-4]} \mathbf{H}_{\mathrm M} \mathbb Q_S(2+2n)[4+4n](-2)[-4] \simeq \mathbf{H}_{\mathrm M} \mathbb Q_S(2n)[4n]/ \end{align*} $$

Note that is follows from the construction of the stable $\mathrm {Sp}$ -operation $\tilde {\operatorname {\mathrm {bo}}}$ (see loc. cit.) that for any integer $n \in \mathbb Z$ , one has

(5.1.1.a) $$ \begin{align} \begin{aligned} \operatorname{\mathrm{bo}}_{4n}:&\mathbf{GW}_S \simeq \mathbf{GW}_S(4n)[8n] \xrightarrow{\operatorname{\mathrm{bo}}_0(4n)[8n]} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(4n)[8n], \\ \operatorname{\mathrm{bo}}_{2+4n}:&\mathbf{GW}_S \simeq \mathbf{GW}_S(4n)[8n] \xrightarrow{\operatorname{\mathrm{bo}}_2(4n)[8n]} \mathbf{H}_{\mathrm{MW}} \mathbb Q_S(2+4n)[4+4n], \end{aligned} \end{align} $$

where the first isomorphism in each line is obtained by the periodicity of Hermitian K-theory.Footnote ¹⁹

Proof. Everything except the statement about $\operatorname {\mathrm {bo}}_0$ follows from Theorem 4.4.6. By definition of $\operatorname {\mathrm {bo}}_0$ , we get the following commutative diagram:

where $(1)$ is the exterior product with the tautological symplectic bundle $\mathfrak H$ on $\mathrm {H}\mathbb {P}^1$ and (2) is the projection on the factor $\widetilde {\mathrm {CH}}{}^0(S).b_1(\mathfrak H)$ , using the symplectic projective bundle theorem for Chow-Witt groups (here $b_1$ denotes the Borel class for Chow-Witt groups). Thus, the statement simply follows from the fact that $\tilde {\operatorname {\mathrm {bo}}}_{2*}=\tilde {\chi }_2=b_1$ – formula (4.4.5.c) (respectively (4.1.7.a)) for the first (respectively second) equality.

5.1.4. Note that, according to the properties of $\tilde {\operatorname {\mathrm {bo}}}_{2n}$ , $\operatorname {\mathrm {bo}}_t$ is compatible with pullbacks: it is a morphism of $\mathbb Z[{1}/{2}]$ -absolute spectra. Using Proposition 4.4.8, we immediately get the commutativity of the following square:

where f is the natural forgetful map and the right-hand vertical map is obtained by the projection on the plus part for the even integer m, $0$ for m odd. In particular, for odd n, the map $\operatorname {\mathrm {bo}}_{2n}=\operatorname {\mathrm {ch}}_{2n} \circ f$ (apply point (1) of Proposition 4.4.8).

5.2.1. Principle of proof of Theorem 5.5.1 . To prove that the Borel character $\operatorname {\mathrm {bo}}_t$ is an isomorphism over any scheme S, by compatibility with pullbacks (see Paragraph 5.1.4), it is sufficient to consider the case $S=\operatorname {\mathrm {Spec}}(\mathbb Z[{1}/{2}])$ . Moreover, we can always use Morel’s decomposition of $\operatorname {\mathrm {SH}}(S)_{\mathbb Q}$ into its plus and minus parts.

That being said, we will prove that $\operatorname {\mathrm {bo}}_t$ is an isomorphism by constructing an explicit inverse $\operatorname {\mathrm {bo}}^{\prime }_t$ using the theory of periodic ring spectra (see Definition below). The advantage of this construction is that $\operatorname {\mathrm {bo}}^{\prime }_t$ will clearly be a morphism of ring spectra. We will construct $\operatorname {\mathrm {bo}}^{\prime }_t$ by separating the plus part (Subsection 5.3) and the minus part (Subsection 5.4).

5.3.1. Given an arbitrary $\mathbb Z[{1}/{2}]$ -scheme S, it follows from [Reference Schlichting33, Th. 6.1] and [Reference Röndigs and Østvær31, Th. 3.4] that one has a canonical distinguished triangle:

$$ \begin{align*}\mathbf{GW}_S(1)[1] \xrightarrow{\eta} \mathbf{GW}_S \xrightarrow f \mathbf{KGL}_S \xrightarrow{h \circ \gamma_{\beta'}} \mathbf{GW}_S(1)[2], \end{align*} $$

where $\eta $ is the Hopf map, f the forgetful map, $h:\mathbf {KGL}_S \rightarrow \mathbf {GW}_S$ is the ‘hyperbolisation’ map and $\gamma _{\beta '}$ is the multiplication by the inverse of the Bott element $\beta $ (Example 5.2.4). Because $\eta _+=0$ and $\mathbf {KGL}_{S-}=0$ , we immediately deduce the following result.

5.3.3. By construction of the Hermitian K-theory spectrum and the forgetful map f (see [Reference Röndigs and Østvær31, Prop. 3.3]), we get the following commutative diagram:

where $\gamma _{\beta ^2}$ is multiplication by $\beta ^2$ and f on the right-hand side is the map forgetting the Hermitian structure. With this notation, we can define the following element in the positive part of Hermitian K-theory:

$$ \begin{align*}\rho_S=\frac 12.\phi([\mathfrak H]) \in \mathbf{GW}^+_{4,2}(S). \end{align*} $$

This element is stable under pullback, so we can erase the base S to simplify notation. It follows from the above commutative square that $f(\rho )=\beta ^2$ .

The same construction can be done replacing in degree $(-4,-2)$ by replacing $\beta $ with $\beta ^{-1}$ . Thus, we get an element $\rho ' \in \mathbf {GW}^+_{-4,-2}(S)$ such that $f(\rho ')=\beta ^{-2}$ .

Let us remark that the forgetful map $f:\mathbf {GW}_S \rightarrow \mathbf {KGL}_S$ is a morphism of ring spectra. This implies that $f(\rho .\rho ')=1$ . Because f is a monomorphism on the plus part, due to (5.3.2.a), we deduce that $\rho $ is invertible. Thus, $(\mathbf {GW}_+,\rho )$ is a periodic absolute $(4,2)$ -spectrum over $\mathbb Z[{1}/{2}]$ . In particular, we get a canonical map by taking the plus part of (5.2.3.a):

$$ \begin{align*}\bigoplus_{n \in \mathbb Z} \mathbb S_{S,+}(2n)[4n] \xrightarrow{\sum_n \rho^n} \mathbf{GW}_{S+}. \end{align*} $$

Note, moreover, that we get the following relations for any integer $n \in \mathbb Z$ :

(5.3.3.a) $$ \begin{align} f(\rho^n)=\beta^{2n}, \quad h(\beta^{n})=\begin{cases} 2.\rho^i & n=2i, \\ 0 & n \text{ odd}. \end{cases} \end{align} $$

The first relation follows because f is a morphism of ring spectra and the second one from the first and the fact that $h \circ \gamma _{\beta '} \circ f=0$ (the above lemma). We immediately deduce from these relations the following lemma.

5.4.1. Let S be a $\mathbb Z[{1}/{2}]$ -scheme. According to our conventions, there exists a natural map

$$ \begin{align*} \mathbf{GW}_S\to \mathbf{W}_S \end{align*} $$

that induces an isomorphism

$$ \begin{align*}\mathbf{GW}_{S,-}\rightarrow \mathbf{W}_S[1/2]. \end{align*} $$

In particular, $\mathbf {W}_S[1/2]$ is $(1,1)$ -periodic with respect to $\eta $ and $(8,4)$ -periodic with respect to $\kappa _- \in \mathbf {W}_{8,4}$ , which is the image of $\kappa $ under the canonical projection: $\mathbf {GW}_S \rightarrow \mathbf {GW}_{S-}$ . The next result generalises the fundamental result of [Reference Ananyevskiy, Levine and Panin4] to an arbitrary base $\mathbb Z[{1}/{2}]$ -scheme.