Proof. The first statement is [Reference Calmès, Harpaz and Nardin4, Remark 8.1.2]; note that $\mathsf {Forg}(\alpha )=\beta ^4$ . The second statement follows from [Reference Calmès, Harpaz and Nardin4, Corollary 8.1.7] and [Reference Calmès, Harpaz and Nardin4, Remark 8.1.8]. The absoluteness statement 3 is [Reference Calmès, Harpaz and Nardin4, Proposition 8.2.1]. Statement 4 follows from [Reference Calmès, Harpaz and Nardin4, Theorem 8.4.2] and statement 5 can be deduced from [Reference Calmès, Harpaz and Nardin4, Proposition 8.3.1]. Statement 6 follows from [Reference Calmès, Harpaz, Land, Nardin, Steimle, Dotto, Hebestreit, Moi and Nikolaus3, Proposition B.2.2] and [Reference Calmès, Harpaz and Nardin4, Corollary 8.1.5].

Proof. Since both motivic spectra are stable under pullback by Theorem 2.1 and Theorem 2.2, it suffices to construct an equivalence $f\colon \mathsf {KQ}^\prime _{\mathbb {Z}}\to \mathsf {KQ}_{\mathbb {Z}}$ . Its main ingredient, a map

$$\begin{align*}\phi\colon \mathbb{Z}\times \mathbf{HGr}_{\mathbb{Z}}\xrightarrow{} \Omega^{\infty+(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{Z}}\end{align*}$$

of motivic spaces over $\mathbb {Z}$ , is given in Lemma 2.4 below. Via the explicit periodicity equivalence $\Sigma ^{4+(4)}\mathsf {KQ}_{\mathbb {Z}}\simeq \mathsf {KQ}_{\mathbb {Z}}$ , one deduces a map

$$\begin{align*}\phi_n\colon \mathbb{Z}\times \mathbf{HGr}_{\mathbb{Z}}\to \Omega^{\infty+(\infty)}\Sigma^{4n+2+(4n+2)}\mathsf{KQ}_{\mathbb{Z}}\end{align*}$$

of motivic spaces over $\mathbb {Z}$ for every natural number n. That these maps define a map of motivic spectra requires, for every $n \in \mathbb {N}$ , the commutativity of the following diagram

(1)

where the vertical arrows depict the respective structure maps of the motivic spectra. Using the adjunction $(\Sigma ^{4+(4)},\Omega ^{4+(4)})$ , each of the two paths in diagram (1) defines a homotopy class $\mathbb {Z}\times \mathbf {HGr}_{\mathbb {Z}}\to \Omega ^{\infty +(\infty )}\Sigma ^{4n+2+(4n+2)}\mathsf {KQ}_{\mathbb {Z}}$ . To conclude that these coincide, observe that the Thom isomorphism [Reference Calmès, Harpaz and Nardin4, Proposition 8.3.1] for $\mathsf {KQ}_{\mathbb {Z}}$ provides, for every special linear vector bundle $V\to X$ of rank r with X smooth over $\mathrm {Spec}(\mathbb {Z})$ , a class

$$\begin{align*}\mathrm{th}(V\to X)\in \mathsf{KQ}^{r+(r)}(\Sigma^{\infty+(\infty)}\mathsf{Th}(V))\cong \mathsf{KQ}^{0+(0)}(X)\end{align*}$$

corresponding to the unit in the latter ring. It is straightforward to check that these classes constitute a normalized $\mathrm {SL}$ -orientation on $\mathsf {KQ}_{\mathbb {Z}}$ in the sense of [Reference Panin and Walter17, Section 5]. Hence the motivic spectrum $\mathsf {KQ}_{\mathbb {Z}}$ obtains a symplectic orientation, that is, a map $\mathsf {MSp}_{\mathbb {Z}}\to \mathsf {KQ}_{\mathbb {Z}}$ which is a homomorphism of commutative monoids in the homotopy category; see [Reference Panin and Walter17, Theorem 1.1].

The theory of Borel classes associated with the symplectic orientation on $\mathsf {KQ}_S$ supplies an isomorphism

where $b_j\in \mathsf {KQ}_S^{2j+(2j)}(\mathbf {HGr}_S)$ , see [Reference Panin and Walter16, Theorem 11.4]. This holds in particular for $S=\mathrm {Spec}(\mathbb {Z})$ . Thus the two homotopy classes under consideration can be identified with elements in the group

where $b_j\in \mathsf {KQ}_{\mathbb {Z}}^{2j+(2j)}(\mathbf {HGr}_{\mathbb {Z}})$ . It follows that the respective power series coefficients for both of the homotopy classes reside in the groups $\mathsf {KQ}^{2m+(2m)}(\mathbb {Z})$ , which by periodicity reduce to the groups $\mathsf {KQ}^{0+(0)}(\mathbb {Z})$ and $\mathsf {KQ}^{2+(2)}(\mathbb {Z})$ . Since the inclusion $\mathbb {Z}\hookrightarrow \mathbb {R}$ induces an injection (even an isomorphism) on these groups, the commutativity of diagram (1) up to homotopy follows from the respective commutativity over $\mathbb {R}$ . The latter follows from the fact that $\mathsf {KQ}_{\mathbb {R}}$ and $\mathsf {KQ}^\prime _{\mathbb {R}}$ are equivalent to the motivic spectrum representing Karoubi’s hermitian K-groups of smooth $\mathbb {R}$ -varieties, as listed in Theorem 2.1 and Theorem 2.2. Since $2$ is invertible in $\mathbb {R}$ , $\phi _n$ is mapped to $\tau _{4n+2}$ in Section 12 of [Reference Panin and Walter15] under the equivalence $S^{2+(2)}\simeq \mathbf {HP}^{1}$ with the quaternionic projective line. As a consequence, diagram (1) commutes up to homotopy for every n.

To provide an actual map of motivic spectra, commuting up to homotopy does not suffice. However, the argument for the commutativity up to homotopy shows that the sequence $(\phi _n)$ is an element in the group $\lim \limits _{n\to \infty } \mathsf {KQ}^{4n+2+(4n+2)}(\mathbb {Z}\times \mathbf {HGr})$ . This group sits in a short exact sequence

(2) $$ \begin{align} 0 \to {\lim_n}^1\mathsf{KQ}^{4n+1+(4n+2)}(\mathbb{Z}\times \mathbf{HGr}) \to [\mathsf{KQ}^\prime,\mathsf{KQ}]_{\mathbf{SH}(\mathbb{Z})} \to \lim\limits_{n\to \infty}\mathsf{KQ}^{4n+2+(4n+2)}(\mathbb{Z}\times \mathbf{HGr}) \to 0 \end{align} $$

by [Reference Panin, Pimenov and Röndigs14, Lemma 1.3.3], where the middle group represents the set of maps from $\mathsf {KQ}^\prime $ to $\mathsf {KQ}$ in the motivic stable homotopy category $\mathbf {SH}(\mathbb {Z})$ . Hence there exists a lift $f\colon \mathsf {KQ}^\prime _{\mathbb {Z}} \to \mathsf {KQ}_{\mathbb {Z}}$ of the sequence $(\phi _n)$ , that is, a map f of motivic spectra whose level $f_n\colon \mathbb {Z}\times \mathbf {HGr} \to \mathsf {KQ}_{4n+2}$ coincides with $\phi _n$ up to homotopy. To prove that this map is an equivalence, consider the diagram

(3)

whose horizontal maps are induced by the structure maps of the motivic spectra $\mathsf {KQ}^\prime _{\mathbb {Z}}$ and $\mathsf {KQ}_{\mathbb {Z}}$ , respectively, and which ends with the respective colimit on the right-hand side. Diagram (3) commutes, since so does diagram (1). The bottom horizontal maps in diagram (3) are all equivalences of motivic spaces by Theorem 2.1. Lemma 2.4 implies that the vertical map $\Omega ^{4n+(4n)}(f_n)$ is an equivalence of motivic spaces for $n>0$ . Hence the map induced on the colimit is an equivalence of motivic spaces. It follows that the map $f\colon \mathsf {KQ}^\prime _{\mathbb {Z}}\to \mathsf {KQ}_{\mathbb {Z}}$ is an equivalence.

Proof. To construct $\phi $ , the theory of Borel classes associated with the symplectic orientation on $\mathsf {KQ}_S$ , as explained in the proof of Theorem 2.3, will be used. This results in an isomorphism

where $b_j\in \mathsf {KQ}_S^{2j+(2j)}(\mathbf {HGr}_S)$ , see [Reference Panin and Walter16, Theorem 11.4]. This holds in particular for $S=\mathrm {Spec}(\mathbb {Z})$ . By [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, Section 3.2] and [Reference Calmès, Harpaz and Nardin4, Corollary 8.1.5] there is an inclusion of groups

$$\begin{align*}\mathbb{Z} \oplus\mathbb{Z}\cong \mathrm{KSp}_0(\mathbb{Z})\oplus\mathbb{Z}\{b_1\}\to \mathsf{KQ}_{\mathbb{Z}}^{2+(2)}(\mathbf{HGr}) \end{align*}$$

where $\mathsf {KQ}^{2+(2)}_{\mathbb {Z}}(\mathbb {Z})\cong \mathrm {KSp}_0(\mathbb {Z})\cong \mathbb {Z}$ is generated by the hyperbolic form of rank $2$ over $\mathbb {Z}$ . Let $\phi \colon \mathbb {Z}\times \mathbf {HGr}_{\mathbb {Z}} \to \Omega ^{\infty +(\infty )}\Sigma ^{2+(2)}\mathsf {KQ}_{\mathbb {Z}}$ be the morphism given by the sequence of elements $(i+b_1)_{i\in \mathbb {Z}}$ in $\mathrm {KSp}_0(\mathbb {Z})\oplus \mathbb {Z} \{b_1\}$ . This provides a unique element in $[\mathbb {Z}\times \mathbf {HGr}_{\mathbb {Z}}, \Omega ^{\infty +(\infty )}\Sigma ^{2+(2)}\mathsf {KQ}_{\mathbb {Z}}]$ , the set of maps in the unstable $\mathbf {A}^1$ -homotopy category $\mathbf {Ho}(\mathbb {Z})$ of $\mathbb {Z}$ .

Having produced $\phi \colon \mathbb {Z}\times \mathbf {HGr}_{\mathbb {Z}}\to \Omega ^{\infty +(\infty )}\Sigma ^{2+(2)}\mathsf {KQ}_{\mathbb {Z}}$ , it remains to prove the assertion on homotopy sheaves. For this purpose, consider the map of homotopy presheaves ${X\mapsto [\Sigma ^n X_+,\phi ]}$ inducing the map on homotopy sheaves. The domain $[\Sigma ^n X_+,\mathbb {Z}\times \mathbf {HGr}_{\mathbb {Z}}]$ of this map is naturally identified with $\pi _n\mathrm {Sing}^{\mathbf {A}^1}\Omega ^\infty \mathrm {KSp}(X)$ in [Reference Kumar10, Theorem 3.5]. The map $\phi $ is then induced by the canonical map

$$\begin{align*}\phi(X)\colon \mathrm{Sing}^{\mathbf{A}^1}\Omega^\infty\mathrm{KSp}(X) \to \mathrm{Sing}^{\mathbf{A}^1}\Omega^{(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{Z}}(X)\simeq \Omega^{(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{Z}}(X)\end{align*}$$

of spectra comparing genuine $(-1)$ -even to (homotopy) $(-1)$ -symmetric forms, the latter being $\mathbf {A}^1$ -invariant [Reference Calmès, Harpaz and Nardin4, Theorem 6.3.1]. Hence $\phi $ arises as the evaluation of a map of presheaves with values in spectra, or rather with values in their associated spaces obtained by applying $\Omega ^\infty $ . Let $i\colon \mathrm {Spec}(\mathbb {F}_2)\hookrightarrow \mathrm {Spec}(\mathbb {Z})$ denote the closed embedding with open complement $j\colon \mathrm {Spec}(\mathbb {Z}[2^{-1}])\hookrightarrow \mathrm {Spec}(Z)$ . The $S^1$ -stabilization

$$\begin{align*}j_\sharp j^\ast \to \mathrm{Id}_{\mathbf{Ho}(\mathbb{Z})} \to i_\ast i^\ast\to \Sigma j_\sharp j^\ast\end{align*}$$

of the unstable localization cofiber sequence [Reference Morel and Voevodsky13, Theorem 3.2.21] applies to give a natural transformation of long exact sequences

$$\begin{align*}\dotsm \to [\Sigma^nX_+,j_\sharp j^\ast \phi] \to[\Sigma^nX_+,\phi] \to[\Sigma^nX_+,i_\ast i^\ast \phi] \to [\Sigma^{n-1}X_+,j_\sharp j^\ast \phi] \to \dotsm. \end{align*}$$

By parts 3 and 5 of Theorem 2.1, for any $f\colon R\to S$ and any vector bundle $V\to S$ of rank r with trivial determinant line bundle the square

commutes. This implies that the corresponding $\mathrm {SL}$ -orientation is stable under base change. In particular, the Borel classes are compatible with pullbacks. Our definition of $\phi $ is the same as Panin and Walter’s identification $\mathbb {Z}\times \mathbf {HGr}\simeq \Omega ^\infty \mathrm {KSp}$ when $2$ is invertible; see [Reference Panin and Walter15, Section 11]. The fact that $\phi $ is compatible with pullbacks, as obtained from the preceding argument, implies that $j^\ast \phi $ is an equivalence. Hence so is $j_\sharp j^\ast \phi $ . It follows that the required connectivity for $\phi $ holds if it holds for $i^\ast \phi $ . Having placed ourselves in the $S^1$ -stable $\mathbf {A}^1$ -homotopy category over the perfect field $\mathbb {F}_2$ , these $\mathbf {A}^1$ -homotopy sheaves are strictly $\mathbf {A}^1$ -invariant by a theorem of Morel [Reference Morel12, Corollary 6.2.9]. For these, it suffices to evaluate on finitely generated fields F over $\mathbb {F}_2$ . The map

$$\begin{align*}i^\ast\phi(F)\colon \mathrm{Sing}^{\mathbf{A}^1}\mathrm{KSp}(F) \to \mathrm{Sing}^{\mathbf{A}^1}\Omega^{(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{F}_2}(F)\end{align*}$$

is induced by the map

$$\begin{align*}i^\ast\phi(\Delta^d_F)\colon \mathrm{KSp}(\Delta^d_F) \to \Omega^{(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{F}_2}(\Delta^d_F)\end{align*}$$

where $\Delta ^d_F=\mathrm {Spec} F[t_0,\dotsc ,t_d]/(\sum t_j=1)$ . Its target is equivalent to $\Omega ^{(\infty )}\Sigma ^{2+(2)}\mathsf {KQ}_{\mathbb {F}_2}(F)$ by $\mathbf {A}^1$ -invariance for $\mathsf {KQ}$ . Since $\Delta ^d_F$ has Krull dimension d, [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, Corollary 1.3.9] for $r=1$ implies that $\pi _ni^\ast \phi (\Delta ^d_F)$ is injective for $n\geq d$ and bijective for $n>d$ . However, the arguments of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, Section 1.3] give more in this special situation. The relevant base ring R is a polynomial ring in d variables over the field F, with trivial involution and canonical duality. Every finitely generated projective R-module is free, by the solution of Serre’s conjecture [Reference Quillen18, Reference Suslin25]. Hence the duality on the resulting derived category of finitely generated projective R-modules interacts with the t-structure in the same way as in the case of the field F; see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and Steimle1, p. 27, Remark 1.18]. Thus $\pi _ni^\ast \phi (\Delta ^d_F)$ is injective for $n\geq 0$ and bijective for $n>0$ , for every d. The same property then holds after realization, which shows that

$$\begin{align*}i^\ast\phi(F)\colon \mathrm{Sing}^{\mathbf{A}^1}\mathrm{KSp}(F) \to \mathrm{Sing}^{\mathbf{A}^1}\Omega^{(\infty)}\Sigma^{2+(2)}\mathsf{KQ}_{\mathbb{F}_2}(F)\end{align*}$$

has the desired connectivity property. This concludes the proof.