$\eta$-periodic motivic stable homotopy theory over Dedekind domains

For existence, see [Jac18, Remark 4.5]. The rest is [Jac18, Theorem 4.4]. ∎

If $p=2$, replace $X$ by $X[\sqrt{-1}]$. We may thus assume that $\mathrm{vcd}_{p}=\mathrm{cd}_{p}$, and if $p=2$ that all residue fields of $X$ are unorderable. By [ILO14, Lemma XVIII-A.2.2] we have $\mathrm{cd}_{p}(X)\leq\dim{X}+\sup_{x\in X}(\dim\mathcal{O}_{X,x}+\mathrm{cd}_{p}(k(x)))$. It hence suffices to show that $\mathrm{cd}_{p}(k(x))+\dim\mathcal{O}_{X,x}\leq\mathrm{cd}_{p}(K(X))$. Since $\mathrm{cd}_{p}(K(X))\geq\mathrm{cd}_{p}(K(\mathcal{O}_{X,x}))$, this follows from [GAV72, Corollary X.2.4]. ∎

Let $N=\mathrm{vcd}_{2}(K(X))+\dim{X}$. Note that for any Hensel local ring $A$ of $X$ we have $\mathrm{vcd}_{2}(A)\leq N$, by Lemma 2.2 and [GAV72, Theorem X.2.1]. Since $A$ is henselian and noetherian, by [Jac18, Lemma 6.2(III)] we have $\cap_{n}I^{n}(A)=0$. Hence by [Jac18, Corollary 4.8] for $n>N$ the map $I^{n}(A)\xrightarrow{2}I^{n+1}(A)$ is an isomorphism. By [Jac17, Proposition 7.1], the divided signatures induce an isomorphism $\operatorname*{colim}_{n}I^{n}(A)\simeq(a_{r\acute{e}t}\mathbb{Z})(A)$. These two results imply that $\sigma/2^{n}(A):I^{n}(A)\to(a_{r\acute{e}t}\mathbb{Z})(A)$ is an isomorphism, for any $n>N$. Since $A$ was arbitrary, the map $\sigma/2^{n}:\underline{I}^{n}\to a_{r\acute{e}t}\mathbb{Z}$ induces an isomorphism on stalks, and hence is an isomorphism. ∎

Let $E=(E_{1},E_{2},\dots)$ be a ${\mathbb{G}_{m}}$-prespectrum in Nisnevich sheaves of spectra. If each $E_{i}$ is connective, we can form a prespectrum $\tau_{\leq 0}^{\mathrm{Nis}}(E)$ with $\tau_{\leq 0}^{\mathrm{Nis}}(E)_{i}\simeq\tau_{\leq 0}^{\mathrm{Nis}}(E_{i})$ the truncation in the usual $t$-structure, and bonding maps

$${\mathbb{G}_{m}}\wedge\tau_{\leq 0}^{\mathrm{Nis}}(E_{i})\to\tau_{\leq 0}^{\mathrm{Nis}}({\mathbb{G}_{m}}\wedge\tau_{\leq 0}^{\mathrm{Nis}}(E_{i}))\simeq\tau_{\leq 0}^{\mathrm{Nis}}({\mathbb{G}_{m}}\wedge E_{i})\to\tau_{\leq 0}^{\mathrm{Nis}}(E_{i+1}).$$

Even if $E$ is a motivic spectrum $\tau_{\leq 0}^{\mathrm{Nis}}(E)$ need not be; however if it is then it represents the truncation $\tau_{\leq 0}(E)\in\mathcal{SH}(D)$ in the homotopy $t$-structure.

In [Spi18, §4.1.1] there is a construction of a specific ${\mathbb{G}_{m}}$-prespectrum $H$ such that (1) $H$ is a motivic spectrum representing $\mathrm{H}\mathbb{Z}/2$ and (2) $\tau_{\leq 0}^{\mathrm{Nis}}(H)\simeq\underline{k}^{M}$, the equivalence being as ${\mathbb{G}_{m}}$-prespectra. The map $\mathrm{H}\mathbb{Z}/2\to(\mathrm{H}\mathbb{Z}/2)/\tau\in\mathcal{SH}(D)$ corresponds to a map $H\to H^{\prime}$ of ${\mathbb{G}_{m}}$-spectra which is immediately seen to be a levelwise zero-truncation. It follows that $H^{\prime}\simeq\tau_{\leq 0}^{\mathrm{Nis}}(H)\simeq\underline{k}^{M}$ as ${\mathbb{G}_{m}}$-prespectra. In particular $\tau_{\leq 0}^{\mathrm{Nis}}(H)\simeq\underline{k}^{M}$ are motivic spectra, and in fact $\underline{k}^{M}\simeq\tau_{\leq 0}(\mathrm{H}\mathbb{Z}/2)\in\mathcal{SH}(D)$. Since $\mathcal{SH}(D)_{\geq 0}$ is closed under smash products, truncation in the homotopy $t$-structure is lax symmetric monoidal on $\mathcal{SH}(D)_{\geq 0}$ and so $\tau_{\leq 0}(\mathrm{H}\mathbb{Z}/2)$ admits a canonical ring structure making $\mathrm{H}\mathbb{Z}/2\to(\mathrm{H}\mathbb{Z}/2)_{\leq 0}$ into a commutative ring map. It remains to show that $\tau_{\leq 0}(\mathrm{H}\mathbb{Z}/2)\simeq\underline{k}^{M}$ is an equivalence of ring spectra. Both of them can be modeled by $\mathcal{E}_{\infty}$-monoids in the ordinary $1$-category of symmetric ${\mathbb{G}_{m}}$-prespectra of sheaves of abelian groups on ${\mathrm{S}\mathrm{m}}_{D}$; i.e. just commutative monoids in the usual sense. The isomorphism between them preserves the product structure by inspection. ∎

Since $(\underline{K}^{W})_{n}=L_{\mathrm{Nis}}\underline{I}^{n}$ is Nisnevich local by construction, to prove it is motivically local we need to establish $\mathbb{A}^{1}$-homotopy invariance, i.e. that $\Omega_{\mathbb{A}^{1}_{+}}\underline{K}^{W}_{n}\simeq\underline{K}^{W}_{n}$. Similarly to prove that we have a spectrum we need to show that $\Omega_{{\mathbb{G}_{m}}}\underline{K}^{W}_{n+1}\simeq\underline{K}^{W}_{n}$. Here we are working in the category $\mathcal{S}\mathrm{hv}^{\mathrm{Nis}}_{\mathcal{SH}}({\mathrm{S}\mathrm{m}}_{D})$ of Nisnevich sheaves of spectra on ${\mathrm{S}\mathrm{m}}_{D}$. For $x\in D$, denote by $p_{x}:D_{x}\to D$ the inclusion of the local scheme. By [Hoy15, Lemmas A.3 and A.4], the functor $p_{x}^{*}$ commutes with $\Omega_{{\mathbb{G}_{m}}}$ and $\Omega_{\mathbb{A}^{1}_{+}}$, and by [BH21, Proposition A.3(1,3)] the family of functors $p_{x}^{*}:\mathcal{S}\mathrm{hv}^{\mathrm{Nis}}_{\mathcal{SH}}({\mathrm{S}\mathrm{m}}_{D})\to\mathcal{S}\mathrm{hv}^{\mathrm{Nis}}_{\mathcal{SH}}({\mathrm{S}\mathrm{m}}_{D_{x}})$ is conservative. Finally by [Bac18b, Corollary 51] we have $p_{x}^{*}\underline{I}^{n}\simeq\underline{I}^{n}$. It follows that we may assume (replacing $D$ by $D_{x}$) that $D$ is the spectrum of a discrete valuation ring or field.

By Lemma 2.10 below, we have $D\simeq\operatorname*{lim}_{\alpha}D_{\alpha}$, where each $D_{\alpha}$ is the spectrum of a discrete valuation ring or field and $\mathrm{vcd}_{2}(K(D_{\alpha}))<\infty$. By [Gro67, Theorem 8.8.2(ii), Proposition 17.7.8(ii)] for $X\in{\mathrm{S}\mathrm{m}}_{D}$ there exists (possibly after shrinking the indexing system) a presentation $X\simeq\operatorname*{lim}_{\alpha}X_{\alpha}$, with $X_{\alpha}\in{\mathrm{S}\mathrm{m}}_{D_{\alpha}}$ and the transition maps being affine. We have a fibered topos [GAV72, §Vbis.7] $X_{\bullet\mathrm{Nis}}$ with $\operatorname*{lim}X_{\bullet\mathrm{Nis}}\simeq X_{\mathrm{Nis}}$. The sheaves $(\underline{I}^{n}|_{X_{\alpha\mathrm{Nis}}})_{\alpha}$ define a section of the fibered topos $X_{\bullet\mathrm{Nis}}$. It follows from [Bac18b, Lemma 49] and [GAV72, Proposition Vbis.8.5.2] that (in the notation of the latter reference) $Q^{*}(\underline{I}^{n}|_{X_{\bullet\mathrm{Nis}}})\simeq\underline{I}^{n}|_{X_{\mathrm{Nis}}}$. Hence by [GAV72, Theorem Vbis.8.7.3] we get

$$H^{i}(X,\underline{I}^{n})\simeq\operatorname*{colim}_{\alpha}H^{i}(X_{\alpha},\underline{I}^{n}).$$

The same holds for cohomology on $X\times\mathbb{A}^{1}$ and $X_{+}\wedge{\mathbb{G}_{m}}$. We may thus assume (replacing $D$ by $D_{\alpha}$) that $\mathrm{vcd}_{2}(K(D))<\infty$.

Let $X\in{\mathrm{S}\mathrm{m}}_{D}$. We need to prove that $(*)$

$$H^{*}(X,\underline{I}^{n})\simeq H^{*}(X\times\mathbb{A}^{1},\underline{I}^{n})\simeq H^{*}(X_{+}\wedge{\mathbb{G}_{m}},\underline{I}^{n}).$$

Since $\underline{k}^{M}_{n}$ satisfies the analog of $(*)$ by Lemma 2.7, the exact sequence $\underline{I}^{n+1}\to\underline{I}^{n}\to\underline{k}_{n}^{M}$ from Theorem 2.1 shows that $(*)$ holds for $\underline{I}^{n}$ if and only if it holds for $\underline{I}^{n+1}$. By Proposition 2.3 (and [GAV72, Theorem X.2.1]), for $n$ sufficiently large we get $\underline{I}^{n}|_{X_{\mathrm{Nis}}}\simeq a_{r\acute{e}t}\mathbb{Z}|_{X_{\mathrm{Nis}}}$. It thus suffices to show that $L_{\mathrm{Nis}}a_{r\acute{e}t}\mathbb{Z}$ satisfies the analog of $(*)$. This follows from the fact that there exists a motivic spectrum $E=H_{\mathbb{A}^{1}}\mathbb{Z}[\rho^{-1}]$ with $E_{i}=L_{\mathrm{Nis}}a_{r\acute{e}t}\mathbb{Z}$ for all $i$ [Bac18a, Proposition 41]. ∎

Let $K=K(R)$. Then $K=\operatorname*{colim}_{\alpha}K_{\alpha}$, where the colimit is over finitely generated subfields $K_{\alpha}\subset K$; this colimit is filtered. Let $R_{\alpha}=R\cap K_{\alpha}$. We shall show that $R_{\alpha}$ is a noetherian valuation ring, and $K(R_{\alpha})=K_{\alpha}$. This will imply the result since $\mathrm{vcd}_{p}(K_{\alpha})<\infty$ uniformly in $p$ by [GAV72, Theorem X.2.1, Proposition X.6.1, Theorem X.5.1]. It is clear that $R_{\alpha}$ is a valuation ring and $K(R_{\alpha})=K_{\alpha}$: if $x\in K_{\alpha}^{\times}$ then one of $x,x^{-1}\in R$ [Sta18, Tag 00IB], and hence one of $x,x^{-1}\in R_{\alpha}$; thus we conclude by [Sta18, Tag 052K]. To show that $R_{\alpha}$ is noetherian we must show that $K_{\alpha}^{\times}/R_{\alpha}^{\times}\simeq\mathbb{Z}$ or $\simeq 0$ [Sta18, Tags 00IE and 00II]. This is clear since there is an injection $K_{\alpha}^{\times}/R_{\alpha}^{\times}\hookrightarrow K^{\times}/R^{\times}$ and the latter group is $\simeq\mathbb{Z}$ or $\simeq 0$. ∎

(1) holds by definition. (2) The functor $U_{\Sigma}:\mathcal{P}_{\Sigma}(\mathrm{Cor}^{\mathrm{fr}}(S))\to\mathcal{P}_{\Sigma}({\mathrm{S}\mathrm{m}}_{S})_{*}$ preserves filtered (in fact sifted) colimits and commutes with $L_{\mathrm{Nis}}$ [EHK⁺21, Proposition 3.2.14]. Consequently $U_{\mathrm{Nis}}:\mathcal{S}\mathrm{hv}^{\mathrm{Nis}}(\mathrm{Cor}^{\mathrm{fr}}(S))\to\mathcal{S}\mathrm{hv}_{\mathrm{Nis}}({\mathrm{S}\mathrm{m}}_{S})_{*}$ also preserves filtered colimits. Being a right adjoint it also preserves limits, and hence commutes with spectrification. Consequently it suffices to prove the following: given $F\in\mathcal{P}_{\Sigma}(\mathrm{Cor}^{\mathrm{fr}}(S))$ and $n\geq 0$, we have $U_{\Sigma}(\Sigma^{n}F)\in\mathcal{P}_{\Sigma}({\mathrm{S}\mathrm{m}}_{S})_{\geq n}$; indeed then

$$U(\Sigma^{\infty}F)\simeq\operatorname*{colim}_{n}\Sigma^{\infty-n}U_{\Sigma}(\Sigma^{n}F)\in\mathcal{S}\mathrm{hv}^{\mathrm{Nis}}_{\mathcal{SH}}({\mathrm{S}\mathrm{m}}_{S})_{\geq 0}$$

by what we have already said. Writing $\Sigma^{n}F$ as an iterated sifted colimit, using semi-additivity of $\mathcal{P}_{\Sigma}(\mathrm{Cor}^{\mathrm{fr}}(S))$ [EHK⁺21, sentence after Lemma 3.2.5] and the fact that $U_{\Sigma}$ commutes with sifted colimits, we find that $U_{\Sigma}(\Sigma^{n}F)$ is given by $B^{n}U_{\Sigma}(F)$, i.e. the iterated bar construction applied sectionwise. The required connectivity is well-known; see e.g. [Seg74, Proposition 1.5]. ∎

Write $\langle t\rangle:(\mathbb{A}^{1}\setminus 0)_{+}\to(\mathbb{A}^{1}\setminus 0)_{+}\in\mathcal{SH}(S)$ for the map induced by the framing $t$ of the identity, and $\tilde{\eta}$ for the induced map $\Sigma^{\infty}_{+}{\mathbb{G}_{m}}\to\mathbbm{1}$. By [EHK⁺20, Example 3.1.6], the map $\langle t\rangle$ is given by $\Sigma^{\infty-2,1}$ of the map of pointed motivic spaces

$$(\mathbb{A}^{1}\setminus 0)_{+}\wedge\mathbb{A}^{1}/(\mathbb{A}^{1}\setminus 0)\to(\mathbb{A}^{1}\setminus 0)_{+}\wedge\mathbb{A}^{1}/(\mathbb{A}^{1}\setminus 0),\quad(t,x)\mapsto(t,tx),$$

and hence $\tilde{\eta}$ is given by $\Sigma^{\infty-2,1}$ of the map

$$\eta^{\prime}:(\mathbb{A}^{1}\setminus 0)_{+}\wedge\mathbb{A}^{1}/(\mathbb{A}^{1}\setminus 0)\to\mathbb{A}^{1}/\mathbb{A}^{1}\setminus 0,\quad(t,x)\mapsto tx.$$

Consider the commutative diagram of pointed motivic spaces (with $\mathbb{A}^{1}$ pointed at $1$)

$$\begin{CD}{\mathbb{G}_{m}}\times{\mathbb{G}_{m}}@>{}>{}>{\mathbb{G}_{m}}\times\mathbb{A}^{1}@>{}>{}>({\mathbb{G}_{m}}\times\mathbb{A}^{1})/({\mathbb{G}_{m}}\times{\mathbb{G}_{m}})\simeq(\mathbb{A}^{1}\setminus 0)_{+}\wedge\mathbb{A}^{1}/(\mathbb{A}^{1}\setminus 0)\\ @V{}V{}V@V{}V{}V@V{\eta^{\prime}}V{}V\\ {\mathbb{G}_{m}}@>{}>{}>\mathbb{A}^{1}@>{}>{}>\mathbb{A}^{1}/{\mathbb{G}_{m}}.\end{CD}$$

All the vertical maps are induced by $(t,x)\mapsto tx$ and the horizontal maps are induced by the canonical inclusions and projections. Stably this splits as [Mor04a, Lemma 6.1.1]

$$\begin{CD}{\mathbb{G}_{m}}\vee{\mathbb{G}_{m}}\vee{\mathbb{G}_{m}^{\wedge 2}}@>{(\operatorname{id},0,0)}>{}>{\mathbb{G}_{m}}@>{}>{}>S^{2,1}\vee\Sigma^{2,1}{\mathbb{G}_{m}}\\ @V{(\operatorname{id},\operatorname{id},\Sigma^{1,1}\eta)}V{}V@V{}V{}V@V{\Sigma^{2,1}\tilde{\eta}}V{}V\\ {\mathbb{G}_{m}}@>{}>{}>0@>{}>{}>S^{2,1}.\end{CD}$$

Since the rows are cofibration sequences, it follows that $\tilde{\eta}\simeq(\operatorname{id},\eta)$, which implies the desired result. ∎

In light of Lemma 3.4, this is a special case of [Hoy16, Proposition 3.2]. ∎

The framed construction of $\mathrm{KO}$ as in [BW21, §A] together with Lemma 3.4 shows that the map $\eta^{*}:\mathrm{HW}\to\Omega_{\mathbb{G}_{m}}\mathrm{HW}$ is on the level of spectral sheaves (i.e. sheaves of abelian groups) induced by multiplication by $\langle t\rangle-1$. The result now follows from Remark 2.9. ∎

The equivalence $\mathcal{SH}^{\mathrm{fr}}(D)\simeq\mathcal{SH}(D)$ restricts to the subcategories of those objects such that $\underline{\pi}_{i}(\mathord{-})_{j}=0$ for $i\neq 0$. These are exactly the objects representable by prespectra valued in sheaves of abelian groups that are motivic spectra when viewed as valued in spectral sheaves. The construction of $\mathrm{HW}$ supplies the sheaf $\underline{W}$ with a structure of framed transfers. Then the subsheaf $\underline{I}^{n}$ admits at most one compatible structure of framed transfers; the existence of the map $\underline{K}^{W}\to\mathrm{HW}$ together with the above discussion shows that this structure exists. This supplies a description of the (unique) lift of $\underline{K}^{W}$ to $\mathcal{SH}^{\mathrm{fr}}(D)$. From this and Lemma 3.4 it follows that the action by $\eta$ on $\underline{K}^{W}$ induces the inclusion $\underline{I}^{*+1}\to\underline{I}^{*}$ on homotopy sheaves. This implies immediately that $\underline{K}^{W}[\eta^{-1}]\to\mathrm{HW}$ induces an isomorphism on homotopy sheaves and hence is an equivalence (see Remark 3.3). For $\underline{K}^{W}/\eta$ the same argument works using Theorem 2.1. ∎

(1, 2) Since the negative homotopy sheaves (in weight $0$) of $\mathrm{KO}$ and $\mathrm{KW}$ coincide [Sch17, Proposition 6.3], we have

$$\tau_{<0}^{\mathrm{Nis}}\omega^{\infty}_{\mathrm{fr}}\mathrm{KO}\simeq\tau_{<0}^{\mathrm{Nis}}\omega^{\infty}_{\mathrm{fr}}\mathrm{KW}=:E.$$

Since $\omega^{\infty}\mathrm{KO}$ and $\omega^{\infty}_{\mathrm{fr}}\mathrm{KW}$ are motivically local, and $\omega^{\infty}_{\mathrm{fr}}\mathrm{KW}$ is $\eta$-periodic, it thus suffices to show that $E$ is motivically local and $\eta$-periodic. Since motivically local, $\eta$-periodic spectral presheaves are closed under limits and colimits, by Remark 3.7 it thus suffices to show that $\tau^{\mathrm{Nis}}_{=0}\omega^{\infty}_{\mathrm{fr}}\mathrm{KW}\simeq\mathrm{HW}$ is motivically local and $\eta$-periodic. This is Lemma 3.8.