---
abstract: 'We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.'
author:
- Markus Spitzweck
bibliography:
- 'ma.bib'
title: Algebraic Cobordism in mixed characteristic
---

Introduction
============

Algebraic cobordism is a theory for smooth schemes over a base scheme $S$ defined by a motivic ring spectrum ${\mathsf{MGL}}_S$ in the stable motivic homotopy category ${\mathsf{SH}}(S)$. It is the motivic counterpart of complex cobordism ${\mathsf{MU}}$. A famous Theorem of Quillen states that the natural map from the Lazard ring $L_*$ classifying formal group laws to the coefficients of ${\mathsf{MU}}$ is an isomorphism, moreover $L_* \cong {{\mathbb Z}}[x_1,x_2,x_3,\ldots]$ with $\deg(x_i)=i$ (here we divide the usual topological grading by $2$).

For an oriented motivic ring spectrum $E$ the geometric part $E_{(2,1)*}$ of the coefficients also carries a formal group law constructed in the exact same way as in topology by evaluating the theory on ${{\mathbb P}}^\infty$ and using that ${{\mathbb P}}^\infty$ is naturally endowed with a multiplication.

Thus there is a classifying map $L_* \to E_{(2,1)*}$. It is known that for $E={\mathsf{MGL}}_k$ for a field $k$ of characteristic $0$ this map is an isomorphism using the Hopkins-Morel isomorphism, see [@hoyois.hopkins-morel Proposition 8.2]. More generally in [@levine.comparison] it is shown that over such fields the Levine-Morel algebraic cobordism $\Omega^*(-)$ is isomorphic to ${\mathsf{MGL}}_k^{(2,1)*}(-)$ on smooth schemes over $k$. If the base field $k$ has positive characteristic the map $L_* \to {\mathsf{MGL}}_{(2,1)*}$ becomes at least an isomorphism after inverting the characteristic, see again [@hoyois.hopkins-morel Proposition 8.2].

The main ingredient in the proof is that the Hopkins-Morel isomorphism yields a computation of the slices of ${\mathsf{MGL}}_S$ with respect to Voevodsky’s slice filtration, that ${\mathsf{MGL}}_S$ is complete with respect to this filtration and that the slices have a simple form, namely they are shifted twists of the motivic Eilenberg-MacLane spectrum.

The facts about the slices of ${\mathsf{MGL}}_S$ hold more generally true over spectra $S$ of Dedekind domains of mixed characteristic (after inverting the positive residue characteristics), using the motivic Eilenberg-MacLane spectrum introduced in [@spitzweck.em]. The main new input of this note is that in this case ${\mathsf{MGL}}_S$ is also complete with respect to the slice filtration (Corollary \[gfrerz\]), a consequence of the fact that ${\mathsf{MGL}}_S$ is connective with respect to the homotopy sheaves, see Proposition \[gdrttt\].

This yields a computation of the geometric part of the homotopy groups of ${\mathsf{MGL}}_S$ (Theorem \[grfe5z4\]), again after inverting the residue characteristics. In our formulation we always assume a Hopkins-Morel isomorphism for the given coefficients, hoping that the Hopkins-Morel isomorphism will be settled completely in the future.

We prove cases of a Conjecture of Voevodsky ([@voevodsky.icm Conjecture 1]), see Theorem \[hte4234t\], comparing the Lazard ring to $({\mathsf{MGL}}_S)_{(2,1)*}$ for $S$ the spectrum of a regular local ring.

We also give applications to some homotopy groups or sheaves of ${\mathsf{MGL}}_S$ outside the geometric diagonal, see section \[dzu634e\], and discuss generalizations of our results to motivic Landweber spectra.

.4cm

We note that the observation that the Hopkins-Morel isomorphism yields the computation of the zero-slice of the sphere spectrum (after inverting suitable primes), see Theorem \[dhu5t3\], was independently made by Oliver Röndigs.

.7cm

[**Acknowledgements:**]{} I would like to thank Peter Arndt, Christian Häsemeyer, Marc Hoyois, Moritz Kerz, Marc Levine, Niko Naumann, Oliver Röndigs, Manfred Stelzer, Florian Strunk, Jörg Wildeshaus and Paul Arne [Ø]{}stv[æ]{}r for very helpful discussions and suggestions on the subject.

Preliminaries
=============

By a base scheme we always mean a separated Noetherian scheme of finite Krull dimension. For a base scheme $S$ we let ${\mathsf{SH}}(S)$ be the stable motivic homotopy category.

We let ${\mathsf{M}\mathbb{Z}}_S \in {\mathsf{SH}}(S)$ be the motivic Eilenberg-MacLane spectrum over $S$ constructed in [@spitzweck.em]. Also we let ${{\mathcal M}}(r) \in {{\mathsf D}}({\mathrm{Sh}}({\mathrm{Sm}}_{S,{\mathit{Zar}}},{{\mathbb Z}}))$ (for notation see [@spitzweck.em]) be the motivic complexes of weight $r \in {{\mathbb Z}}$, so as a ${\mathbb{G}_{m,S}}$-spectrum ${\mathsf{M}\mathbb{Z}}_S$ has ${{\mathcal M}}(r)[r]$ in level $r$. If $S$ is the spectrum of a Dedekind domain of mixed characteristic we note that ${{\mathcal M}}(0)= S^0 \underline{{{\mathbb Z}}}$, thus for $X \in {\mathrm{Sm}}_S$ we have $H^{0,0}(X,{{\mathbb Z}})={{\mathbb Z}}^{\pi_0(X)}$. Also ${{\mathcal M}}(1) \cong {{\mathcal O}}^*[-1]$, so $H^{1,1}(X,{{\mathbb Z}}) \cong {{\mathcal O}}^*(X)$ and $H^{2,1}(X,{{\mathbb Z}}) \cong {\mathrm{Pic}}(X)$. We have ${{\mathcal M}}(r) \cong 0$ for $r < 0$.

For general $S$ we denote by ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)$ the algebraic cobordism spectrum. There is a natural map $L_* \to ({\mathsf{MGL}}_S)_{2*,*}$, where $L_*$ denotes the Lazard ring. Fixing generators $x_i \in L_i$ there is a map $$\Phi_S \colon {\mathsf{MGL}}_S/(x_1,x_2,\ldots){\mathsf{MGL}}_S \to {\mathsf{M}\mathbb{Z}}_S,$$ see [@spitzweck.em §11.1], which is an isomorphism after inverting all positive residue characteristics of $S$, see [@spitzweck.em Theorem 11.3].

For any ring or abelian group $R$ we let $M_R \in {\mathsf{SH}}(S)$ be the Moore spectrum on $R$ and ${\mathsf{M}R}_S$ the version of ${\mathsf{M}\mathbb{Z}}_S$ with $R$-coeffcients.

Slices
======

For $i \in {{\mathbb Z}}$ denote by $f_i$ resp. $l_i$ the $i$-th colocalization resp. localization functor for Voevodsky’s motivic slice filtration on ${\mathsf{SH}}(S)$. For any $E \in {\mathsf{SH}}(S)$ and $k \ge n$ we set $E\left<n,k \right>:=l_{k+1}(f_n(E))$. Thus we have exact triangles $$f_{k+1}(E) \to f_n(E) \to E\left<n,k \right> \to f_{k+1}(E)[1]$$ and $s_n(E) = E\left<n,n \right>$.

We note that all these functors commute with homotopy colimits.

\[dhu5t3\] Let $X$ be an essentially smooth scheme over a Dedekind domain of mixed characteristic and $R$ a localization of ${{\mathbb Z}}$ such that $\Phi_X \wedge M_R$ is an isomorphism (e.g. if every positive residue characteristic of $X$ is invertible in $R$). Then $$s_0 M_R \cong s_0 ({\mathsf{MGL}}_X \wedge M_R) \cong {\mathsf{M}R}_X.$$ More generally $$s_n ({\mathsf{MGL}}_X \wedge M_R) \cong \Sigma^{2n,n} {\mathsf{M}R}_X \otimes L_n.$$

The first isomorphism of the first line follows from [@spitzweck.rel Corollary 3.3]. From the assumption that $\Phi_X \wedge M_R$ is an isomorphism it follows that the map ${\mathsf{MGL}}_X \wedge M_R \to {\mathsf{M}R}_X$ induces an isomorphism on zero-slices and that ${\mathsf{M}R}_X$ is effective. Moreover $l_1 {\mathsf{M}\mathbb{Z}}_X \cong {\mathsf{M}\mathbb{Z}}_X$, since negative weight motivic cohomology vanishes in our situation. Thus the second isomorphism of the first line follows. The second line is a version of [@spitzweck.rel Theorem 4.7] with $R$-coefficients.

It is then also possible to determine the slices of motivic Landweber spectra with $R$-coeffcients, see [@spitzweck.slices], for example of ${\mathsf{KGL}}_X \wedge M_R$.

Subcategories of the stable motivic homotopy category
=====================================================

Fix a base scheme $S$. We let ${\mathsf{SH}}(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}(S)$ with respect to the homotopy $t$-structure, see e.g. [@hoyois.hopkins-morel §2.1]. Thus ${\mathsf{SH}}(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^{p,q} \Sigma^\infty_+ X$ for $X \in {\mathrm{Sm}}_S$ and $p-q \ge n$.

For each $E \in {\mathsf{SH}}(S)$ we let ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ be the presheaf $$X \mapsto {\mathrm{Hom}}_{{\mathsf{SH}}(S)}(\Sigma^{p,q} \Sigma^\infty_+ X,E)$$ on ${\mathrm{Sm}}_S$. Let ${\underline{\pi}}_{p,q}(E)$ be the sheafification of ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ with respect to the Nisnevich topology. We also set $\pi_{p,q}(E):={\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(S)=E_{p,q}$.

We let ${\mathsf{SH}}(S)_{h \ge n}$ be the full subcategory of ${\mathsf{SH}}(S)$ of objects $E$ such that ${\underline{\pi}}_{p,q}(E)=0$ for $p-q< n$.

The categories ${\mathsf{SH}}(S)_{h \ge n}$ are closed under homotopy colimits and extensions in ${\mathsf{SH}}(S)$.

Th functors ${\underline{\pi}}_{p,q}$ respect sums. Moreover the long exact sequences of homotopy sheaves associated to an exact triangle in ${\mathsf{SH}}(S)$ show that ${\mathsf{SH}}(S)_{h \ge n}$ is closed under cofibers and extensions. This shows the claim.

\[gegrsy\] Let $i \colon Z \hookrightarrow S$ be a closed inclusion of base schemes. Then $i_*({\mathsf{SH}}(Z)_{h \ge n}) \subset {\mathsf{SH}}(S)_{h \ge n}$.

Let $E \in {\mathsf{SH}}(Z)_{h \ge n}$. Let $Y$ be the spectrum of the henselization of a local ring of a scheme from ${\mathrm{Sm}}_S$. Then $Y_Z:=Y \times_S Z$ is also the spectrum of a henselian local ring, and ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(i_*E)(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(Y_Z)=0$ for $p-q < n$ (the first isomorphism holds since $i_*$ commutes with homotopy colimits).

We let ${\mathsf{SH}}^{{S^1}}_s(S)$ be the homotopy category of presheaves of ${{S^1}}$-spectra on ${\mathrm{Sm}}_S$ localized with respect to the Nisnevich topology, and ${\mathsf{SH}}^{{S^1}}(S)$ the further ${{\mathbb A}}^1$-localization of that category.

We let ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}^{{S^1}}_s(S)$ with respect to the standard $t$-structure, and for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we let $E_{\ge n}$ and $E_{\le n}$ be the corresponding truncations. We let $E_{=n} := (E_{\ge n})_{\le n}$.

As above for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have the presheaves ${\underline{\pi}}^{\mathrm{pre}}_p(E)$ and the sheaves ${\underline{\pi}}_p(E)$. For $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have $E \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ if and only if ${\underline{\pi}}_k(E)=0$ for $k < n$.

Note that ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^n \Sigma^\infty_+ X$, $X \in {\mathrm{Sm}}_S$, thus the canonical functor $\sigma \colon {\mathsf{SH}}^{{S^1}}_s(S) \to {\mathsf{SH}}(S)$ sends ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ to ${\mathsf{SH}}(S)_{\ge n}$.

We have ${\mathsf{SH}}(S)_{h \ge n} \subset {\mathsf{SH}}(S)_{\ge n}$. If $S$ is the spectrum of a field then the inclusion is an equality.

Let $E \in {\mathsf{SH}}(S)_{h \ge n}$. For any $i \in {{\mathbb N}}$ let $E_i$ be the image of $\Sigma^{i,i} E$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. By assumption we have $E_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$. Thus $\Sigma^{-i,-i} \sigma(E_i) \in {\mathsf{SH}}(S)_{\ge n}$. The proof of the first statement concludes by noting that $E \cong {\mathrm{hocolim}}_{i \to \infty} \Sigma^{-i,-i} \sigma(E_i)$.

The second statement is [@hoyois.hopkins-morel Theorem 2.3].

\[hterge\] Let $E \in {\mathsf{SH}}^{{S^1}}_s(S)$. Then $E \to {\mathrm{holim}}_{n \to \infty} E_{\le n}$ is an isomorphism.

We show that for all $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_n(E) \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})$. Fix $n \in {{\mathbb Z}}$ and let $X \in {\mathrm{Sm}}_S$ be of dimension $d$. We are ready if we show $${\underline{\pi}}_n(E)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \; \; (*).$$ For $m > n+d$ we have ${\underline{\pi}}^{\mathrm{pre}}_n(E_{=m}[j])(Y)=0$ for $Y \in X_{\mathit{Nis}}$ and $j \ge 0$, so homing out of $\Sigma^\infty_+ Y$ into the exact triangle $$E_{=m} \to E_{\le m} \to E_{\le (m-1)} \to E_{=m}[1]$$ shows that ${\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le (m-1)})(Y)$. Using the Milnor exact sequence this shows that $${\underline{\pi}}^{\mathrm{pre}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})|_{X_{\mathit{Nis}}}$$ for $m \ge n+d$. Sheafifiying proves $(*)$.

\[dwrzjj\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)$. Suppose for each $n \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that $E_i \in {\mathsf{SH}}(S)_{h \ge n}$ for $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \cong 0$.

Fix $q \in {{\mathbb Z}}$ and let $F_i$ be the image of $\Sigma^{q,q} E_i$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. We are ready if we show ${\mathrm{holim}}_{i \to \infty} F_i \cong 0$. By assumption for every $n \in {{\mathbb N}}$ there is a $N \in {{\mathbb N}}$ such that $F_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ for each $i \ge N$. By Lemma \[hterge\] we have $F_i \cong {\mathrm{holim}}_{k \to \infty} (F_i)_{\le k}$. Thus $${\mathrm{holim}}_{i \to \infty} F_i \cong {\mathrm{holim}}_i {\mathrm{holim}}_k (F_i)_{\le k} \cong {\mathrm{holim}}_k {\mathrm{holim}}_i (F_i)_{\le k}
\cong {\mathrm{holim}}_k 0 \cong 0.$$

We also have the

Let $E \in {\mathsf{SH}}(S)_{h \ge n}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(X)=0$$ for $p-q<n-d$.

\[gr455\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)_{h \ge n}$. Suppose for each $p,q \in {{\mathbb Z}}$ and $d \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that for $X \in {\mathrm{Sm}}_S$ of dimension $d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(X)$$ is an isomorphism for all $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \in {\mathsf{SH}}(S)_{h \ge n}$. (Here the latter homotopy limit is computed in ${\mathsf{SH}}(S)$.)

Let $p,q \in {{\mathbb Z}}$, $d \in {{\mathbb N}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Choose $N \in {{\mathbb N}}$ such that for any $Y \in {\mathrm{Sm}}_S$ of dimension $\le d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y)$$ is an isomorphism for all $i \ge N$. We claim that $${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$$ for all $i \ge N$. For every $Y \in X_{\mathit{Nis}}$ we have the Milnor short exact sequence $$0 \to \text{$\lim_i$}^1 {\underline{\pi}}^{\mathrm{pre}}_{p+1,q}(E_i)(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathrm{holim}}_i E_i)(Y) \to \lim_i {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y) \to 0.$$ The $\lim^1$-term vanishes because the inverse system of abelian groups stabilizes by assumption. Sheafifying we see that ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$ for $i \ge N$, in particular ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} = 0$ in case $p-q<n$. Since this is true for all $X \in {\mathrm{Sm}}_S$ we conclude ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k) =0$ for $p-q < n$.

Connectivity of algebraic cobordism
===================================

\[vftjt\] Let $X$ be a smooth scheme over a Dedekind domain of mixed characteristic or over a field. Then for any abelian group $A$ we have ${\mathsf{M}A}_X \in {\mathsf{SH}}(X)_{h \ge 0}$.

This follows from the fact that the motivic complexes ${{\mathcal M}}(r)$ have vanishing $i$-th cohomology sheaf for $i>r$, see [@geisser.dede Corollary 4.4].

\[fhtrth\] Let $S$ be the spetrum of a discrete valuation ring of mixed characteristic, $j \colon \eta \to S$ the inclusion of the generic point. Then for any abelian group $A$ we have $j_* {\mathsf{M}A}_\eta \in {\mathsf{SH}}(S)_{h \ge 0}$.

Let $i \colon s \to S$ be the inclusion of the closed point. We have an exact triangle $$i_!i^! {\mathsf{M}A}_S \to {\mathsf{M}A}_S \to j_* {\mathsf{M}A}_\eta \to i_!i^! {\mathsf{M}A}_S[1]$$ and an isomorphism $i^! {\mathsf{M}A}_S \cong {\mathsf{M}A}_s(-1)[-2] \in {\mathsf{SH}}(s)_{h \ge -1}$, see [@spitzweck.em Theorem 7.4]. We conclude with Proposition \[gegrsy\] and Lemma \[vftjt\].

\[gfewet\] Let the situation be as in Proposition \[fhtrth\]. Then $$j_*{\mathsf{MGL}}_\eta\left<0,n\right> \wedge M_A \in {\mathsf{SH}}(S)_{h \ge 0}$$ for all $n \ge 0$.

We can assume $A={{\mathbb Z}}$. Since $\eta$ is of characteristic $0$ we have $s_n {\mathsf{MGL}}_\eta \cong \Sigma^{2n,n} {\mathsf{M}\mathbb{Z}}\otimes L_n$. Moreover we have exact triangles $$s_n {\mathsf{MGL}}_\eta \to {\mathsf{MGL}}_\eta\left<0,n\right> \to {\mathsf{MGL}}_\eta \left<0,n-1 \right> \to s_n {\mathsf{MGL}}_\eta [1].$$ Applying $j_*$ to these triangles and using Proposition \[fhtrth\] one concludes by induction on $n$.

\[rergr\] Let the situation be as in Proposition \[fhtrth\]. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n+1 \right>)(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n \right>)(X)$$ is an isomorphism for $n \ge p-q+d$.

Consider the exact triangle $$j_* s_{n+1} {\mathsf{MGL}}_\eta \to j_* {\mathsf{MGL}}_\eta \left< 0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to s_{n+1} {\mathsf{MGL}}_\eta [1].$$ We have $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* s_{n+1} {\mathsf{MGL}}_\eta)(X) = H_{\mathrm{mot}}^{2(n+1)-p,n+1-q}(X_\eta, L_{n+1}).$$ The latter group vanishes for $2(n+1)-p> n+1 -q+d$, showing the claim.

\[eweggg\] Let the situation be as in Proposition \[fhtrth\]. Then $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{{\mathsf{h}}\ge 0}$.

Consider the inverse system $$\cdots \to j_* {\mathsf{MGL}}_\eta \left<0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to \cdots \to j_* s_0 {\mathsf{MGL}}_\eta$$ in ${\mathsf{SH}}(S)$. Since $j_*$ preserves homotopy limits the homotopy limit over this system is $j_* {\mathsf{MGL}}_\eta$, using [@hoyois.hopkins-morel Corollary 2.4 and Lemma 8.10 or Theorem 8.12]. By Lemma \[gfewet\] every object of this system is in ${\mathsf{SH}}(S)_{h \ge 0}$. Moreover by Lemma \[rergr\] the assumptions of Proposition \[gr455\] are satisfied. Thus this Proposition implies the claim.

\[greerg\] Let the situation be as in Proposition \[fhtrth\] and let $i \colon s \to S$ be the inclusion of the closed point. Then $i^! {\mathsf{MGL}}_S \in {\mathsf{SH}}(s)_{\ge -1}$.

Note first that $i^*$ sends ${\mathsf{SH}}(S)_{\ge 0}$ to ${\mathsf{SH}}(s)_{\ge 0}$. We have ${\mathsf{MGL}}\in {\mathsf{SH}}(S)_{\ge 0}$ and by Lemma \[eweggg\] also $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{h \ge 0}
\subset {\mathsf{SH}}(S)_{\ge 0}$. Applying $i^*$ to the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1]$$ shows the claim.

\[gferghd\] Let $S$ be the spectrum of a discrete valuation ring of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$.

Let the notation be as above. The claim follows from the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1],$$ Lemma \[eweggg\], Proposition \[greerg\] and Proposition \[gegrsy\].

\[gdrttt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$.

The henselization of a local ring of a scheme in ${\mathrm{Sm}}_S$ lies over a local ring of $S$, thus the claim follows from Lemma \[gferghd\].

Compare the following result to [@voevodsky.open Conjecture 15].

\[gfrerz\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ we have $$f_n {\mathsf{MGL}}_S \wedge M_A \cong {\mathrm{holim}}_{k \to \infty} {\mathsf{MGL}}_S \left< n,k \right> \wedge M_A.$$

Under the assumption we have ${\mathsf{f}}_k {\mathsf{MGL}}_S \wedge M_A \in {\mathsf{SH}}(S)_{h \ge k-1}$, since this is a homotopy colimit of objects of the form $\Sigma^{2i,i} {\mathsf{MGL}}_S \wedge M_A$ with $i \ge k$, see the proof of [@spitzweck.rel Theorem 4.7], using Proposition \[gdrttt\]. Thus by Corollary \[dwrzjj\] we have ${\mathrm{holim}}_{k \to \infty} f_k {\mathsf{MGL}}_S \wedge M_A \cong 0$ implying the claim.

A similar result holds for motivic Landweber spectra using the same argument as in the proof of [@hoyois.hopkins-morel Lemma 8.11]. For example ${\mathsf{KGL}}_S \wedge M_A$ is complete with respect to the slice filtration.

The geometric part of algebraic cobordism
=========================================

\[jewet45\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the inverse system of abelian groups $({\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left< 0,k \right> \wedge M_A)(X))_k$ eventually becomes constant for $k \to \infty$.

This follows from the exact triangle $$s_k {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left<0,k \right> \wedge M_A \to {\mathsf{MGL}}_S \left< 0,k-1 \right> \wedge M_A \to s_k {\mathsf{MGL}}_S \wedge M_A [1]$$ and $s_k {\mathsf{MGL}}_S \wedge M_A \cong \Sigma^{2k,k} {\mathsf{M}A}\otimes L_k$ since ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(\Sigma^{2k+j,k} {\mathsf{M}A})(X)=0$, $j \ge 0$, for $k$ big enough.

\[dsgtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the canonical map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to \lim_k {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A)(X)$$ is an isomorphism.

This follows from Corollary \[gfrerz\], the Milnor short exact sequence and Lemma \[jewet45\].

\[xfhtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $n \in {{\mathbb Z}}$. Then for $k \ge n+1$ and any $R$-module $A$ the natural map $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A$$ is an isomorphism.

This follows from the exact sequence $$\pi_{2n,n} s_{k+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to$$ $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A \to
\pi_{2n-1,n} s_k {\mathsf{MGL}}_S \wedge M_A$$ and the fact the the two outer terms are $0$ for $k \ge n+1$.

\[sgtjtr\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ the canonical map $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A$$ is an isomorphism.

This follows from Corollary \[dsgtjt\] and Lemma \[xfhtjt\].

\[grfe5z4\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for every $n \in {{\mathbb Z}}$ and $R$-module $A$ there is a canonical isomorphism $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \cong L_n \otimes A \oplus L_{n+1} \otimes \mathrm{Pic}(S) \otimes A.$$

We have the exact sequence $$\pi_{2n+1,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,n+1 \right> \wedge M_A$$ $$\to \pi_{2n,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n-1,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A.$$ The two outer terms are $0$. Also $\pi_{0,0} \Sigma^{2,1} {\mathsf{M}A}_S \cong \mathrm{Pic}(S) \otimes A$. Moreover there is a canonical map $L_n \otimes A \to \pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A$ splitting the resulting short exact sequence, whence the claim follows form Corollary \[sgtjtr\].

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic and $R$ the localization of ${{\mathbb Z}}$ obtained by inverting all positive residue characteristics of $S$. Then $$(\pi_{2n,n} {\mathsf{MGL}}_S) \otimes R \cong (L_n \oplus L_{n+1} \otimes \mathrm{Pic}(S)) \otimes R.$$

We have the following case of a Conjecture of Voevodsky (see [@voevodsky.icm Conjecture 1]):

\[hte4234t\] Let $S={\mathrm{Spec}}(R)$, where $R$ is a (regular) Noetherian local ring which is regular over some discrete valuation ring of mixed characteristic. Then the natural map $$L_* \to ({\mathsf{MGL}}_S)_{2*,*}$$ becomes an isomorphism after inverting the residue characteristic of the closed point of $S$.

By Popescu’s Theorem $R$ is a filtered colimit of smooth algebras over a discrete valuation ring $V$ of mixed characteristic. Thus we are reduced to the case where $R$ is the local ring of a scheme $X \in {\mathrm{Sm}}_{{\mathrm{Spec}}(V)}$ by a colimit argument. Let $p$ be the residue characteristic of the closed point of ${\mathrm{Spec}}(V)$. By the same type of argument as above and the vanishing of $(p,q)$-motivic cohomology of such local rings for $p>q$ we have $$(MGL_S)_{2n,n} [1/p] \cong (s_n {\mathsf{MGL}}_S)_{2n,n} [1/p] \cong L_n [1/p],$$ using that for a fixed dimension only a fixed finite number of slices of ${\mathsf{MGL}}_S[1/p]$ contribute to the value of $\pi^{\mathrm{pre}}_{2n,n}({\mathsf{MGL}}_S [1/p])$ on schemes of that dimension.

More generally we have

Let $S$ be as in the previous Theorem and $E \in {\mathsf{SH}}(S)$ a motivic Landweber spectrum modelled on $E^{\mathrm{top}}_{2*}$. Then the natural map $$E^{\mathrm{top}}_{2*} \to E_{2*,*}$$ is an isomorphism after inverting the residue characteristic of the closed point of $S$.

This follows from the definition of motivic Landweber spectrum.

Some other parts of algebraic cobordism {#dzu634e}
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We have the following vanishing result:

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have ${\underline{\pi}}_{p,q} {\mathsf{MGL}}_S \wedge M_A \cong 0$ for $p< 2q$ or $p<q$. In particular we have ${\mathsf{MGL}}_S \wedge M_R \in {\mathsf{SH}}(S)_{h \ge 0}$.

Let $p,q \in {{\mathbb Z}}$ satisfying the condition of the statement. Let $d \in {{\mathbb N}}$. Then there is a $N \ge q$ such that for any scheme of dimension $\le d$ and $k \ge N$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to
{\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S\left< 0,k \right> \wedge M_A)(X)$$ is an isomorphism. The assertion then follows by an induction argument on $i$ showing that ${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left< q,q+i \right> \wedge M_A)=0$.

Generalizing the argument given in the last proof we get

\[dh5t43\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have $${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \wedge M_A) \cong \lim_k {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A)
\cong {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<\max(0,q),n \right> \wedge M_A)$$ for $n \ge p-q$ or $n \ge p-2q$.

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{K}^M_{-n} \otimes A$, where $\underline{K}^M_{-n}$ is the $(-n)$-th Milnor-$K$-theory sheaf defined via the degree $(-n,-n)$-motivic cohomology.

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{L}_n \otimes A$, where the latter sheaf is the constant sheaf on $L_n \otimes A$.

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong {{\mathcal O}}^* \otimes L_{n+1} \otimes A$.

By Lemma \[dh5t43\] we have $${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong
{\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A).$$ The long exact sequence of sheaves associated to the exact triangle $$s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A \to s_n {\mathsf{MGL}}_S \wedge M_A
\to s_{n+1} {\mathsf{MGL}}_S \wedge M_A [1]$$ together with $${\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A [-1])= {\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A)=0$$ and $${\underline{\pi}}_{0,0}(\Sigma^{1,1} {\mathsf{M}A}_S) \cong {{\mathcal O}}^* \otimes A$$ gives the result.

Similarly we get

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequence $$\underline{K}^M_{1-n} \otimes A \to {\underline{\pi}}_{n+1,n}({\mathsf{MGL}}_S \wedge M_A) \to {{\mathcal H}}_{\mathrm{mot}}^{-n-1,-n}(-,A) \to 0,$$ where the latter group denotes the motivic cohomology sheaf in degees $(-n-1,-n)$ and $A$-coefficients.

We also have

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequences $$H^{3,2}(S) \otimes A \otimes L_{n+2} \to \pi_{2n+1,n} {\mathsf{MGL}}_S \to H^{1,1}(S,A) \otimes L_{n+1} \to 0.$$ If $A$ is torsionfree the first map is also injective.

Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $X$ be an essentially smooth scheme over $S$. Then for any $R$-module $A$, $n \in {{\mathbb Z}}$ and $i \ge 2$ we have ${\mathsf{MGL}}_S^{2n+i,n}(X,A)=0$.

This follows from the above considerations and the fact that for $X \in {\mathrm{Sm}}_S$ we have $H^{p,q}(X)=0$ for $p \ge 2q+2$, since motivic cohomology is computed as hypercohomology over $S$ of the Bloch-Levine cycle complexes.

We leave assertions about the groups $\pi_{2n-1,n} {\mathsf{MGL}}_S \wedge M_A$, $\pi_{n,n} {\mathsf{MGL}}_S \wedge M_A$ and $\pi_{n-1,n} {\mathsf{MGL}}_S \wedge M_A$ to the interested reader.

Fakult[ä]{}t f[ü]{}r Mathematik, Universit[ä]{}t Osnabrück, Germany.\
e-mail: markus.spitzweck@uni-osnabrueck.de
