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Motivic and real étale stable homotopy theory

Published online by Cambridge University Press:  20 March 2018

Tom Bachmann*
Fakultät Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany email
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Let $S$ be a Noetherian scheme of finite dimension and denote by $\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\in {\mathcal{O}}^{\times }(S)$. Here $\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\unicode[STIX]{x1D70C}$ in $\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\text{r}\acute{\text{e}}\text{t}}$, by which we mean the small real étale site of $S$, comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the $\unicode[STIX]{x1D70C}$-local sphere (over $\mathbb{R}$). As further applications we show that $D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$ (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$ for $i=1,2$ and establish some new rigidity results.

Research Article
© The Author 2018 

1 Introduction

For a scheme $S$ we denote by $\mathbf{SH}(S)$ the motivic stable homotopy category [Reference Morel and VoevodskyMV99, Reference AyoubAyo07]. We recall that this is a triangulated category which is the homotopy category of a stable model category that (roughly) is obtained from the homotopy theory of (smooth, pointed) schemes by making the ‘Riemann sphere’ $\mathbb{P}_{S}^{1}$ into an invertible object.

If $\unicode[STIX]{x1D6FC}:k{\hookrightarrow}\mathbb{C}$ is an embedding of a field $k$ into the complex numbers, then we obtain a complex realisation functor $R_{\unicode[STIX]{x1D6FC},\mathbb{C}}:\mathbf{SH}(k)\rightarrow \mathbf{SH}$ (where now $\mathbf{SH}$ denotes the classical stable homotopy category) connecting the world of motivic stable homotopy theory to classical stable homotopy theory [Reference Morel and VoevodskyMV99, § 3.3.2]. This functor is induced from the functor which sends a smooth scheme $S$ over $k$ to its topological space of complex points $S(\mathbb{C})$ (this depends on $\unicode[STIX]{x1D6FC}$ ). Similarly if $\unicode[STIX]{x1D6FD}:k{\hookrightarrow}\mathbb{R}$ is an embedding into the real numbers, then there is a real realisation functor $R_{\unicode[STIX]{x1D6FD},\mathbb{R}}:\mathbf{SH}(k)\rightarrow \mathbf{SH}$ induced from $S\mapsto S(\mathbb{R})$ [Reference Morel and VoevodskyMV99, § 3.3.3] [Reference Heller and OrmsbyHO16, Proposition 4.8].

These functors serve as a good source of inspiration and a convenient test of conjectures in stable motivic homotopy theory. For example, in order for a morphism $f:E\rightarrow F$ to be an equivalence it is necessary that $R_{\unicode[STIX]{x1D6FC},\mathbb{C}}(f)$ and $R_{\unicode[STIX]{x1D6FD},\mathbb{R}}(f)$ are equivalences, for all such embeddings $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ . On the other hand, this criterion is clearly not sufficient; there are fields without any real or complex embeddings!

It is thus a very natural question to ask how far these functors are from being an equivalence, or what their ‘kernel’ is. The aim of this article is to give some kind of complete answer to this question in the case of real realisation. We begin with the simplest formulation of our result. Write $R_{\mathbb{R}}$ for the (unique) real realisation functor for the field $k=\mathbb{R}$ . The first clue comes from the observation that $R_{\mathbb{R}}(\mathbb{G}_{m})=\mathbb{R}\setminus 0\simeq \{\pm 1\}=S^{0}$ . That is to say $R_{\mathbb{R}}$ identifies $\mathbb{G}_{m}$ and $S^{0}$ . We can even do better. Write $\unicode[STIX]{x1D70C}^{\prime }:S^{0}\rightarrow \mathbb{G}_{m}$ for the map of pointed motivic spaces corresponding to $-1\in \mathbb{R}^{\times }$ . Then one may check easily that $R_{\mathbb{R}}(\unicode[STIX]{x1D70C}^{\prime })$ is an equivalence between $S^{0}\simeq R_{\mathbb{R}}(S^{0})$ and $R_{\mathbb{R}}(\mathbb{G}_{m})$ .

We prove that $\mathbf{SH}(\mathbb{R})[{\unicode[STIX]{x1D70C}^{\prime }}^{-1}]\simeq \mathbf{SH}$ via real realisation. That is to say $R_{\mathbb{R}}$ is in some sense the universal functor turning $\unicode[STIX]{x1D70C}^{\prime }$ into an equivalence. More precisely, the functor $R_{\mathbb{R}}:\mathbf{SH}(\mathbb{R})\rightarrow \mathbf{SH}$ has a right adjoint $R^{\ast }$ (e.g. by Neeman’s version of Brown representability) and we show that $R^{\ast }$ is fully faithful with image consisting of the $\unicode[STIX]{x1D70C}^{\prime }$ -stable motivic spectra, i.e. those $E\in \mathbf{SH}(\mathbb{R})$ such that $E(X\wedge \mathbb{G}_{m})\xrightarrow[{}]{\unicode[STIX]{x1D70C}^{\ast }}E(X)$ is an equivalence for all $X\in Sm(\mathbb{R})$ .

Of course, our description of $\mathbf{SH}(\mathbb{R})[{\unicode[STIX]{x1D70C}^{\prime }}^{-1}]$ is just an explicit description of a certain Bousfield localisation of $\mathbf{SH}(\mathbb{R})$ . Moreover the element $\unicode[STIX]{x1D70C}^{\prime }$ exists not only over $\mathbb{R}$ but already over $\mathbb{Z}$ , so we are lead to study more generally the category $\mathbf{SH}(S)[{\unicode[STIX]{x1D70C}^{\prime }}^{-1}]$ , for more or less arbitrary base schemes $S$ . Actually, for some formulas it is nicer to consider $\unicode[STIX]{x1D70C}:=-\unicode[STIX]{x1D70C}^{\prime }\in [S,\unicode[STIX]{x1D6F4}^{\infty }\mathbb{G}_{m}]$ and we shall write this from now on. Of course $\mathbf{SH}(S)[{\unicode[STIX]{x1D70C}^{\prime }}^{-1}]=\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]$ . In this generality we can no longer expect that $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]\simeq \mathbf{SH}$ . Indeed as we have said before in general there is no real realisation! As a first attempt, one might guess that if $X$ is a scheme over $\mathbb{R}$ , then $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]\simeq \mathbf{SH}(S(\mathbb{R}))$ , where the right-hand side denotes some form of parametrised homotopy theory [Reference May and SigurdssonMS06]. This cannot be quite true unless $S$ is proper, because the category $\mathbf{SH}(S(\mathbb{R}))$ will then not be compactly generated. The way out is to use semi-algebraic topology. For this we have to recall that if $S$ is a scheme, then there exists a topological space $R(S)$ [Reference ScheidererSch94, (0.4.2)]. Its points are pairs $(x,\unicode[STIX]{x1D6FC})$ with $x\in S$ and $\unicode[STIX]{x1D6FC}$ an ordering of the residue field $k(x)$ . This is given a topology incorporating all of these orderings. Write $\operatorname{Shv}(RS)$ for the category of sheaves on this topological space.

Now, given any topos ${\mathcal{X}}$ , there is a naturally associated stable homotopy category $\text{SH}({\mathcal{X}})$ . If ${\mathcal{X}}\simeq \operatorname{Set}$ then $\text{SH}({\mathcal{X}})$ is just the ordinary stable homotopy category. In general, if ${\mathcal{X}}\simeq \operatorname{Shv}({\mathcal{C}})$ where ${\mathcal{C}}$ is a Grothendieck site, then $\text{SH}(X)$ is the local homotopy category of presheaves of spectra on ${\mathcal{C}}$ .

With this preparation out of the way, we can state our main result as follows.

Theorem (See Theorem 35). Let $S$ be a Noetherian scheme of finite dimension. Then there is a canonical equivalence of categories

$$\begin{eqnarray}\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]\simeq \text{SH}(\operatorname{Shv}(RS)).\end{eqnarray}$$

A more detailed formulation is given later in this introduction. For now let us mention one application. We go back to $S=\operatorname{Spec}(\mathbb{R})$ . In this case Proposition 36 in § 10 assures us that the equivalence from the above theorem does indeed come from real realisation. But given $E\in \mathbf{SH}(\mathbb{R})$ , its $\unicode[STIX]{x1D70C}$ -localisation can be calculated quite explicitly (see Lemma 15). From this one concludes that $\unicode[STIX]{x1D70B}_{i}(R_{\mathbb{R}}E)=\operatorname{colim}_{n}\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{n}(\mathbb{R})$ , where the colimit is along multiplication by $\unicode[STIX]{x1D70C}$ in the second grading of the bigraded homotopy sheaves of $E$ . (Recall that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{n}(\mathbb{R})=[\unicode[STIX]{x1D7D9}[i],E\wedge \mathbb{G}_{m}^{\wedge n}]$ so $\unicode[STIX]{x1D70C}$ indeed induces $\unicode[STIX]{x1D70C}:\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{n}(\mathbb{R})\rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{n+1}(\mathbb{R})$ .)

This may seem slightly esoteric, but actually $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1},2^{-1}]=\mathbf{SH}(S)[\unicode[STIX]{x1D702}^{-1},2^{-1}]$ and so our computations apply, after inverting two, to the more conventional $\unicode[STIX]{x1D702}$ -localisation as well. As a corollary, we obtain the following.

Theorem. The motivic stable 2-local, $\unicode[STIX]{x1D702}$ -local stems over $\mathbb{R}$ agree with the classical stable 2-local stems:

$$\begin{eqnarray}\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}_{\unicode[STIX]{x1D702},2})_{j}(\mathbb{R})=\unicode[STIX]{x1D70B}_{i}^{s}\otimes _{\mathbb{ Z}}\mathbb{Z}[1/2].\end{eqnarray}$$

Some more applications will be described later in this introduction.

Overview of the proof. The proof uses a different description of the category $\operatorname{Shv}(RS)$ . Namely, there is a topology on all schemes called the real étale topology and abbreviated rét-topology [Reference ScheidererSch94, (1.2)]. (The covers are families of étale morphisms which induce a jointly surjective family on the associated real spaces $R(\bullet )$ .) We write $Sm(S)_{\text{r}\acute{\text{e}}\text{t}}$ for the site of all smooth schemes over $S$ with this topology, and $S_{\text{r}\acute{\text{e}}\text{t}}$ for the site of all étale schemes over $S$ with this topology. Then $\operatorname{Shv}(S_{\text{r}\acute{\text{e}}\text{t}})\simeq \operatorname{Shv}(RS)$ [Reference ScheidererSch94, Theorem (1.3)].

Write $\mathbf{SH}(S)$ for the motivic stable homotopy category, $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]$ for the $\unicode[STIX]{x1D70C}$ -local motivic stable homotopy category, $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}$ for the rét-local motivic stable homotopy category (i.e. the category obtained from the site $Sm(S)_{\text{r}\acute{\text{e}}\text{t}}$ by precisely the same construction as is used to build $\mathbf{SH}(S)$ from $Sm(S)_{\operatorname{Nis}}$ ), and $\mathbf{SH}^{S^{1}}(S)$ for the motivic $S^{1}$ -stable homotopy category. We trust that $\mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}$ , $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ and so on have evident meanings. Write $\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})$ for the rét-local stable homotopy category on the small real étale site. This is just the homotopy category of the category of presheaves of spectra on $S_{\text{r}\acute{\text{e}}\text{t}}$ with the local model structure. Similarly $\text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})$ means the rét-local presheaves of spectra on $Sm(S)$ . Then for example $\mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}$ is the $\mathbb{A}^{1}$ -localisation of $\text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})$ .

The canonical functor $e:\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})$ (extending a (pre)sheaf on the small site to the large site) is fully faithful by general results (see Corollary 6). It is moreover $t$ -exact: for $E\in \text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})$ we have $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(eE)=e\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)$ . Here $\text{}\underline{\unicode[STIX]{x1D70B}}_{\ast }$ denotes the homotopy sheaves.

If $F$ is a sheaf on the small real étale site of a scheme $Y$ , then $H^{p}(Y\times \mathbb{A}^{1},F)=H^{p}(Y,F)$ and $H^{p}(Y_{+}\wedge \mathbb{G}_{m},F)=H^{p}(Y,F)$ . If $Y$ is of finite type over $\mathbb{R}$ and $F$ is locally constant, then this follows by comparison of real étale cohomology with Betti cohomology of the real points [Reference DelfsDel91, Theorem II.5.7]. For the general case, see Theorem 8.

Now the category $\mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ is obtained from $\text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})$ by $(\mathbb{A}^{1},\unicode[STIX]{x1D70C})$ -localisation. It follows from $t$ -exactness of $e$ , the descent spectral sequence, and the above result about rét-cohomology that the composite $\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ is still fully faithful.

The category $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ is obtained from $\mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ by $\otimes$ -inverting $\mathbb{G}_{m}$ . However in the latter category we have $\mathbb{G}_{m}\simeq \unicode[STIX]{x1D7D9}$ (via $\unicode[STIX]{x1D70C}$ !), so $\mathbb{G}_{m}$ is already invertible, and inverting it has no effect: $\mathbf{SH}^{S^{1}}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]\simeq \mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ . We have thus shown that

$$\begin{eqnarray}\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]\end{eqnarray}$$

is fully faithful.

The next step is to show that it is essentially surjective. This follows from the proper base change theorem by a clever argument of Cisinski–Déglise. Of course this first requires that we know that $\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})$ and $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ satisfy proper base change. For $\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}})$ this is a consequence of the proper base change theorem in real étale cohomology established by Scheiderer, see Theorem 9. For $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ this would follow from the axiomatic six functors formalism of Voevodsky/Ayoub/Cisinski–Déglise, see § 5. It is in fact not very hard to show directly that $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ satisfies the six functors formalism. Instead we shall show (without assuming the six functors formalism) that $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]\simeq \mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]$ , and that this latter category satisfies the six functors formalism.

The next step is thus to show that the localisation functor $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]\rightarrow \mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ is an equivalence. It clearly has dense image, so it suffices to show that it is fully faithful. Using the fact that $\mathbf{SH}(S)[\unicode[STIX]{x1D70C}^{-1}]$ satisfies continuity and gluing (which follows quite easily from the same statement for $\mathbf{SH}(S)$ ), we may reduce to the case where $S$ is the spectrum of a field $k$ . The case where $\text{char}(k)>0$ is easily dealt with (note that such fields are never orderable), so we may assume that $k$ has characteristic zero and so in particular is perfect.

The $\unicode[STIX]{x1D70C}$ -localisation can be described rather explicitly. For $E\in \mathbf{SH}(k)$ , consider the directed system

$$\begin{eqnarray}E\xrightarrow[{}]{\unicode[STIX]{x1D70C}}E\wedge \mathbb{G}_{m}\xrightarrow[{}]{\unicode[STIX]{x1D70C}}E\wedge \mathbb{G}_{m}\wedge \mathbb{G}_{m}\xrightarrow[{}]{\unicode[STIX]{x1D70C}}\cdots \,.\end{eqnarray}$$

Then $\operatorname{hocolim}_{n}E\wedge \mathbb{G}_{m}^{\wedge n}$ is a model for the $\unicode[STIX]{x1D70C}$ -localisation $E[\unicode[STIX]{x1D70C}^{-1}]$ of $E$ (see Lemma 15). It follows that its homotopy sheaves are given by

$$\begin{eqnarray}\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E[\unicode[STIX]{x1D70C}^{-1}])=\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{\ast }[\unicode[STIX]{x1D70C}^{-1}]=:\operatorname{colim}_{n}\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{n}.\end{eqnarray}$$

Here the colimit is along multiplication by $\unicode[STIX]{x1D70C}$ . (Let us remark here that the homotopy sheaves in $\mathbf{SH}(k)$ are bigraded, and so, technically, are those in $\mathbf{SH}(k)[\unicode[STIX]{x1D70C}^{-1}]$ . However inverting $\unicode[STIX]{x1D70C}$ means that up to canonical isomorphism, the homotopy sheaf is independent of the second index, so we suppress it.) It then follows from the descent spectral sequence that in order to prove that the functor $\mathbf{SH}(k)[\unicode[STIX]{x1D70C}^{-1}]\rightarrow \mathbf{SH}(k)^{\text{r}\acute{\text{e}}\text{t}}[\unicode[STIX]{x1D70C}^{-1}]$ is an equivalence, it is enough to prove that if $F_{\ast }$ is a homotopy module (element in the heart of $\mathbf{SH}(k)$ ) such that $\unicode[STIX]{x1D70C}:F_{n}\rightarrow F_{n+1}$ is an isomorphism for all $n$ (we call such a homotopy module $\unicode[STIX]{x1D70C}$ -stable), then $H_{\text{r}\acute{\text{e}}\text{t}}^{n}(X,F_{\ast })=H_{\operatorname{Nis}}^{n}(X,F_{\ast })$ for all $X$ smooth over $k$ . In particular, we need to show that $F_{\ast }$ is a sheaf in the real étale topology. This is actually sufficient, because Nisnevich, Zariski and real étale cohomology of real étale sheaves all agree [Reference ScheidererSch94, Proposition 19.2.1].

This ties in with work of Jacobson and Scheiderer. Recall that $\text{}\underline{\unicode[STIX]{x1D70B}}_{0}(\unicode[STIX]{x1D7D9})_{\ast }=\text{}\underline{K}_{\ast }^{MW}$ , i.e. the zeroth stable motivic homotopy sheaf is unramified Milnor–Witt $K$ -theory. A theorem of Jacobson [Reference JacobsonJac17] together with work of Morel implies that $\text{}\underline{K}_{\ast }^{MW}[\unicode[STIX]{x1D70C}^{-1}]=\operatorname{colim}_{n}\text{}\underline{I}^{n}=a_{\text{r}\acute{\text{e}}\text{t}}\text{}\underline{\mathbb{Z}}$ ; here $\text{}\underline{I}$ is the sheaf of fundamental ideals. Finally if $F_{\ast }$ is a general $\unicode[STIX]{x1D70C}$ -stable homotopy module, we use properties of transfers for homotopy modules together with the structure of $F_{\ast }$ as a module over $\text{}\underline{K}_{\ast }^{MW}[\unicode[STIX]{x1D70C}^{-1}]=a_{\text{r}\acute{\text{e}}\text{t}}\text{}\underline{\mathbb{Z}}$ to show that $F_{\ast }$ is a sheaf in the real étale topology. This concludes the overview of the proof.

Throughout the article we actually establish all our results for both the stable motivic homotopy category $\mathbf{SH}(S)$ and the stable $\mathbb{A}^{1}$ -derived category $D_{\mathbb{A}^{1}}(S)$ . The proofs in the latter case are essentially always the same as in the former, so we do not tend to give them. (In fact in some cases proofs just for the latter category would be simpler.)

Overview of the article. In § 2 we recall some results from local homotopy theory, including the existence and basic properties of the homotopy $t$ -structure, a general compact generation criterion and a fully faithfulness result.

In § 3 we recall the real étale topology and establish some supplements.

In § 4 we recall some results about motivic stable homotopy categories and transfers for finite étale morphisms. In particular we establish the base change and projection formulas for these.

In § 5 we recall the formalism of pre-motivic and motivic categories and how it can be used to establish that a category satisfies the six functors formalism.

In § 6 we carefully prove some basic facts about monoidal Bousfield localisation.

We judge these five sections as preliminary and the results as not very original. The ‘real work’ is contained in the next three sections. In § 7 we review Jacobson’s theorem on the colimit of the powers of the sheaf of fundamental ideals and use it together with our results on transfers to prove that $\unicode[STIX]{x1D70C}$ -stable homotopy modules are sheaves in the real étale topology.

Section 8 contains various preliminary observations and reductions.

Finally in § 9 we carry out the proof as outlined above.

The remaining three sections contain some applications. In § 10 we show that our functor $\mathbf{SH}(\mathbb{R})\rightarrow \mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]\simeq \text{SH}(\operatorname{Spec}(\mathbb{R})_{\text{r}\acute{\text{e}}\text{t}})\simeq \mathbf{SH}$ coincides with the real realisation functor. It follows that the $\unicode[STIX]{x1D70C}$ -inverted stable homotopy sheaves of $E\in \mathbf{SH}(\mathbb{R})$ are just the stable homotopy groups of its real realisation.

In § 11 we collect some consequences for the $\unicode[STIX]{x1D702}$ -inverted sphere. We use that $\unicode[STIX]{x1D7D9}[1/2,1/\unicode[STIX]{x1D70C}]\simeq \unicode[STIX]{x1D7D9}[1/2,1/\unicode[STIX]{x1D702}]$ . Since the classical stable stems $\unicode[STIX]{x1D70B}_{i}^{s}=\mathbb{Z}/2$ for $i=1,2$ are $2$ -torsion, it follows that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/2,1/\unicode[STIX]{x1D702}])(\mathbb{R})=0$ for $i=1,2$ . Since the $\unicode[STIX]{x1D70C}$ -local homotopy sheaves are unramified sheaves in the real étale topology, this (more or less) implies that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/2,1/\unicode[STIX]{x1D702}])=0$ for $i=1,2$ . This reproves a result of Röndigs [Reference RöndigsRön16].

A different but related question is to determine rational motivic stable homotopy theory. By a recent result of Ananyevskiy et al. [Reference Ananyevskiy, Levine and PaninALP17] we have $\mathbf{SH}(k)_{\mathbb{Q}}^{-}\simeq \mathbf{DM}_{W}(k,\mathbb{Q})$ , where the right-hand side denotes a category of rational Witt-motives. Our results show easily that $\mathbf{DM}_{W}(k,\mathbb{Z}[1/2])\simeq D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq D(\operatorname{Spec}(k)_{\text{r}\acute{\text{e}}\text{t}},\mathbb{Z}[1/2])$ and more generally that $D_{\mathbb{A}^{1}}(k,\mathbb{Z})[1/\unicode[STIX]{x1D70C}]\simeq D(\operatorname{Spec}(k)_{\text{r}\acute{\text{e}}\text{t}})$ . By the same proof as in classical rational stable homotopy theory we have $\mathbf{SH}(k)_{\mathbb{Q}}^{-}\simeq D_{\mathbb{A}^{1}}(k,\mathbb{Q})^{-}$ , and so we consider our results as one version of an integral strengthening of the result of Ananyevskiy–Levine–Panin.

In § 12 we collect some applications to the rigidity problem. A sheaf $F$ on $Sm(k)$ is called rigid if for every essentially smooth, Henselian local scheme $X$ with closed point $x$ we have $F(X)=F(x)$ . For example, sheaves with transfers in the sense of Voevodsky which are of torsion prime to the characteristic of the perfect base field are rigid (see [Reference Suslin and VoevodskySV96, Theorem 4.4]). Our results imply that the homotopy sheaves of any $E\in \mathbf{SH}(k)[\unicode[STIX]{x1D70C}^{-1}]$ are real étale sheaves extended from the small real étale site of $k$ . One might already call this a rigidity result, but it is also not hard to see (and we show) that all such sheaves are rigid in the above sense. As an application, we show that the motivic stable homotopy sheaves $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9})_{0}[1/e]$ are all rigid, where $e$ is the exponential characteristic. This ties up a loose end of the author’s PhD thesis.

Notation. If $S$ is a scheme, we denote the motivic stable homotopy category by $\mathbf{SH}(S)$ . We denote the $S^{1}$ -stable motivic homotopy category (i.e. where $\mathbb{G}_{m}$ has not been inverted yet) by $\mathbf{SH}^{S^{1}}(S)$ . If ${\mathcal{X}}$ is a topos or site, we denote by $\text{SH}({\mathcal{X}})$ the associated stable homotopy category, see § 2. In particular $\text{SH}(S_{\text{r}\acute{\text{e}}\text{t}}),\text{SH}(Sm(S)_{\text{r}\acute{\text{e}}\text{t}})$ and $\mathbf{SH}(S)^{\text{r}\acute{\text{e}}\text{t}}$ should be carefully distinguished: the first is the stable homotopy category of the small rét-site on $S$ , the second is the stable homotopy category of the site of all smooth schemes, with the rét-topology, and the latter is the rét-localisation of the motivic stable homotopy category. This last category is $\mathbb{A}^{1}$ -local and $\mathbb{G}_{m}$ -stable, whereas the second category is neither, and these notions do not even make sense for the first category.

The classical stable homotopy category will still be denoted by $\mathbf{SH}$ .

We denote the unit of a monoidal category ${\mathcal{C}}$ by $\unicode[STIX]{x1D7D9}_{{\mathcal{C}}}$ or just by $\unicode[STIX]{x1D7D9}$ , if ${\mathcal{C}}$ is clear from the context. Thus if ${\mathcal{C}}$ is a stable homotopy category of some sort, then $\unicode[STIX]{x1D7D9}$ is the sphere spectrum.

2 Recollections on local homotopy theory

If $({\mathcal{C}},\unicode[STIX]{x1D70F})$ is a Grothendieck site, we can consider the associated category $\operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ of sheaves (a topos), the category $\operatorname{sPre}({\mathcal{C}})$ of simplicial presheaves on ${\mathcal{C}}$ , as well as the categories ${\mathcal{S}}{\mathcal{H}}({\mathcal{C}})$ of presheaves of spectra and $C({\mathcal{C}})$ of presheaves of complexes of abelian groups on ${\mathcal{C}}$ . The latter three categories carry various local model structures, in particular the injective and the projective one [Reference JardineJar15]. We denote the homotopy category of ${\mathcal{S}}{\mathcal{H}}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ by $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ and the homotopy category of $C({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ by $D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ .

It is also possible to model $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ and so on by sheaves. For this, let $\operatorname{sShv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ denote the category of sheaves of simplicial sets, and similarly let ${\mathcal{S}}{\mathcal{H}}^{\text{s}}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ be the category of sheaves of spectra, and let $C^{\text{s}}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ be the category of sheaves of chain complexes. (Here we mean sheaves in the 1-categorical sense, so this category is equivalent to the category of chain complexes of sheaves of abelian groups, and similarly for the spectra.) These also afford local model structures, and $\operatorname{Ho}(\operatorname{sShv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}}))\simeq \operatorname{Ho}(\operatorname{sPre}({\mathcal{C}}_{\unicode[STIX]{x1D70F}}))$ , and so on.

Given a functor $f^{\ast }:{\mathcal{C}}\rightarrow {\mathcal{D}}$ , there is an induced restriction functor $f_{\ast }:\operatorname{Pre}({\mathcal{D}})\rightarrow \operatorname{Pre}({\mathcal{C}})$ , where $\operatorname{Pre}({\mathcal{C}})$ denotes the category of presheaves (of sets) on ${\mathcal{C}}$ (and similarly for ${\mathcal{D}}$ ). The functor $f_{\ast }$ has a left adjoint $f^{\ast }:\operatorname{Pre}({\mathcal{C}})\rightarrow \operatorname{Pre}({\mathcal{D}})$ . It is in fact the left Kan extension of $f^{\ast }:{\mathcal{C}}\rightarrow {\mathcal{D}}$ .

If ${\mathcal{C}},{\mathcal{D}}$ are sites the functor $f^{\ast }$ is called continuous if $f_{\ast }:\operatorname{Pre}({\mathcal{D}})\rightarrow \operatorname{Pre}({\mathcal{C}})$ preserves sheaves. In this case the induced functor $f_{\ast }:\operatorname{Shv}({\mathcal{D}})\rightarrow \operatorname{Shv}({\mathcal{C}})$ has a left adjoint still denoted $f^{\ast }:\operatorname{Shv}({\mathcal{C}})\rightarrow \operatorname{Shv}({\mathcal{D}})$ . If this induced functor is left exact (commutes with finite limits) then $f$ is called a geometric morphism.

More generally, an adjunction $f^{\ast }:\operatorname{Shv}({\mathcal{C}})\leftrightarrows \operatorname{Shv}({\mathcal{D}}):f_{\ast }$ (where $f^{\ast }\vdash f_{\ast }$ does not necessarily come from a functor $f^{\ast }:{\mathcal{C}}\rightarrow {\mathcal{D}}$ ) is called a geometric morphism if $f^{\ast }$ preserves finite limits.

If $f:{\mathcal{C}}\rightarrow {\mathcal{D}}$ is any functor, then there are induced adjunctions $f^{\ast }:\operatorname{sPre}({\mathcal{C}})\leftrightarrows \operatorname{sPre}({\mathcal{D}}):f_{\ast }$ , and similarly for spectra and chain complexes. Similarly if $f^{\ast }:\operatorname{Shv}({\mathcal{C}})\leftrightarrows \operatorname{Shv}({\mathcal{D}}):f_{\ast }$ is any adjunction, then there are induced adjunctions $f^{\ast }:\operatorname{sShv}({\mathcal{C}})\leftrightarrows \operatorname{sShv}({\mathcal{D}}):f_{\ast }$ , and so on. If $f^{\ast }\vdash f_{\ast }$ is a geometric morphism in either of the above senses, then the induced adjunctions on presheaves (sheaves) of simplicial sets, spectra, and chain complexes are Quillen adjunctions in the local model structure [Reference JardineJar15, § 5.3] [Reference Cisinski and DégliseCD09, Theorem 1.18].

The above discussion allows us to prove the following useful result.

Lemma 1. Let $f^{\ast }:\operatorname{Shv}({\mathcal{C}})\leftrightarrows \operatorname{Shv}({\mathcal{D}}):f_{\ast }$ be a geometric morphism such that $f^{\ast }$ is fully faithful and $f_{\ast }$ preserves colimits. Then the induced functors

$$\begin{eqnarray}Lf^{\ast }:\text{SH}({\mathcal{C}})\rightarrow \text{SH}({\mathcal{D}})\end{eqnarray}$$


$$\begin{eqnarray}Lf^{\ast }:D({\mathcal{C}})\rightarrow D({\mathcal{D}})\end{eqnarray}$$

are fully faithful.

The same result also holds for $Lf^{\ast }:\operatorname{Ho}(\operatorname{sPre}({\mathcal{C}}))\rightarrow \operatorname{Ho}(\operatorname{sPre}({\mathcal{D}}))$ , with the same proof.

Proof. We give the proof for the derived categories, it is the same for spectra.

Since $f_{\ast }$ preserves colimits it affords a right adjoint $f^{!}$ . Then $f_{\ast }\vdash f^{!}$ is a geometric morphism in the opposite direction (note that $f_{\ast }$ preserves finite limits, and in fact all limits, since it is a right adjoint) and consequently $f_{\ast }$ is bi-Quillen. It follows that $f_{\ast }:C^{\text{s}}({\mathcal{D}})\rightarrow C^{\text{s}}({\mathcal{C}})$ preserves weak equivalences, and consequently coincides (up to weak equivalence) with its derived functor.

Now to show that $Lf^{\ast }$ is fully faithful we need to show that $Rf_{\ast }Lf^{\ast }\simeq \operatorname{id}$ . But $Rf_{\ast }\simeq f_{\ast }$ since $f_{\ast }$ is bi-Quillen. Let $E\in C^{\text{s}}({\mathcal{C}})$ be cofibrant. Then $Lf^{\ast }E\simeq f^{\ast }E$ and consequently $Rf_{\ast }Lf^{\ast }E\simeq f_{\ast }f^{\ast }E$ . Since $f^{\ast }$ is fully faithful we have $f_{\ast }f^{\ast }E\cong E$ . This concludes the proof.◻

We will also make use of $t$ -structures. We shall use homological notation for $t$ -structures [Reference LurieLur16, Definition]. Briefly, a $t$ -structure on a triangulated category ${\mathcal{C}}$ consists of two (strictly full) subcategories ${\mathcal{C}}_{{\geqslant}0}$ and ${\mathcal{C}}_{{\leqslant}0}$ , satisfying various axioms. We put ${\mathcal{C}}_{{\geqslant}n}={\mathcal{C}}_{{\geqslant}0}[n]$ and ${\mathcal{C}}_{{\leqslant}n}={\mathcal{C}}_{{\leqslant}0}[n]$ . One then has ${\mathcal{C}}_{{\geqslant}n+1}\subset {\mathcal{C}}_{{\geqslant}n}$ and ${\mathcal{C}}_{{\leqslant}n}\subset {\mathcal{C}}_{{\leqslant}n+1}$ and $[{\mathcal{C}}_{{\geqslant}n+1},{\mathcal{C}}_{{\leqslant}n}]=0$ . In fact $E\in {\mathcal{C}}_{{\geqslant}n+1}$ if and only if for all $F\in {\mathcal{C}}_{{\leqslant}n}$ we have $[E,F]=0$ , and vice versa. The inclusion ${\mathcal{C}}_{{\geqslant}n}{\hookrightarrow}{\mathcal{C}}$ has a right adjoint which we denote $E\mapsto E_{{\geqslant}n}$ , and the inclusion ${\mathcal{C}}_{{\leqslant}n}{\hookrightarrow}{\mathcal{C}}$ has a left adjoint which we denote $E\mapsto E_{{\leqslant}n}$ . The adjunctions furnish map $E_{{\geqslant}n+1}\rightarrow E\rightarrow E_{{\leqslant}n}$ and this extends to a distinguished triangle in a unique and functorial way. The intersection ${\mathcal{C}}^{\heartsuit }:={\mathcal{C}}_{{\geqslant}0}\cap {\mathcal{C}}_{{\leqslant}0}$ called the heart. It is an abelian category. We put $\unicode[STIX]{x1D70B}_{0}^{{\mathcal{C}}}(E)=(E_{{\leqslant}0})_{{\geqslant}0}\simeq (E_{{\geqslant}0})_{{\leqslant}0}\in {\mathcal{C}}^{\heartsuit }$ and $\unicode[STIX]{x1D70B}_{i}^{{\mathcal{C}}}(E)=\unicode[STIX]{x1D70B}_{0}^{{\mathcal{C}}}(E[i])$ . Then $\unicode[STIX]{x1D70B}_{\ast }^{{\mathcal{C}}}$ is a homological functor on ${\mathcal{C}}$ . The $t$ -structure is called non-degenerate if $\unicode[STIX]{x1D70B}_{i}^{{\mathcal{C}}}(E)=0$ implies that $E\simeq 0$ .

By a $t$ -category we mean a triangulated category with a fixed $t$ -structure.

Suppose that $({\mathcal{C}},\unicode[STIX]{x1D70F})$ is a site. Let for $E\in {\mathcal{S}}{\mathcal{H}}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ and $i\in \mathbb{Z}$ the sheaf $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)\in \operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ be defined as the sheaf associated with the presheaf ${\mathcal{C}}\ni X\mapsto \unicode[STIX]{x1D70B}_{i}(E(X))$ . Here we view $E$ as a presheaf of spectra. By definition, local weak equivalences of spectra induce isomorphisms on $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}$ , so $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)$ is well defined for $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ . This is a sheaf of abelian groups. Put

$$\begin{eqnarray}\displaystyle \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\geqslant}0} & = & \displaystyle \{E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}}):\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=0\text{ for }i<0\},\nonumber\\ \displaystyle \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\leqslant}0} & = & \displaystyle \{E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}}):\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=0\text{ for }i>0\}.\nonumber\end{eqnarray}$$

We define similarly for $E\in D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ the sheaf $\text{}\underline{h}_{i}(E)$ , and then the subcategories $D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\geqslant}0},D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\leqslant}0}.$

Lemma 2. If $({\mathcal{C}},\unicode[STIX]{x1D70F})$ is a Grothendieck site, then the above construction provides $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ with a non-degenerate $t$ -structure. The functor $\text{}\underline{\unicode[STIX]{x1D70B}}_{0}:\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})^{\heartsuit }\rightarrow \operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ is an equivalence of categories. Moreover let $F\in \operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})\simeq \text{SH}({\mathcal{C}})^{\heartsuit }$ . Then for $X\in {\mathcal{C}}$ there is a natural isomorphism $[\unicode[STIX]{x1D6F4}^{\infty }X_{+},F[n]]=H_{\unicode[STIX]{x1D70F}}^{n}(X,F)$ .

Similar statements hold for $D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ in place of $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ .

Proof. For derived categories, this result is classical. For $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ , the result is also fairly well known, but the author does not know an explicit reference, so we sketch a proof.

Note that there is a Quillen adjunction (in the local model structures)

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}^{\infty }:\operatorname{sPre}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{\ast }\leftrightarrows {\mathcal{S}}{\mathcal{H}}({\mathcal{C}}_{\unicode[STIX]{x1D70F}}):\unicode[STIX]{x1D6FA}^{\infty }.\end{eqnarray}$$

By direct computation using the above adjunction, we find that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D6FA}^{\infty }E)=\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)$ , for $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ and $i\geqslant 0$ .

By [Reference LurieLur16, Proposition and Remark] the category $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ admits a $t$ -structure,Footnote 1 where $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\leqslant}0}$ if and only if $\unicode[STIX]{x1D6FA}^{\infty }(E)\simeq \ast$ , and the subcategory $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\geqslant}0}$ is generated under homotopy colimits and extensions by $\unicode[STIX]{x1D6F4}^{\infty }{\mathcal{C}}_{+}$ . We first need to show that this is the $t$ -structure we want, i.e. that the positive and negative parts are determined by vanishing of homotopy sheaves. Since $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D6FA}^{\infty }E)=\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)$ , this is correct for the negative part. I claim that if $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\geqslant}0}$ , then $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=0$ for $i<0$ . If $X\in \operatorname{sPre}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{\ast }$ , then $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D6F4}^{\infty }X)=0$ for $i<0$ by direct computation. It thus remains to show that the subcategory of $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ with $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=0$ for $i<0$ is closed under homotopy colimits and extensions. For extensions this is clear. Homotopy colimits are generated by pushouts and filtered colimits [Reference LurieLur09, Propositions and], so we need only deal with cones and filtered colimits. For cones this is again clear, and for filtered colimits it holds because homotopy groups of spectra commute with filtered colimits, and hence the same is true for homotopy sheaves (see the proof of Corollary 3 for more details on this). This proves the claim. Conversely, let $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ with $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=0$ for $i<0$ . Consider the decomposition $E_{{\geqslant}0}\rightarrow E\rightarrow E_{{<}0}$ . Then $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E_{{\geqslant}0})=0$ for $i<0$ , so $0=\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)=\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E_{{<}0})$ for $i<0$ . It follows that $E_{{<}0}\simeq 0$ and so $E\simeq E_{{\geqslant}0}\in \text{SH}(E)_{{\geqslant}0}$ .

The $t$ -structure is non-degenerate because it is defined in terms of homotopy sheaves, and homotopy sheaves detect weak equivalences by definition.

We have an adjunction

$$\begin{eqnarray}M:\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})\leftrightarrows D({\mathcal{C}}_{\unicode[STIX]{x1D70F}}):U.\end{eqnarray}$$

By construction $U$ is $t$ -exact and thus $M$ is right $t$ -exact. Consider the induced adjunction

$$\begin{eqnarray}M^{\heartsuit }:\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})^{\heartsuit }\leftrightarrows D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})^{\heartsuit }:U.\end{eqnarray}$$

By direct computation using the classical Hurewicz isomorphism (and the above adjunction), $\text{}\underline{\unicode[STIX]{x1D70B}}_{0}(UME)=\text{}\underline{\unicode[STIX]{x1D70B}}_{0}(E)$ if $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})_{{\geqslant}0}$ . It follows that $UM^{\heartsuit }\simeq \operatorname{id}$ . Since $U$ is faithful by definition, from this we deduce that $M^{\heartsuit }U\simeq \operatorname{id}$ as well. Thus $\text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})^{\heartsuit }\simeq D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})^{\heartsuit }\simeq \operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ , the latter equivalence being classical. Finally if $X\in {\mathcal{C}}$ and $F\in \operatorname{Shv}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ then $[\unicode[STIX]{x1D6F4}^{\infty }X_{+},F[n]]=[\unicode[STIX]{x1D6F4}^{\infty }X_{+},UF[n]]=H_{\unicode[STIX]{x1D70F}}^{n}(X,F)$ , the first equality by definition and the second by adjunction and the same result in $D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ .◻

Corollary 3. Let $({\mathcal{C}},\unicode[STIX]{x1D70F})$ be a Grothendieck site.

  1. (1) Let $X\in {\mathcal{C}}$ . If $\unicode[STIX]{x1D70F}$ -cohomology on $X$ commutes with filtered colimits of sheaves and the $\unicode[STIX]{x1D70F}$ -cohomological dimension of $X$ is finite, then $\unicode[STIX]{x1D6F4}^{\infty }X_{+}\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ is a compact object.

  2. (2) For any collection $E_{i}\in \text{SH}({\mathcal{C}})$ and $j\in \mathbb{Z}$ we have $\text{}\underline{\unicode[STIX]{x1D70B}}_{j}(\bigoplus _{i}E_{i})=\bigoplus _{i}\text{}\underline{\unicode[STIX]{x1D70B}}_{j}(E_{i})$ .

Similarly for $D({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ .

Proof. Let us show that (1) reduces to (2). For $E\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ there is a conditionally convergent spectral sequence

$$\begin{eqnarray}H_{\unicode[STIX]{x1D70F}}^{p}(X,\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}E)\Rightarrow [X,E[p+q]].\end{eqnarray}$$

Under our assumptions on the cohomological dimension of $X$ , it converges strongly to the right-hand side. Under the assumption of commutation of cohomology with filtered colimits, by spectral sequence comparison, it thus suffices to show that for $E_{i}\in \text{SH}({\mathcal{C}}_{\unicode[STIX]{x1D70F}})$ we have $\text{}\underline{\unicode[STIX]{x1D70B}}_{n}(\bigoplus _{i}E_{i})=\bigoplus _{i}\text{}\underline{\unicode[STIX]{x1D70B}}_{n}E_{i}$ .

Now we prove (2). For $E\in {\mathcal{S}}{\mathcal{H}}({\mathcal{C}})$ write $\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(E)(X)=\unicode[STIX]{x1D70B}_{j}(E(X))$ ; this defines a presheaf of abelian groups on ${\mathcal{C}}$ . By definition $\text{}\underline{\unicode[STIX]{x1D70B}}_{j}(E)=a_{\unicode[STIX]{x1D70F}}\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(E)$ . Let $\{E_{i}\}_{i}\in {\mathcal{S}}{\mathcal{H}}({\mathcal{C}})$ . Then $\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(\bigoplus _{i}E_{i})=\bigoplus _{i}\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(E_{i})$ , since homotopy groups of spectra commute with filtered colimits. We may assume that all the $E_{i}$ are cofibrant, so their presheaf direct sum coincides with the derived direct sum. In this case it remains to show that

$$\begin{eqnarray}{a_{\unicode[STIX]{x1D70F}}\bigoplus }_{i}\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(E_{i})\cong \bigoplus _{i}a_{\unicode[STIX]{x1D70F}}\text{}\underline{\unicode[STIX]{x1D70B}}_{j}^{p}(E_{i}).\end{eqnarray}$$

(Note that here we write $\bigoplus _{i}$ for both direct sums of presheaves and direct sums of sheaves, depending on whether the terms on the right are presheaves or sheaves.) But this holds for any collection of presheaves on any site (both sides satisfy the same universal property).

The proof for $D$ is the same.◻

We can enhance the functoriality of the $\text{SH}$ construction as follows. Recall that a triangulated functor $F:{\mathcal{C}}\rightarrow {\mathcal{D}}$ between $t$ -categories is called right (respectively left) $t$ -exact if $F({\mathcal{C}}_{{\geqslant}0})\subset {\mathcal{D}}_{{\geqslant}0}$ (respectively $F({\mathcal{C}}_{{\leqslant}0})\subset {\mathcal{D}}_{{\leqslant}0}$ ). The functor is called $t$ -exact if it is both left and right $t$ -exact.

Lemma 4. Let $f^{\ast }:\operatorname{Shv}({\mathcal{C}})\leftrightarrows \operatorname{Shv}({\mathcal{D}}):f_{\ast }$ be a geometric morphism, where $\operatorname{Shv}({\mathcal{D}})$ has enough points. Then in the adjunction

$$\begin{eqnarray}Lf^{\ast }:\text{SH}({\mathcal{C}})\leftrightarrows \text{SH}({\mathcal{D}}):Rf_{\ast }\end{eqnarray}$$

the left adjoint $Lf^{\ast }$ is $t$ -exact, the right adjoint $Rf_{\ast }$ is left $t$ -exact, and the induced functors

$$\begin{eqnarray}(Lf^{\ast })^{\heartsuit }:\text{SH}({\mathcal{C}})^{\heartsuit }\leftrightarrows \text{SH}({\mathcal{D}})^{\heartsuit }:(Rf_{\ast })^{\heartsuit }\end{eqnarray}$$

coincide (under the identification from Lemma 2) with $f^{\ast }\vdash f_{\ast }$ .

Similar statements hold for $D$ in place of $\text{SH}$ .

The author contends that the assumption that ${\mathcal{D}}$ has enough points is not really necessary. See also [Reference LurieLur09, Remark].

Proof. Certainly $Rf_{\ast }$ is left $t$ -exact if $Lf^{\ast }$ is $t$ -exact by adjunction, and $(Rf_{\ast })^{\heartsuit }$ is right adjoint to $(Lf^{\ast })^{\heartsuit }$ , so it suffices to prove the claims for $Lf^{\ast }$ .

Since ${\mathcal{D}}$ has enough points, it is then enough to assume that $\operatorname{Shv}({\mathcal{D}})=\operatorname{Set}$ . (Indeed let $p:\operatorname{Set}\rightarrow \operatorname{Shv}({\mathcal{D}})$ be a point; we will have

$$\begin{eqnarray}p^{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(Lf^{\ast }E)=\unicode[STIX]{x1D70B}_{i}(Lp^{\ast }Lf^{\ast }E)=p^{\ast }f^{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{i}E\end{eqnarray}$$

for all $E\in \text{SH}({\mathcal{C}})$ by applying the reduced case to $p$ and $fp$ which are points of ${\mathcal{D}}$ and ${\mathcal{C}}$ , respectively. Since ${\mathcal{D}}$ has enough points it follows that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(Lf^{\ast }E)=f^{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)$ , as was to be shown.)

Let $p^{\ast }:\operatorname{Shv}({\mathcal{C}})\leftrightarrows \operatorname{Set}:p_{\ast }$ be a point of ${\mathcal{C}}$ . Then $p^{\ast }$ corresponds to a pro-object in ${\mathcal{C}}$ , which is to say that there is a filtered family $X_{\unicode[STIX]{x1D6FC}}\in {\mathcal{C}}$ such that for $F\in \operatorname{Shv}({\mathcal{C}})$ we have $p^{\ast }(F)=\operatorname{colim}_{\unicode[STIX]{x1D6FC}}F(X_{\unicode[STIX]{x1D6FC}})$ [Reference Gabber and KellyGK15, Proposition 1.4 and Remark 1.5].

It follows that for $E\in {\mathcal{S}}{\mathcal{H}}^{\text{s}}({\mathcal{C}})$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{i}(p^{\ast }E)=\unicode[STIX]{x1D70B}_{i}(\operatorname{colim}_{\unicode[STIX]{x1D6FC}}E(X_{\unicode[STIX]{x1D6FC}}))\cong \operatorname{colim}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70B}_{i}(E(X_{\unicode[STIX]{x1D6FC}}))=p^{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E),\end{eqnarray}$$

where the isomorphism in the middle holds because homotopy groups commute with filtered colimits of spectra. In particular $p^{\ast }$ preserves weak equivalences and so $p^{\ast }\simeq Lp^{\ast }$ . Thus the previous equation is precisely what we intended to prove.◻

3 Recollections on real étale cohomology

If $X$ is a scheme, let $R(X)$ be the set of pairs $(x,p)$ where $x\in X$ and $p$ is an ordering of the residue field $k(x)$ . For a ring $A$ we put $\operatorname{Sper}(A)=R(\operatorname{Spec}(A))$ . A family of morphisms $\{\unicode[STIX]{x1D6FC}_{i}:X_{i}\rightarrow X\}_{i\in I}$ is called a real étale covering if each $\unicode[STIX]{x1D6FC}$ is étale and $R(X)=\bigcup _{i}\unicode[STIX]{x1D6FC}(R(X_{i}))$ . (Note that for $(x,p)\in X_{i}$ the extension $k(x)/k(\unicode[STIX]{x1D6FC}(x))$ defines by restriction an ordering of $k(\unicode[STIX]{x1D6FC}(x))$ .) The real étale coverings define a topology on all schemes [Reference ScheidererSch94, (1.1)] called the real étale topology. We often abbreviate this name to ‘rét-topology’.

For a scheme $X$ , we let $X_{\text{r}\acute{\text{e}}\text{t}}$ denote the small real étale site on $X$ and $Sm(X)_{\text{r}\acute{\text{e}}\text{t}}$ the site of smooth (separated, finite type) schemes over $X$ with the real étale topology. If $f:X\rightarrow Y$ is any morphism of schemes, we get the usual base change functors $f^{\ast }:Y_{\text{r}\acute{\text{e}}\text{t}}\rightarrow X_{\text{r}\acute{\text{e}}\text{t}}$ and $f^{\ast }:Sm(Y)\rightarrow Sm(X)$ . Also the natural inclusion $e:X_{\text{r}\acute{\text{e}}\text{t}}\rightarrow Sm(X)$ induces an adjunction $e^{p}:\operatorname{Pre}(X_{\text{r}\acute{\text{e}}\text{t}})\leftrightarrows \operatorname{Pre}(Sm(X)):r=e_{\ast }$ .

Lemma 5. If $X$ is a scheme, the above adjunction induces a geometric morphism $e:\operatorname{Shv}(X_{\text{r}\acute{\text{e}}\text{t}})\leftrightarrows \operatorname{Shv}(Sm(X)_{\text{r}\acute{\text{e}}\text{t}}):r$ where $e$ is fully faithful and $r$ preserves colimits.

Proof. The functor $r$ is restriction and $e$ is left Kan extension. Since $e$ preserves covers, $r$ preserves sheaves. Moreover $r$ commutes with taking the associated sheaf, because every cover of $Y\in X_{\text{r}\acute{\text{e}}\text{t}}$ in $Sm(X)$ comes from a cover in $X_{\text{r}\acute{\text{e}}\text{t}}$ (because étale morphisms are stable under composition). It follows that $r$ commutes with colimits. Since $e:X_{\text{r}\acute{\text{e}}\text{t}}\rightarrow Sm(X)_{\text{r}\acute{\text{e}}\text{t}}$ preserves pullbacks (and $X_{\text{r}\acute{\text{e}}\text{t}}$ has pullbacks!), the adjunction is a geometric morphism [Sta17, Tag 00X6]. In order to see that $e$ is fully faithful, i.e. $F\rightarrow reF$ an isomorphism for every $F\in \operatorname{Shv}(X_{\text{r}\acute{\text{e}}\text{t}})$ , we note that for the presheaf adjunction $e^{p}:\operatorname{Pre}(X_{\text{r}\acute{\text{e}}\text{t}})\leftrightarrows \operatorname{Pre}(Sm(k)):r$ we have $re^{p}F=F$ . Indeed this holds for $F$ representable by definition, every sheaf is a colimit of representables, and $e^{p}$ and $f$ both commute with taking colimits. Finally note that for a sheaf $F$ we have $eF=a_{\text{r}\acute{\text{e}}\text{t}}e^{p}F$ and thus $reF=ra_{\text{r}\acute{\text{e}}\text{t}}e^{p}F=a_{\text{r}\acute{\text{e}}\text{t}}re^{p}F=a_{\text{r}\acute{\text{e}}\text{t}}F=F$ , where we have used again that $r$ commutes with taking the associated sheaf.◻

Corollary 6. If $X$ is a scheme, the induced derived functor $Le:\text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \text{SH}(Sm(X)_{\text{r}\acute{\text{e}}\text{t}})$ is $t$ -exact and fully faithful. Similarly for $D$ in place of $\text{SH}$ .

Proof. The functor is fully faithful by Lemmas 5 and 1. It is $t$ -exact by Lemma 4.◻

Lemma 7. If $f:X\rightarrow Y$ is a morphism of schemes, then the induced functor $f^{\ast }:Y_{\text{r}\acute{\text{e}}\text{t}}\rightarrow X_{\text{r}\acute{\text{e}}\text{t}}$ is the left adjoint of a geometric morphism of sites. Moreover the derived functor

$$\begin{eqnarray}Lf^{\ast }:\text{SH}(Y_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})\end{eqnarray}$$

is $t$ -exact, and similarly for $Lf^{\ast }:D(Y_{\text{r}\acute{\text{e}}\text{t}})\rightarrow D(X_{\text{r}\acute{\text{e}}\text{t}})$ .

Proof. The ‘moreover’ part follows from Lemma 4.

Since $f^{\ast }:Y_{\text{r}\acute{\text{e}}\text{t}}\rightarrow X_{\text{r}\acute{\text{e}}\text{t}}$ preserves covers $f_{\ast }:\operatorname{Pre}(X_{\text{r}\acute{\text{e}}\text{t}})\rightarrow \operatorname{Pre}(Y_{\text{r}\acute{\text{e}}\text{t}})$ preserves sheaves and the morphism is continuous. It is a geometric morphism of sites because $f^{\ast }$ preserves pullbacks [Sta17, Tag 00X6].◻

If $X$ is a scheme, there is the natural map $X\rightarrow X\times \mathbb{A}^{1}$ corresponding to the point $0\in \mathbb{A}^{1}$ . Similarly there is the natural map $X\coprod X\rightarrow X\times (\mathbb{A}^{1}\setminus 0)$ corresponding to the points $\pm 1\in \mathbb{A}^{1}\setminus 0$ .

Theorem 8. Let $X$ be a scheme and $F\in \operatorname{Shv}(X_{\text{r}\acute{\text{e}}\text{t}})$ . Then for any $p\geqslant 0$ the natural maps $X\rightarrow X\times \mathbb{A}^{1}$ and $X\coprod X\rightarrow X\times (\mathbb{A}^{1}\setminus 0)$ induce isomorphisms

$$\begin{eqnarray}\displaystyle & H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X\times \mathbb{A}^{1},F)\rightarrow H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X,F) & \displaystyle \nonumber\\ \displaystyle & H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X\times (\mathbb{A}^{1}\setminus 0),F)\rightarrow H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X,F)\oplus H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X,F). & \displaystyle \nonumber\end{eqnarray}$$

Proof. The first statement is homotopy invariance, see [Reference ScheidererSch94, Example 16.7.2].

For the second statement, we follow closely that proof. Let $f:X\coprod X\rightarrow X\times (\mathbb{A}^{1}\setminus 0)$ be the canonical map. It suffices to show that $R^{n}f_{\ast }F=0$ for $n>0$ and $R^{0}f_{\ast }F=F$ , where we identify $F$ with its pullback to $X\coprod X$ and $X\times (\mathbb{A}^{1}\setminus 0)$ for notational convenience. All of these statements are local on $X$ , so we may assume that $X$ is affine.

Then one may assume that $F$ is constructible (since $\text{r}\acute{\text{e}}\text{t}$ -cohomology commutes with filtered colimits of sheaves, and all sheaves on a spectral space are filtered colimits of constructible sheaves; see again [Reference ScheidererSch94, Example 16.7.2]). Next, writing $X=\operatorname{Spec}(A)$ as the inverse limit of the filtering system $\operatorname{Spec}(A^{\prime })$ , with $A^{\prime }\subset A$ finitely generated over $\mathbb{Z}$ , and using Proposition (A.9) of [Reference ScheidererSch94, Example 16.7.2], we may assume that $X$ is of finite type over $\mathbb{Z}$ .

But $\operatorname{Sper}(\mathbb{Z})=\operatorname{Sper}(\mathbb{Q})=\operatorname{Sper}(\mathbb{R})$ , whence $H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X,F)=H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X\times _{\mathbb{Z}}\mathbb{R},F)$ , so we may assume that $X$ is of finite type over $\mathbb{R}$ .

We may further assume that $F=M_{Z}$ is the constant sheaf on a closed, constructible subset of $X$ (Proposition (A.6) of [Reference ScheidererSch94, Example 16.7.2]).

It is thus enough to prove the analog of our result for an affine semi-algebraic space $X$ over $\mathbb{R}$ and $F=M$ a constant sheaf. But then $H_{\text{r}\acute{\text{e}}\text{t}}^{\ast }(X,M)=H_{\text{sing}}^{\ast }(X(\mathbb{R}),M)$ [Reference DelfsDel91, Theorem II.5.7] and so on, so this is obvious.◻

Theorem 9 (Proper base change).

Consider a cartesian square of schemes

with $f$ proper and $Y$ finite-dimensional Noetherian. Then for any $E\in \text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})$ (respectively $E\in D(X_{\text{r}\acute{\text{e}}\text{t}})$ ) the canonical map

$$\begin{eqnarray}g^{\ast }Rf_{\ast }(E)\rightarrow Rf_{\ast }^{\prime }{g^{\prime }}^{\ast }(E)\end{eqnarray}$$

is a weak equivalence.

Proof. We prove the claim for $\text{SH}$ , the proof we give will work just as well for $D$ . We proceed in several steps.

Step 0. If $g$ is étale, then the claim follows from the observation that $f^{\ast }g_{\#}=g_{\#}^{\prime }{f^{\prime }}^{\ast }$ .

Step 1. If $f:X\rightarrow Y$ is any morphism and $E\in \text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})$ , then there is a conditionally convergent spectral sequence

$$\begin{eqnarray}E_{2}^{pq}=R^{p}f_{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}E\Rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{-p-q}(Rf_{\ast }E).\end{eqnarray}$$

For this, let $E\in \operatorname{Spt}(X_{\text{r}\acute{\text{e}}\text{t}})$ also denote a fibrant model. Then $Rf_{\ast }E\simeq f_{\ast }E$ and for $U\in Y_{\text{r}\acute{\text{e}}\text{t}}$ we have $f_{\ast }(E)(U)=E(f^{\ast }U)$ . Since $E$ is fibrant there is a conditionally convergent descent spectral sequence

$$\begin{eqnarray}H^{p}(f^{\ast }U,\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}(E))\Rightarrow \unicode[STIX]{x1D70B}_{-p-q}(E(f^{\ast }U)).\end{eqnarray}$$

By varying $U$ , this yields a presheaf of spectral sequences on $Y_{\text{r}\acute{\text{e}}\text{t}}$ . Equivalently, this is a spectral sequence of presheaves. Taking the associated sheaf on both sides we obtain a conditionally convergent spectral sequence

$$\begin{eqnarray}a_{\text{r}\acute{\text{e}}\text{t}}H_{\text{r}\acute{\text{e}}\text{t}}^{p}(f^{\ast }\bullet ,\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}(E))\Rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{-p-q}(f_{\ast }E).\end{eqnarray}$$

It remains to see that $a_{\text{r}\acute{\text{e}}\text{t}}H_{\text{r}\acute{\text{e}}\text{t}}^{p}(f^{\ast }\bullet ,F)=R^{p}f_{\ast }F$ , for any sheaf $F$ on $X_{\text{r}\acute{\text{e}}\text{t}}$ . For this we view $F\in D(X_{\text{r}\acute{\text{e}}\text{t}})^{\heartsuit }$ . Then by definition $R^{p}f_{\ast }F=\text{}\underline{\unicode[STIX]{x1D70B}}_{-p}Rf_{\ast }F$ . Repeating the above argument with $D(X_{\text{r}\acute{\text{e}}\text{t}})$ in place of $\text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})$ , we obtain a conditionally convergent spectral sequence

$$\begin{eqnarray}a_{\text{r}\acute{\text{e}}\text{t}}H_{\text{r}\acute{\text{e}}\text{t}}^{p}(f^{\ast }\bullet ,\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}F)\Rightarrow R^{p+q}f_{\ast }F.\end{eqnarray}$$

Since $\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}F=0$ for $q\neq 0$ this spectral sequence converges strongly, yielding the desired identification.

Step 2. If $f$ is proper and of relative dimension at most $n$ , then for $F\in \operatorname{Shv}(X_{\text{r}\acute{\text{e}}\text{t}})$ and $p>n$ we have $R^{p}f_{\ast }F=0$ .

Indeed in this situation, by the proper base change theorem in real étale cohomology [Reference ScheidererSch94, Theorem 16.2], for any real closed point $y\rightarrow Y$ we get $(R^{p}f_{\ast }F)_{y}=H_{\text{r}\acute{\text{e}}\text{t}}^{p}(X_{y},F|_{X_{y}})$ . Since real closed fields are the stalks of the rét-topology, in order for a sheaf $G\in \operatorname{Shv}(Y_{\text{r}\acute{\text{e}}\text{t}})$ to be zero it is necessary and sufficient that $G_{y}=0$ for all such $y$ . But real étale cohomological dimension is bounded by Krull dimension [Reference ScheidererSch94, Theorem 7.6], so we find that $R^{p}f_{\ast }F=0$ for $p>n$ , as claimed.

Conclusion of proof. Since isomorphism in $\text{SH}(Y_{\text{r}\acute{\text{e}}\text{t}}^{\prime })$ is local on $Y^{\prime }$ , it is an easy consequence of step 0 that we may assume that $Y^{\prime }$ is quasi-compact (e.g. affine). Then $f^{\prime }$ is of bounded relative dimension (being of finite type).

Now let $E\in \text{SH}(X_{\text{r}\acute{\text{e}}\text{t}})$ . By $t$ -exactness of $g^{\ast }$ and ${g^{\prime }}^{\ast }$ we get from step 1 conditionally convergent spectral sequences

$$\begin{eqnarray}g^{\ast }R^{p}f_{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}E\Rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{-p-q}(g^{\ast }Rf_{\ast }E)\end{eqnarray}$$


$$\begin{eqnarray}R^{p}f_{\ast }^{\prime }{g^{\prime }}^{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}E\Rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{-p-q}(Rf_{\ast }^{\prime }{g^{\prime }}^{\ast }E).\end{eqnarray}$$

The exchange transformation $g^{\ast }Rf_{\ast }(E)\rightarrow Rf_{\ast }^{\prime }{g^{\prime }}^{\ast }(E)$ induces a morphism of spectral sequences (i.e. respecting the differentials and filtrations). By proper base change for sheaves, we have $g^{\ast }R^{p}f_{\ast }\cong R^{p}f_{\ast }^{\prime }{g^{\prime }}^{\ast }$ . Thus the two spectral sequences are isomorphic. By step 2 the second one converges strongly, and hence so does the first. Thus the result follows from spectral sequence comparison.◻


The only place in the above proof where we have used the assumption on $Y$ is in step 1, namely in the construction of the conditionally convergent spectral sequence

$$\begin{eqnarray}R^{p}f_{\ast }\text{}\underline{\unicode[STIX]{x1D70B}}_{-q}E\Rightarrow \text{}\underline{\unicode[STIX]{x1D70B}}_{-p-q}(Rf_{\ast }E).\end{eqnarray}$$

The author does not know how to construct such a spectral sequence in general. He nonetheless contends that the proper base change theorem should be true without assumptions on $Y$ , but perhaps a different proof is needed.


In the above proof we deduce proper base change for spectra and unbounded complexes from proper base change for bounded complexes. Since we are dealing with hypercomplete toposes, this is not tautological; see for example [Reference LurieLur09, Counterexample and Remark]. The crucial property which seems to make the proof work is encapsulated in step 2 and might be phrased as ‘a proper morphism is locally of finite relative rét-cohomological dimension’. The same is true in étale (instead of real étale) cohomology and this seems to be what the proof of proper base change for unbounded étale complexes [Reference Cisinski and DégliseCD13, Theorem 1.2.1] ultimately rests on, in the guise of [Reference Cisinski and DégliseCD13, Lemma 1.1.7]. This fails for a general proper morphism of topological spaces (consider for example an infinite product of compact positive dimensional spaces mapping to the point).

4 Recollections on motivic homotopy theory

We denote the stable motivic homotopy category over a base scheme $X$ [Reference AyoubAyo07] by $\mathbf{SH}(X)$ , and the stable $\mathbb{A}^{1}$ -derived category over $X$ [Reference Cisinski and DégliseCD12, § 5.3] by $D_{\mathbb{A}^{1}}(X)$ . We write $\unicode[STIX]{x1D7D9}_{X}\in \mathbf{SH}(X)$ for the monoidal unit. If the context is clear we may just write $\unicode[STIX]{x1D7D9}$ .

Let $f:Y\rightarrow X$ be a finite étale morphism of schemes. Then in the category $\mathbf{SH}(X)$ we have an induced morphism $f:f_{\#}\unicode[STIX]{x1D7D9}_{Y}\rightarrow \unicode[STIX]{x1D7D9}_{X}$ and consequently $D(f):D(\unicode[STIX]{x1D7D9}_{X})\rightarrow D(f_{\#}\unicode[STIX]{x1D7D9}_{Y})$ . Here $DE:=\text{}\underline{\operatorname{Hom}}(E,\unicode[STIX]{x1D7D9})$ . Now in fact whenever $f:Y\rightarrow X$ is smooth proper then $D(f_{\#}\unicode[STIX]{x1D7D9}_{Y})\simeq f_{\ast }\unicode[STIX]{x1D7D9}_{Y}$ [Reference Cisinski and DégliseCD12, Proposition 2.4.31] and if $f$ is étale then $f_{\ast }(\unicode[STIX]{x1D7D9}_{Y})\simeq f_{\#}(\unicode[STIX]{x1D7D9}_{Y})$ [Reference Cisinski and DégliseCD12, Example 2.4.3(2), Definition 2.4.24 and Proposition 2.4.31]. Let us write $\unicode[STIX]{x1D6FC}_{X,Y}:f_{\#}\unicode[STIX]{x1D7D9}_{Y}\rightarrow D(f_{\#}\unicode[STIX]{x1D7D9}_{Y})$ for this canonical isomorphism. We can then form the commutative diagram

where $\text{tr}_{f}$ is defined so that the diagram commutes. This is the duality transfer of $f$ as defined in [Reference Röndigs and ØstværRØ08, § 2.3].

Now suppose that $k$ is a perfect field. Recall that then $\mathbf{SH}(k)$ has a $t$ -structure. To define it, for $E\in \mathbf{SH}(k)$ denote by $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{j}$ the Nisnevich sheaf associated with the presheaf $X\mapsto [\unicode[STIX]{x1D6F4}^{\infty }X_{+}[i],E\wedge \mathbb{G}_{m}^{\wedge j}]$ . Then $E\in \mathbf{SH}(k)_{{\geqslant}0}$ if and only if $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(E)_{j}=0$ for all $i<0$ and all $j\in \mathbb{Z}$ . This indeed defines a $t$ -structure [Reference MorelMor03, § 5.2], and then its heart can be described explicitly: it is equivalent to the category of homotopy modules [Reference MorelMor03, Theorem 5.2.6].

Let $F_{\ast }\in \mathbf{SH}(k)$ is a homotopy module, which we identify with an element in the heart of the homotopy $t$ -structure. Given a finite étale morphism $f:Y\rightarrow X$ of essentially $k$ -smooth schemes, write $s:X\rightarrow \operatorname{Spec}(k)$ for the structure map. We then define $\text{tr}_{f}:F_{n}(Y)\rightarrow F_{n}(X)$ as

$$\begin{eqnarray}\text{tr}_{f}(F):=\text{tr}_{f}^{\ast }:[f_{\#}\unicode[STIX]{x1D7D9}_{Y},s^{\ast }F\wedge \mathbb{G}_{m}^{\wedge n}]\rightarrow [\unicode[STIX]{x1D7D9}_{X},s^{\ast }F\wedge \mathbb{G}_{m}^{\wedge n}].\end{eqnarray}$$

This transfer has the usual properties, of which we recall two.

Proposition 10 (Base change).

Let $k$ be a perfect field, $g:V\rightarrow X$ be a morphism of essentially $k$ -smooth schemes and $f:Y\rightarrow X$ finite étale. Consider the following cartesian square.

Then for any homotopy module $F_{\ast }$ , we have $g^{\ast }\text{tr}_{f}=\text{tr}_{p}q^{\ast }:F_{\ast }(Y)\rightarrow F_{\ast }(V)$ .

Proof. Note that $p:W\rightarrow V$ is finite étale, so this makes sense. By continuity (of $F$ ), we may assume that $X$ and $V$ are smooth (and hence so are $Y$ and $W$ ). Write $s:X\rightarrow \operatorname{Spec}(k)$ for the structure map.

If $t:A\rightarrow B$ is any map in $\mathbf{SH}(X)$ , then the canonical diagram

commutes, since $g^{\ast }$ is a functor. Applying this to $\text{tr}_{f}:\unicode[STIX]{x1D7D9}_{X}\rightarrow f_{\#}\unicode[STIX]{x1D7D9}_{Y}$ it is enough to prove that $g^{\ast }(\text{tr}_{f})=\text{tr}_{p}$ under the canonical identifications.

Let $f_{+}:f_{\#}\unicode[STIX]{x1D7D9}_{Y}\simeq \unicode[STIX]{x1D6F4}_{X}^{\infty }Y_{+}\rightarrow \unicode[STIX]{x1D6F4}_{X}^{\infty }X_{+}=\unicode[STIX]{x1D7D9}_{X}$ be the canonical map (so that $\text{tr}_{f}=D(f_{+})$ via $\unicode[STIX]{x1D6FC}_{X,Y}$ ), and similarly for $p_{+}$ . Then $g^{\ast }(f_{+})\simeq p_{+}$ and consequently $g^{\ast }(D(f_{+}))\simeq D(p_{+})$ . It thus remains to show that $\unicode[STIX]{x1D6FC}_{\bullet ,\bullet }$ is natural, i.e. that $g^{\ast }\unicode[STIX]{x1D6FC}_{X,Y}=\unicode[STIX]{x1D6FC}_{V,W}:\unicode[STIX]{x1D6F4}_{V}^{\infty }W_{+}\rightarrow D(\unicode[STIX]{x1D6F4}_{V}^{\infty }W_{+})$ .

For this we use the notation of [Reference Cisinski and DégliseCD12, Example 2.4.3(2), Definition 2.4.24 and Proposition 2.4.31]. The isomorphism $\unicode[STIX]{x1D6FC}_{X,Y}:f_{\#}\unicode[STIX]{x1D7D9}\rightarrow D(f_{\#}\unicode[STIX]{x1D7D9})$ is factored into the isomorphisms $D(f_{\#}\unicode[STIX]{x1D7D9})\rightarrow f_{\ast }\unicode[STIX]{x1D7D9}$ , the Thom transformation $f_{\#}\unicode[STIX]{x1D6FA}_{f}\unicode[STIX]{x1D7D9}\rightarrow f_{\ast }\unicode[STIX]{x1D7D9}$ [Reference Cisinski and DégliseCD12, Definition 2.4.21] and $\unicode[STIX]{x1D6FA}_{f}\unicode[STIX]{x1D7D9}\rightarrow \unicode[STIX]{x1D7D9}$ . All of these are natural in the required sense.◻

Lemma 11 (Commutation of transfer with external product).

Let $f:X^{\prime }\rightarrow X$ and $g:Y^{\prime }\rightarrow Y$ be finite étale. Then

$$\begin{eqnarray}\displaystyle s_{X\times Y\#}(\text{tr}_{f\times g}) & = & \displaystyle s_{X\#}(\text{tr}_{f})\wedge s_{Y\#}(\text{tr}_{g}):\unicode[STIX]{x1D6F4}^{\infty }(X^{\prime }\times Y^{\prime })_{+}\simeq \unicode[STIX]{x1D6F4}^{\infty }X_{+}^{\prime }\wedge \unicode[STIX]{x1D6F4}^{\infty }Y_{+}^{\prime }\nonumber\\ \displaystyle & \rightarrow & \displaystyle \unicode[STIX]{x1D6F4}^{\infty }(X\times Y)_{+}\simeq \unicode[STIX]{x1D6F4}^{\infty }X_{+}\wedge \unicode[STIX]{x1D6F4}^{\infty }Y_{+}.\nonumber\end{eqnarray}$$

Here we write $s_{X}:X\rightarrow \operatorname{Spec}(k)$ for the canonical map, and similarly for $Y,X\times Y$ .

Proof. Write $p_{X}:X\times Y\rightarrow X$ and $p_{Y}:X\times Y\rightarrow Y$ for the projections. I claim that the following diagram commutes up to natural isomorphism.

To prove the claim, first note that there is, for $T\in \mathbf{SH}(X),U\in \mathbf{SH}(Y)$ , a natural map $s_{X\times Y\#}(p_{X}^{\ast }T\wedge p_{Y}^{\ast }U)\rightarrow s_{X\#}T\wedge s_{y\#}U$ , which can be obtained by adjunctions, using that the pullback functors are monoidal, and that $s_{X\times Y}=s_{X}\circ p_{X}$ (and similarly for $Y$ ). Then to prove that the comparison map is an isomorphism it suffices to consider $T=\unicode[STIX]{x1D6F4}^{\infty }X^{\prime },U=\unicode[STIX]{x1D6F4}^{\infty }Y^{\prime }$ for $X^{\prime }\rightarrow X$ smooth any $Y^{\prime }\rightarrow Y$ smooth (note that all our functors are left adjoints and so commute with arbitrary sums, and objects of the forms $T,U$ are generators). But then the claim boils down to

$$\begin{eqnarray}X^{\prime }\times _{k}Y^{\prime }\cong (X^{\prime }\times Y)\times _{X\times Y}(X\times Y^{\prime })\end{eqnarray}$$

which is clear.

To prove the lemma, we now specialise to $f:X^{\prime }\rightarrow X$ and $g:Y^{\prime }\rightarrow Y$ finite étale. Then

$$\begin{eqnarray}\text{tr}_{f\times g}=s_{X\times Y\#}(D\unicode[STIX]{x1D6F4}_{X\times Y}^{\infty }(f\times g)_{+}).\end{eqnarray}$$

Note that

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{X\times Y}^{\infty }(f\times g)_{+}=p_{X}^{\ast }\unicode[STIX]{x1D6F4}_{X}^{\infty }f_{+}\wedge p_{Y}^{\ast }\unicode[STIX]{x1D6F4}_{Y}^{\infty }g_{+}.\end{eqnarray}$$

Since $p_{X}^{\ast },p_{Y}^{\ast }$ are monoidal we compute

$$\begin{eqnarray}\text{tr}_{f\times g}=s_{X\times Y\#}p_{X}^{\ast }D\unicode[STIX]{x1D6F4}_{X}^{\infty }f_{+}\wedge p_{Y}^{\ast }D\unicode[STIX]{x1D6F4}_{Y}^{\infty }g_{+}=s_{X\#}D\unicode[STIX]{x1D6F4}_{X}^{\infty }f_{+}\wedge s_{Y\#}D\unicode[STIX]{x1D6F4}_{Y}^{\infty }g_{+},\end{eqnarray}$$

where in the last equality we have used the claim. Since $s_{X\#}D\unicode[STIX]{x1D6F4}_{X}^{\infty }f_{+}=\text{tr}_{f}$ by definition (and similarly for $Y$ ), this is what we wanted to prove.◻

Recall also the homotopy module $\text{}\underline{K}_{\ast }^{MW}=\text{}\underline{\unicode[STIX]{x1D70B}}_{0}(\unicode[STIX]{x1D7D9})_{\ast }$ of Milnor–Witt K-theory [Reference MorelMor12, ch. 3]. Every homotopy module $F_{\ast }$ is a module over $\text{}\underline{K}_{\ast }^{MW}$ in the sense that there are natural pairings $\text{}\underline{K}_{\ast }^{MW}(X)\otimes F_{\ast }(X)\rightarrow F_{\ast +\ast }(X)$ .

Corollary 12 (Projection formula).

Let $k$ be a perfect field, $f:Y\rightarrow X$ a finite étale morphism of essentially $k$ -smooth schemes, and $F_{\ast }$ a homotopy module. Then for $a\in \text{}\underline{K}_{\ast }^{MW}(Y)$ and $b\in F_{\ast }(X)$ we have $\text{tr}_{f}(af^{\ast }b)=\text{tr}_{f}(a)b$ . Similarly for $a\in \text{}\underline{K}_{\ast }^{MW}(X)$ and $b\in F_{\ast }(Y)$ we have $\text{tr}_{f}(f^{\ast }(a)b)=a\text{tr}_{f}(b)$ .

Proof. The usual proof works, see for example [Reference Calmès and FaselCF17, Proof of Corollary 3.5]. We review it. We only show the first statement, the second is similar. Consider the cartesian square

where $\unicode[STIX]{x1D6FF}_{X}:X\rightarrow X\times X$ is the diagonal and similarly for $Y$ . We have the map $\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6F4}^{\infty }Y_{+}\wedge \unicode[STIX]{x1D6F4}^{\infty }X_{+}\rightarrow \text{}\underline{K}_{\ast }^{MW}\wedge F\rightarrow F$ , where $\text{}\underline{K}_{\ast }^{MW}\wedge F\rightarrow F$ is the module structure and the first map is the tensor product of $\unicode[STIX]{x1D6F4}^{\infty }Y_{+}\rightarrow \text{}\underline{K}_{\ast }^{MW}$ (corresponding to $a$ ) and $\unicode[STIX]{x1D6F4}^{\infty }X_{+}\rightarrow F$ (corresponding to $b$ ). This defines an element $\unicode[STIX]{x1D6FD}\in F(Y\times X)$ . We have $\text{tr}_{f}((\operatorname{id}\times f)\unicode[STIX]{x1D6FF}_{Y})^{\ast }\unicode[STIX]{x1D6FD}=\text{tr}_{f}(af^{\ast }b)$ and $\unicode[STIX]{x1D6FF}_{X}^{\ast }\text{tr}_{f\times \operatorname{id}_{Y}}\unicode[STIX]{x1D6FD}=\text{tr}_{f}(a)b$ (the latter since $\text{tr}_{\operatorname{id}}=\operatorname{id}$ and $\text{tr}_{f\times g}(x\otimes y)=\text{tr}_{f}(x)\otimes \text{tr}_{g}(y)$ by Lemma 11). These two elements are equal by the base change formula, i.e. Proposition 10.◻

5 Recollections on pre-motivic categories

The six functors formalism [Reference Cisinski and DégliseCD12, § A.5] is a very strong, and very general, duality theory. As such it is no surprise that proving that any theory satisfies it requires some work. Fortunately it is now possible to reduce this to establishing a few axioms.

Let ${\mathcal{S}}$ be a base category of schemes. Recall that a pre-motivic category ${\mathcal{M}}$ over ${\mathcal{S}}$ consists of [Reference Cisinski and DégliseCD13, Definition A.1.1] a pseudofunctor ${\mathcal{M}}$ on ${\mathcal{S}}$ , taking values in triangulated, closed symmetric monoidal categories. Often these categories will be obtained as the homotopy categories of a pseudofunctor taking values in suitable Quillen model categories and left Quillen functors. For $f:X\rightarrow Y\in {\mathcal{S}}$ , the functor ${\mathcal{M}}(f):{\mathcal{M}}(Y)\rightarrow {\mathcal{M}}(X)$ is denoted $f^{\ast }$ . For any $f$ , the functor $f^{\ast }$ has a triangulated right adjoint $f_{\ast }$ (which is not required to be monoidal). If $f$ is smooth, then $f^{\ast }$ has a triangulated left adjoint $f_{\#}$ (also not required to be monoidal). Moreover, ${\mathcal{M}}$ needs to satisfy smooth base change and the smooth projection formula, in the following sense.


be a cartesian square in ${\mathcal{S}}$ , with $p$ smooth. Then smooth base change means that the natural transformation $q_{\#}g^{\ast }\rightarrow f^{\ast }p_{\#}$ is required to be a natural isomorphism.

Finally, let $f:Y\rightarrow X$ be a smooth morphism in ${\mathcal{S}}$ . Then the smooth projection formula means that, for $E\in {\mathcal{M}}(X)$ and $F\in {\mathcal{M}}(Y)$ we have $f_{\#}(F\otimes f^{\ast }E)\simeq f_{\#}(F)\otimes E$ , via the canonical map.

Here are some further properties a pre-motivic category can satisfy. We say ${\mathcal{M}}$ satisfies the homotopy property if for every $X\in {\mathcal{S}}$ the natural map $p_{\#}\unicode[STIX]{x1D7D9}\rightarrow \unicode[STIX]{x1D7D9}\in {\mathcal{M}}(X)$ is an isomorphism, where $p:\mathbb{A}^{1}\times X\rightarrow X$ is the canonical map.

Let now $q:\mathbb{P}^{1}\times X\rightarrow X$ be the canonical map. We say that ${\mathcal{M}}$ satisfies the stability property if the cone of the canonical map $q_{\#}\unicode[STIX]{x1D7D9}\rightarrow \unicode[STIX]{x1D7D9}\in {\mathcal{M}}(X)$ is a $\otimes$ -invertible object. In this case we write $\unicode[STIX]{x1D7D9}(1)=fib(q_{\#}\unicode[STIX]{x1D7D9}\rightarrow \unicode[STIX]{x1D7D9})[-2]$ and then as usual $E(n)=E\otimes \unicode[STIX]{x1D7D9}(1)^{\otimes n}$ for $n\in \mathbb{Z},E\in {\mathcal{M}}(X)$ .

Finally, let $X\in {\mathcal{S}}$ , $j:U\rightarrow X\in {\mathcal{S}}$ an open immersion, and $i:Z\rightarrow {\mathcal{S}}$ a complementary closed immersion. Then for $E\in {\mathcal{M}}(U)$ there are the adjunction maps

$$\begin{eqnarray}j_{\#}j^{\ast }E\rightarrow E\rightarrow i_{\ast }i^{\ast }E.\end{eqnarray}$$

We say that ${\mathcal{M}}$ satisfies the localisation property if these maps are always part of a distinguished triangle.

One then has the following fundamental result. It was discovered by Voevodsky, first worked out in detail by Ayoub, and then formalised by Cisinski–Déglise.

Theorem 13 (Ayoub, Cisinski–Déglise).

Let ${\mathcal{S}}$ be the category of Noetherian schemes of finite dimension and ${\mathcal{M}}$ a pre-motivic category which satisfies the homotopy property, the stability property, and the localisation property. Then if ${\mathcal{M}}(X)$ is a well-generated triangulated category for every $X$ , ${\mathcal{M}}$ satisfies the full six functors formalism.

Proof. This is proved for ‘adequate categories of schemes’ in [Reference Cisinski and DégliseCD12, Theorem 2.4.50], of which Noetherian finite dimensional schemes are an example. ◻

One further property we will make use of is continuity. This can be formulated as follows. Let $\{S_{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}\in A}$ be an inverse system in ${\mathcal{S}}$ , where all the transition morphisms are affine and the limit $S:=\lim _{\unicode[STIX]{x1D6FC}}S_{\unicode[STIX]{x1D6FC}}$ exists in ${\mathcal{S}}$ . Write $p_{\unicode[STIX]{x1D6FC}}:S\rightarrow S_{\unicode[STIX]{x1D6FC}}$ for the canonical projection. Let $E\in {\mathcal{M}}(S_{\unicode[STIX]{x1D6FC}_{0}})$ for some $\unicode[STIX]{x1D6FC}_{0}\in A$ and write for $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{0},$ $E_{\unicode[STIX]{x1D6FC}}=(S_{\unicode[STIX]{x1D6FC}}\rightarrow S_{\unicode[STIX]{x1D6FC}_{0}})^{\ast }E$ . We say that ${\mathcal{M}}$ satisfies the continuity property if for every affine inverse system $S_{\unicode[STIX]{x1D6FC}}$ as above, every $E$ and every $i\in \mathbb{Z}$ the canonical map

$$\begin{eqnarray}\operatorname{colim}_{\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{0}}[\unicode[STIX]{x1D7D9}(i),E_{\unicode[STIX]{x1D6FC}}]_{{\mathcal{M}}(S_{\unicode[STIX]{x1D6FC}})}\rightarrow [\unicode[STIX]{x1D7D9}(i),p_{\unicode[STIX]{x1D6FC}_{0}}^{\ast }E]_{{\mathcal{M}}(S)}\end{eqnarray}$$

is an isomorphism.

We in particular use the following consequence of continuity and localisation.

Corollary 14. Suppose that ${\mathcal{M}}$ be a pre-motivic category over ${\mathcal{S}}$ (where ${\mathcal{S}}$ contains all Henselizations of its schemes), coming from a pseudofunctor valued in model categories. Assume that ${\mathcal{M}}$ satisfies continuity and localisation.

Let $E\in {\mathcal{M}}(X)$ , where $X$ is Noetherian of finite dimension. Then $E\simeq 0$ if and only if for every morphism $f:\operatorname{Spec}(k)\rightarrow X$ with $k$ a field we have $f^{\ast }E\simeq 0$ .

Proof. By localisation, we may assume that $X$ is reduced (see for example [Reference Cisinski and DégliseCD12, Proposition 2.3.6(1)]). By [Reference Cisinski and DégliseCD12, Proposition 4.3.9] (this result requires ${\mathcal{M}}$ to come from a model category) we may assume that $X$ is (Henselian) local with closed point $x$ and open complement $U$ . By localisation, it suffices to show that $E|_{x}\simeq 0$ and $E|_{U}\simeq 0$ . The former holds by assumption, and the latter by induction on the dimension. This concludes the proof.◻


The pseudofunctors $X\mapsto \mathbf{SH}(X)$ and $X\mapsto D_{\mathbb{A}^{1}}(X)$ satisfy the six functors formalism and continuity (for the base category of Noetherian finite dimensional schemes) [Reference AyoubAyo07, Reference Cisinski and DégliseCD12].

6 Recollections on monoidal bousfield localisation

Let ${\mathcal{M}}$ be a monoidal model category and $\unicode[STIX]{x1D6FC}:Y^{\prime }\rightarrow Y\in {\mathcal{M}}$ a morphism. We wish to ‘monoidally invert $\unicode[STIX]{x1D6FC}$ ’, by which we mean passing to a model category $L_{\unicode[STIX]{x1D6FC}}^{\otimes }{\mathcal{M}}$ obtained by localising ${\mathcal{M}}$ and such that for every $T\in L_{\unicode[STIX]{x1D6FC}}^{\otimes }{\mathcal{M}}$ the induced map $\unicode[STIX]{x1D6FC}_{T}:T\otimes ^{L}Y^{\prime }\rightarrow T\otimes ^{L}Y$ is a weak equivalence. We will also write $L_{\unicode[STIX]{x1D6FC}}^{\otimes }{\mathcal{M}}=:{\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}]$ and even $\operatorname{Ho}({\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}])=:\operatorname{Ho}({\mathcal{M}})[\unicode[STIX]{x1D6FC}^{-1}]$ .

The monoidal $\unicode[STIX]{x1D6FC}$ -localisation exists very generally. Suppose that $Y^{\prime }$ and $Y$ are cofibrant, and that ${\mathcal{M}}$ admits a set of cofibrant homotopy generators $G$ (for example ${\mathcal{M}}$ combinatorial [Reference BarwickBar10, Corollary 4.33]). Let $H_{\unicode[STIX]{x1D6FC}}=\{Y^{\prime }\otimes T\xrightarrow[{}]{\unicode[STIX]{x1D6FC}\otimes \operatorname{id}}Y\otimes T\mid T\in G\}$ . When no confusion can arise, we will denote $H_{\unicode[STIX]{x1D6FC}}$ just by $H$ . Then the Bousfield localisation $L_{H}{\mathcal{M}}$ , if it exists (for example if ${\mathcal{M}}$ is left proper and combinatorial) is ${\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}]$ . We will call $H_{\unicode[STIX]{x1D6FC}}$ -local objects $\unicode[STIX]{x1D6FC}$ -local. As a further sanity check, the model category $L_{H}{\mathcal{M}}$ is still monoidal as follows from [Reference BarwickBar10, Proposition 4.47].

The situation simplifies somewhat if $Y^{\prime }$ and $Y$ are invertible and ${\mathcal{M}}$ is stable. Then we may as well assume that $Y^{\prime }=\unicode[STIX]{x1D7D9}$ . Given $T\in {\mathcal{M}}$ cofibrant we can consider the directed system

$$\begin{eqnarray}T\cong T\otimes \unicode[STIX]{x1D7D9}\xrightarrow[{}]{\operatorname{id}\otimes \unicode[STIX]{x1D6FC}}T\otimes Y\cong T\otimes Y\otimes \unicode[STIX]{x1D7D9}\rightarrow T\otimes Y^{\otimes 2}\rightarrow \cdots\end{eqnarray}$$

and its homotopy colimit $T[\unicode[STIX]{x1D6FC}^{-1}]:=\operatorname{hocolim}_{n}T\otimes X^{\otimes n}$ . More generally, if $T$ is not cofibrant, we can either first cofibrantly replace it, or use the derived tensor product. Either way, we denote the result still by $T[\unicode[STIX]{x1D6FC}^{-1}]$ . The main point of this section is to show that under suitable conditions, $T[\unicode[STIX]{x1D6FC}^{-1}]$ is the $\unicode[STIX]{x1D6FC}$ -localisation of $T$ .

Clearly this is only a reasonable expectation under some compact generation assumption. More generally, one would expect a transfinite iteration of $\unicode[STIX]{x1D6FC}$ . Since all our applications will be in compactly generated situations, we refrain from giving the more general argument.

Recall that by a set of compact homotopy generators $G$ for ${\mathcal{M}}$ we mean a set of (usually cofibrant) objects $G\subset Ob({\mathcal{M}})$ such that ${\mathcal{M}}$ is generated by the objects in $G$ under homotopy colimits, and such that for any directed system $X_{1}\rightarrow X_{2}\rightarrow \cdots \in {\mathcal{M}}$ and $T\in G$ , the canonical map $\operatorname{hocolim}_{i}\text{Map}^{d}(T,X_{i})\rightarrow \text{Map}^{d}(T,\operatorname{hocolim}_{i}X_{i})$ is an equivalence.

Lemma 15. Let $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D7D9}\rightarrow Y$ be a map between objects in a symmetric monoidal, stable model category such that $Y$ is invertible (in the homotopy category). Assume that ${\mathcal{M}}$ has a set of compact homotopy generators $G$ , and that ${\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}]$ exists.

Then for each $U\in {\mathcal{M}}$ the object $U[\unicode[STIX]{x1D6FC}^{-1}]$ is $\unicode[STIX]{x1D6FC}$ -local and $\unicode[STIX]{x1D6FC}$ -locally weakly equivalent to $U$ . In other words, $U\mapsto U[\unicode[STIX]{x1D6FC}^{-1}]$ is an $\unicode[STIX]{x1D6FC}$ -localisation functor.

Also $G$ defines a set of compact homotopy generators for ${\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}]$ .

Proof. We first show that the images of $G$ in $\operatorname{Ho}({\mathcal{M}}[\unicode[STIX]{x1D6FC}^{-1}])$ are compact homotopy generators. Generation is clear, and for homotopy compactness it is enough to show that a filtered homotopy colimit of $\unicode[STIX]{x1D6FC}$ -local objects is $\unicode[STIX]{x1D6FC}$ -local. But this follows from homotopy compactness of $T\otimes Y^{\otimes n}$ (for $T\in G$ and $n\in \{0,1\}$ ) and definition of $\unicode[STIX]{x1D6FC}$ -locality.

In a model category ${\mathcal{N}}$ with compact homotopy generators, if $T_{1}\rightarrow T_{2}\rightarrow \cdots \,$ is a directed system of weak equivalences then $\operatorname{hocolim}_{i}T_{i}$ is weakly equivalent to $T_{1}$ . (This follows from the same result in the category of simplicial sets.) Thus $U[\unicode[STIX]{x1D6FC}^{-1}]$ is $\unicode[STIX]{x1D6FC}$ -locally weakly equivalent to $U$ .

It remains to see that $U[\unicode[STIX]{x1D6FC}^{-1}]$ is $\unicode[STIX]{x1D6FC}$ -local. This follows from the next two lemmas.◻

In the above lemma, we have defined an object $X$ to be $\unicode[STIX]{x1D6FC}$ -local if for all $T\in {\mathcal{M}}$ the induced map $\unicode[STIX]{x1D6FC}^{\ast }:\text{Map}^{d}(T\otimes ^{L}Y,X)\rightarrow \text{Map}^{d}(T,X)$ is an equivalence, because this is the way Bousfield localisation works. Another intuitively appealing property would be for the canonical map $X\rightarrow X\otimes Y$ to be an equivalence. As the next lemma shows, these two notions agree in our case.

Lemma 16. Let ${\mathcal{M}}$ be a symmetric monoidal model category and $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D7D9}\rightarrow Y$ a morphism with $Y$ invertible.

Call an object $X\in {\mathcal{M}}$ $\unicode[STIX]{x1D6FC}^{\prime }$ -local if $X\rightarrow X\otimes ^{L}Y$ is a weak equivalence. Then $X$ is $\unicode[STIX]{x1D6FC}$ -local if and only if $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local, if and only if $X$ is $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ -local.

Proof. We shall show that (1) $X$ is $\unicode[STIX]{x1D6FC}$ -local if and only if it is $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ -local, (2) $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local if and only if it is $(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{\prime }$ -local, (3) $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local if it is $\unicode[STIX]{x1D6FC}$ -local and (4) $X$ is $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ -local if it is $(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{\prime }$ -local.

All tensor products and mapping spaces will be derived in this proof.

(1) Consider the string of maps

$$\begin{eqnarray}\text{Map}(T\otimes Y^{\otimes 3},X)\rightarrow \text{Map}(T\otimes Y^{\otimes 2},X)\rightarrow \text{Map}(T\otimes Y,X)\rightarrow \text{Map}(T,X).\end{eqnarray}$$

If $X$ is $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ -local, then the composite of any two consecutive maps is an equivalence, and hence all maps are equivalences by 2-out-of-6. Consequently $X$ is $\unicode[STIX]{x1D6FC}$ -local. The converse is clear.

(2) Consider the string of maps

$$\begin{eqnarray}X\rightarrow X\otimes Y\rightarrow X\otimes Y^{\otimes 2}\rightarrow X\otimes Y^{\otimes 3}.\end{eqnarray}$$

If $X$ is $(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{\prime }$ -local then so is $Z\otimes X$ for any $Z$ , since (derived) tensor product preserves weak equivalences. It follows that $X\otimes Y$ is $(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{\prime }$ -local, and hence the composite of any two consecutive maps is an equivalence. Again by 2-out-of-6 this implies that $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local. The converse is clear.

(3) An object $X$ is $\unicode[STIX]{x1D6FC}$ -local if (and only if) for all $T\in {\mathcal{M}}$ the map $\text{Map}(T\otimes Y,X)\rightarrow \text{Map}(T,X)$ is a weak equivalence (of simplicial sets). In particular $T\rightarrow T\otimes Y$ is an $\unicode[STIX]{x1D6FC}$ -local weak equivalence for all $T$ . It also follows that $X\otimes Y$ is $\unicode[STIX]{x1D6FC}$ -local if $X$ is (here we use invertibility of $Y$ ). Since $X\rightarrow X\otimes Y$ is an $\unicode[STIX]{x1D6FC}$ -local weak equivalence, it is a weak equivalence if $X$ (and hence $X\otimes Y$ ) is $\unicode[STIX]{x1D6FC}$ -local. Thus $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local if it is $\unicode[STIX]{x1D6FC}$ -local.

(4) For any simplicial set $K$ we have $[K,\text{Map}(T,X)]=[K\otimes T,X]$ (using a framing if the model category is not simplicial). It follows that $X$ is $\unicode[STIX]{x1D6FC}$ -local if and only if for all $T\in {\mathcal{M}}$ the map $\unicode[STIX]{x1D6FC}^{\ast }:[T\otimes Y,X]\rightarrow [T,X]$ is an isomorphism. In particular, this property can be checked entirely in the homotopy category of ${\mathcal{M}}$ , in which we will work from now on.

Suppose, for now, that $X$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local. (We will find that our strategy does not work, but it will work for $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ , and this is all that is left to prove.) We can choose an inverse equivalence $\unicode[STIX]{x1D6FD}:X\otimes Y\rightarrow X$ . We consider the map $\overline{\unicode[STIX]{x1D6FD}}:[T,X]\rightarrow [T\otimes Y,X]$ sending $f:T\rightarrow X$ to $T\otimes Y\xrightarrow[{}]{f\otimes \operatorname{id}}X\otimes Y\xrightarrow[{}]{\unicode[STIX]{x1D6FD}}X$ . We would like to say that $\overline{\unicode[STIX]{x1D6FD}}$ is inverse to $\unicode[STIX]{x1D6FC}^{\ast }$ . Given $f:T\rightarrow X$ we get the following commutative diagram.

Consequently $\unicode[STIX]{x1D6FC}_{\ast }\unicode[STIX]{x1D6FC}^{\ast }\overline{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D6FC}_{\ast }:[T,X]\rightarrow [T,X\otimes Y]$ and thus $\unicode[STIX]{x1D6FC}^{\ast }\overline{\unicode[STIX]{x1D6FD}}=\operatorname{id}$ (note that $\unicode[STIX]{x1D6FC}_{\ast }$ means composition with $X\rightarrow X\otimes Y$ , which is an isomorphism).

The problem is with showing that $\overline{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FC}^{\ast }=\operatorname{id}$ . For this we fix $f:T\otimes Y\rightarrow X$ and consider the following diagram.

If it commutes for all such $f$ , then $\overline{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FC}^{\ast }=\operatorname{id}$ . But this is not clear; the two paths differ by a switch of $Y$ .

However, in any symmetric monoidal category, the switch isomorphism on the square of an invertible object is the identity [Reference DuggerDug14, Propositions 4.20 and 4.21]. Consequently our argument works for $\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC}$ , and this is what we set out to prove.◻


The assumption that $Y$ is invertible is necessary in general for the above result. For example, if ${\mathcal{M}}$ is a cartesian symmetric monoidal model category, then there cannot be any $\unicode[STIX]{x1D6FC}^{\prime }$ -local objects unless $\ast =\unicode[STIX]{x1D7D9}\rightarrow Y$ is already an equivalence.

Lemma 17. Notations and assumptions as in Lemma 15.

For any (cofibrant) $X\in {\mathcal{M}}$ , the object $X[\unicode[STIX]{x1D6FC}^{-1}]$ is $\unicode[STIX]{x1D6FC}$ -local.

Proof. By the previous lemma, it suffices to show that $X[\unicode[STIX]{x1D6FC}^{-1}]$ is $(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{\prime }$ -local. Clearly $X[\unicode[STIX]{x1D6FC}^{-1}]\simeq X[(\unicode[STIX]{x1D6FC}\otimes \unicode[STIX]{x1D6FC})^{-1}]$ , i.e. we may assume without loss of generality that $Y$ is a square, and so its switch isomorphism (in the homotopy category) is the identity.

Since tensor product commutes with colimits (in each variable) we have $X[1/f]\simeq X\otimes ^{L}\unicode[STIX]{x1D7D9}[1/f]$ , and we can simplify notation by assuming without loss of generality that $X=\unicode[STIX]{x1D7D9}$ .

What we need to prove is that the following diagram induces an equivalence on homotopy colimits.

Because of the domains and codomains, it is tempting to guess that $f_{i}\simeq h_{i}\simeq f_{i}^{\prime }$ . Here we write $f\simeq g$ to mean that the maps become equal in the homotopy category. We claim that this guess is correct. Then if $T$ is any homotopy compact object, applying $[T,\bullet ]$ to our diagram we get a diagram of abelian groups which we need to show induces an isomorphism on colimits. Homotopic maps become equal when applying $[T,\bullet ]$ , and then the desired result follows from an easy diagram chase. By compact generation and stability, this will conclude the proof.

It remains to prove the claim. For this we may work entirely in the homotopy category, which we will do from now on. It is easy to see that indeed $f_{i}=h_{i}$ . For general $Y$ , it would not be true that $f_{i}^{\prime }=f_{i}$ ; one may check that the maps differ by appropriate switches of $Y$ . However, we have assumed that the switch on $Y$ is the identity, so indeed $f_{i}=f_{i}^{\prime }$ as well.◻


The stability assumption was used in the above proof in the following form: if $A\rightarrow B$ is any morphism in ${\mathcal{M}}$ and $[T,A]\rightarrow [T,B]$ is an isomorphism for all homotopy compact $T$ , then $A\rightarrow B$ is a weak equivalence. This fails for example in the homotopy category of spaces.

The stability assumption is in fact necessary for the above result. The author learned the following counterexample from Marc Hoyois: let ${\mathcal{M}}$ be the model category of small, stable $\infty$ -categories, $Y=\unicode[STIX]{x1D7D9}$ the category of finite spectra and $\unicode[STIX]{x1D6FC}=2$ , i.e. the functor which sends a finite spectrum $s$ to $s\oplus s$ . Then ${\mathcal{C}}\in {\mathcal{M}}$ is $\unicode[STIX]{x1D6FC}^{\prime }$ -local only if it is trivial. Indeed for $c\in {\mathcal{C}}$ the map $[c,c]\rightarrow [c\oplus c,c\oplus c]$ needs to be an isomorphism, which forces $c\simeq 0$ . But one may show that $\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D6FC}]$ is not the zero category, and so is not $\unicode[STIX]{x1D6FC}^{\prime }$ -local (let alone $\unicode[STIX]{x1D6FC}$ -local).

See [Reference HoyoisHoy17, Theorem 3.8] for a criterion that can be applied in unstable situations.

7 The theorem of Jacobson and $\unicode[STIX]{x1D70C}$ -stable homotopy modules

Throughout this section, $k$ is a field of characteristic zero. Recall that the real étale topology is finer than the Nisnevich topology; in particular every real étale sheaf is a Nisnevich sheaf.

Theorem 18 (Jacobson [Reference JacobsonJac17, Theorem 8.5]).

There is a canonical isomorphism (in $\operatorname{Shv}_{\operatorname{Nis}}(Sm(k))$ )

$$\begin{eqnarray}\operatorname{colim}_{n}\text{}\underline{I}^{n}\rightarrow a_{\text{r}\acute{\text{e}}\text{t}}\text{}\underline{\mathbb{Z}},\end{eqnarray}$$

where the transition maps $\text{}\underline{I}^{n}\rightarrow \text{}\underline{I}^{n+1}$ are given by multiplication with $2=\langle 1,1\rangle \in \text{}\underline{I}$ .

Here $\text{}\underline{I}$ denotes the sheaf of fundamental ideals on $Sm(k)_{\operatorname{Nis}}$ , i.e. the sheaf associated with the presheaf $X\mapsto I(X)$ , where $I(X)$ is the fundamental ideal of the Witt ring of $X$ [Reference Knebusch and OrzechKne77]. We similarly write $\text{}\underline{W}$ for the sheaf of Witt rings, etc.

Let us recall the construction of the isomorphism in Jacobson’s theorem. If $\unicode[STIX]{x1D719}\in W(K)$ , where $K$ is a field, and $p$ is an ordering of $K$ , then there is the signature $\unicode[STIX]{x1D70E}_{p}(\unicode[STIX]{x1D719})\in \mathbb{Z}$ . If $\unicode[STIX]{x1D719}\in W(X)$ , define $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719}):R(X)\rightarrow \mathbb{Z}$ as follows. For $(x,p)\in R(X)$ put $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719})(x,p)=\unicode[STIX]{x1D70E}_{p}(\unicode[STIX]{x1D719}|_{x})$ . Then one shows that $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719})$ is a continuous function from $R(X)$ to $\mathbb{Z}$ , i.e. an element of $H_{\text{r}\acute{\text{e}}\text{t}}^{0}(X,\mathbb{Z})$ .

Next if $\unicode[STIX]{x1D719}\in I(k)$ then $\unicode[STIX]{x1D70E}_{p}(\unicode[STIX]{x1D719})\in 2\mathbb{Z}$ . Consequently if $\unicode[STIX]{x1D719}\in I(X)$ also $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719})\in 2H_{\text{r}\acute{\text{e}}\text{t}}^{0}(X,\mathbb{Z})$ . We may thus define $\tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719})/2$ and in this way we obtain $\tilde{\unicode[STIX]{x1D70E}}:I(X)\rightarrow H_{\text{r}\acute{\text{e}}\text{t}}^{0}(X,\mathbb{Z})$ . Similarly we get $\tilde{\unicode[STIX]{x1D70E}}:I^{n}(X)\rightarrow H_{\text{r}\acute{\text{e}}\text{t}}^{0}(X,\mathbb{Z})$ with $\tilde{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D719})/2^{n}$ for $\unicode[STIX]{x1D719}\in I^{n}(X)$ . For each $n$ there is a commutative diagram as follows.

Consequently there is an induced map $\tilde{\unicode[STIX]{x1D70E}}:\operatorname{colim}_{n}I^{n}(X)\rightarrow H_{\text{r}\acute{\text{e}}\text{t}}^{0}(X,\mathbb{Z})$ . The claim is that this is an isomorphism after sheafifying, i.e. for $X$ local.

Corollary 19. Let $\text{}\underline{K}_{n}^{MW}$ denote the $n$ th unramified Milnor–Witt K-theory sheaf. Then there is a canonical isomorphism $\operatorname{colim}_{n}\text{}\underline{K}_{n}^{MW}\rightarrow a_{\text{r}\acute{\text{e}}\text{t}}\mathbb{Z}$ . Here the colimit is along multiplication with $\unicode[STIX]{x1D70C}:=-[-1]\in K_{1}^{MW}(k)$ .

Proof. Recall the element $h\in K_{0}^{MW}(k)$ with the following properties: $\text{}\underline{K}_{n}^{MW}/h=\text{}\underline{I}^{n}$ [Reference MorelMor04, Théorème 2.1] and for $a\in K_{1}^{MW}(k)$ we have $a^{2}h=0$ [Reference MorelMor12, Corollary 3.8] (this relation is the analogue of the fact that in a graded commutative ring $R_{\ast }$ with $a\in R_{1}$ we have $a^{2}=-a^{2}$ by graded commutativity, so $2a^{2}=0$ ). Consequently $\unicode[STIX]{x1D70C}^{2}h=0$ and so $\operatorname{colim}_{n}\text{}\underline{K}_{n}^{MW}\rightarrow \operatorname{colim}_{n}\text{}\underline{I}^{n}$ is an isomorphism. It remains to note that the image of $\unicode[STIX]{x1D70C}$ in $K_{1}^{MW}/h(k)\cong I(k)$ is given by $-(\langle -1\rangle -1)=2\in I(k)\subset W(k)$ , so the induced transition maps in the colimit are precisely those used in Jacobson’s theorem.◻

Note that the sheaves $\text{}\underline{I}^{n}$ f