aa r X i v : . [ m a t h . K T ] J a n MOTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY
TOM BACHMANN
Abstract.
Let S be a Noetherian scheme of finite dimension and denote by ρ ∈ [ , G m ] SH ( S ) the(additive inverse of the) morphism corresponding to − ∈ O × ( S ). Here SH ( S ) denotes the motivicstable homotopy category. We show that the category obtained by inverting ρ in SH ( S ) is canonicallyequivalent to the (simplicial) local stable homotopy category of the site S r ´ et , by which we mean the small real ´etale site of S , comprised of ´etale schemes over S with the real ´etale topology.One immediate application is that SH ( R )[ ρ − ] is equivalent to the classical stable homotopy cate-gory. In particular this computes all the stable homotopy sheaves of the ρ -local sphere (over R ). Asfurther applications we show that D A ( k, Z [1 / − ≃ DM W ( k )[1 /
2] (improving a result of Ananyevskiy-Levine-Panin), reprove R¨ondigs’ result that π i ( [1 /η, / i = 1 , Contents
1. Introduction 12. Recollections on Local Homotopy Theory 53. Recollections on Real ´Etale Cohomology 74. Recollections on Motivic Homotopy Theory 105. Recollections on Pre-Motivic Categories 126. Recollections on Monoidal Bousfield Localization 137. The Theorem of Jacobson and ρ -stable Homotopy Modules 158. Preliminary Observations 189. Main Theorems 2010. Real Realisation 2111. Application 1: The η -inverted Sphere 2212. Application 2: Some Rigidity Results 24References 251. Introduction
For a scheme S we denote by SH ( S ) the motivic stable homotopy category [MV99, Ayo07]. We recallthat 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 “Riemannsphere” P S into an invertible object.If α : k ֒ → C is an embedding of a field k into the complex numbers, then we obtain a complexrealisation functor R α, C : SH ( k ) → SH (where now SH denotes the classical stable homotopy category)connecting the world of motivic stable homotopy theory to classical stable homotopy theory [MV99,Section 3.3.2]. This functor is induced from the functor which sends a smooth scheme S over k to itstopological space of of complex points S ( C ) (this depends on α ). Similarly if β : k ֒ → R is an embeddinginto the real numbers, then there is a real realisation functor R β, R : SH ( k ) → SH induced from S S ( R )[MV99, Section 3.3.3] [HO16, Proposition 4.8].These functors serve as a good source of inspiration and a convenient test of conjectures in stablemotivic homotopy theory. For example, in order for a morphism f : E → F to be an equivalence it isnecessary that R α, C ( f ) and R β, R ( f ) are equivalences, for all such embeddings α, β . 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, orwhat their “kernel” is. The aim of this article is to give some kind of complete answer to this questionin the case of real realisation. We begin with the simplest formulation of our result. Write R R for the(unique) real realisation functor for the field k = R . The first clue comes from the observation that R R ( G m ) = R \ ≃ {± } = S . That is to say R R identifies G m and S . We can even do better. Write ρ ′ : S → G m for the map of pointed motivic spaces corresponding to − ∈ R × . Then one may checkeasily that R R ( ρ ′ ) is an equivalence between S ≃ R R ( S ) and R R ( G m ).We prove that SH ( R )[ ρ ′− ] ≃ SH via real realisation. That is to say R R is in some sense the universalfunctor turning ρ ′ into an equivalence. More precisely, the functor R R : SH ( R ) → SH has a right adjoint R ∗ (e.g. by Neeman’s version of Brown representability) and we show that R ∗ is fully faithful with imageconsisting of the ρ ′ -stable motivic spectra, i.e. those E ∈ SH ( R ) such that E ( X ∧ G m ) ρ ∗ −→ E ( X ) is anequivalence for all X ∈ Sm ( R ).Of course, our description of SH ( R )[ ρ ′− ] is just an explicit description of a certain Bousfield locali-sation of SH ( R ). Moreover the element ρ ′ exists not only over R but already over Z , so we are lead tostudy more generally the category SH ( S )[ ρ ′− ], for more or less arbitrary base schemes S . Actually, forsome formulas it is nicer to consider ρ := − ρ ′ ∈ [ S, Σ ∞ G m ] and we shall write this from now on. Ofcourse SH ( S )[ ρ ′− ] = SH ( S )[ ρ − ]. In this generality we can of longer expect that SH ( S )[ ρ − ] ≃ SH .Indeed as we have said before in general there is no real realisation! As a first attempt, one might guessthat if X is a scheme over R , then SH ( S )[ ρ − ] ≃ SH ( S ( R )), where the right hand side denotes someform of parametrised homotopy theory [MS06]. This cannot be quite true unless S is proper, becausethe category SH ( S ( R )) will then not be compactly generated. The way out is to use semi-algebraictopology. For this we have to recall that if S is a scheme, then there exists a topological space R ( S )[Sch94, (0.4.2)]. Its points are pairs ( x, α ) with x ∈ S and α an ordering of the residue field k ( x ). Thisis given a topology incorporating all of these orderings. Write Shv ( RS ) for the category of sheaves onthis topological space.Now, given any topos X , there is a naturally associated stable homotopy category SH( X ). If X ≃
Set then SH( X ) is just the ordinary stable homotopy category. In general, if X ≃
Shv ( C ) where C is aGrothendieck site, then SH( X ) is the local homotopy category of presheaves of spectra on C .With this preparation out of the way, we can state our main result: Theorem ((see Theorem 35)) . Let S be a Noetherian scheme of finite dimension. Then there is acanonical equivalence of categories SH ( S )[ ρ − ] ≃ SH(
Shv ( RS )) . A more detailed formulation is given later in this introduction. For now let us mention one application.We go back to S = Spec ( R ). In this case Proposition 36 in Section 10 assures us that the equivalence fromthe above theorem does indeed come from real realisation. But given E ∈ SH ( R ), its ρ -localisation can becalculated quite explicitly (see Lemma 15). From this one concludes that π i ( R R E ) = colim n π i ( E ) n ( R ),where the colimit is along multiplication by ρ in the second grading of the bigraded homotopy sheavesof E . (Recall that π i ( E ) n ( R ) = [ [ i ] , E ∧ G ∧ nm ] so ρ indeed induces ρ : π i ( E ) n ( R ) → π i ( E ) n +1 ( R ).)This may seem slightly esoteric, but actually SH ( S )[ ρ − , − ] = SH ( S )[ η − , − ] and so our com-putations apply, after inverting two, to the more conventional η -localisation as well. As a corollary, weobtain the following. Theorem.
The motivic stable 2-local, η -local stems over R agree with the classical stable 2-local stems: π i ( η, ) j ( R ) = π si ⊗ Z Z [1 / . Some more applications will be described later in this introduction.Overview of the proof. The proof uses a different description of the category
Shv ( RS ). Namely, there isa topology on all schemes called the real ´etale topology and abbreviated r´et-topology [Sch94, (1.2)]. (Thecovers are families of ´etale morphisms which induce a jointly surjective family on the associated realspaces R ( • ).) We write Sm ( S ) r ´ et for the site of all smooth schemes over S with this topology, and S r ´ et for the site of all ´etale schemes over S with this topology. Then Shv ( S r ´ et ) ≃ Shv ( RS ) [Sch94, Theorem(1.3)].Write SH ( S ) for the motivic stable homotopy category, SH ( S )[ ρ − ] for the ρ -local motivic stablehomotopy category, SH ( S ) r ´ et for the r´et-local motivic stable homotopy category (i.e. the categoryobtained from the site Sm ( S ) r ´ et by precisely the same construction as is used to build SH ( S ) from Sm ( S ) Nis ), and SH S ( S ) for the motivic S -stable homotopy category. We trust that SH S ( S ) r ´ et , SH ( S ) r ´ et [ ρ − ] and so on have evident meanings. Write SH( S r ´ et ) for the r´et-local stable homotopycategory on the small real ´etale site. This is just the homotopy category of the category of presheaves ofspectra on S r ´ et with the local model structure. Similarly SH( Sm ( S ) r ´ et ) means the r´et-local presheavesof spectra on Sm ( S ). Then for example SH S ( S ) r ´ et is the A -localisation of SH( Sm ( S ) r ´ et ). OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 3
The canonical functor e : SH( S r ´ et ) → SH( Sm ( S ) r ´ et ) (extending a (pre)sheaf on the small site to thelarge site) is fully faithful by general results (see Corollary 6). It is moreover t -exact: for E ∈ SH( S r ´ et )we have π i ( eE ) = eπ i ( E ). Here π ∗ denotes the homotopy sheaves.If F is a sheaf on the small real ´etale site of a scheme Y , then H p ( Y × A , F ) = H p ( Y, F ) and H p ( Y + ∧ G m , F ) = H p ( Y, F ). If Y is of finite type over R and F is locally constant, then this follows bycomparison of real ´etale cohomology with Betti cohomology of the real points [Del91, Theorem II.5.7].For the general case, see Theorem 8.Now the category SH S ( S ) r ´ et [ ρ − ] is obtained from SH( Sm ( S ) r ´ et ) by ( A , ρ )-localisation. It followsfrom t -exactness of e , the descent spectral sequence, and the above result about r´et-cohomology that thecomposite SH( S r ´ et ) → SH( Sm ( S ) r ´ et ) → SH S ( S ) r ´ et [ ρ − ] is still fully faithful.The category SH ( S ) r ´ et [ ρ − ] is obtained from SH S ( S ) r ´ et [ ρ − ] by ⊗ -inverting G m . However in thelatter category we have G m ≃ (via ρ !), so G m is already invertible, and inverting it has no effect: SH S ( S ) r ´ et [ ρ − ] ≃ SH ( S ) r ´ et [ ρ − ]. We have thus shown thatSH( S r ´ et ) → SH ( S ) r ´ et [ ρ − ]is fully faithful.The next step is to show that it is essentially surjective. This follows from the proper base changetheorem by a clever argument of Cisinski-D´eglise. Of course this first requires that we know that SH( S r ´ et )and SH ( S ) r ´ et [ ρ − ] satisfy proper base change. For SH( S r ´ et ) this is a consequence of the proper basechange theorem in real ´etale cohomology established by Scheiderer, see Theorem 9. For SH ( S ) r ´ et [ ρ − ]this would follow from the axiomatic six functors formalism of Voevodsky/Ayoub/Cisinski-D´eglise, seeSection 5. It is in fact not very hard to show directly that SH ( S ) r ´ et [ ρ − ] satisfies the six functorsformalism. Instead we shall show (without assuming the six functors formalism) that SH ( S ) r ´ et [ ρ − ] ≃ SH ( S )[ ρ − ], and that this latter category satisfies the six functors formalism.The next step is thus to show that the localisation functor SH ( S )[ ρ − ] → SH ( S ) r ´ et [ ρ − ] is an equiva-lence. It clearly has dense image, so it suffices to show that it is fully faithful. Using the fact that SH ( S )[ ρ − ] satisfies continuity and gluing (which follows quite easily from the same statement for SH ( S )), we may reduce to the case where S is the spectrum of a field k . The case where char ( k ) > k has characteristiczero and so in particular is perfect .The ρ -localisation can be described rather explicitly. For E ∈ SH ( k ), consider the directed system E ρ −→ E ∧ G m ρ −→ E ∧ G m ∧ G m ρ −→ . . . . Then hocolim n E ∧ G ∧ nm is a model for the ρ -localisation E [ ρ − ] of E (see Lemma 15). It follows that itshomotopy sheaves are given by π i ( E [ ρ − ]) = π i ( E ) ∗ [ ρ − ] =: colim n π i ( E ) n . Here the colimit is along multiplication by ρ . (Let us remark here that the homotopy sheaves in SH ( k )are bigraded , and so, technically, are those in SH ( k )[ ρ − ]. However inverting ρ means that up to canonicalisomorphism, the homotopy sheaf is independent of the second index, so we suppress it.) It then followsfrom the descent spectral sequence that in order to prove that the functor SH ( k )[ ρ − ] → SH ( k ) r ´ et [ ρ − ]is an equivalence, it is enough to prove that if F ∗ is a homotopy module (element in the heart of SH ( k ))such that ρ : F n → F n +1 is an isomorphism for all n (we call such a homotopy module ρ -stable ), then H nr ´ et ( X, F ∗ ) = H nNis ( X, F ∗ ) for all X smooth over k . In particular, we need to show that F ∗ is a sheaf inthe real ´etale topology. This is actually sufficient, because Nisnevich, Zariski and real ´etale cohomologyof real ´etale sheaves all agree [Sch94, Proposition 19.2.1].This ties in with work of Jacobson and Scheiderer. Recall that π ( ) ∗ = K MW ∗ , i.e. the zeroth stablemotivic homotopy sheaf is unramified Milnor-Witt K -theory. A theorem of Jacobson [Jac17] togetherwith work of Morel implies that K MW ∗ [ ρ − ] = colim n I n = a r ´ et Z ; here I is the sheaf of fundamentalideals. Finally if F ∗ is a general ρ -stable homotopy module, we use properties of transfers for homotopymodules together with the structure of F ∗ as a module over K MW ∗ [ ρ − ] = a r ´ et Z to show that F ∗ is asheaf in the real ´etale topology. This concludes the overview of the proof.Throughout the article we actually establish all our results for both the stable motivic homotopycategory SH ( S ) and the stable A -derived category D A ( S ). The proofs in the latter case are essentiallyalways the same as in the former, so we do not tend to give them. (In fact in some cases proofs just forthe latter category would be simpler.) TOM BACHMANN
Overview of the article. In Section 2 we recall some results from local homotopy theory, including theexistence and basic properties of the homotopy t -structure, a general compact generation criterion anda fully faithfulness result.In Section 3 we recall the real ´etale topology and establish some supplements.In Section 4 we recall some results about motivic stable homotopy categories and transfers for finite´etale morphisms. In particular we establish the base change and projection formulas for these.In Section 5 we recall the formalism of pre-motivic and motivic categories and how it can be used toestablish that a category satisfies the six functors formalism.In Section 6 we carefully prove some basic facts about monoidal Bousfield localization.We judge these five sections as preliminary and the results as not very original. The “real work” iscontained in the next three sections. In Section 7 we review Jacobson’s theorem on the colimit of thepowers of the sheaf of fundamental ideals and use it together with our results on transfers to prove that ρ -stable homotopy modules are sheaves in the real ´etale topology.Section 8 contains various preliminary observations and reductions.Finally in Section 9 we carry out the proof as outlined above.The remaining three sections contain some applications. In Section 10 we show that our functor SH ( R ) → SH ( R )[ ρ − ] ≃ SH(
Spec ( R ) r ´ et ) ≃ SH coincides with the real realisation functor. It followsthat the ρ -inverted stable homotopy sheaves of E ∈ SH ( R ) are just the stable homotopy groups of itsreal realisation.In Section 11 we collect some consequences for the η -inverted sphere. We use that [1 / , /ρ ] ≃ [1 / , /η ]. Since the classical stable stems π si = Z / i = 1 , π i ( [1 / , /η ])( R ) = 0 for i = 1 ,
2. Since the ρ -local homotopy sheaves are unramified sheaves inthe real ´etale topology, this (more or less) implies that π i ( [1 / , /η ]) = 0 for i = 1 ,
2. This reproves aresult of R¨ondigs [R¨on16].A different but related question is to determine rational motivic stable homotopy theory. By a recentresult of Ananyevskiy-Levine-Panin [ALP17] we have SH ( k ) − Q ≃ DM W ( k, Q ), where the right handside denotes a category of rational Witt-motives. Our results show easily that DM W ( k, Z [1 / ≃ D A ( k, Z [1 / − ≃ D ( Spec ( k ) r ´ et , Z [1 / D A ( k, Z )[1 /ρ ] ≃ D ( Spec ( k ) r ´ et ). Bythe same proof as in classical rational stable homotopy theory we have SH ( k ) − Q ≃ D A ( k, Q ) − , and so weconsider our results as one version of an integral strengthening of the result of Ananyevskiy-Levine-Panin.In Section 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 ). Forexample, sheaves with transfers in the sense of Voevodsky which are of torsion prime to the characteristicof the perfect base field are rigid (see [SV96, Theorem 4.4]). Our results imply that the homotopy sheavesof any E ∈ SH ( k )[ ρ − ] are real ´etale sheaves extended from the small real ´etale site of k . One mightalready call this a rigidity result, but it is also not hard to see (and we show) that all such sheaves are rigidin the above sense. As an application, we show that the motivic stable homotopy sheaves π i ( ) [1 /e ] areall rigid, where e is the exponential characteristic. This ties up a loose end of the author’s PhD thesis.Acknowledgements. This paper owes a huge debt to a number of people. In May 2016 Denis-CharlesCisinski taught a mini-course on motives in Essen. As a result the author realised that he could extendhis theorem from perfect base fields to fairly general base schemes. In particular he learned the patternof the proof that D ( S et , Z /p ) is equivalent to DM h ( S, Z /p ). In a lot of ways our proof is a variant of thatone. A similar strategy is followed in the Cisinski-D´eglise article [CD13] on which we also rely heavilyboth in spirit and in practice.Just like in that article, many of our results are relatively straightforward consequences of difficulttheorems in real ´etale cohomology established by Scheiderer and other semi-algebraic topologists.The importance of the Voevodsky/Ayoub/Cisinski-D´eglise approach to the six functors formalism forour article also cannot be overstated.Discussions with Fabien Morel, Oliver R¨ondigs, Marc Hoyois, Marco Schlichting, and Markus Spitzweckabout early versions of this work were also influential to its current form.The author would further like to thank Denis-Charles Cisinski for carefully reading a draft of thiswork and pointing out several mistakes, and Elden Elmanto, Daniel Harrer, Marc Levine and MariaYakerson for providing comments.Notation. If S is a scheme, we denote the motivic stable homotopy category by SH ( S ). We denote the S -stable motivic homotopy category (i.e. where G m has not been inverted yet) by SH S ( S ). If X is atopos or site, we denote by SH( X ) the associated stable homotopy category, see Section 2. In particularSH( S r ´ et ) , SH( Sm ( S ) r ´ et ) and SH ( S ) r ´ et should be carefully distinguished: the first is the stable homotopycategory of the small r´et-site on S , the second is the stable homotopy category of the site of all smooth OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 5 schemes, with the r´et-topology, and the latter is the r´et-localization of the motivic stable homotopycategory. This last category is A -local and G m -stable, whereas the second category is neither, and thesenotions do not even make sense for the first category.The classical stable homotopy category will still be denoted by SH .We denote the unit of a monoidal category C by C or just by , if C is clear from the context. Thusif C is a stable homotopy category of some sort, then is the sphere spectrum.2. Recollections on Local Homotopy Theory
If ( C , τ ) is a Grothendieck site, we can consider the associated category Shv ( C τ ) of sheaves (a topos),the category sP re ( C ) of simplicial presheaves on C , as well as the categories SH ( C ) of presheaves ofspectra and C ( C ) of presheaves of complexes of abelian groups on C . The latter three categories carryvarious local model structures, in particular the injective and the projective one [Jar15]. We denote thehomotopy category of SH ( C τ ) by SH( C τ ) and the homotopy category of C ( C τ ) by D ( C τ ).It is also possible to model SH( C τ ) and so on by sheaves . For this, let sShv ( C τ ) denote the categoryof sheaves of simplicial sets, and similarly SH s ( C τ ) the category of sheaves of spectra, and C s ( C τ )the category of sheaves of chain complexes. (Here we mean sheaves in the 1-categorical sense, so thiscategory is equivalent to the category of chain complexes of sheaves of abelian groups, and similarly forthe spectra.) These also afford local model structures, and Ho ( sShv ( C τ )) ≃ Ho ( sP re ( C τ )), and so on.Given a functor f ∗ : C → D , there is an induced restriction functor f ∗ : P re ( D ) → P re ( C ), where P re ( C ) denotes the category of presheaves (of sets) on C (and similarly for D ). The functor f ∗ has a leftadjoint f ∗ : P re ( C ) → P re ( D ). It is in fact the left Kan extension of f ∗ : C → D .If C , D are sites the functor f ∗ is called continuous if f ∗ : P re ( D ) → P re ( C ) preserves sheaves. In thiscase the induced functor f ∗ : Shv ( D ) → Shv ( C ) has a left adjoint still denoted f ∗ : Shv ( C ) → Shv ( D ).If this induced functor is left exact (commutes with finite limits) then f is called a geometric morphism .More generally, an adjunction f ∗ : Shv ( C ) ⇆ Shv ( D ) : f ∗ (where f ∗ ⊢ f ∗ does not necessarily comefrom a functor f ∗ : C → D ) is called a geometric morphism if f ∗ preserves finite limits.If f : C → D is any functor, then there are induced adjunctions f ∗ : sP re ( C ) ⇆ sP re ( D ) : f ∗ , andsimilarly for spectra and chain complexes. Similarly if f ∗ : Shv ( C ) ⇆ Shv ( D ) : f ∗ is any adjunction,then there are induced adjunctions f ∗ : sShv ( C ) ⇆ sShv ( D ) : f ∗ , and so on. If f ∗ ⊢ f ∗ is a geometricmorphism in either of the above senses, then the induced adjunctions on presheaves (sheaves) of simplicialsets, spectra, and chain complexes are Quillen adjunctions in the local model structure [Jar15, Section5.3] [CD09a, Theorem 1.18].The above discussion allows us to prove the following useful result. Lemma 1.
Let f ∗ : Shv ( C ) ⇆ Shv ( D ) : f ∗ be a geometric morphism such that f ∗ is fully faithful and f ∗ preserves colimits.Then the induced functors Lf ∗ : SH( C ) → SH( D ) and Lf ∗ : D ( C ) → D ( D ) are fully faithful. The same result also holds for Lf ∗ : Ho ( sP re ( C )) → Ho ( sP re ( D )), with the same proof. Proof.
We give the proof for the derived categories, it is the same for spectra.Since f ∗ preserves colimits it affords a right adjoint f ! . Then f ∗ ⊢ f ! is a geometric morphism in theopposite direction (note that f ∗ preserves finite limits, and in fact all limits, since it is a right adjoint)and consequently f ∗ is bi -Quillen. It follows that f ∗ : C s ( D ) → C s ( C ) preserves weak equivalences, andconsequently coincides (up to weak equivalence) with its derived functor.Now to show that Lf ∗ is fully faithful we need to show that Rf ∗ Lf ∗ ≃ id. But Rf ∗ ≃ f ∗ since f ∗ isbi-Quillen. Let E ∈ C s ( C ) be cofibrant. Then Lf ∗ E ≃ f ∗ E and consequently Rf ∗ Lf ∗ E ≃ f ∗ f ∗ E . Since f ∗ is fully faithful we have f ∗ f ∗ E ∼ = E . This concludes the proof. (cid:3) We will also make use of t -structures. We shall use homological notation for t -structures [Lur16,Definition 1.2.1.1]. Briefly, a t -structure on a triangulated category C consists of two (strictly full)subcategories C ≥ and C ≤ , satisfying various axioms. We put C ≥ n = C ≥ [ n ] and C ≤ n = C ≤ [ n ]. Onethen has C ≥ n +1 ⊂ C ≥ n and C ≤ n ⊂ C ≤ n +1 and [ C ≥ n +1 , C ≤ n ] = 0. In fact E ∈ C ≥ n +1 if and only if forall F ∈ C ≤ n we have [ E, F ] = 0, and vice versa. The inclusion C ≥ n ֒ → C has a right adjoint whichwe denote E E ≥ n , and the inclusion C ≤ n ֒ → C has a left adjoint which we denote E E ≤ n . Theadjunctions furnish map E ≥ n +1 → E → E ≤ n and this extends to a distinguished triangle in a unique TOM BACHMANN and functorial way. The intersection C ♥ := C ≥ ∩ C ≤ called the heart . It is an abelian category. We put π C ( E ) = ( E ≤ ) ≥ ≃ ( E ≥ ) ≤ ∈ C ♥ and π C i ( E ) = π C ( E [ i ]). Then π C∗ is a homological functor on C . The t -structure is called non-degenerate if π C i ( E ) = 0 implies that E ≃ t -category we mean a triangulated category with a fixed t -structure.Suppose that ( C , τ ) is a site. Let for E ∈ SH ( C τ ) and i ∈ Z the sheaf π i ( E ) ∈ Shv ( C τ ) be definedas the sheaf associated with the presheaf C ∋ X π i ( E ( X )). Here we view E as a presheaf of spectra.By definition, local weak equivalences of spectra induce isomorphisms on π i , so π i ( E ) is well-defined for E ∈ SH( C τ ). This is a sheaf of abelian groups . PutSH( C τ ) ≥ = { E ∈ SH( C τ ) : π i ( E ) = 0 for i < } SH( C τ ) ≤ = { E ∈ SH( C τ ) : π i ( E ) = 0 for i > } . We define similarly for E ∈ D ( C τ ) the sheaf h i ( E ), and then the subcategories D ( C τ ) ≥ , D ( C τ ) ≤ . Lemma 2. If ( C , τ ) is a Grothendieck site, then the above construction provides SH( C τ ) with a non-degenerate t -structure. The functor π : SH( C τ ) ♥ → Shv ( C τ ) is an equivalence of categories. Moreoverlet F ∈ Shv ( C τ ) ≃ SH( C ) ♥ . Then for X ∈ C there is a natural isomorphism [Σ ∞ X + , F [ n ]] = H nτ ( X, F ) .Similar statements hold for D ( C τ ) in place of SH( C τ ) .Proof. For derived categories, this result is classical. For SH( C τ ), the result is also fairly well known, butthe author does not know an explicit reference, so we sketch a proof.Note that there is a Quillen adjunction (in the local model structures)Σ ∞ : sP re ( C τ ) ∗ ⇆ SH ( C τ ) : Ω ∞ . By direct computation using the above adjunction, we find that π i (Ω ∞ E ) = π i ( E ), for E ∈ SH( C τ ) and i ≥ C τ ) admits a t -structure , where E ∈ SH( C τ ) ≤ if and only if Ω ∞ ( E ) ≃ ∗ , and the subcategory SH( C τ ) ≥ is generated under homotopycolimits and extensions by Σ ∞ C + . We first need to show that this is the t -structure we want, i.e. that thepositive and negative parts are determined by vanishing of homotopy sheaves. Since π i (Ω ∞ E ) = π i ( E ),this is correct for the negative part. I claim that if E ∈ SH( C τ ) ≥ , then π i ( E ) = 0 for i <
0. If X ∈ sP re ( C τ ) ∗ , then π i (Σ ∞ X ) = 0 for i < E ∈ SH( C τ ) with π i ( E ) = 0 for i < E ∈ SH( C τ ) with π i ( E ) = 0 for i <
0. Consider thedecomposition E ≥ → E → E < . Then π i ( E ≥ ) = 0 for i <
0, so 0 = π i ( E ) = π i ( E < ) for i <
0. Itfollows that E < ≃ E ≃ E ≥ ∈ SH( E ) ≥ .The t -structure is non-degenerate because it is defined in terms of homotopy sheaves, and homotopysheaves detect weak equivalences by definition.We have an adjunction M : SH( C τ ) ⇆ D ( C τ ) : U. By construction U is t -exact and thus M is right t -exact. Consider the induced adjunction M ♥ : SH( C τ ) ♥ ⇆ D ( C τ ) ♥ : U. By direct computation using the classical Hurewicz isomorphism (and the above adjunction), π ( U M E ) = π ( E ) if E ∈ SH( C τ ) ≥ . It follows that U M ♥ ≃ id. Since U is faithful by definition, from this we deducethat M ♥ U ≃ id as well. Thus SH( C τ ) ♥ ≃ D ( C τ ) ♥ ≃ Shv ( C τ ), the latter equivalence being classical.Finally if X ∈ C and F ∈ Shv ( C τ ) then [Σ ∞ X + , F [ n ]] = [Σ ∞ X + , U F [ n ]] = H nτ ( X, F ), the first equalityby definition and the second by adjunction and the same result in D ( C τ ). (cid:3) Corollary 3.
Let ( C , τ ) be a Grothendieck site.(1) Let X ∈ C . If τ -cohomology on X commutes with filtered colimits of sheaves and the τ -cohomologicaldimension of X is finite, then Σ ∞ X + ∈ SH( C τ ) is a compact object.(2) For any collection E i ∈ SH( C ) and j ∈ Z we have π j ( L i E i ) = L i π j ( E i ) .Similarly for D ( C τ ) . The author would like to thank Saul Glasman for pointing out this reference.
OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 7
Proof.
Let us show that (1) reduces to (2). For E ∈ SH( C τ ) there is a conditionally convergent spectralsequence H pτ ( X, π − q E ) ⇒ [ X, E [ p + q ]] . Under our assumptions on the cohomological dimension of X , it converges strongly to the right handside. Under the assumption of commutation of cohomology with filtered colimits, by spectral sequencecomparison, it thus suffices to show that for E i ∈ SH( C τ ) we have π n ( L i E i ) = L i π n E i .Now we prove (2). For E ∈ SH ( C ) write π pj ( E )( X ) = π j ( E ( X )); this defines a presheaf of abeliangroups on C . By definition π j ( E ) = a τ π pj ( E ). Let { E i } i ∈ SH ( C ). Then π pj ( L i E i ) = L i π pj ( E i ),since homotopy groups of spectra commute with filtered colimits. We may assume that all the E i arecofibrant, so their presheaf direct sum coincides with the derived direct sum. In this case it remains toshow that a τ M i π pj ( E i ) ∼ = M i a τ π pj ( E i ) . (Note that here we write L i for both direct sums of presheaves and direct sums of sheaves, depending onwhether the terms on the right are presheaves or sheaves.) But this holds for any collection of presheaveson any site (both sides satisfy the same universal property).The proof for D is the same. (cid:3) We can enhance the functoriality of the SH construction as follows. Recall that a triangulated functor F : C → D between t -categories is called right (respectively left) t -exact if F ( C ≥ ) ⊂ D ≥ (respectively F ( C ≤ ) ⊂ D ≤ ). The functor is called t -exact if it is both left and right t -exact. Lemma 4.
Let f ∗ : Shv ( C ) ⇆ Shv ( D ) : f ∗ be a geometric morphism, where Shv ( D ) has enough points.Then in the adjunction Lf ∗ : SH( C ) ⇆ SH( D ) : Rf ∗ the left adjoint Lf ∗ is t -exact, the right adjoint Rf ∗ is left t -exact, and the induced functors ( Lf ∗ ) ♥ : SH( C ) ♥ ⇆ SH( D ) ♥ : ( Rf ∗ ) ♥ coincide (under the identification from Lemma 2) with f ∗ ⊢ f ∗ .Similar statements hold for D in place of SH . The author contends that the assumption that D has enough points is not really necessary. See also[Lur09, Remark 6.5.1.4]. Proof.
Certainly Rf ∗ is left t -exact if Lf ∗ is t -exact by adjunction, and ( Rf ∗ ) ♥ is right adjoint to ( Lf ∗ ) ♥ ,so it suffices to prove the claims for Lf ∗ .Since D has enough points, it is then enough to assume that Shv ( D ) = Set . (Indeed let p : Set → Shv ( D ) be a point; we will have p ∗ π i ( Lf ∗ E ) = π i ( Lp ∗ Lf ∗ E ) = p ∗ f ∗ π i E for all E ∈ SH( C ) by applying the reduced case to p and f p which are points of D and C , respectively.Since D has enough points it follows that π i ( Lf ∗ E ) = f ∗ π i ( E ), as was to be shown.)Let p ∗ : Shv ( C ) ⇆ Set : p ∗ be a point of C . Then p ∗ corresponds to a pro-object in C , which is to saythat there is a filtered family X α ∈ C such that for F ∈ Shv ( C ) we have p ∗ ( F ) = colim α F ( X α ) [GK15,Proposition 1.4 and Remark 1.5].It follows that for E ∈ SH s ( C ) we have π i ( p ∗ E ) = π i (colim α E ( X α )) ∼ = colim α π i ( E ( X α )) = p ∗ π i ( E ) , where the isomorphism in the middle holds because homotopy groups commute with filtered colimits ofspectra. In particular p ∗ preserves weak equivalences and so p ∗ ≃ Lp ∗ . Thus the previous equation isprecisely what we intended to prove. (cid:3) Recollections on Real ´Etale Cohomology If X is a scheme, let R ( X ) be the set of pairs ( x, p ) where x ∈ X and p is an ordering of the residuefield k ( x ). For a ring A we put Sper ( A ) = R ( Spec ( A )). A family of morphisms { α i : X i → X } i ∈ I iscalled a real ´etale covering if each α is ´etale and R ( X ) = ∪ i α ( R ( X i )). (Note that for ( x, p ) ∈ X i theextension k ( x ) /k ( α ( x )) defines by restriction an ordering of k ( α ( x )).) The real ´etale coverings define atopology on all schemes [Sch94, (1.1)] called the real ´etale topology . We often abbreviate this name to“r´et-topology”. TOM BACHMANN
For a scheme X , we let X r ´ et denote the small real ´etale site on X and Sm ( X ) r ´ et the site of smooth(separated, finite type) schemes over X with the real ´etale topology. If f : X → Y is any morphism ofschemes, we get the usual base change functors f ∗ : Y r ´ et → X r ´ et and f ∗ : Sm ( Y ) → Sm ( X ). Also thenatural inclusion e : X r ´ et → Sm ( X ) induces an adjunction e p : P re ( X r ´ et ) ⇆ P re ( Sm ( X )) : r = e ∗ . Lemma 5. If X is a scheme, the above adjunction induces a geometric morphism e : Shv ( X r ´ et ) ⇆ Shv ( Sm ( X ) r ´ et ) : 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 preservessheaves. Moreover r commutes with taking the associated sheaf, because every cover of Y ∈ X r ´ et in Sm ( X ) comes from a cover in X r ´ et (because ´etale morphisms are stable under composition). Itfollows that r commutes with colimits. Since e : X r ´ et → Sm ( X ) r ´ et preserves pullbacks (and X r ´ et haspullbacks!), the adjunction is a geometric morphism [Sta17, Tag 00X6]. In order to see that e is fullyfaithful, i.e. F → reF an isomorphism for every F ∈ Shv ( X r ´ et ), we note that for the presheaf adjunction e p : P re ( X r ´ et ) ⇆ P re ( 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 notethat for a sheaf F we have eF = a r ´ et e p F and thus reF = ra r ´ et e p F = a r ´ et re p F = a r ´ et F = F , where wehave used again that r commutes with taking the associated sheaf. (cid:3) Corollary 6. If X is a scheme, the induced derived functor Le : SH( X r ´ et ) → SH( Sm ( X ) r ´ et ) is t -exactand fully faithful. Similarly for D in place of SH .Proof. The functor is fully faithful by Lemmas 5 and 1. It is t -exact by Lemma 4. (cid:3) Lemma 7. If f : X → Y is a morphism of schemes, then the induced functor f ∗ : Y r ´ et → X r ´ et is theleft adjoint of a geometric morphism of sites. Moreover the derived functor Lf ∗ : SH( Y r ´ et ) → SH( X r ´ et ) is t -exact, and similarly for Lf ∗ : D ( Y r ´ et ) → D ( X r ´ et ) .Proof. The “moreover” part follows from Lemma 4.Since f ∗ : Y r ´ et → X r ´ et preserves covers f ∗ : P re ( X r ´ et ) → P re ( Y r ´ et ) preserves sheaves and themorphism is continuous. It is a geometric morphism of sites because f ∗ preserves pullbacks [Sta17, Tag00X6]. (cid:3) If X is a scheme, there is the natural map X → X × A corresponding to the point 0 ∈ A . Similarlythere is the natural map X ` X → X × ( A \
0) corresponding to the points ± ∈ A \ Theorem 8.
Let X be a scheme and F ∈ Shv ( X r ´ et ) . Then for any p ≥ the natural maps X → X × A and X ` X → X × ( A \ induce isomorphisms H pr ´ et ( X × A , F ) → H pr ´ et ( X, F ) H pr ´ et ( X × ( A \ , F ) → H pr ´ et ( X, F ) ⊕ H pr ´ et ( X, F ) . Proof.
The first statement is homotopy invariance, see [Sch94, Example 16.7.2].For the second statement, we follow closely that proof. Let f : X ` X → X × ( A \
0) be the canonicalmap. It suffices to show that R n f ∗ F = 0 for n > R f ∗ F = F , where we identify F with itspullback to X ` X and X × ( A \
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 r ´ et -cohomology commutes with filtered colimitsof sheaves, and all sheaves on a spectral space are filtered colimits of constructible sheaves; see again loc. cit. ). Next, writing X = Spec ( A ) as the inverse limit of the filtering system Spec ( A ′ ), with A ′ ⊂ A finitely generated over Z , and using Proposition (A.9) of loc. cit. , we may assume that X is of finite typeover Z .But Sper ( Z ) = Sper ( Q ) = Sper ( R ), whence H pr ´ et ( X, F ) = H pr ´ et ( X × Z R , F ), so we may assume that X is of finite type over R .We may further assume that F = M Z is the constant sheaf on a closed, constructible subset of X (Proposition (A.6) of loc. cit. ).It is thus enough to prove the analog of our result for an affine semi-algebraic space X over R and F = M a constant sheaf. But then H ∗ r ´ et ( X, M ) = H ∗ sing ( X ( R ) , M ) [Del91, Theorem II.5.7] and so on,so this is obvious. (cid:3) OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 9
Theorem 9 ((Proper Base Change)) . Consider a cartesian square of schemes X ′ g ′ −−−−→ X f ′ y y f Y ′ g −−−−→ Y, with f proper and Y finite-dimensional Noetherian. Then for any E ∈ SH( X r ´ et ) (respectively E ∈ D ( X r ´ et ) ) the canonical map g ∗ Rf ∗ ( E ) → Rf ′∗ g ′∗ ( E ) is a weak equivalence.Proof. We prove the claim for SH, the proof we give will work just as well for D . We proceed in severalsteps.Step 0. If g is ´etale, then the claim follows from the observation that f ∗ g = g ′ f ′∗ .Step 1. If f : X → Y is any morphism and E ∈ SH( X r ´ et ), then there is a conditionally convergentspectral sequence E pq = R p f ∗ π − q E ⇒ π − p − q ( Rf ∗ E ) . For this, let E ∈ Spt ( X r ´ et ) also denote a fibrant model. Then Rf ∗ E ≃ f ∗ E and for U ∈ Y r ´ et we have f ∗ ( E )( U ) = E ( f ∗ U ). Since E is fibrant there is a conditionally convergent descent spectral sequence H p ( f ∗ U, π − q ( E )) ⇒ π − p − q ( E ( f ∗ U )) . By varying U , this yields a presheaf of spectral sequences on Y r ´ et . Equivalently, this is a spectral sequenceof presheaves. Taking the associated sheaf on both sides we obtain a conditionally convergent spectralsequence a r ´ et H pr ´ et ( f ∗ • , π − q ( E )) ⇒ π − p − q ( f ∗ E ) . It remains to see that a r ´ et H pr ´ et ( f ∗ • , F ) = R p f ∗ F , for any sheaf F on X r ´ et . For this we view F ∈ D ( X r ´ et ) ♥ . Then by definition R p f ∗ F = π − p Rf ∗ F . Repeating the above argument with D ( X r ´ et ) inplace of SH( X r ´ et ), we obtain a conditionally convergent spectral sequence a r ´ et H pr ´ et ( f ∗ • , π − q F ) ⇒ R p + q f ∗ F. Since π − q F = 0 for q = 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 ∈ Shv ( X r ´ et ) and p > n we have R p f ∗ F = 0.Indeed in this situation, by the proper base change theorem in real ´etale cohomology [Sch94, Theorem16.2], for any real closed point y → Y we get ( R p f ∗ F ) y = H pr ´ et ( X y , F | X y ). Since real closed fields are thestalks of the r´et-topology, in order for a sheaf G ∈ Shv ( Y r ´ et ) to be zero it is necessary and sufficient that G y = 0 for all such y . But real ´etale cohomological dimension is bounded by Krull dimension [Sch94,Theorem 7.6], so we find that R p f ∗ F = 0 for p > n , as claimed.Conclusion of proof. Since isomorphism in SH( Y ′ r ´ et ) is local on Y ′ , it is an easy consequence of step 0that we may assume that Y ′ is quasi-compact (e.g. affine). Then f ′ is of bounded relative dimension(being of finite type).Now let E ∈ SH( X r ´ et ). By t -exactness of g ∗ and g ′∗ we get from step 1 conditionally convergentspectral sequences g ∗ R p f ∗ π − q E ⇒ π − p − q ( g ∗ Rf ∗ E )and R p f ′∗ g ′∗ π − q E ⇒ π − p − q ( Rf ′∗ g ′∗ E ) . The exchange transformation g ∗ Rf ∗ ( E ) → Rf ′∗ g ′∗ ( E ) induces a morphism of spectral sequences (i.e.respecting the differentials and filtrations). By proper base change for sheaves, we have g ∗ R p f ∗ ∼ = R p f ′∗ g ′∗ . 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. (cid:3) Remark. The only place in the above proof where we have used the assumption on Y is in step 1, namelyin the construction of the conditionally convergent spectral sequence R p f ∗ π − q E ⇒ π − p − q ( Rf ∗ E ) . The author does not know how to construct such a spectral sequence in general. He nonetheless contendsthat the proper base change theorem should be true without assumptions on Y , but perhaps a differentproof is needed. Remark. In the above proof we deduce proper base change for spectra and unbounded complexes fromproper base change for bounded complexes. Since we are dealing with hypercomplete toposes, this is nottautological; see for example [Lur09, Counterexample 6.5.4.2 and Remark 6.5.4.3]. The crucial propertywhich seems to make the proof work is encapsulated in step 2 and might be phrased as “a propermorphism is locally of finite relative r´et-cohomological dimension”. The same is true in ´etale (instead ofreal ´etale) cohomology and this seems to be what the proof of proper base change for unbounded ´etalecomplexes [CD13, Theorem 1.2.1] ultimately rests on, in the guise of [CD13, Lemma 1.1.7]. This fails fora general proper morphism of topological spaces (consider for example an infinite product of compactpositive dimensional spaces mapping to the point).4.
Recollections on Motivic Homotopy Theory
We denote the stable motivic homotopy category over a base scheme X [Ayo07] by SH ( X ), and thestable A -derived category over X [CD09b, Section 5.3] by D A ( X ). We write X ∈ SH ( X ) for themonoidal unit. If the context is clear we may just write .Let f : Y → X be a finite ´etale morphism of schemes. Then in the category SH ( X ) we have an inducedmorphism f : f Y → X and consequently D ( f ) : D ( X ) → D ( f Y ). Here DE := Hom( E, ). Nowin fact whenever f : Y → X is smooth proper then D ( f Y ) ≃ f ∗ Y [CD09b, Proposition 2.4.31]and if f is ´etale then f ∗ ( Y ) ≃ f ( Y ) [CD09b, Example 2.4.3(2), Definition 2.4.24 and Proposition2.4.31]. Let us write α X,Y : f Y → D ( f Y ) for this canonical isomorphism. We can then form thecommutative diagram D ( f Y ) α X,Y ←−−−− f YD ( f ) x tr f x D ( X ) α X,X ←−−−− X , where tr f is defined so that the diagram commutes. This is the duality transfer of f as defined in [RØ08,Section 2.3].Now suppose that k is a perfect field. Recall that then SH ( k ) has a t -structure. To define it, for E ∈ SH ( k ) denote by π i ( E ) j the Nisnevich sheaf associated with the presheaf X [Σ ∞ X + [ i ] , E ∧ G ∧ jm ].Then E ∈ SH ( k ) ≥ if and only if π i ( E ) j = 0 for all i < j ∈ Z . This indeed defines a t -structure[Mor03, Section 5.2], and the its heart can be described explicitly: it is equivalent to the category of homotopy modules [Mor03, Theorem 5.2.6].Let F ∗ ∈ SH ( k ) is a homotopy module, which we identify with an element in the heart of thehomotopy t -structure. Given a finite ´etale morphism f : Y → X of essentially k -smooth schemes, write s : X → Spec ( k ) for the structure map. We then define tr f : F n ( Y ) → F n ( X ) as tr f ( F ) := tr ∗ f : [ f Y , s ∗ F ∧ G ∧ nm ] → [ X , s ∗ F ∧ G ∧ nm ] . This transfer has the usual properties, of which we recall two.
Proposition 10 ((Base Change)) . Let k be a perfect field, g : V → X be a morphism of essentially k -smooth schemes and f : Y → X finite ´etale. Consider the cartesian square W q −−−−→ Y p y y f V −−−−→ g X. Then for any homotopy module F ∗ , we have g ∗ tr f = tr p q ∗ : F ∗ ( Y ) → F ∗ ( V ) .Proof. Note that p : W → V is finite ´etale, so this makes sense. By continuity (of F ), we may assumethat X and V are smooth (and hence so are Y and W ). Write s : X → Spec ( k ) for the structure map.If t : A → B is any map in SH ( X ), then the canonical diagram F ∗ ( B ) = [ B, s ∗ F ] ◦ t −−−−→ [ A, s ∗ F ] = F ∗ ( A ) g ∗ y g ∗ y F ∗ ( g ∗ B ) = [ g ∗ B, g ∗ s ∗ F ] ◦ g ( t ) −−−−→ [ g ∗ A, g ∗ s ∗ F ] = F ∗ ( g ∗ A )commutes, since g ∗ is a functor. Applying this to tr f : X → f Y it is enough to prove that g ∗ ( tr f ) = tr p under the canonical identifications. OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 11
Let f + : f Y ≃ Σ ∞ X Y + → Σ ∞ X X + = X be the canonical map (so that tr f = D ( f + ) via α X,Y ), andsimilarly for p + . Then g ∗ ( f + ) ≃ p + and consequently g ∗ ( D ( f + )) ≃ D ( p + ). It thus remains to show that α • , • is natural, i.e. that g ∗ α X,Y = α V,W : Σ ∞ V W + → D (Σ ∞ V W + ).For this we use the notation of [CD09b, Example 2.4.3(2), Definition 2.4.24 and Proposition 2.4.31].The isomorphism α X,Y : f → D ( f ) is factored into the isomorphisms D ( f ) → f ∗ , the Thomtransformation f Ω f → f ∗ [CD09b, Definition 2.4.21] and Ω f → . All of these are natural in therequired sense. (cid:3) Lemma 11 ((Commutation of Transfer with External Product)) . Let f : X ′ → X and g : Y ′ → Y befinite ´etale. Then s X × Y ( tr f × g ) = s X ( tr f ) ∧ s Y ( tr g ) : Σ ∞ ( X ′ × Y ′ ) + ≃ Σ ∞ X ′ + ∧ Σ ∞ Y ′ + → Σ ∞ ( X × Y ) + ≃ Σ ∞ X + ∧ Σ ∞ Y + . Here we write s X : X → Spec ( k ) for the canonical map, and similarly for Y, X × Y . Proof.
Write p X : X × Y → X and p Y : X × Y → Y for the projections. I claim that the followingdiagram commutes up to natural isomorphism: SH ( X ) × SH ( Y ) p ∗ X ∧ p ∗ Y −−−−−→ SH ( X × Y ) s X ∧ s Y y s X × Y y SH ( k ) SH ( k ) . To prove the claim first note that there is, for T ∈ SH ( X ) , U ∈ SH ( Y ), a natural map s X × Y ( p ∗ X T ∧ p ∗ Y U ) → s X T ∧ s y U , which can be obtained by adjunctions, using that the pullback functors aremonoidal, and that s X × Y = s X ◦ p X (and similarly for Y ). Then to prove that the comparison map isan isomorphism it suffices to consider T = Σ ∞ X ′ , U = Σ ∞ Y ′ for X ′ → X smooth any Y ′ → Y smooth(note that all our functors are left adjoints and so commute with arbitrary sums, and objects of theforms T, U are generators). But then the claim boils down to X ′ × k Y ′ ∼ = ( X ′ × Y ) × X × Y ( X × Y ′ )which is clear.To prove the lemma, we now specialise to f : X ′ → X and g : Y ′ → Y finite ´etale. Then tr f × g = s X × Y ( D Σ ∞ X × Y ( f × g ) + ) . Note that Σ ∞ X × Y ( f × g ) + = p ∗ X Σ ∞ X f + ∧ p ∗ Y Σ ∞ Y g + . Since p ∗ X , p ∗ Y are monoidal we compute tr f × g = s X × Y p ∗ X D Σ ∞ X f + ∧ p ∗ Y D Σ ∞ Y g + = s X D Σ ∞ X f + ∧ s Y D Σ ∞ Y g + , where in the last equality we have used the claim. Since s X D Σ ∞ X f + = tr f by definition (and similarlyfor Y ), this is what we wanted to prove. (cid:3) Recall also the homotopy module K MW ∗ = π ( ) ∗ of Milnor-Witt K-theory [Mor12, Chapter 3]. Everyhomotopy module F ∗ is a module over K MW ∗ in the sense that there are natural pairings K MW ∗ ( X ) ⊗ F ∗ ( X ) → F ∗ + ∗ ( X ). Corollary 12 ((Projection Formula)) . Let k be a perfect field, f : Y → X a finite ´etale morphism ofessentially k -smooth schemes, and F ∗ a homotopy module. Then for a ∈ K MW ∗ ( Y ) and b ∈ F ∗ ( X ) wehave tr f ( af ∗ b ) = tr f ( a ) b . Similarly for a ∈ K MW ∗ ( X ) and b ∈ F ∗ ( Y ) we have tr f ( f ∗ ( a ) b ) = atr f ( b ) .Proof. The usual proof works, see for example [CF17, Proof of Corollary 3.4]. We review it. We onlyshow the first statement, the second is similar. Consider the cartesian square Y (id × f ) δ Y −−−−−−→ Y × X f y y f × id X −−−−→ δ X X × X, where δ X : X → X × X is the diagonal and similarly for Y . We have the map β : Σ ∞ Y + ∧ Σ ∞ X + → K MW ∗ ∧ F → F , where K MW ∗ ∧ F → F is the module structure and the first map is the tensor productof Σ ∞ Y + → K MW ∗ (corresponding to a ) and Σ ∞ X + → F (corresponding to b ). This defines an element β ∈ F ( Y × X ). We have tr f ((id × f ) δ Y ) ∗ β = tr f ( af ∗ b ) and δ ∗ X tr f × id Y β = tr f ( a ) b (the latter since tr id = id and tr f × g ( x ⊗ y ) = tr f ( x ) ⊗ tr g ( y ) by Lemma 11). These two elements are equal by the basechange formula, i.e. Proposition 10. (cid:3) Recollections on Pre-Motivic Categories
The six functors formalism [CD09b, Section 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 isnow possible to reduce this to establishing a few axioms.Let S be a base category of schemes. Recall that a pre-motivic category M over S consists of [CD13,Definition A.1.1] a pseudofunctor M on S , taking values in triangulated, closed symmetric monoidalcategories. Often these categories will be obtained as the homotopy categories of a pseudofunctor takingvalues in suitable Quillen model categories and left Quillen functors. For f : X → Y ∈ S , the functor M ( f ) : M ( Y ) → M ( X ) is denoted f ∗ . For any f , the functor f ∗ has a triangulated right adjoint f ∗ (which is not required to be monoidal). If f is smooth, then f ∗ has a triangulated left adjoint f (also not required to be monoidal). Moreover, M needs to satisfy smooth base change and the smoothprojection formula, in the following sense.Let Y q −−−−→ X g y y f T p −−−−→ S be a cartesian square in S , with p smooth. Then smooth base change means that the natural transfor-mation q g ∗ → f ∗ p is required to be a natural isomorphism.Finally, let f : Y → X be a smooth morphism in S . Then the smooth projection formula means that,for E ∈ M ( X ) and F ∈ M ( Y ) we have f ( F ⊗ f ∗ E ) ≃ f ( F ) ⊗ E , via the canonical map.Here are some further properties a pre-motivic category can satisfy. We say M satisfies the homotopyproperty if for every X ∈ S the natural map p → ∈ M ( X ) is an isomorphism, where p : A × X → X is the canonical map.Let now q : P × X → X be the canonical map. We say that M satisfies the stability property if the cone of the canonical map q → ∈ M ( X ) is a ⊗ -invertible object. In this case we write (1) = f ib ( q → )[ −
2] and then as usual E ( n ) = E ⊗ (1) ⊗ n for n ∈ Z , E ∈ M ( X ).Finally, let X ∈ S , j : U → X ∈ S an open immersion, and i : Z → S a complementary closedimmersion. Then for E ∈ M ( U ) there are the adjunction maps j j ∗ E → E → i ∗ i ∗ E. We say that 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 indetail by Ayoub, and then formalised by Cisinski-D´eglise. Theorem 13 ((Ayoub, Cisinski-D´eglise)) . Let S be the category of Noetherian schemes of finite dimen-sion and M a pre-motivic category which satisfies the homotopy property, the stability property, and thelocalisation property. Then if M ( X ) is a well-generated triangulated category for every X , M satisfiesthe full six functors formalism.Proof. This is proved for “adequate categories of schemes” in [CD09b, Theorem 2.4.50], of which Noe-therian finite dimensional schemes are an example. (cid:3)
One further property we will make use of is continuity . This can be formulated as follows. Let { S α } α ∈ A be an inverse system in S , where all the transition morphisms are affine and the limit S := lim α S α existsin S . Write p α : S → S α for the canonical projection. Let E ∈ M ( S α ) for some α ∈ A and write for α > α , E α = ( S α → S α ) ∗ E . We say that M satisfies the continuity property if for every affine inversesystem S α as above, every E and every i ∈ Z the canonical mapcolim α>α [ ( i ) , E α ] M ( S α ) → [ ( i ) , p ∗ α E ] M ( S ) is an isomorphism.We in particular use the following consequence of continuity and localisation. Corollary 14.
Suppose that M be a pre-motivic category over S (where S contains all Henselizationsof its schemes), coming from a pseudofunctor valued in model categories. Assume that M satisfiescontinuity and localisation.Let E ∈ M ( X ) , where X is Noetherian of finite dimension. Then E ≃ if and only if for everymorphism f : Spec ( k ) → X with k a field we have f ∗ E ≃ . OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 13
Proof.
By localisation, we may assume that X is reduced (see for example [CD09b, Proposition 2.3.6(1)]).By [CD09b, Proposition 4.3.9] (this result requires M to come from a model category) we may assumethat X is (Henselian) local with closed point x and open complement U . By localisation, it suffices toshow that E | x ≃ E | U ≃
0. The former holds by assumption, and the latter by induction on thedimension. This concludes the proof. (cid:3)
Example. The pseudofunctors X SH ( X ) and X D A ( X ) satisfy the six functors formalism andcontinuity (for the base category of Noetherian finite dimensional schemes) [Ayo07, CD09b].6. Recollections on Monoidal Bousfield Localization
Let M be a monoidal model category and α : Y ′ → Y ∈ M a morphism. We wish to “monoidallyinvert α ”, by which we mean passing to a model category L ⊗ α M obtained by localizing M and such thatfor every T ∈ L ⊗ α M the induced map α T : T ⊗ L Y ′ → T ⊗ L Y is a weak equivalence. We will also write L ⊗ α M =: M [ α − ] and even Ho ( M [ α − ]) =: Ho ( M )[ α − ].The monoidal α -localisation exists very generally. Suppose that Y ′ and Y are cofibrant, and that M admits a set of cofibrant homotopy generators G (for example M combinatorial [Bar10, Corollary 4.33]).Let H α = { Y ′ ⊗ T α ⊗ id −−−→ Y ⊗ T | T ∈ G } . When no confusion can arise, we will denote H α just by H .Then the Bousfield localisation L H M , if it exists (for example if M is left proper and combinatorial) is M [ α − ]. We will call H α -local objects α -local. As a further sanity check, the model category L H M isstill monoidal as follows from [Bar10, Proposition 4.47].The situation simplifies somewhat if Y ′ and Y are invertible and M is stable. Then we may as wellassume that Y ′ = . Given T ∈ M cofibrant we can consider the directed system T ∼ = T ⊗ id ⊗ α −−−→ T ⊗ Y ∼ = T ⊗ Y ⊗ → T ⊗ Y ⊗ → . . . and its homotopy colimit T [ α − ] := hocolim n T ⊗ X ⊗ n . More generally, if T is not cofibrant, we caneither first cofibrantly replace it, or use the derived tensor product. Either way, we denote the resultstill by T [ α − ]. The main point of this section is to show that under suitable conditions, T [ α − ] is the α -localization of T .Clearly this is only a reasonable expectation under some compact generation assumption. Moregenerally, one would expect a transfinite iteration of α . Since all our applications will be in compactlygenerated situations, we refrain from giving the more general argument.Recall that by a set of compact homotopy generators G for M we mean a set of (usually cofibrant)objects G ⊂ Ob ( M ) such that M is generated by the objects in G under homotopy colimits, and such thatfor any directed system X → X → · · · ∈ M and T ∈ G , the canonical map hocolim i M ap d ( T, X i ) → M ap d ( T, hocolim i X i ) is an equivalence. Lemma 15.
Let α : → Y be a map between objects in a symmetric monoidal, stable model categorysuch that Y is invertible (in the homotopy category). Assume that M has a set of compact homotopygenerators G , and that M [ α − ] exists.Then for each U ∈ M the object U [ α − ] is α -local and α -locally weakly equivalent to U . In otherwords, U U [ α − ] is an α -localization functor.Also G defines a set of compact homotopy generators for M [ α − ] .Proof. We first show that the images of G in Ho ( M [ α − ]) are compact homotopy generators. Generationis clear, and for homotopy compactness it is enough to show that a filtered homotopy colimit of α -localobjects is α -local. But this follows from homotopy compactness of T ⊗ Y ⊗ n (for T ∈ G and n ∈ { , } )and definition of α -locality.In a model category N with compact homotopy generators, if T → T → . . . is a directed system ofweak equivalences then hocolim i T i is weakly equivalent to T . (This follows from the same result in thecategory of simplicial sets.) Thus U [ α − ] is α -locally weakly equivalent to U .It remains to see that U [ α − ] is α -local. This follows from the next two lemmas. (cid:3) In the above lemma, we have defined an object X to be α -local if for all T ∈ M the induced map α ∗ : M ap d ( T ⊗ L Y, X ) → M ap d ( T, X ) is an equivalence, because this is the way Bousfield localizationworks. Another intuitively appealing property would be for the canonical map X → X ⊗ Y to be anequivalence. As the next lemma shows, these two notions agree in our case. Lemma 16.
Let M be a symmetric monoidal model category and α : → Y a morphism with Y invertible.Call an object X ∈ M α ′ -local if X → X ⊗ L Y is a weak equivalence. Then X is α -local if and onlyif X is α ′ -local, if and only if X is α ⊗ α -local. Proof.
We shall show that (1) X is α -local if and only if it is α ⊗ α -local, (2) X is α ′ -local if and only ifit is ( α ⊗ α ) ′ -local, (3) X is α ′ -local if it is α -local and (4) X is α ⊗ α -local if it is ( α ⊗ α ) ′ -local.All tensor products and mapping spaces will be derived in this proof.(1) Consider the string of maps M ap ( T ⊗ Y ⊗ , X ) → M ap ( T ⊗ Y ⊗ , X ) → M ap ( T ⊗ Y, X ) → M ap ( T, X ) . If X is α ⊗ α -local, then the composite of any two consecutive maps is an equivalence, and hence allmaps are equivalences by 2-out-of-6. Consequently X is α -local. The converse is clear.(2) Consider the string of maps X → X ⊗ Y → X ⊗ Y ⊗ → X ⊗ Y ⊗ . If X is ( α ⊗ α ) ′ -local then so is Z ⊗ X for any Z , since (derived) tensor product preserves weak equivalences.It follows that X ⊗ Y is ( α ⊗ α ) ′ -local, and hence the composite of any two consecutive maps is anequivalence. Again by 2-out-of-6 this implies that X is α ′ -local. The converse is clear.(3) An object X is α -local if (and only if) for all T ∈ M the map M ap ( T ⊗ Y, X ) → M ap ( T, X ) isa weak equivalence (of simplicial sets). In particular T → T ⊗ Y is an α -local weak equivalence for all T . It also follows that X ⊗ Y is α -local if X is (here we use invertibility of Y ). Since X → X ⊗ Y is an α -local weak equivalence, it is a weak equivalence if X (and hence X ⊗ Y ) is α -local. Thus X is α ′ -localif it is α -local.(4) For any simplicial set K we have [ K, M ap ( T, X )] = [ K ⊗ T, X ] (using a framing if the modelcategory is not simplicial). It follows that X is α -local if and only if for all T ∈ M the map α ∗ :[ T ⊗ Y, X ] → [ T, X ] is an isomorphism. In particular, this property can be checked entirely in thehomotopy category of M , in which we will work from now on.Suppose, for now, that X is α ′ -local. (We will find that our strategy does not work, but it will workfor α ⊗ α , and this is all that is left to prove.) We can choose an inverse equivalence β : X ⊗ Y → X .We consider the map β : [ T, X ] → [ T ⊗ Y, X ] sending f : T → X to T ⊗ Y f ⊗ id −−−→ X ⊗ Y β −→ X . We wouldlike to say that β is inverse to α ∗ . Given f : T → X we get a commutative diagram T ⊗ Y f ⊗ id −−−−→ X ⊗ Y α x α x T f −−−−→ X. Consequently α ∗ α ∗ β = α ∗ : [ T, X ] → [ T, X ⊗ Y ] and thus α ∗ β = id (note that α ∗ means compositionwith X → X ⊗ Y , which is an isomorphism).The problem is with showing that βα ∗ = id. For this we fix f : T ⊗ Y → X and consider the diagram T ⊗ Y ⊗ Y f ⊗ id −−−−→ X ⊗ Y id ⊗ α ⊗ id x α x T ⊗ ⊗ Y f −−−−→ X. If it commutes for all such f , then βα ∗ = 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 invertibleobject is the identity [Dug14, Propositions 4.20 and 4.21]. Consequently our argument works for α ⊗ α ,and this is what we set out to prove. (cid:3) Remark. The assumption that Y is invertible is necessary in general for the above result. For example, if M is a cartesian symmetric monoidal model category, then there cannot be any α ′ -local objects unless ∗ = → Y is already an equivalence. Lemma 17.
Notations and assumptions as in Lemma 15.For any (cofibrant) X ∈ M , the object X [ α − ] is α -local.Proof. By the previous lemma, it suffices to show that X [ α − ] is ( α ⊗ α ) ′ -local. Clearly X [ α − ] ≃ X [( α ⊗ α ) − ], i.e. we may assume without loss of generality that Y is a square, and so its switchisomorphism (in the homotopy category) is the identity.Since tensor product commutes with colimits (in each variable) we have X [1 /f ] ≃ X ⊗ L [1 /f ], andwe can simplify notation by assuming without loss of generality that X = . OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 15
What we need to prove is that the following diagram induces an equivalence on homotopy colimits: f −−−−→ G f −−−−→ G ⊗ G f −−−−→ G ⊗ G ⊗ G −−−−→ . . . h y h y h y h y G f ′ −−−−→ G ⊗ G f ′ −−−−→ G ⊗ G ⊗ G f ′ −−−−→ G ⊗ G ⊗ G −−−−→ . . . Because of the domains and codomains, it is tempting to guess that f i ≃ h i ≃ f ′ i . Here we write f ≃ 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, • ] to our diagram we get a diagram of abeliangroups which we need to show induces an isomorphism on colimits. Homotopic maps become equal whenapplying [ T, • ], and then the desired result follows from an easy diagram chase. By compact generationand stability, this will conclude the proof.It remains to prove the claim. For this we may work entirely in the homotopy category, which we willdo 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 = f i ;one may check that the maps differ by appropriate switches of Y . However, we have assumed that theswitch on Y is the identity, so indeed f i = f ′ i as well. (cid:3) Remark. The stability assumption was used in the above proof in the following form: if A → B is anymorphism in M and [ T, A ] → [ T, B ] is an isomorphism for all homotopy compact T , then A → B is aweak 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 followingcounterexample from Marc Hoyois: let M be the model category of small, stable ∞ -categories, Y = the category of finite spectra and α = 2, i.e. the functor which sends a finite spectrum s to s ⊕ s . Then C ∈ M is α ′ -local only if it is trivial. Indeed for c ∈ C the map [ c, c ] → [ c ⊕ c, c ⊕ c ] needs to be anisomorphism, which forces c ≃
0. But one may show that [1 /α ] is not the zero category, and so is not α ′ -local (let alone α -local).See [Hoy16, Theorem 3.7] for a criterion that can be applied in unstable situations.7. The Theorem of Jacobson and ρ -stable Homotopy Modules Throughout this section, k is a field of characteristic zero. Recall that the real ´etale topology is finerthan the Nisnevich topology; in particular every real ´etale sheaf is a Nisnevich sheaf. Theorem 18 ((Jacobson [Jac17], Theorem 8.5)) . There is a canonical isomorphism (in
Shv
Nis ( Sm ( k )) ) colim n I n → a r ´ et Z , where the transition maps I n → I n +1 are given by multiplication with h , i ∈ I . Here I denotes the sheaf of fundamental ideals on Sm ( k ) Nis , i.e. the sheaf associated with the presheaf X I ( X ), where I ( X ) is the fundamental ideal of the Witt ring of X [Kne77]. We similarly write W for the sheaf of Witt rings, etc.Let us recall the construction of the isomorphism in Jacobson’s theorem. If φ ∈ W ( K ), where K is a field, and p is an ordering of K , then there is the signature σ p ( φ ) ∈ Z . If φ ∈ W ( X ), define σ ( φ ) : R ( X ) → Z as follows. For ( x, p ) ∈ R ( X ) put σ ( φ )( x, p ) = σ p ( φ | x ). Then one shows that σ ( φ ) is a continuous function from R ( X ) to Z , i.e. an element of H r ´ et ( X, Z ).Next if φ ∈ I ( k ) then σ p ( φ ) ∈ Z . Consequently if φ ∈ I ( X ) also σ ( φ ) ∈ H r ´ et ( X, Z ). We may thusdefine ˜ σ ( φ ) = σ ( φ ) / σ : I ( X ) → H r ´ et ( X, Z ). Similarly we get ˜ σ : I n ( X ) → H r ´ et ( X, Z ) with ˜ σ ( φ ) = σ ( φ ) / n for φ ∈ I n ( X ). For each n there is a commutative diagram I n ( X ) ˜ σ −−−−→ H r ´ et ( X, Z ) y (cid:13)(cid:13)(cid:13) I n +1 ( X ) ˜ σ −−−−→ H r ´ et ( X, Z ) . Consequently there is an induced map ˜ σ : colim n I n ( X ) → H r ´ et ( X, Z ). The claim is that this is anisomorphism after sheafifying, i.e. for X local. Corollary 19.
Let K MWn denote the n -th unramified Milnor-Witt K-theory sheaf. Then there is acanonical isomorphism colim n K MWn → a r ´ et Z . Here the colimit is along multiplication with ρ := − [ − ∈ K MW ( k ) . Proof.
Recall the element h ∈ K MW ( k ) with the following properties: K MWn /h = I n [Mor04, Theoreme2.1] and for a ∈ K MW ( k ) we have a h = 0 [Mor12, Corollary 3.8] (this relation is the analogue of thefact that in a graded commutative ring R ∗ with a ∈ R we have a = − a by graded commutativity, so2 a = 0). Consequently ρ h = 0 and so colim n K MWn → colim n I n is an isomorphism. It remains to notethat the image of ρ in K MW /h ( k ) ∼ = I ( k ) is given by − ( h− i −
1) = 2 ∈ I ( k ) ⊂ W ( k ), so the inducedtransition maps in the colimit are precisely those used in Jacobson’s theorem. (cid:3) Note that the sheaves I n form a homotopy module, namely the homotopy module of Witt K -theory[Mor12, Examples 3.33 and 3.33] [Mor04, Theoreme 2.1]; see also [GSZ16]. Consequently they havetransfers for finite separable field extensions. The sheaf a r ´ et Z also has transfers for finite (separable)field extensions. Indeed if l/k is finite then Sper ( l ) → Sper ( k ) is a finite-sheeted local homeomorphism[Sch85, 3.5.6 Remark (ii)] and hence we transfer by “taking sums over the values at the preimages”. Lemma 20.
The isomorphism colim n I n → a r ´ et Z is compatible with transfers on fields.Proof. It suffices to prove that for a field k , the total signature W ( k ) → H r ´ et ( k, Z ) is compatible withtransfer. Let l/k be a finite extension and R/k a real closure. There is a commutative diagram W ( l ) −−−−→ W ( R ⊗ k l ) tr y tr y W ( k ) −−−−→ W ( R )by the base change formula, i.e. Proposition 10. Note that Sper ( R ⊗ k l ) is the fibre of Sper ( l ) → Sper ( k )over the ordering corresponding to the inclusion k ⊂ R . Consequently we also have the commutativediagram H r ´ et ( l, Z ) −−−−→ H r ´ et ( R ⊗ k l, Z ) tr y tr y H r ´ et ( k, Z ) −−−−→ H r ´ et ( R, Z ) . Since the signature maps are determined by pulling back to a real closure, this means that we mayassume that k is real closed. (Since both sides we are trying to prove equal are additive, we may stillassume that l is a field.) But then either l = k or l = k [ √− (cid:3) We will make good use of the following observation.
Corollary 21.
Let l , . . . , l r /k be finite extensions such that ` i Spec ( l i ) → Spec ( k ) is a r´et-cover. Then tr : ⊕ i H r ´ et ( l i , Z ) → H r ´ et ( k, Z ) is surjective.Proof. The map ` i Sper ( l i ) → Sper ( k ) is a surjective local homeomorphism of compact, Hausdorff,totally disconnected spaces [Sch85, Theorem 3.5.1 and Remarks 3.5.6]. The result thus follows from thenext lemma. (cid:3) Lemma 22.
Let φ : X → Y be a surjective local homeomorphism of compact, Hausdorff, totally discon-nected spaces. Then φ has finite fibers, and the “summing over preimages” transfer H ( X, Z ) → H ( Y, Z ) is surjective.Proof. The claim that φ has finite fibers is well-known. We include a proof for convenience of the reader:since φ is a local homeomorphism the fibers are discrete, since Y is Hausdorff they are closed, and since X is compact they are compact. Now observe that a compact discrete space is finite.We now prove the surjectivity of the transfer. First we make the following claim: if X is a compact,Hausdorff, totally disconnected space, then given x ∈ U ⊂ X with U open, there exists x ∈ V ⊂ U suchthat V is clopen in X . Indeed for y = x let U y be a clopen neighbourhood of y disjoint from x . Then ∪ y ∈ X \ U U y is an open cover of the compact (since closed) complement X \ U . Let U , . . . , U n be a finitesubcover. Then V = X \ ∪ i U i works.Now consider the morphism φ : X → Y . For y ∈ Y choose a clopen neighbourhood U y of y ∈ Y such that there exists a clopen set V y ⊂ X with φ ( V y ) = U y and φ : V y → U y a homeomorphism. Wewill say in this situation that φ splits strongly over U y . We note that such V y , U y exist: since φ is alocal homeomorphism, there exists V ′ y ⊂ X such that U ′ y := φ ( V y ) is an open neighbourhood of y and φ : V ′ y → U ′ y is a homeomorphism. By the claim, we may assume that V ′ y is clopen. Now choose a clopenneighbourhood U y ⊂ U ′ y , using the claim again. Then V y := φ − ( U y ) ∩ V ′ y is clopen in X and mapshomeomorphically to U y . OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 17
We obtain in this way an open cover { U y } y ∈ Y of Y . Since Y is compact, we can choose a finitesubcover U , . . . , U n . Using that all the U i are clopen we can refine further until we have found a disjointclopen cover (replace U i by U i \ ( U ∪ U ∪ · · · ∪ U i − )) over which φ splits strongly. (Note that if φ splitsstrongly over a clopen U ⊂ Y , then it also splits strongly over any clopen U ′ ⊂ U .)Since H ( Y, Z ) is the set of continuous functions from Y to Z , it suffices to prove that the indicatorfunction χ U i : Y → Z of the clopen subset U i is in the image of transfer (because 1 = P i χ U i , and thetransfer is additive). But φ is strongly split over U i by construction, so there exists some clopen subset U ⊂ X such that φ : U → U i is a homeomorphism. Then χ U ∈ H ( X, Z ) and this is taken by transferto χ U i , as follows from the explicit description of transfer in terms of “summing over preimages”. (cid:3) We will want to show that certain presheaves are sheaves in the r´et-topology. We find it easiest tofirst develop a criterion for this. We start with the following result, which is surely well known.
Lemma 23.
Let τ be a topology on a category C and F a presheaf on C which is τ -separated. Let X ∈ C and U • , V • → X be τ -coverings. Suppose that V • refines U • , i.e. we are given a morphism f : V • → U • over X . Then if F satisfies the sheaf condition with respect to V • , it also satisfies the sheaf conditionwith respect to U • .Proof. The proof can be extracted from the proof of [Sta17, Tag 00VX]. We repeat the argument forconvenience. For simplicity, suppose that U • and V • use the same indexing set I , and that the refinementis of the form V i → U i . We are given s i ∈ F ( U i ) for each i , such that s i | U i × X U j = s j | U i × X U j , and weneed to show that there is a (necessarily unique) s ∈ F ( X ) with s | U i = s i .Let t i = f ∗ s i . Then t • is a compatible family for the covering V • , and hence there is s ∈ F ( X ) with s | V i = t i for all i . We need to show that also s | U i = s i . For this, fix i ∈ I and consider the coverings U ′• , V ′• → U i obtained by base change. Then V ′• refines U ′• . We find that s i | U i × X U i = s i | U i × X U i byassumption, and hence f ∗ ( s i | U i × X U i ) = f ∗ ( s i | U i × X U i ) = t i | U i × X V i = s | U i × X V i by construction. Butnow because F is separated in the τ -topology and V ′• → U i is a cover we conclude that s i = s | U i , asneeded. (cid:3) Corollary 24.
Let F be a sheaf on Sm ( k ) Nis . Then F is a sheaf in the r´et-topology if and only if F satisfies the sheaf condition for every r´et-cover f : U → X , where X is (essentially) smooth, Henselianlocal and f is finite ´etale.Proof. For this proof, we call a morphism with the properties of f a fr´et-cover.The condition is clearly necessary; we show the converse.(*) We first claim that every r´et-cover U • → X with X smooth Henselian local can be refined by afr´et-cover. We can certainly refine U • by an affine cover, so assume that each U i is affine. Then by [Sta17,Tag 04GJ] each U i splits as U ′ i ` U ′′ i with U ′ i → X finite ´etale and U ′′ i → X not hitting the closed point m of X (note that U i → X is everywhere quasi-finite). I claim that U ′• is also a r´et-cover. Indeed ´etalemorphisms induce open maps on real spectra [Sch94, Proposition 1.8] and U ′• covers R ( m ) ⊂ R ( X ) byconstruction. But the only open subset of R ( X ) containing R ( m ) is all of R ( X ), by [ABR12, PropositionsII.2.1 and II.2.4]. Finally the real spectrum of any ring is quasi-compact [ABR12, II.1.5] whence we canalways refine by a finite subcover, and then taking the disjoint union we refine by a singleton cover.If X ∈ Sm ( k ), we write F X := F | X Nis for the restriction to the small site. Write Hom for the internalmapping presheaf functor in this category. Recall that Hom(
V, F X )( V ′ ) = F ( V × X V ′ ); in particularthis functor preserves sheaves.Let U • → X be a r´et-cover. To show that F ( X ) → F ( U • ) ⇒ F ( U • × X U • ) is an equaliser diagram(respectively the first map is injective), it is sufficient to show that F X → Hom( U • , F X ) ⇒ Hom( U • × X U • , F X ) is an equaliser diagram of sheaves (respectively the first map is an injection of sheaves), sincelimits of sheaves are computed in presheaves. But finite limits (respectively injectivity) are detected onstalks, whence in both situations we may assume that X is Henselian local. (**)Now we show that F is r´et-separated. Let U • → X be a r´et-cover. By the above, to show that F ( X ) → F ( U • ) is injective we may assume that X is Henselian local. Then U • → X is refined by afr´et-cover V → X , by (*). But then F ( X ) → F ( U • ) → F ( V ) is injective since F satisfies the sheafcondition for V → X by assumption, so F ( X ) → F ( U • ) is injective and F is r´et-separated.Finally let U • → X be any r´et-cover. We wish to show that F satisfies the sheaf condition for thiscover. By (**), we may assume that X is Henselian local. Then U • → X is refined by a fr´et-cover V → X and F satisfies the sheaf condition with respect to V → X by assumption, so it satisfies thesheaf condition with respect to U • → X by Lemma 23. (cid:3) Remark. Using [ABR12, Corollary II.1.15], the claim (*) can be extended as follows: Every r´et-cover U • → X with X arbitrary is refined by a cover V ′• → V • → X , where V • → X is a Nisnevich cover andeach V ′ i → V i is a fr´et-cover. Theorem 25.
Let F ∗ be a homotopy module such that ρ : F n → F n +1 is an isomorphism for all n . Then F ∗ consists of r´et-sheaves.Proof. We apply Corollary 24. Hence let φ : U → X be a r´et-cover with φ finite ´etale and X essentiallysmooth, Henselian local. We need to show that F satisfies the sheaf condition with respect to this cover.Note that U is then a finite disjoint union of essentially smooth, Henselian local schemes, by [Sta17, Tag04GH (1)].We now use the transfer tr : F ∗ ( U ) → F ∗ ( X ) from Section 4. Any homotopy module is a moduleover K MW ∗ and satisfies the projection formula with respect to this module structure. It follows fromCorollary 19 and our assumption that ρ acts invertibly on F ∗ that F ∗ is a module over a r ´ et Z , and satisfiesthe projection formula with respect to that module structure.We know that for a Henselian local ring A with residue field κ , we have H r ´ et ( A, Z ) = H r ´ et ( κ, Z ). Thisfollows from [ABR12, Propositions II.2.2 and II.2.4] (the author learned this argument from [KSW16,proof of Lemma 6.4]). Consequently by Corollary 21, Proposition 10 and stability of r´et-covers underbase change, there exists a ∈ H r ´ et ( U, Z ) such that tr ( a ) = 1.Now suppose given b ∈ F ∗ ( X ) such that b | U = 0. Then b = 1 b = tr ( a ) b = tr ( a · b | U ) = 0 by theprojection formula (i.e. Corollary 12). Consequently F ∗ ( X ) → F ∗ ( U ) is injective.Write p , p : U × X U → U for the two projections and suppose given b ∈ F ∗ ( U ) such that p ∗ b = p ∗ b .We have to show that there is c ∈ F ∗ ( X ) such that b = c | U . I claim that c := tr ( ab ) works. Indeedwe have tr ( ab ) | U = φ ∗ ( tr φ ( ab )) = tr p ( p ∗ ( ab )) by Proposition 10. Now p ∗ b = p ∗ b by assumption, andso tr p ( p ∗ ( a ) p ∗ ( b )) = tr p ( p ∗ ( a ) p ∗ ( b )) = tr p ( p ∗ a ) b by the projection formula again. Finally tr p ( p ∗ a ) = φ ∗ tr φ ( a ) = φ ∗ (cid:3) Preliminary Observations
We are now almost ready to prove our main theorems. This section collects some preliminary obser-vations and reductions.Lemma 15 from Section 6 applies in particular to SH ( S ) and D A ( S ) for a Noetherian base scheme S . We will be particularly interested in the case Y = G m and α = ρ : S → G m the additive inverse ofthe morphism corresponding to -1. What the lemma says is that the ρ -localization can be computed asthe obvious colimit.We write SH ( S ) r ´ et for the real ´etale localisation of SH ( S ) and D A ( S ) r ´ et for the real ´etale localisationof D A ( S ). There is possibly a slight confusion as to what this means, since it could mean the localisationat desuspensions of real ´etale (hyper-) covers, or the category obtained by the same procedure as SH ( S )but replacing the Nisnevich topology by the real ´etale one from the start. This does not actually makea difference: Lemma 26.
Let M be a monoidal model category, T ∈ M cofibrant and H a set of maps. There is anisomorphism of Quillen model categories Spt ( L H M , T ) = L H ′ Spt ( M , T ) , provided that all the localisations exist (e.g. M left proper and combinatorial). Here H ′ = ∪ i ∈ Z Σ ∞ + i H and Spt ( N , U ) denotes the model category of (non-symmetric) U -spectra in N with the local modelstructure.Proof. We follow [Hov01]. Recall that
Spt ( N , U ) denotes the category of sequences ( X , X , X , . . . )together with bonding maps X i ⊗ U → X i +1 , and morphisms the compatible sequences of morphisms.This is firstly provided with a global model structure Spt ( N , U ) gl in which a map ( X • ) → ( Y • ) is afibration or weak equivalence if and only if X i → Y i is for all i . This is also called a levelwise fibrationor weak equivalence. The local model structure is then obtained by localisation at a set of maps whichis not important to us, because it only depends on a choice of set of generators of M , and for L H M wecan just choose the same generators.Since in any model category L H L H N = L H ∪ H N , it is enough to show that L H ′ Spt ( M , T ) gl = Spt ( L H M , T ) gl . Note that an acyclic fibration in Spt ( L H M , T ) gl is the same as a levelwise acyclic H -local fibration in M , i.e. a levelwise acyclic fibration. Consequently the cofibrations in Spt ( L H M , T ) gl are the same as in Spt ( M , T ) gl , whereas the former has more weak equivalences. Thus the formeris a Bousfield localisation of the latter and hence it is enough to show that L H ′ Spt ( M , T ) gl and OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 19
Spt ( L H M , T ) gl have the same fibrant objects. An object of Spt ( L H M , T ) gl is fibrant if and only ifit is levelwise H -locally fibrant. An object E of L H ′ Spt ( M , T ) gl is fibrant if and only if it is levelwisefibrant and H ′ -local, which means that for each α : X → Y ∈ H and every n ∈ Z the map M ap d (Σ ∞ + n α, E ) : M ap d (Σ ∞ + n Y, E ) → M ap d (Σ ∞ + n Y, E )is a weak equivalence. By adjunction, this is the same as
M ap d ( α, E n ) being an equivalence, i.e. all E n being H -local. This concludes the proof. (cid:3) Write SH S ( S ) for the S -stable homotopy category (i.e. obtained from motivic spaces by justinverting S , but not G m ). Lemma 27.
There are canonical Quillen equivalences SH S ( S )[ ρ − ] ≃ SH ( S )[ ρ − ] and similarly forthe real ´etale topology.Proof. By Lemma 26 we know that
Spt ( SH S ( S )[ ρ − ] , G m ) = Spt ( SH S ( S ) , G m )[ ρ − ] ≃ SH ( S )[ ρ − ].But the map ρ : S → G m is invertible in SH S ( S )[ ρ − ] and thus Spt ( SH S ( S )[ ρ − ] , G m ) ≃ SH S ( S )[ ρ − ],i.e. inverting an invertible object has no effect [Hov01, Theorem 5.1] (cid:3) We also observe the following:
Proposition 28.
The pseudofunctor X SH ( X )[ ρ − ] satisfies the full six functors formalism (onNoetherian schemes of finite dimension), compact generation, and continuity.Proof. If i : Z → X is a closed immersion then the functor i ∗ : SH ( Z ) → SH ( X ) commutes withfiltered homotopy colimits (being right adjoint to a functor preserving compact objects) and satisfies i ∗ ( X ⊗ G m ) ≃ i ∗ ( X ) ⊗ G m [CD09b, A.5.1 (6) and (3)]. It follows from the explicit description of ρ -localisation in Lemma 15 that i ∗ commutes with L : SH ( X ) → SH ( X )[ ρ − ]. Thus SH ( X )[ ρ − ]satisfies localisation, by [CD09b, Proposition 2.3.19]. Since SH ( X )[ ρ − ] clearly satisfies the homotopyand stability properties, it satisfies the six functors formalism by Theorem 13.Since SH ( X ) is compactly generated so is SH ( X )[ ρ − ], by the last sentence of Lemma 15.For any morphism f : X → Y the functor f ∗ : SH ( Y ) → SH ( X ) commutes with (filtered) homo-topy colimits (being a left adjoint), and consequently it commutes with ρ -localisation, as above. Thuscontinuity for SH ( X )[ ρ − ] follows from continuity for SH ( X ). (cid:3) For completeness, we include the following rather formal observation. It is not used in the remainderof this text (except that it is restated as part of Theorem 35).
Proposition 29.
The canonical functor SH ( S ) r ´ et → SH ( S ) r ´ et [ ρ − ] is an equivalence. In other words, ρ is a weak equivalence in SH ( S ) r ´ et .Proof. I claim that in SH ( S ) r ´ et there is a splitting G m ≃ ∨ ∆ such that the composite ρ −→ G m ≃ ∨ ∆ → is the identity. It will follow from Lemma 30 below that ∆ ≃
0, proving this lemma.Call a ∈ O × ( X ) totally positive if for every real closed field r and morphism α : Spec ( r ) → X we have α ∗ ( a ) >
0. Note that in particular any square of a unit is totally positive.This defines a sub-presheaf G + ⊂ R A \ of the presheaf represented by A \
0. Define G − analogouslyusing totally negative units. I claim that a r ´ et R A \ = a r ´ et G + ` a r ´ et G − . We may prove this on stalks,which are Henselian rings with real closed residue fields [Sch94, (3.7.3)]. If A is such a ring and a ∈ A × ,then the reduction ¯ a ∈ A/m is a unit and so either positive or negative. It follows that either ¯ a or − ¯ a isa square, whence either a or − a is a square ( A being Henselian of characteristic zero). Consequently a is either totally positive or totally negative, proving the claim.We may thus define a map a r ´ et G m → a r ´ et S = a r ´ et ( ∗ ` ∗ ) by mapping a r ´ et G + to the base pointand a r ´ et G − to the other point. Since − a r ´ et S → a r ´ et G m → a r ´ et S of the required form. The stable splitting follows. (cid:3) Lemma 30.
Let C be an additive symmetric monoidal category in which ⊗ distributes over ⊕ .If G ∈ C is an invertible object, such that G ∼ = ⊕ ∆ , then ∆ ∼ = 0 .Proof. The object G is rigid (being invertible) and hence ∆ is rigid (being a summand of G ). We have ∼ = D ( G ) ⊗ G ∼ = ( D ⊕ D ∆) ⊗ ( ⊕ ∆) ∼ = ⊕ ∆ ⊕ D (∆) ⊕ D (∆) ⊗ ∆. Thus in order to prove the claim wemay assume that G = . Now the splitting ∼ = ⊕ ∆ corresponds to morphisms e −→ ⊕ ∆ f −→ with f e = id and ef ∈ End ( ⊕ ∆) the projection. Fixing an isomorphism z −→ ⊕ ∆ we get correspondingelements z − e, f z ∈ [ , ]. We have id = f e = f ( zz − ) e = ( f z )( z − e ). But End ( ) is commutative[Bal10, sentence before Proposition 2.2] so id = ( z − e )( f z ) and consequently ef = zz − = id and∆ = 0. (cid:3) Main Theorems
Proposition 31.
Let k be a field of characteristic zero. The functor L : SH ( k )[ ρ − ] → SH ( k ) r ´ et [ ρ − ] is an equivalence.Proof. It is enough to show that all objects in SH ( k )[ ρ − ] are r´et-local. Let U • → X be a r´et-hypercover,and let ˆ X be its homotopy colimit (in SH ( k )). We need to show that if E ∈ SH ( k )[ ρ − ], then [ ˆ X, E ] =[
X, E ]. We have conditionally convergent descent spectral sequences(1) H pNis ( X, π − q ( E ) − i ) ⇒ [Σ ∞ X + ∧ G ∧ im , E [ p + q ]](2) [ ˆ X, π − q ( E ) − i [ p ]] ⇒ [ ˆ X ∧ G ∧ im , E [ p + q ]] . Here we display the E -pages on the left hand side. We moreover have the conditionally convergenthomotopy colimit spectral sequence(3) [ U ∗ , π − q ( E ) − i [ p ]] ⇒ [ ˆ X, π − q ( E ) − i [ p + ∗ ]] . Here the left hand side is the E -page. We have [ U n , π − q ( E ) − i [ p ]] = H pNis ( U n , π − q ( E ) − i ) = H pr ´ et ( U n , π − q ( E ) − i );indeed since E is ρ -local each π − q ( E ) − i is a r´et-sheaf, by Theorem 25, and for any r´et-sheaf F we have H pret ( U n , F ) = H pNis ( U n , F ) [Sch94, Proposition 19.2.1]. It follows that spectral sequence (3) convergesstrongly (because the dimension of X is finite) and identifies with the descent spectral sequence in r´et-cohomology for the cover U • . In particular, it converges to H p + ∗ r ´ et ( X, π − q ( E ) − i ). Thus we find that[ ˆ X, π − q ( E ) − i [ p ]] = H pr ´ et ( X, π − q ( E ) − i ). Using [Sch94, Proposition 19.2.1] again, we conclude that theevident map from spectral sequence (1) to spectral sequence (2) induces an isomorphism on the E -pages,and moreover both converge strongly (again for cohomological dimension reasons). Thus the inducedmap on targets is an isomorphism, which is what we wanted to show. (cid:3) Corollary 32.
The proposition holds for all fields.Proof.
I claim that if k has positive characteristic, then ρ is nilpotent in SH ( k ). By base change, itsuffices to prove this when k = F p . That is to say, we wish to show that ρ is nilpotent in K MW ∗ ( F p ), orequivalently that colim n K MWn ( F p ) = 0. By the same argument as in the proof of Corollary 19 we knowthat colim n K MWn ( F p )] = colim n I n ( F p ). Thus our claim follows from nilpotence of the fundamentalideal of F p , which is well known [MH73, III (5.9)]. (cid:3) Corollary 33.
Let S be a Noetherian scheme of finite dimension. The functor L : SH ( S )[ ρ − ] → SH ( S ) r ´ et [ ρ − ] is an equivalence.In particular SH ( S ) r ´ et [ ρ − ] satisfies the full six functors formalism. Our initial proof of this statement contained a mistake; a correction and vast simplification has kindlybeen communicated by Denis-Charles Cisinski.
Proof.
It suffices to prove that all objects of SH ( S )[ ρ − ] are r´et-local. Thus let X ∈ Sm ( S ) and U • → X a r´et-hypercover. We need to show that α : hocolim ∆ Σ ∞ U • → Σ ∞ X is an equivalence in SH ( S )[ ρ − ]. (See also Lemma 26.) Since SH ( S )[ ρ − ] satisfies the six functorsformalism by Proposition 28, it follows from Corollary 14 that suffices to show that if f : Spec ( k ) → S is a morphism (with k a field), then f ∗ α is an equivalence. But f ∗ is a left adjoint so commutes withhomotopy colimits (and Σ ∞ ), so f ∗ α is isomorphic to the maphocolim ∆ Σ ∞ f ∗ U • → Σ ∞ f ∗ X in SH ( k )[ ρ − ]. Since r´et-covers are stable by pullback, this is an equivalence by Corollary 32. (cid:3) Proposition 34.
Let S be a Noetherian scheme of finite dimension. Then the canonical functor SH( S r ´ et ) → SH ( S ) r ´ et [ ρ − ] is an equivalence.Proof. The functor SH( S r ´ et ) → SH( Sm ( S ) r ´ et ) is fully faithful and t -exact by Corollary 6. The imageof SH( S r ´ et ) in SH( Sm ( S ) r ´ et ) consists of A -local and ρ -local objects, by the descent spectral sequenceand Theorem 8 (and Corollary 6, which implies that the homotopy sheaves of LeE are the exten-sions of the homotopy sheaves of E ). Consequently SH( S r ´ et ) → SH S ( S ) r ´ et [ ρ − ] is fully faithful. But SH S ( S ) r ´ et [ ρ − ] → SH ( S ) r ´ et [ ρ − ] is an equivalence by Lemma 27. We have thus established that thefunctor is fully faithful. We need to show it is essentially surjective. OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 21
The category SH ( S ) r ´ et [ ρ − ] is generated by objects of the form p ∗ ( ) where p : T → S is projec-tive [CD09b, Proposition 4.2.13]. Since the functor e : SH( S r ´ et ) → SH ( S ) r ´ et [ ρ − ] has a right adjointit commutes with arbitrary sums, and hence it identifies SH( S r ´ et ) with a localising subcategory of SH ( S ) r ´ et [ ρ − ]. It thus suffices to show that e commutes with p ∗ , where p : T → S is a projective mor-phism. This is exactly the same as the proof of [CD13, Proposition 4.4.3]. It boils down to the properbase change theorem holding both in SH ( S ) r ´ et [ ρ − ] (where it follows from the six functors formalismwhich we have already established by showing that SH ( S ) r ´ et [ ρ − ] ≃ SH ( S )[ ρ − ]) and in SH( S r ´ et ); thelatter is Theorem 9. (cid:3) Remark. If S is the spectrum of a field, the above proof can be simplified greatly, by arguing as in[Bac16, Section 5]. See in particular Lemma 21, Corollary 26 and Proposition 28 of loc. cit. This waywe no longer need to use the proper base change theorems, and thus also do not need to know that SH ( X ) r ´ et [ ρ − ] satisfies the six functors formalism.One may also extract from loc. cit. a proof of Proposition 31 not relying on Theorem 25. Thus if thebase is a field, Sections 3, 5, and 7 can be dispensed with.. In summary, we have thus established the following result. Theorem 35.
Let S be a Noetherian scheme of finite dimension. In the following two diagrams, allfunctors are the canonical ones, and are equivalences of categories: SH ( S r ´ et ) a −−−−→ SH S ( S ) r ´ et [ ρ − ] ←−−−− SH S ( S )[ ρ − ] y b y b ′ y SH ( S ) r ´ et c −−−−→ SH ( S ) r ´ et [ ρ − ] d ←−−−− SH ( S )[ ρ − ] D A ( S r ´ et ) −−−−→ D S A ( S ) r ´ et [ ρ − ] ←−−−− D S A ( S )[ ρ − ] y y y D A ( S ) r ´ et −−−−→ D A ( S ) r ´ et [ ρ − ] ←−−−− D A ( S )[ ρ − ] . In particular all these categories satisfy the full six functors formalism, and continuity.Proof.
The functor d is an equivalence by Corollary 33, b and b ′ are equivalences by Corollary 27, c isan equivalence by Proposition 29 and ba is an equivalence by Proposition 34. It follows that a is anequivalence, and so are the two unlabelled functors.By Proposition 28, SH ( • )[ ρ − ] satisfies the full six functors formalism, and hence so do all the otherpseudofunctors, being equivalent.We have provided the proofs for SH , the ones for D A are exactly the same. (cid:3) Real Realisation
In this section we work over the field R of real numbers. We then have a composite R : SH ( R ) L ρ −−→ SH ( R ) r ´ et [ ρ − ] ≃ SH S ( R ) r ´ et [ ρ − ] r −→ SH s . Here by SH s we mean the model of the stable homotopy category SH built from simplicial sets. Ofcourse SH s ≃ SH canonically (and this may be an equality depending, on our favourite model of SH ).Also r denotes the functor induced by the right adjoint of e : P re ( R r ´ et ) → P re ( Sm ( R )) from Section 3.Following Heller-Ormsby [HO16, Section 4.4], there is also the real realisation functor LR : SH ( R ) → SH t . Here SH t is the model of SH built from topological spaces. The functor LR is defined by startingwith the functor R : Sm ( R ) → T op, X X ( R ) assigning a smooth scheme over R its set of real pointswith the strong topology. We then get a functor R : sP re ( Sm ( R )) ∗ → T op by left Kan extension, i.e.demanding that R (∆ n + ∧ X + ) = ∆ n + ∧ X ( R ) + and that R preserves colimits. Using the projective modelstructure on sP re ( Sm ( R )) ∗ this functor is left Quillen and then one promotes it to LR : SH ( R ) → SH t in the usual way.Fortunately the two potential real realisation functors are the same. To state this result, recall thatthere is an adjunction | • | : sSet ⇆ T op : Sing t , and then by passing to homotopy categories of spectra one obtains the adjoint equivalence L | • | : SH s ⇆ SH t : RSing t . Proposition 36.
The two functors L | R ( • ) | , LR ( • ) : SH ( R ) → SH t are canonically isomorphic. Proof.
The functor R takes multiplication by ρ into a weak equivalence. Consequently it remainsleft Quillen in the ρ -local model structure and hence LR canonically factors through the localisation SH ( R ) → SH ( R )[ ρ − ] . Since SH ( R )[ ρ − ] ≃ SH S ( R ) r ´ et [ ρ − ] the obvious functor R ′ : Spt ( Sm ( R )) → Spt t is left Quillen in the ( ρ, r ´ et, A )-local model structure. (Here we have used twice the followingwell-known observation: if L : M ⇆ N : R is a Quillen adjunction and H is a set of maps betweencofibrant objects in M which is taken by L into weak equivalences, then L : L H M ⇆ N : R is also aQuillen adjunction. This follows from [Hir09, Propositions 8.5.4 and 3.3.16].)We now have the following diagram (which we do not know to be commutative so far): SH S ( R ) r ´ et [ ρ − ] R ′ −−−−→ SH tr y (cid:13)(cid:13)(cid:13) SH s |•| −−−−→ SH t . Here all the functors are derived; we omit the “L” and “R”. The functor r is an equivalence with inverse e by Theorem 35. Thus for E ∈ SH S ( R ) r ´ et [ ρ − ] we have a canonical isomorphism R ′ E ≃ R ′ erE andso to prove the proposition it suffices to exhibit a canonical isomorphism of functors R ′ e ≃ | • | .But this isomorphism exists on the level of underived functors, and then passes to the homotopycategories. Indeed if E ∈ Spt s then R ′ eE and | E | are both computed by applying functors (of the samenames) levelwise to E , so we may just as well show that for E ∈ sSet ∗ we have R ′ eE ∼ = | E | . But now R ′ , e and | • | all preserve colimits, so we may just deal with E = ∆ n + . But then R ′ eE ∆ n + = | ∆ n + | holdsbasically by definition. (cid:3) A similar result can be obtained for the A -derived category. We have r : D A ( R )[ ρ − ] → D ( Spec ( R ) r ´ et ) ≃ D ( Ab ) . There is also R : D A ( R ) → D ( Ab ) which is obtained by (derived) left Kan extension fromthe functor Sm ( R ) → C ( Ab ) which sends a smooth scheme X the singular complex of its real points C ∗ ( X ( R )). Then there is a commutative diagram SH ( R ) R −−−−→ SH C ∗ y C ∗ y D A ( R ) R −−−−→ D ( Ab ) . Proposition 37.
The functors rL ρ , R : D A ( R ) → D ( Ab ) are canonically isomorphic.Proof. As before R factors through L ρ as R ′ and we may show that r, R ′ : D A ( R )[ ρ − ] → D ( Ab ) arecanonically isomorphic. The functor r is an equivalence with inverse e , so it is enough to show that R ′ e : D ( Ab ) → D A ( R )[ ρ − ] ≃ D S A ( R )[ ρ − ] → D ( Ab ) is canonically isomorphic to the identity. This isthe same argument as before. (cid:3) Let us make explicit the following consequence.
Corollary 38.
Let E ∈ SH ( R ) . Then π i ( E )( R )[ ρ − ] = π i ( RE ) and h A i ( E )( R )[ ρ − ] = H i ( RE ) . Here R : SH ( R ) → SH denotes any one of the (canonically isomorphic) real realisation functors we haveconsidered and h A i ( E ) := h i ( F E ) where F : SH ( R ) → D A ( R ) is the canonical functor.Proof. Combine Lemma 15 (saying that L ρ E = E [ ρ − ] = hocolim n E ∧ G ∧ nm ) with compactness of theunits of SH , D A and the above two propositions. (cid:3) Application 1: The η -inverted Sphere From now on, k will denote a perfect field. Since essentially all our results concern the ρ -invertedsituation, they are really only interesting if k has characteristic zero, so this is not a big restriction.Recall that the motivic Hopf map η : A \ → P defines an element of the same name in motivicstable homotopy theory η : Σ ∞ G m → . Here we use that Σ ∞ ( A \ ≃ Σ ∞ G m ∧ Σ ∞ P . The element η ∈ π ( ) − is non-nilpotent, and so inverting it is very natural. The category SH ( k )[ η − ] can beconstructed very similarly to SH ( k )[ ρ − ]. In particular the localisation functor L : SH ( k ) → SH ( k )[ η − ]is just the evident colimit, see Lemma 15. It is typically denoted E E η or E E [1 /η ]. One may OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 23 similarly invert other endomorphisms of the sphere spectrum. If 0 = n ∈ Z then there is a correspondingautomorphism of , and we denote the localisation by E E [1 /n ].At least after inverting 2, inverting η is essentially the same as inverting ρ : Lemma 39.
The endomorphism ring K MW ∗ ( k )[1 /
2] = [ [1 / , [1 / ∧ G ∧∗ m ] splits canonically into twosummands K MW ∗ ( k )[1 /
2] = K + ⊕ K − . In fact K − = K MW ∗ ( k )[1 / , /η ] and K + is characterised by thefact that ηK + = 0 .In K − we have the equality ηρ = 2 , whereas in K + we have ρ = 0 . In particular K MW ∗ ( k )[1 / , /η ] = K − = K MW ∗ ( k )[1 / , /ρ ] . Proof.
This is well known, see for example [Mor12, Section 3.1]. We summarise: For a ∈ k × let h a i =1 + η [ a ] ∈ K MW ( k ). Put ǫ = −h− i . Then ǫ = 1 and so after inverting 2, K MW ∗ ( k ) splits into theeigenspaces for ǫ . One puts h = 1 − ǫ and then has ηh = 0. On K + we have ǫ = −
1, so h = 2 andconsequently η = 0 (since 2 is invertible).By definition ρ = − [ −
1] and consequently ηρ = 1 + ǫ . Thus on K − where ǫ = 1 we find ηρ = 2 asclaimed, and in particular η is invertible on K − .Finally ρ h = 0 in K MW ∗ ( k ) and thus 2 ρ = 0 in K + . (This is just another expression of the fact that K + ∼ = K M ( k )[1 /
2] is graded-commutative and ρ has degree 1, so ρ = − ρ .) But since 2 is invertible in K + we find ρ = 0 (in K + ). This concludes the proof. (cid:3) Oliver R¨ondigs has studied the homotopy sheaves π ( η ) and π ( η ) and proved that they vanish[R¨on16]. (Note that π i ( E η ) ∗ is independent of ∗ , because multiplication by η is an isomorphism, so weshall suppress the second index.) He argues that π i ( ) ∗ → π i ( [1 / ∗ is injective for i = 1 , π i ( [1 /η, / i = 1 ,
2. We can deduce this as an easy corollary from our work:
Proposition 40.
Let k be a perfect field. Then π i ( [1 /η, / for i = 1 , .Proof. By Lemma 39 we know that SH ( k )[1 / , /η ] = SH ( k )[1 / , /ρ ]. By Theorem 35, we have SH ( k )[1 / , /ρ ] = SH( Spec ( k ) r ´ et )[1 / k has characteristiczero, which we shall assume from now on.The sheaves π i ( [1 / , /ρ ]) are unramified [Mor05, Lemma 6.4.4], so it suffices to show that π i ( [1 / , /ρ ])( K ) =0 for i = 1 , K (of characteristic zero). Since k was also arbitrary, we may just as well showthe result for k = K , simplifying notation. We are dealing with r´et-sheaves by Theorem 25, and so if Spec ( l ) a Spec ( l ) a · · · a Spec ( l n ) → Spec ( k )is a r´et-cover, the canonical map π i ( [1 / , /ρ ])( k ) → Y m π i ( [1 / , /ρ ])( l m )is injective. Consequently we may assume that k is real closed. But then SH( Spec ( k ) r ´ et ) = SH is justthe ordinary stable homotopy category, so it suffices to show: π si [1 /
2] = 0 for i = 1 ,
2, where π si are theclassical stable homotopy groups. But π s = Z / π s is well known, so we are done. (cid:3) In classical algebraic topology, it is well known that rational stable homotopy theory is the sameas rational homology theory: SH Q ≃ D ( Q ). In motivic stable homotopy theory, the situation is notso simple. As is well known (and follows for example from Lemma 39) there is a splitting SH ( k ) Q = SH ( k ) + Q × SH ( k ) − Q . The + part has been identified with rational motivic homology theory by Cisinski-D´eglise [CD09b, Section 16]: SH ( k ) + Q ≃ DM ( k, Q ).The - part has been only identified recently with an appropriate category of rational Witt motivesby Ananyevskiy-Levine-Panin [ALP17]: SH ( k ) − Q = DM W ( k, Q ). Here the category DM W ( k, Q ) maybe conveniently defined as the homotopy category of modules over the (strict ring spectrum model ofthe) homotopy module of rational Witt theory. That is to say there is the homotopy module W Q with( W Q ) i = W ⊗ Z Q for all i . This is the same as K − ⊗ Z [1 / Q , or equivalently π ( η, Q ). Then correspondingto this homotopy module there is a strict ring spectrum, the Eilenberg-MacLane spectrum EM W Q .Finally we may form the model category EM W Q -Mod and its homotopy category Ho ( EM W Q -Mod ) =: DM W ( k, Q ).More generally, one may define DM W ( k, Z [1 / W Q = π ( Q [1 /η ]) in the above con-struction with W [1 /
2] = π ( [1 /η, / π i ( η, Q ) = 0for i >
0. We can deduce this and more from our general theory. We write H A Z for the image of thetensor unit in D A ( k ) under the “forgetful” functor D A ( k ) → SH ( k ). Proposition 41.
We have π i ( H A Z [1 /ρ ]) = 0 for i > , and consequently π i ( H A Z [1 / , /η ]) = 0 for i > . Similarly π i ( Q [1 /ρ ]) = π i ( Q [1 /η ]) = 0 for i > .Thus we have the equivalences D A ( k, Z [1 / − ≃ DM W ( k, Z [1 / ≃ D ( Spec ( k ) r ´ et , Z [1 / SH ( k ) − Q ≃ DM W ( k, Q ) ≃ D ( Spec ( k ) r ´ et , Q ) . Proof.
As in the proof of Proposition 40 we have D A ( k )[1 / − := D A ( k )[1 / , /η ] = D A ( k )[1 / , /ρ ] . By Theorem 35, this is the same as D ( Spec ( k ) r ´ et )[1 /
2] = D ( Spec ( k ) r ´ et , Z [1 / SH ( k ) − Q := SH ( k ) Q [1 /η ] = SH ( k ) Q [1 /ρ ] = SH( Spec ( k ) r ´ et ) Q , and the latter category is equivalent to D ( Spec ( k ) r ´ et ) Q by classical stable rational homotopy theory.From this we can read off π ∗ ( H A Z [1 / , /η ]) and so on. The main point is that π n ( H A Z [1 / , /η ]) =0 for n >
0. It suffices to check this on fields, so we may as well check it for k ( k being arbitrary), andwe have π n ( H A Z [1 / , /η ])( k ) = [ [ n ] , ] D ( Spec ( k ) r ´ et , Z [1 / = H − nr ´ et ( k, Z [1 / D A ( k, Z [1 / − ≃ DM W ( k, Z [1 / H A Z [1 /ρ, / → EM W [1 /
2] is a weak equivalence. The result follows. (cid:3)
Let us also make explicit the following observation.
Corollary 42.
Let k be a real closed field or Q . Then π ∗ ( [1 /ρ ])( k ) = π s ∗ and in particular π ∗ ( [1 /η, / k ) = π s ∗ ⊗ Z Z [1 / . Here π s ∗ denotes the classical stable homotopy groups.Proof. This follows immediately from
Shv ( Spec ( k ) r ´ et ) = Set , Lemma 39 and Theorem 35. (cid:3)
Application 2: Some Rigidity Results
In this section we establish some rigidity results. We work with ρ -stable sheaves. These sheaves are h -torsion (because ρ h = 0), explaining to some extent why we do not need the usual torsion assumptions.There are various notions of rigidity for sheaves. We shall call a presheaf F on Sm ( k ) rigid if forevery essentially smooth, Henselian local scheme X with residue field x , the natural map F ( X ) → F ( x )is an isomorphism. This notion goes back to perhaps Gillet-Thomason [GT84] and Gabber [Gab92]. Lemma 43.
Let F ∈ Shv ( Spec ( k ) r ´ et ) . Then eF ∈ Shv ( Sm ( k ) r ´ et ) is rigid.Proof. Extension e and pullback are both left Kan extensions. From this it is easy to show that theycommute, and so we find that ( eF ) | X r ´ et = f ∗ F ∈ Shv ( X r ´ et ), where X is (essentially) smooth over k with structural morphism f .If char ( k ) > Spec ( k ) r ´ et and Sm ( k ) r ´ et are both the trivial site, so we may assume that k is of characteristic zero and consequently perfect. In this case, for an essentially k -smooth Henselianlocal scheme X with closed point i : x → X , there exists a retraction s : X → x . (Write k ( x ) /k as k ( T , . . . , T n )[ U ] /P with P ∈ k ( T , . . . , T n )[ U ] separable; this is possible because k ( x ) /k is separable, k being perfect. Lift the elements T i to O X arbitrarily and then use Hensel’s lemma to produce a root of P in O X .)It is thus enough to prove: if F ∈ Shv ( x r ´ et ) then H ( x, F ) = H ( X, s ∗ F ). It follows from [Sch94, Dis-cussion after Proposition 19.2.1] and [ABR12, Propositions II.2.2 and II.2.4] that for any G ∈ Shv ( X r ´ et )we have H ( X, G ) = H ( x, i ∗ G ). Consequently H ( X, s ∗ F ) = H ( x, i ∗ s ∗ F ) = H ( x, F ), because si = idby construction. (cid:3) Corollary 44. If E ∈ SH ( k )[ ρ − ] then all the homotopy sheaves π i ( E ) are rigid.Proof. By Theorem 35 and Corollary 6 we know that all the homotopy sheaves of E are of the form eF ,with F ∈ Shv ( Spec ( k ) r ´ et ). Thus the claim follows immediately from Lemma 43. (cid:3) Corollary 45.
Let k be a perfect field of finite virtual 2-´etale cohomological dimension and exponentialcharacteristic e = 2 . Then the homotopy sheaves π i ( ) [1 /e ] are rigid.Proof. We will first assume that e = 1, and explain the necessary changes in positive characteristic atthe end. OTIVIC AND REAL ´ETALE STABLE HOMOTOPY THEORY 25
For i = 0 we have π ( ) = GW and this sheaf is known to be rigid [Gil17, Theorem 2.4]. We considerthe arithmetic square [RSØ16, Lemma 3.9] −−−−→ [1 / y y ∧ −−−−→ ∧ [1 / . Since rigid sheaves are stable under extension, kernel and cokernel, the five lemma implies that it isenough to show that π ∗ ( [1 / , π ∗ ( ∧ ) and π ∗ ( ∧ [1 / are rigid. Since rigid sheaves are stable bycolimit, the case of ∧ [1 /
2] follows from ∧ .By [HKO11, Theorem 1] and [R¨on16, proof of Theorem 8.1], we know that ∧ is the target of theconvergent motivic Adams spectra sequence. The homotopy sheaves at the E page are all sheaves withtransfers in the sense of Voevodsky and torsion prime to the characteristic, and hence rigid, for exampleby [HY07, Paragraph after Lemma 1.6]. Since rigid sheaves are stable by extension etc., it follows thatthe E ∞ page is rigid, and finally so are the homotopy sheaves of ∧ .By motivic Serre finiteness [ALP17, Theorem 6] (beware that their indexing convention for motivichomotopy groups differs from ours!), π i ( [1 / is torsion for i >
0. By design, it is of odd torsion primeto the exponential characteristic. Consequently all of the l -torsion subsheaves of π i ( [1 / + ) are rigidby the same argument as before, and so is the colimit π i ( [1 / + ) .It remains to deal with π i ( [1 / − ) . But this is just the same as π i ( [1 / , /ρ ]) and so is rigid byCorollary 44.This concludes the proof if e = 1. If e > e . (cid:3) Remark. We appeal to [ALP17] in order to know that π i ( ) ⊗ Q = 0 for i >
0. This can also be deducedfrom Proposition 41, using that SH ( k ) + Q = DM ( k, Q ).There is another (older) notion of rigidity first considered by Suslin [Sus83]. This corresponds to (1)in the next result. It is a slightly silly property in our situation, but (2) is a replacement in spirit. It isrelated to important results in semialgebraic topology due to Coste-Roy, Delfs [Del91, see in particularCorollary II.6.2] and Scheiderer [Sch94]. Proposition 46.
Let E ∈ SH ( k )[ ρ − ] and i ∈ Z .(1) If ¯ L/ ¯ K is an extension of algebraically closed fields over k , then π i ( E )( ¯ K ) = π i ( E )( ¯ L ) = 0 . (2) If L r /K r is an extension of real closed fields over k , then also π i ( E )( K r ) = π i ( E )( L r ) . Proof.
As before, by Theorem 35 and Corollary 6 we know that all the homotopy sheaves of E are of theform eF , with F ∈ Shv ( Spec ( k ) r ´ et ). For such sheaves we have eF ( ¯ K ) = 0 = eF ( ¯ L ), so (1) holds. Sincepullback Spec ( K r ) r ´ et → Spec ( L r ) r ´ et induces an isomorphism of sites, (2) also follows immediately. (Seealso the first paragraph of the proof of Lemma 43.) 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