η -periodic motivic stable homotopy theory over Dedekind domains
aa r X i v : . [ m a t h . K T ] J un η -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKINDDOMAINS TOM BACHMANN
Abstract.
We construct well-behaved extensions of the motivic spectra representing generalized mo-tivic cohomology and connective Balmer–Witt K -theory (among others) to mixed characteristic Dedekindschemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of η -periodicmotivic stable homotopy theory established in [BH20] from fields to arbitrary base schemes, and usethis to determine (among other things) the η -periodized algebraic symplectic and SL-cobordism groupsof mixed characteristic Dedekind schemes containing 1 / Contents
1. Introduction 12. The sheaves I n η -periodic framed spectra 54. Main results 75. Applications 10References 131. Introduction
Let D be the prime spectrum of a Dedekind domain, perhaps of mixed characteristic, or a field.Consider the motivic spectrum KO D ∈ SH ( D ) (see e.g. [BH17, §§ SH ( D )), representing Hermitian K -theory [Hor05]. From this we can buildthe following related spectra KW D = KO D [ η − ]kw D = τ ≥ KW D HW D = τ ≤ kw D H W Z D = f (HW D ) K WD = τ ≤ H W Z D H˜ Z D = τ eff ≤ ( D ) . Here by τ ≤ , τ ≥ we mean the truncation in the homotopy t -structure on SH ( D ), and by τ eff ≤ we meanthe truncation in SH ( D ) eff (see e.g. [BH17, § B]). If f : D ′ → D is any morphism there are naturalinduced base change maps f ∗ (kw D ) → kw D ′ , and so on. It thus makes sense to ask if the spectra aboveare stable under base change, i.e., if the base change maps are equivalences. This is true for KO (andKW), since this spectrum can be built out of (orthogonal or symplectic) Grassmannians [ST15, PW10],which are stable under base change. Our main result is the following. Theorem 1.1 (see Theorem 4.4) . All of the above spectra are stable under base change among Dedekinddomains or fields, provided that they contain / . Over fields, the above definitions of spectra coincide with other definitions that can be found in theliterature (see [Bac17, DF17, ARØ17, Mor12]; this is proved in Lemma 4.2). In other words, we constructwell-behaved extensions to motivic stable homotopy theory over Dedekind domains of certain motivicspectra which so far have mainly been useful over fields. In fact, we show that all of the above spectra
Date : June 4, 2020. (which we have built above out of KO by certain universal properties) admit more explicit (and socalculationally useful) descriptions. For example we show that π ( K W ) ∗ = I ∗ , π i ( K W ) = 0 for i = 0;here I is the Nisnevich sheaf associated with the presheaf of fundamental ideals in the Witt rings.Using the above result, together with the fact that equivalences (and connectivity) of motivic spectraover D can be checked after pullback to the residue fields of D [BH17, Proposition B.3], one obtainsessentially for free the following extension of [BH20]. Corollary 1.2 (see Theorem 4.10) . For D as above, there is a fiber sequence [ η − ] (2) → kw (2) → Σ kw (2) ∈ SH ( D ) . Using this, we also extend many of the other results of [BH20] to Dedekind domains.
Overview.
The main observation allowing us to prove the above results is the following. Recallthat there is an equivalence SH ( D ) ≃ SH fr ( D ), where the right hand side means the category ofmotivic spectra with framed transfers [EHK +
17, Hoy18]. This supplies us with an auxiliary functor σ ∞ : SH S fr ( D ) → SH fr ( D ). The Hopf map η : G m → already exists in SH S fr ( D ). This readilyimplies that we can make sense of the category SH S fr ( D )[ η − ] of η -periodic S -spectra with framedtransfers, and that there is an equivalence SH S fr ( D )[ η − ] ≃ SH fr ( D )[ η − ] . The significance of this is that the left hand side no longer involves P -stabilization, and hence is mucheasier to control. In the end this allows us to relate all our spectra in the list above to a spectrum ko fr which is known to be stable under base change. To do so we employ (1) work of Jeremy Jacobson [Jac18]on the Gersten conjecture for Witt rings in mixed characteristic, and (2) work of Markus Spitzweck[Spi12] on stability under base change of H Z . Organization. In § K W with the expected homotopysheaves. In § SH S fr ( D )[ η − ], allowing us among other things to constructa spectrum kw with the expected homotopy sheaves. We prove our main theorems in §
4. We first givealternative, more explicit definitions of the spectra in our list and deduce stability under base change.Then we show that the spectra we constructed satisfy the expected universal properties. We establish thefundamental fiber sequence of η -periodic motivic stable homotopy theory as an easy corollary. Finallyin § § Notation and terminology.
By a Dedekind scheme we mean a finite disjoint union of spectra ofDedekind domains or fields, that is, a regular noetherian scheme of Krull dimension ≤ E ∈ SH ( S ) we denote by π i ( E ) j the homotopy sheaves (see e.g. [BH20, § t -structure.We denote by a Nis , a ´ et , and a r ´ et respectively the associated sheaves of sets in the Nisnevich, ´etale andreal ´etale topologies. We write L Nis for the Nisnevich localization of presheaves of spaces or spectra.Unless specified otherwise, all cohomology is with respect to the Nisnevich topology.All schemes are assumed quasi-compact and quasi-separated.We denote by S pc the ∞ -category of spaces, and by SH the ∞ -category of spectra. Acknowledgements.
I would like to thank Shane Kelly for help with Lemma 2.10. To the best of myknowledge, the first person suggesting to study ko fr was Marc Hoyois.2. The sheaves I n X , denote by W the Nisnevich sheaf of commutative discrete rings obtained bysheafification from the presheaf of Witt rings [Kne77, § I.5]. The canonical map W → a ´ et W ≃ Z / rank map , and its kernel is the ideal sheaf I ⊂ W . We write I ∗ for the sheaf of commutativegraded rings given by the powers of I . Somewhat anachronistically we put k Mn = a Nis H n ´ et ( − , Z / -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 3 Theorem 2.1 (Jacobson [Jac18]) . Assume that / ∈ X . Then there is a unique map of sheaves (ofrings) I ∗ → k M ∗ given in degree zero by the rank and in degree one locally by ( h a i − ( a ) . This mapannihilates I ∗ +1 ⊂ I ∗ and induces an isomorphism of sheaves I ∗ /I ∗ +1 ≃ k M ∗ . Proof.
For existence, see [Jac18, Remark 4.5]. The rest is [Jac18, Theorem 4.4]. (cid:3)
Still assuming that 1 / ∈ X , the canonical map σ : W → a r ´ et W ≃ a r ´ et Z is the global signature . One may show that σ ( I ) ⊂ a r ´ et Z (indeed locally I( − ) consists of diagonal formsof even rank [MH73, Corollary I.3.4], and the signature is thus a sum of an even number of terms ± σ ( I n ) ⊂ n a r ´ et Z . Since a r ´ et Z is torsion-free, there are thus induced maps σ/ n : I n → a r ´ et Z . For a scheme X , denote by K ( X ) the product of the residue fields of its minimal points. Lemma 2.2.
Let X be a noetherian scheme with /p ∈ X . Then vcd p ( X ) ≤ dim X + vcd p ( K ( X )) .Proof. If p = 2, replace X by X [ √− p = cd p , and if p = 2 thatall residue fields of X are unorderable. By [ILO14, Lemma XVIII-A.2.2] we have cd p ( X ) ≤ dim X +sup x ∈ X (dim O X,x + cd p ( k ( x ))). It hence suffices to show that cd p ( k ( x )) + dim O X,x ≤ cd p ( K ( X )). Sincecd p ( K ( X )) ≥ cd p ( K ( O X,x )), this follows from [GAV72, Corollary X.2.4]. (cid:3)
Proposition 2.3 (Jacobson) . Assume that X is noetherian and / ∈ X . Then for n > vcd ( K ( X )) +dim X the divided signature σ/ n : I n → a r ´ et Z is an isomorphism of sheaves on X Nis .Proof.
Let N = vcd ( K ( X )) + dim X . Note that for any Hensel local ring A of X we have vcd ( A ) ≤ N ,by Lemma 2.2 and [GAV72, Theorem X.2.1]. Since A is henselian and noetherian, by [Jac18, Lemma5.2(III)] we have ∩ n I n ( A ) = 0. Hence by [Jac18, Corollary 4.8] for n > N the map I n ( A ) −→ I n +1 ( A ) is anisomorphism. By [Jac17, Proposition 7.1], the divided signatures induce an isomorphism colim n I n ( A ) ≃ ( a r ´ et Z )( A ). These two results imply that σ/ n ( A ) : I n ( A ) → ( a r ´ et Z )( A ) is an isomorphism, for any n > N . Since A was arbitrary, the map σ/ n : I n → a r ´ et Z induces an isomorphism on stalks, and henceis an isomorphism. (cid:3) D . A G m -prespectrum E over D means a sequence of objects ( E , E , . . . )with E i ∈ Fun(Sm op D , SH ), together with maps E i → Ω G m E i +1 . Such a prespectrum can in particularbe exhibited by defining E i as a presheaf of abelian groups. See e.g. [CD09, §
6] for details as wellas symmetric (monoidal) variants. A G m -prespectrum E is called a motivic spectrum if each E i ismotivically local, and the structure maps E i → Ω G m E i +1 are equivalences. Example . There is a G m -prespectrum H Z / Z / i = Σ i τ Nis ≥− i L ´ et Z / . In other words π ∗ (H Z / ∗ ≃ k M ∗ [ τ ] . The prespectrum H Z / Example . Let R ∗ be a Nisnevich sheaf of commutative graded (discrete) rings, and t ∈ R ( A \ R ∗ defines a commutative monoid ˜ R ∗ in symmetric sequences (of Nisnevich sheaves) with trivialsymmetric group actions, and t defines a class [ t ] in the summand R ( G m ), making ˜ R ∗ into a commutativemonoid under the free commutative monoid on G m . In other words, ˜ R ∗ is a commutative monoid insymmetric G m -prespectra [CD09, second half of § R ∗ . Definition 2.6.
Applying Example 2.5 to the sheaf of graded rings I ∗ and the class h t i− ∈ I ( D × G m ),we obtain a G m -prespectrum K W over D with K Wn = I n . Similarly we obtain a G m -prespectrum k M ,and in fact a morphism of commutative monoids in symmetric G m -prespectra K W → k M (coming fromthe ring map I ∗ → k M ∗ of Theorem 2.1). TOM BACHMANN
From now on we view the category of Nisnevich sheaves of abelian groups as embedded into Nisnevichsheaves of spectra, and view all sheaves of abelian groups as sheaves of spectra, so that for X ∈ Sm D wehave ( K W ) i ( X ) = L Nis I i , and similarly for k M . Lemma 2.7.
There is a commutative ring map H Z / → k M inducing an equivalence of G m -prespectra k M ≃ (H Z / /τ . In particular k M is a motivic spectrum.Proof. Let E = ( E , E , . . . ) be a G m -prespectrum in Nisnevich sheaves of spectra. We can form aprespectrum τ Nis ≤ ( E ) with τ Nis ≤ ( E ) i ≃ τ Nis ≤ ( E i ) the truncation in the usual t -structure, and bondingmaps G m ∧ τ Nis ≤ ( E i ) → τ Nis ≤ ( G m ∧ τ Nis ≤ ( E i )) ≃ τ Nis ≤ ( G m ∧ E i ) → τ Nis ≤ ( E i +1 ) . Even if E is a motivic spectrum τ Nis ≤ ( E ) need not be; however if it is then it represents the truncation τ ≤ ( E ) ∈ SH ( D ) in the homotopy t -structure.In [Spi12, § G m -prespectrum H such that (1) H is a motivicspectrum representing H Z / τ Nis ≤ ( H ) ≃ k M , the equivalence being as G m -prespectra. The mapH Z / → (H Z / /τ ∈ SH ( D ) corresponds to a map H → H ′ of G m -spectra which is immediately seento be a levelwise zero-truncation. It follows that H ′ ≃ τ Nis ≤ ( H ) ≃ k M as G m -prespectra. In particular τ Nis ≤ ( H ) ≃ k M are motivic spectra, and in fact k M ≃ τ ≤ (H Z / ∈ SH ( D ). Since SH ( D ) ≥ is closedunder smash products, truncation in the homotopy t -structure is lax symmetric monoidal on SH ( D ) ≥ and so τ ≤ (H Z /
2) admits a canonical ring structure making H Z / → (H Z / ≤ into a commutative ringmap. It remains to show that τ ≤ (H Z / ≃ k M is an equivalence of ring spectra. Both of them can bemodeled by E ∞ -monoids in the ordinary 1-category of symmetric G m -prespectra of sheaves of abeliangroups on Sm D ; i.e. just commutative monoids in the usual sense. The isomorphism between thempreserves the product structure by inspection. (cid:3) Corollary 2.8.
Let D be a Dedekind scheme with / ∈ D . The G m -prespectrum K W is a motivicspectrum over D .Proof. Since ( K W ) n = L Nis I n is Nisnevich local by construction, to prove it is motivically local we needto establish homotopy invariance, i.e. that Ω A K Wn ≃ K Wn . Similarly to prove that we have a spectrumwe need to show that Ω G m K Wn +1 ≃ K Wn . Here we are working in the category S hv Nis SH (Sm D ) of Nisnevichsheaves of spectra on Sm D . For x ∈ D , denote by p x : D x → X the inclusion of the local scheme. By[Hoy15, Lemmas A.3 and A.4], the functor p ∗ x commutes with Ω G m and Ω A , and by [BH17, PropositionA.3(1,3)] the family of functors p ∗ x : S hv Nis SH (Sm D ) → S hv Nis SH (Sm D x ) is conservative. Finally by [Bac18b,Corollary 51] we have p ∗ x I n ≃ I n . It follows that we may assume (replacing D by D x ) that D is thespectrum of a discrete valuation ring or field.By Lemma 2.10 below, we have D ≃ lim α D α , where each D α is the spectrum of a discrete valuationring or field and vcd ( K ( D α )) < ∞ . By [Gro67, Theorem 8.8.2(ii), Proposition 17.7.8(ii)] for X ∈ Sm D there exists (possibly after shrinking the indexing system) a presentation X ≃ lim α X α , with X α ∈ Sm D α and the transition maps being affine. We have a fibered topos [GAV72, § VII.7] X • Nis with lim X • Nis ≃ X Nis . The sheaves ( I n | X α Nis ) α define a section of the fibered topos X • Nis . It follows from [Bac18b,Lemma 49] and [GAV72, Proposition VII.8.4] that Q ∗ ( I n | X • Nis ) ≃ I n | X Nis . Hence by [GAV72, TheoremVII.8.7.3] we get H i ( X, I n ) ≃ colim α H i ( X α , I n ) . The same holds for cohomology on X × A and X + ∧ G m . We may thus assume (replacing D by D α )that vcd ( K ( D )) < ∞ .Let X ∈ Sm D . We need to prove that ( ∗ ) H ∗ ( X, I n ) ≃ H ∗ ( X × A , I n ) ≃ H ∗ ( X + ∧ G m , I n ) . Since k Mn satisfies the analog of ( ∗ ) by Lemma 2.7, the exact sequence I n +1 → I n → k Mn from Theorem2.1 shows that ( ∗ ) holds for I n if and only if it holds for I n +1 . By Proposition 2.3 (and [GAV72, TheoremX.2.1]), for n sufficiently large we get I n | X Nis ≃ a r ´ et Z | X Nis . It thus suffices to show that L Nis a r ´ et Z satisfiesthe analog of ( ∗ ). This follows from the fact that there exists a motivic spectrum E = H A Z [ ρ − ] with E i = L Nis a r ´ et Z for all i [Bac18a, Proposition 41]. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 5 Remark . Theorem 2.1 shows that Ω G m L Nis I ≃ −→ Ω G m L Nis W ; indeed the cofiber of this map is givenby Ω G m Z / ≃ π ( K W ) ∗ ≃ I ∗ and π i = 0 for i = 0; here I n = W for n <
0. It also follows thatΩ G m L Nis W ≃ L Nis W , and that this spectral sheaf is homotopy invariant.Recall that a noetherian valuation ring is a ring which is either a discrete valuation ring or a field[Sta18, Tag 00II]. Lemma 2.10.
Let R be a noetherian valuation ring. Then there is a filtered system R α of noetherianvaluation rings with vcd( K ( R α )) < ∞ (i.e. there exists N with vcd p < N for all p ) and colim α R α ≃ R .Proof. Let K = K ( R ). Then K = colim α K α , where the colimit is over finitely generated subfields K α ⊂ K ; this colimit is filtered. Let R α = R ∩ K α . We shall show that R α is a noetherian valuation ring,and K ( R α ) = K α . This will imply the result since vcd p ( K α ) < ∞ uniformly in p by [GAV72, TheoremX.2.1, Proposition X.6.1, Theorem 5.1]. It is clear that R α is a valuation ring and K ( R α ) = K α : if x ∈ K × α then one of x, x − ∈ R [Sta18, Tag 00IB], and hence one of x, x − ∈ R α ; thus we conclude by[Sta18, Tag 052K]. To show that R α is noetherian we must show that K × α /R × α ≃ Z or ≃ K × α /R × α ֒ → K × /R × and the latter group is ≃ Z or ≃ (cid:3) η -periodic framed spectra + § ∞ -category Cor fr ( S ) under Sm S . Wedenote by S pc fr ( S ) and SH S fr ( S ) the motivic localizations of respectively Fun(Cor fr ( S ) op , S pc) andFun(Cor fr ( S ) op , SH ), and we put SH fr ( S ) = SH S fr ( S )[ G ∧− m ] ≃ S pc fr ( S )[( P ) − ]. The free-forgetfuladjunction SH ( S ) ⇆ SH fr ( S )is an adjoint equivalence [Hoy18, Theorem 18]. We obtain the diagram S pc fr ( S ) SH S fr ( S ) SH ( S ) . Σ ∞ S Σ ∞ fr σ ∞ fr Ω ∞ S ω ∞ fr Ω ∞ fr S hv Nis SH (Sm S ) = L Nis
Fun(Sm op S , SH ) and S hv Nis SH (Cor fr ( S )) = L Nis
Fun(Cor fr ( S ) op , SH ) . On either category we consider the t -structure with non-negative part generated [Lur16, Proposition1.4.4.11] by the smooth schemes. Lemma 3.1. (1) E ∈ Fun(Cor fr ( S ) op , SH ) is Nisnevich local (or homotopy invariant, or motivicallylocal) if and only if the underlying spectral presheaf U E ∈ Fun(Sm op S , SH ) is.(2) The forgetful functor U : S hv Nis SH (Cor fr ( S )) → S hv Nis SH (Sm S ) is t -exact.Proof. (1) holds by definition. (2) The functor U Σ : P Σ (Cor fr ( S )) → P Σ (Sm S ) ∗ preserves filtered(in fact sifted) colimits and commutes with L Nis [EHK +
17, Proposition 3.2.14]. Consequently U Nis : S hv Nis (Cor fr ( S )) → S hv Nis (Sm S ) ∗ also preserves filtered colimits. Being a right adjoint it also preserveslimits, and hence commutes with spectrification. Consequently it suffices to prove the following: given F ∈ P Σ (Cor fr ( S )) and n ≥
0, we have U Σ (Σ n F ) ∈ P Σ (Sm S ) ≥ n ; indeed then U (Σ ∞ F ) ≃ colim n Σ ∞− n U Σ (Σ n F ) ∈ S hv Nis SH (Sm S ) ≥ by what we have already said. Writing Σ n F as an iterated sifted colimit, using semi-additivity of P Σ (Cor fr ( S )) [EHK +
17, sentence after Lemma 3.2.5] and the fact that U Σ commutes with sifted colimits,we find that U Σ (Σ n F ) is given by B n U Σ ( F ), i.e. the iterated bar construction applied sectionwise. Therequired connectivity is well-known; see e.g. [Seg74, Proposition 1.5]. (cid:3) TOM BACHMANN
We denote by τ Nis ≥ i , τ Nis ≤ i and τ Nis= i the truncation functors corresponding to the above t -structures.Note that the t -structure we have constructed on S hv Nis SH (Sm S ) coincides with the usual one [Lur18,Definition 1.3.2.5], and in particular S hv Nis SH (Sm S ) ♥ is just the category of Nisnevich sheaves of abeliangroups on Sm S [Lur18, Proposition 1.3.2.7(4)]. Remark . The proof of Lemma 3.1 also shows the following: if F ∈ S hv Nis (Cor fr ( S )) and X ∈ Sm S ,then F ( X ) ∈ S pc is a commutative monoid (Cor fr ( S ) being semiadditive) and Σ ∞ F ∈ S hv Nis SH (Cor fr ( S ))has underlying sheaf of spectra corresponding to the group completion of F . Remark . If S has finite Krull dimension, then the above t -structure is non-degenerate [BH17, Propo-sition A.3].3.3. The unit t ∈ O ( A \ × defines a framing of the identity on A \ A \ ∗ . We denote by η : G m → ∈ SH S fr ( S )the corresponding map (obtain by precomposition with G m ֒ → G m ∨ ≃ A \ Lemma 3.4.
There is a homotopy σ ∞ ( η ) ≃ η , where on the right hand side we mean the usual motivicstable Hopf map.Proof. Write h t i : ( A \ + → ( A \ + ∈ SH ( S ) for the map induced by the framing t of the identity,and ˜ η for the induced map Σ ∞ + G m → . By [EHK +
18, Example 3.1.6], the map h t i is given by Σ ∞− , of the map of pointed motivic spaces( A \ + ∧ A / ( A \ → ( A \ + ∧ A / ( A \ , ( t, x ) ( t, tx ) , and hence ˜ η is given by Σ ∞− , of the map η ′ : ( A \ + ∧ A / ( A \ → A / A \ , ( t, x ) tx. Consider the commutative diagram of pointed motivic spaces (with A pointed at 1) G m × G m −−−−→ G m × A −−−−→ ( G m × A ) / ( G m × G m ) ≃ ( A \ + ∧ A / ( A \ y y η ′ y G m −−−−→ A −−−−→ A / G m . All the vertical maps are induced by ( t, x ) tx and the horizontal maps are induced by the canonicalinclusions and projections. The rows are cofibration sequences. Stably this splits as [Mor04a, Lemma6.1.1] G m ∨ G m ∨ G ∧ m (id , , −−−−−→ G m −−−−→ S , ∨ Σ , G m (id , id , Σ , η ) y y Σ , ˜ η y G m −−−−→ −−−−→ S , . It follows that ˜ η ≃ (id , η ), which implies the desired result. (cid:3) We call E ∈ SH S fr ( S ) η -periodic if the canonical map η ∗ : E → Ω G m E is an equivalence. Write SH S fr ( S )[ η − ] ⊂ SH S fr ( S )for the full subcategory on η -periodic spectra. These are the local objects of a symmetric monoidallocalization of SH S fr ( S ). Lemma 3.5.
There is a canonical equivalence SH S fr ( S )[ η − ] ≃ SH ( S )[ η − ] .Proof. In light of Lemma 3.4, this is a special case of [Hoy16, Proposition 3.2]. (cid:3) D with 1 / ∈ D . Consider ω ∞ fr (KW) ∈ SH S fr ( D ) ⊂S hv Nis SH (Cor fr ( D )). This object is η -periodic by construction. Definition 3.6.
We put HW = τ Nis=0 ω ∞ fr (KW) ∈ S hv Nis SH (Cor fr ( D )) . Remark . Multiplication by β i induces τ Nis=4 i ω ∞ fr (KW) ≃ HW, and the other homotopy sheaves vanish[Sch17, Proposition 6.3] [Bal05, Theorem 1.5.22].
Lemma 3.8. HW is motivically local and η -periodic. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 7 Proof.
The framed construction of KO as in [BW20, § B] together with Lemma 3.4 shows that the map η ∗ : HW → Ω G m HW is on the level of spectral sheaves (i.e. sheaves of abelian groups) induced bymultiplication by h t i −
1. The result now follows from Remark 2.9. (cid:3)
Thus HW defines an object of SH ( D )[ η − ].3.5. By construction, the forgetful functor SH S fr ( D )[ η − ] → SH ( D )[ η − ] ⊂ SH ( D ) sends HW to thespectrum ( W , W , . . . ), with the canonical structure maps. It follows that there is a canonical morphism(of ring spectra) K W → HW, given in degree n by the inclusion I n ֒ → W . Lemma 3.9.
The canonical maps K W → HW and K W → k M induce equivalences K W [ η − ] ≃ HW and K W /η ≃ k M .Proof. The equivalence SH fr ( D ) ≃ SH ( D ) restricts to the subcategories of those objects such that π i ( − ) j = 0 for i = 0. These are exactly the objects representable by prespectra valued in sheaves ofabelian groups that are motivic spectra when viewed as valued in spectral sheaves. The constructionof HW supplies the sheaf W with a structure of framed transfers. Then the subsheaf I n admits atmost one compatible structure of framed transfers; the existence of the map K W → HW together withthe above discussion shows that this structure exists. This supplies a description of the (unique) liftof K W to SH fr ( D ). From this and Lemma 3.4 it follows that the action by η on K W induces theinclusion I ∗ +1 → I ∗ on homotopy sheaves. This implies immediately that K W [ η − ] → HW induces anisomorphism on homotopy sheaves and hence is an equivalence (see Remark 3.3). For K W /η the sameargument works using Theorem 2.1. (cid:3) Definition 3.10.
We put kw = τ Nis ≥ ω ∞ KW ∈ S hv Nis SH (Cor fr ( D ))The canonical map KO → KW ∈ SH ( D ) induces τ Nis ≥ ω ∞ KO → kw . Lemma 3.11. (1) The objects τ Nis ≥ ω ∞ fr KO , kw ∈ S hv Nis SH (Cor fr ( D )) are motivically local.(2) kw is η -periodic.(3) The canonical map Σ ∞ S Ω ∞ fr KO → τ Nis ≥ ω ∞ fr KO is an equivalence.(4) The canonical map Σ ∞ S Ω ∞ fr KO → kw is an η -periodization.Proof. (1, 2) Since the negative homotopy sheaves (in weight 0) of KO and KW coincide [Sch17, Propo-sition 6.3], we have τ Nis < ω ∞ fr KO ≃ τ Nis < ω ∞ fr KW =: E. Since ω ∞ KO and ω ∞ fr KW are motivically local, and ω ∞ fr KW is η -periodic, it thus suffices to show that E is motivically local and η -periodic. Since motivically local, η -periodic spectral presheaves are closedunder limits and colimits, by Remark 3.7 it thus suffices to show that τ Nis=0 ω ∞ fr KW ≃ HW is motivicallylocal and η -periodic. This is Lemma 3.8.(3) It follows from Remark 3.2 that the functor Σ ∞ S Ω ∞ S sends the spectral sheaf E on Cor fr ( D ) to L mot E ′ , where E ′ ( X ) ≃ E ( X ) ≥ . We thus obtain Σ ∞ S Ω ∞ S ≃ L mot τ Nis ≥ . Since Ω ∞ fr ≃ Ω ∞ S ◦ ω ∞ fr , the resultfollows.(4) KO → KW ∈ SH ( S ) is an η -periodization. It follows from [Hoy16, Lemma 3.3] and [BH17,Lemma 12.1] that ω ∞ fr preserves η -periodizations. Hence ω ∞ fr KO → ω ∞ fr KW is an η -periodization. Thusit suffices to show that τ Nis < ω ∞ fr KO → τ Nis < ω ∞ fr KWis an η -periodization. Since this is an equivalence of η -periodic objects (see the proof of (1,2)), this isclear. (cid:3) Thus in particular kw defines an object of SH ( D )[ η − ].4. Main results D with 1 / ∈ D . In the previous two sections we have defined ring spectra K W , k M and HW in SH ( D ). TOM BACHMANN
Definition 4.1.
Define commutative ring spectra H W Z and H˜ Z in SH ( D ) as pullbacks in the followingdiagram with cartesian squares H˜ Z −−−−→ H W Z −−−−→ K W y y y H Z −−−−→ H Z / −−−−→ k M . Put ko fr := Σ ∞ fr Ω ∞ fr KO.
Lemma 4.2. If D is the spectrum of a field, then K W , k M , HW , H W Z , H˜ Z and kw coincide with theirusual definitions, and ko fr ≃ ko . Over a field, the spectra K W , k M are defined in [Mor12, Example 3.33], the spectra H W Z , H˜ Z aredefined in [Bac17, Notation p. 12] (another definition of H˜ Z was given in [DF17], the two definitionsare shown to coincide in [BF17]), the spectrum ko is defined in [ARØ17], and the spectra HW , kw aredefined in [BH20, § Proof.
We first treat ko. Since very effective covers are stable under pro-smooth base change [BH17,Lemma B.1], and ko fr is stable under arbitrary base change (see the proof of Theorem 4.4 below), we mayassume that the base field is perfect. In this case S pc fr ( k ) gp ≃ SH ( k ) veff [EHK +
17, Theorem 3.5.14(i)]and hence Σ ∞ fr Ω ∞ fr coincides with the very effective cover functor. Thus ko ≃ ko fr as needed.The claim for kw follows via [BH20, Lemma 6.9].For K W , k M , the claim is true essentially by construction. This implies the claim for HW via Lemma3.9.The claim for H W Z now follows from [Bac17, Theorem 17] (see also [BH20, (6.5)]). For H˜ Z considerthe commutative diagram E −−−−→ K MW −−−−→ K W y y y H Z −−−−→ K M −−−−→ k M . The left hand square is defined to be cartesian, so that by [Bac17, Theorem 17], E coincides with theusual definition of H˜ Z . The right hand square is cartesian by [Mor04b, Theorem 5.3]. Hence the outerrectangle is cartesian and E ≃ H˜ Z as defined above. This concludes the proof. (cid:3) Lemma 4.3.
Over a Dedekind scheme containing / we have π ∗ (kw) ≃ W [ β ] and kw /β ≃ HW .Proof. Immediate from Remarks 3.7 and 3.3. (cid:3)
Theorem 4.4.
The spectra K W , k M , HW , H W Z , H˜ Z , ko fr and kw are stable under base change amongDedekind schemes containing / .Proof. For ko fr this follows from the facts that (1) Ω ∞ KO ∈ S pc( D ) is motivically equivalent to theorthogonal Grassmannian [ST15, Theorem 1.1], which is stable under under base change, and that (2)the forgetful functor S pc fr ( D ) → S pc( D ) commutes with base change [Hoy18, Lemma 16]. The case ofkw follows from this and Lemma 3.11, which shows that kw ≃ ko fr [ η − ]. The case of HW now followsfrom Lemma 4.3.The spectra H Z and H Z / k M ≃ (H Z / /τ (seeLemma 2.7). To show that K W is stable under base change it suffices to show the same about K W [ η − ]and K W /η (see e.g. [BH20, Lemma 2.16]). In light of Lemma 3.9, this thus follows from the cases ofHW and k M , which we have already established.Finally the stability under base change of H W Z and H˜ Z reduces by definition to the same claim aboutH Z , H Z / , k M and K W , which we have aready dealt with.This concludes the proof. (cid:3) Definition 4.5.
Let S be a scheme with 1 / ∈ S . Define spectra K W , k M , HW , H W Z , H˜ Z , ko fr , kw ∈SH ( S ) by pullback along the unique map S → Spec( Z [1 / Remark . (1) By Theorem 4.4, if S is a Dedekind scheme, the new and old definitions agree.(2) The proof of Theorem 4.4 shows that for S arbitrary we have ko fr S ≃ Σ ∞ fr Ω ∞ fr KO S and hencekw S ≃ (Σ ∞ fr Ω ∞ fr KO S )[ η − ]. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 9 (3) Since formation of homotopy sheaves is compatible with essentially smooth base change, andso is W (see e.g. [Bac18b, Corollary 51]), we find that if S → D is essentially smooth then π ∗ (kw S ) ≃ W [ β ], and similarly for the homotopy sheaves of the other spectra. Lemma 4.7.
Suppose that S has finite dimension. Let E ∈ SH ( S ) (respectively E ∈ SH ( S ) eff ). Inorder that E ∈ SH ( S ) ≤ (respectively E ∈ SH ( S ) eff ≤ ) it suffices that π i ( E ) ∗ = 0 for i > and ∗ ∈ Z (respectively i > and ∗ = 0 ).Proof. We gave the proof for SH ( S ), the one for SH ( S ) eff is analogous. Let C ⊂ SH ( S ) denote thesubcategory on those spectra F with Map(Σ F, E ) = ∗ . We need to show that SH ( S ) ≥ ⊂ C . Bydefinition, for this it is enough to show that (a) C is closed under colimits, (b) C is closed under extensions,and (c) Σ ∞ + X ∧ G ∧ im ∈ C for every X ∈ Sm S and i ∈ Z . (a) is clear, and (b) follows from the fact thatgiven any fiber sequence of spaces ∗ → S → ∗ we must have S ≃ ∗ . To prove (c), it is enough toshow that Ω ∞ ( G ∧ im ∧ E ) ≃ ∗ ∈ S hv S pc Nis (Sm S ). By [BH17, Proposition A.3], this can be checked onhomotopy sheaves. (cid:3) We denote by τ ≥ , τ ≤ (respectively τ eff ≥ , τ eff ≤ ) the truncation functors for the homotopy t -structure on SH ( S ) (respectively SH ( S ) eff ) and f for the effective cover functor; see e.g. [BH17, § B] for a uniformtreatment.
Remark . We will repeatedly use [BH17, Proposition B.3], which states the following: if E ∈ SH ( S ),where S is finite dimensional, then E ∈ SH ( S ) ≥ (respectively SH ( S ) eff , SH ( S ) eff ≥ , respectively { } )if and only if for every point s ∈ S with inclusion i s : s ֒ → S we have i ∗ s ( E ) ∈ SH ( s ) ≥ (respectively i ∗ s ( E ) ∈ SH ( s ) eff , i ∗ s ( E ) ∈ SH ( s ) eff ≥ , i ∗ s ( E ) ≃ Corollary 4.9.
Let D be a Dedekind scheme containing / , and S → D pro-smooth (e.g. essentiallysmooth). The canonical maps exhibit equivalences in SH ( S )kw ≃ τ ≥ KWHW ≃ τ ≤ [ η − ] ≃ τ ≤ kwH W Z ≃ f HW ≃ f K W H˜ Z ≃ τ eff ≤ ≃ τ eff ≤ ko fr K W ≃ τ ≤ H W Z . Proof.
By [BH17, Lemma B.1], pro-smooth base change commutes with truncation in the homotopy t -structure, in the effective homotopy t -structure, and also with effective covers. We may thus assumethat S = D .Consider the cofiber sequence kw → KW → E . To prove that kw ≃ τ ≥ KW, it is enough to showthat kw ∈ SH ( D ) ≥ and E ∈ SH ( D ) < . By Remark 4.8 we may check the first claim after base changeto fields, and hence by Theorem 4.4 for this part we may assume that D is the spectrum of a field, wherethe claim holds by definition. The second claim follows via Lemma 4.7 from our definition of kw as aconnective cover in the Nisnevich topology (see Definition 3.10).Consider the fiber sequence F → [ η − ] → HW. To prove that τ ≤ [ η − ] ≃ HW it suffices to showthat F ∈ SH ( D ) > and HW ∈ SH ( D ) ≤ . As before the first claim can be checked over fields where itholds by definition, and the second one follows from Lemma 4.7 and the definition of HW (see Definition3.6). The argument for τ ≤ kw ≃ HW is similar.Since the map K W → HW of § π ∗ ( − ) (by construction), we have f HW ≃ f K W . We have H W Z ∈ SH ( D ) eff by checking over fields; it thus remains to show that f H W Z ≃ f K W . By the defining fiber square, for this it is enough to show that f H Z / ≃ f k M . Thisfollows from the cofibration sequence Σ , − H Z / τ −→ H Z / → k M together with the fact that f H Z / ≃ Z ∈ SH ( D ) veff by checking over fields. Knowledge of π ∗ (H Z ) ≃ Z , π ∗ ( k M ) = Z / π ∗ ( K W ) ≃ W implies via Lemma 4.7 that H˜ Z ∈ SH ( D ) eff ≤ . It hence remains to show that the fibers of → H˜ Z and ko fr → H˜ Z are in SH ( D ) eff > , which may again be checked over fields.Consider the fiber sequence F → H W Z → K W . We need to show that F ∈ SH ( D ) ≥ and K W ∈SH ( D ) ≤ . As before the first claim can be checked over fields where it holds by [Bac17, Lemma 18],and the second claim follows from Lemma 4.7 and the knowledge of the homotopy sheaves of K W , i.e.Remark 2.9. This concludes the proof. (cid:3) §
3] the stable Adams operation ψ : KO[1 / → KO[1 / ∈ SH ( D ). ViaCorollary 4.9 this induces ψ : kw (2) → kw (2) ∈ SH ( D ). Theorem 4.10.
Let D be a Dedekind scheme with / ∈ D .(1) The map ψ − id : kw (2) → kw (2) factors uniquely (up to homotopy) through β : Σ kw (2) → kw (2) ,yielding ϕ : kw (2) → Σ kw (2) ∈ SH ( D ) . (2) The unit map → kw (2) factors uniquely (up to homotopy) through the fiber of ϕ .(3) The resulting sequence [ η − ] (2) → kw ϕ −→ Σ kw (2) ∈ SH ( D ) is a fiber sequence.In particular for any scheme S with / ∈ S there is a canonical fiber sequence η − ] (2) → kw (2) ϕ −→ Σ kw (2) ∈ SH ( S ) . Proof. (1,2) We can repeat the arguments from [BH20, Corollary 7.2]. It suffices to show that (a) [ η − ] → kw is 1-connected, (b) kw /β ∈ SH ( D ) ≤ , and (c) [ , Σ n kw] SH ( D ) = 0 for n ∈ { , } . (a) canbe checked over fields, hence holds by [BH20, Lemma 7.1]. (b) follows from Lemma 4.3 (showing thatkw /β ≃ HW) and Corollary 4.9 (showing that HW ∈ SH ( D ) ≤ ). (c) follows from knowledge of thehomotopy sheaves of kw together with the descent spectral sequence, using that dim D ≤ [ η − ] → fib( ϕ ) is an equivalence. This can be checkedover fields, where it is [BH20, Theorem 7.8].The last claim follows by pullback along S → Spec( Z [1 / (cid:3) Applications
Throughout we assume that 2 is invertible on all schemes.5.1. Recall that kw ∗ MSL ≃ kw ∗ [ p , p , . . . ] with | p i | = 4 i . The canonical orientation MSL → kwinduces MSL → kw → τ ≤ kw ≃ HW and hencekw ∗ MSL → kw ∗ kw → kw ∗ HW . Proposition 5.1.
The images of the p i induce equivalences of right modules kw ∧ kw (2) ≃ _ n ≥ kw (2) { p i } and kw ∧ HW (2) ≃ _ n ≥ HW (2) { p i } . Proof.
We may assume that S = Spec( Z [1 / (cid:3) Proposition 5.2.
Let S be essentially smooth over a Dedekind scheme. We have π ∗ ( S [ η − ]) ≃ W ∗ = 0 W [1 / ⊗ π s ∗ ⊕ coker(8 n : W (2) → W (2) ) ∗ = 4 n − > W [1 / ⊗ π s ∗ ⊕ ker(8 n : W (2) → W (2) ) ∗ = 4 n > W [1 / ⊗ π s ∗ else . Here π s ∗ denotes the classical stable stems.Proof. We first show that π ( [ η − ]) → π (HW) ≃ W is an isomorphism. We may do so after ⊗ Z (2) and ⊗ Z [1 / W [1 / ≃ a r ´ et Z [1 /
2] [Jac17, Corollary 7.1]). Theproof of [BH20, Theorem 8.1] now goes through unchanged. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 11
Lemma 5.3.
Suppose that D is a localization of Z . Then H ∗ ( D, W ) = 0 for ∗ > .Proof. [MH73, Corollary IV.3.3] shows that for any Dedekind scheme D there is a natural exact sequenceof abelian groups ( ∗ ) 0 → W ( D ) → M x ∈ D (0) W ( x ) → M x ∈ D (1) W ( x, ω x/D ) . The last map is surjective if D has only one point, and hence these sequences constitute a resolution ofthe presheaf W on D Nis . The terms of this resolution are acyclic [Mor12, Lemma 5.42] , and hence thisresolution can be used to compute cohomology. In order to show that H ∗ ( D, W ) = 0 for ∗ > ∗ ) is surjective. Clearly if this is true for D then it also holds forany localization of D . It hence suffices to prove surjectivity for Z ; this is [MH73, Theorem IV.2.1]. (cid:3) Example . Lemma 5.3 shows that the descent spectral sequence for π ∗ kw (or π ∗ KW) collapses over Z [1 /
2] (and localizations of this base). Thus π ∗ kw Z [1 / ≃ W( Z [1 / β ] . From this we can read off as in the proof of Proposition 5.2 that (using that W( Z [1 / / ≃ Z [1 / π ∗ ( Z [1 / [ η − ]) ≃ W( Z [1 / ∗ = 0 π s ∗ [1 / ⊕ coker(8 n : W( Z [1 / (2) → W( Z [1 / (2) ) ∗ = 4 n − > π s ∗ [1 / ⊕ ker(8 n : W( Z [1 / (2) → W( Z [1 / (2) ) ∗ = 4 n > π s ∗ [1 /
2] else . Remark . We often use the above result in conjunction with the isomorphism (see e.g. [BW20, proofof Theorem 5.8]) W( Z [1 / ≃ Z [ g ] / ( g , g ) . (Here g = h i − Z [1 / red ≃ Z and W( Z [1 / (2) is a local ring.5.4. Proposition 5.6. (1) We have π ∗ MSp Z [1 / [ η − ] ≃ W( Z [1 / y , y , . . . ] . (2) The generators from (1) induce for a scheme S essentially smooth over a Dedekind scheme D with / ∈ D π ∗ MSp S [ η − ] ≃ W [ y , y , . . . ] and π ∗ MSL S [ η − ] ≃ W [ y , y , . . . ] . (3) Over any S we have we have MSp / ( y , y , . . . ) ≃ MSL .(4) There exist generators y , y , . . . such that MSL / ( y , y , . . . ) ≃ kw . Proof.
We implicitly invert η throughout.We first prove (1) and the part of (2) about MSp, localized at 2. Note that kw ∗ MSp (2) ≃ W( Z [1 / (2) [ b , b , . . . ]is degreewise finitely generated over the local ring W( Z [1 / (2) (see Remark 5.5). Moreover base changealong Spec( C ) → Spec( Z [1 / Z [1 / (2) → W( C ) of passing to the residue field.Consider the morphism ϕ : kw ∗ MSp (2) → kw ∗− MSp (2) . We deduce from the above discussion and [BH20, Lemma 8.4] that ϕ ⊗ W( Z [1 / (2) F is surjective,and hence ϕ is split surjective. Then ker( ϕ ) ⊗ W( Z [1 / (2) F ≃ ker( ϕ ⊗ W( Z [1 / (2) F ) is a polyno-mial ring on generators ¯ y i . Write ˜ y i ∈ π ∗ MSp (2) for arbitrary lifts of these generators. The proof of[BH20, Corollary 8.6] shows that if Spec( k ) → Spec( Z [1 / π ∗ (MSp (2) )( k ) ≃ W( k ) (2) [˜ y , ˜ y , . . . ]. We shall show that if S is henselian local and essentially smooth over D then π ∗ (MSp (2) )( S ) ≃ π ∗ (MSp (2) )( s ), where s is the closed point; this will imply our claims. Using the fun-damental fiber sequence, for this it is enough to show that W( S ) ≃ W( s ), which holds by [Jac18, Lemma4.1].Note that by real realization, π ∗ (MSp[1 / Z [1 / ≃ π ∗ (MSp[1 / R ) ≃ Z [ y ′ , y ′ , . . . ]. Let J =ker( π ∗ (MSp)( Z [1 / → π ∗ (HW)( Z [1 / M = ( J/J ) n . We shall prove that M ≃ W( Z [1 / The proof of this result does not use the stated assumption that X is smooth over a field. argument is the same as in [BH20, Theorem 8.7]: such isomorphisms exist for M (2) and M [1 /
2] by whatwe have already done, which implies that M is an invertible W( Z [1 / Z [1 / red ≃ Z (see Remark 5.5) and hence all such invertible modules are trivial. Write y n ∈ π n (MSp)( Z [1 / M . We shall prove that the proposition holds with these choices of y i .Let S be as in (2). To show that π ∗ (MSp S ) ≃ W [ y , y , . . . ], it is enough that the map is anisomorphism after ⊗ Z (2) and after ⊗ Z [1 / SH ( S )[1 / , /η ] ≃ SH ( S r ´ et )[1 / E ∈ SH ( Z [1 / r ´ et )[1 /
2] with π ∗ E ≃ Z [1 / , y , y , . . . ], then for any morphism f : S → Z [1 /
2] we have π ∗ ( f ∗ E ) ≃ W [1 / , y , y , . . . ]. This follows from the facts that (a) Sper( Z [1 / ≃ ∗ , so our assumptionimplies that π ∗ E ≃ a r ´ et Z [1 / , y , y , . . . ], (b) f ∗ is (in this setting) t -exact, and (c) W [1 / ≃ a r ´ et Z [1 / Z [1 /
2] (and hence in general), MSp → MSL annihilates y i for i odd. Indeedwe can check this after ⊗ Z (2) and ⊗ Z [1 / R where we already know it. For the former case, we first verify as above that ϕ : kw ∗ MSL (2) → kw ∗− MSL (2) is surjective, and hence π ∗ MSL (2) → kw ∗ MSL (2) is injective; the claim follows easily fromthis. It follows that we may form MSp / ( y , y , . . . ) → MSL. To check that this is an equivalence wemay pull back to fields, and so we are reduced to [BH20, Corollary 8.9]. We have now proved (3), whichimplies the missing half of (2).It remains to establish (4). We claim that π MSL Z [1 / → π kw Z [1 / is surjective, and hence anisomorphism since both groups are free W( Z [1 / ⊗ Z (2) and ⊗ Z [1 / ⊗ W( Z [1 / (2) F , i.e. over C , in which case the claim holds by [BH20, Lemma 8.10]. In thelatter case, via real realization we reduce to R , and so again the claim holds by [BH20, Lemma 8.10].The upshot is that we may choose y in such a way that its image in π kw is the generator β . Then as inthe proof of [BH20, Corollary 8.11] we modify the other generators to be annihilated in kw to obtain amap MSL / ( y , y , . . . ) → kw. This map is an equivalence since it is so after pullback to fields, by [BH20,Corollary 8.11]. (cid:3) Proposition 5.7.
Let S be the set of primes not invertible on S . The spectra kw , HW , H˜ Z [ S − ] , H W Z , K W ∈SH ( S ) are cellular.Proof. We can argue as in [BH20, Proposition 8.12]:For kw this follows from Proposition 5.6(3) and cellularity of MSp. Hence HW ≃ kw /β is cellular.H Z [ S − ] is cellular by [Spi12, Corollary 11.4]; in particular H Z / k M ≃ (H Z / /τ are cellular. Thus K W /η ≃ k M and K W [ η − ] ≃ HW are cellular and thus so is K W . Cellularity of H˜ Z [ S − ] and H W Z now follows from the defining fiber squares (in which all the other objects are cellular by what we havealready proved). (cid:3) Proposition 5.8. (1) There exist generators x i ∈ π i (kw ∧ HW (2) ) such that x m x n = 2 (cid:18) m + nn (cid:19) x m + n . (2) We have HW ∧ HW (2) ≃ _ n ≥ Σ n HW / n. Proof.
We may work over Z [1 / π ∗ HW = W( Z [1 / ≃ Z [ g ] / ( g , g ) . (1) Let K = ker(W( Z [1 / → W( R )). Then K is generated by g and hence annihilated by 2. Basechange along Z [1 / → R induces a surjection π ∗ (kw ∧ HW (2) )( Z [1 / → π ∗ (kw ∧ HW (2) )( R ); pick the x i over Z [1 /
2] as arbitrary generators lifting the appropriate generators over R (which exist by [BH20,Lemma 8.15]). Then the desired relation holds without the factor of 2 over R , and hence with the factorof 2 over Z [1 / K being annihilated by 2.(2) The image of β in π (kw ∧ HW (2) ) is given by ax for some a ∈ W( Z [1 / a = 8 u for some unit u . The result now follows from (1), as in the proofof [BH20, Corollary 8.16]. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER DEDEKIND DOMAINS 13 Proposition 5.9.
For arbitrary S we have kw ∗ (2) kw ≃ kw ∗ (2) J ′ ϕ K , in the sense that ϕ need not be central and so the multiplicative structure is more complicated than apower series ring. We have ϕβ = 9 βϕ + 8 .Proof. Stability under base change implies that [BH20, Lemma 8.18] holds over any base; hence theadditive structure of kw ∗ (2) kw can be determined as in [BH20, proof of Corollary 8.19]. The interactionof β and ϕ may be determine over Z [1 / Z [1 / ֒ → W( Q ) we are reduced to S = Spec( Q ),which was already dealt with in [BH20, Corollary 8.19]. (cid:3) Example . Suppose that S is essentially smooth over a Dedekind scheme, H ∗ ( S, W ) = 0 for ∗ > S ) is generated by 1-dimensional forms (e.g. S henselian local or a localization of Z ). Thenkw ∗ ≃ W( S )[ β ] and π ( ) → π (kw) is surjective, whence ϕ commutes with W( S ). It follows thatProposition 5.9 yields a complete description of kw ∗ (2) kw. References [ARØ17] Alexey Ananyevskiy, Oliver R¨ondigs, and Paul Arne Østvær. On very effective hermitian k -theory. arXivpreprint arXiv:1712.01349 , 2017.[Bac17] Tom Bachmann. The generalized slices of hermitian k-theory. Journal of Topology , 10(4):1124–1144, 2017.arXiv:1610.01346.[Bac18a] Tom Bachmann. Motivic and real ´etale stable homotopy theory.
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