η -periodic motivic stable homotopy theory over fields
aa r X i v : . [ m a t h . K T ] J un η -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS TOM BACHMANN AND MICHAEL J. HOPKINS
Abstract.
Over any field of characteristic = 2, we establish a 2-term resolution of the η -periodic,2-local motivic sphere spectrum by shifts of the connective 2-local Witt K -theory spectrum. Thisis curiously similar to the resolution of the K (1)-local sphere in classical stable homotopy theory.As applications we determine the η -periodized motivic stable stems and the η -periodized algebraicsymplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivicspectrum representing Hermitian K -theory and establish new completeness results for certain motivicspectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proofof the homotopy fixed point theorem for the Hermitian K -theory of fields. Contents
1. Introduction 21.1. Inverting the motivic Hopf map 21.2. Main results 21.3. Proof sketch 31.4. Organization 41.5. Acknowledgements 51.6. Conventions 51.7. Table of notation 62. Preliminaries and Recollections 62.1. Filtered modules 62.2. Derived completion of abelian groups 82.3. Witt groups 92.4. The homotopy t -structure 92.5. Completion and localization 102.6. Very effective spectra 112.7. Inverting ρ j spectrum 233.8. Geometric operations 254. Cobordism spectra 274.1. Summary 274.2. Real realization 284.3. Homology 304.4. Adams action 345. Some completeness results 355.1. Summary 355.2. Main result 355.3. Remaining proofs 356. The HW-homology of kw 376.1. Summary 376.2. The motivic dual Steenrod algebra 376.3. Spectra employed in the proof 40 Date : June 4, 2020. π ∗ (kw ∧ HW) ∧ η -periodic sphere 488.2. Homotopy groups of MSp[ η − ] 498.3. Homotopy groups of MSL[ η − ] 528.4. MSp, MSL and kw 528.5. Cellularity results 538.6. HW ∧ HW (2) ∗ kw (2) Introduction
Inverting the motivic Hopf map.
Let k be a field. We are interested in studying the motivicstable homotopy category SH ( k ) [Mor03, § k of smooth k -varieties by (1) freely adjoining (homotopy) colimits, (2) enforcing Nisnevich descent,(3) making the affine line A contractible, (4) passing to pointed objects and (5) making the operation ∧ P invertible: SH ( k ) = L A , Nis P (Sm k ) ∗ [( P ) − ] . Of particular relevance for us is the motivic Hopf map. Recall that the geometric Hopf map is themorphism of pointed varieties(1.1) A \ → P obtained from the construction of P as a quotient. There are A -equivalences [MV99, § P ≃ S ∧ G m and A \ ≃ P ∧ G m ;here G m := ( A \ , SH ( k ) andso are “spheres”. We may thus desuspend (1.1) by P to obtain the motivic Hopf map η : G m → ∈ SH ( k );here we write for the sphere spectrum. This is completely analogous to the classical stable Hopf map η top : S → ∈ SH . However, while classically we have η = 0, in SH ( k ) no power of η is null-homotopic; this is a theorem of Morel [Mor04a, Corollary 6.4.5]. In slightly more sophisticated language,denote by SH ( k )[ η − ] ⊂ SH ( k )the subcategory of η -periodic spectra ; in other words given E ∈ SH ( k ) we have E ∈ SH ( k )[ η − ] if andonly if G m ∧ E η ∧ id −−−→ ∧ E is an equivalence. The inclusion SH ( k )[ η − ] ⊂ SH ( k ) admits a left adjoint, given by the η -periodization E E [ η − ] = colim( E η −→ G ∧− m ∧ E η −→ . . . ) . Morel’s result shows that SH ( k )[ η − ] = 0; in fact he proves that π ( [ η − ]) ≃ W( k ) = 0 , where W( k ) is the Witt ring of symmetric bilinear forms .1.2.
Main results.
Our aim is to study the category SH ( k )[ η − ]; for example we would like to de-termine the higher homotopy groups π ∗ ( [ η − ]) for ∗ >
0. It turns out (see § SH ( k )[1 /η, /
2] is quite easy to understand. For many purposes it thus suffices to study the category SH ( k )[ η − ] (2) ⊂ SH ( k )[ η − ] of those η -periodic spectra E such that E p −→ E is an equivalence for allodd integers p . Our main result is the following. Herekw = KO[ η − ] ≥ is the spectrum of connective Balmer–Witt K -theory ; see § -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 3 Theorem 1.1 (see Theorem 7.8) . Let k be a field of characteristic = 2 . In SH ( k ) there exists a fibersequence [ η − ] (2) → kw (2) ϕ −→ Σ kw (2) . Here ϕ is the unique (up to homotopy) map making the following diagram commute Σ kw (2) kw (2) kw (2) βψ − ϕ , where ψ is the third Adams operation and β is the Bott element. It is well-known that π ∗ kw = W( k )[ β ]. On the other hand the construction of well-behaved Adamsoperations on kw is one of the major technical challenges of our work (see § K (1)-local sphere in classical stable homotopy theory(see e.g. [Hop98, § Remark . As an immediate consequence, if (for example) k = C then β lifts to π [ η − ] (2) . Hornbostelhas shown [Hor18, Theorem 3.2] that in this case π ∗ (cof( [ η − ] (2) → kw (2) )[ β − ]) ≃ π ∗ Σ kw (2) [ β − ] . Ormsby–R¨ondigs [OR19, p. 13] construct (still over C ) a map Σ kw → [ η − ] (2) inducing π ∗ (cof( [ η − ] (2) → kw (2) )) ≃ π ∗ Σ kw (2) . These are weak forms of our main result. In fact Ormsby–R¨ondigs [OR19, Conjecture 4.11] explicitlyask a question equivalent to our main theorem.The following are some consequences which can be obtained by more or less immediate computation.
Corollary 1.3 (see Theorems 8.1, 8.7 and 8.8) . Let k be a field of characteristic = 2 .(1) π ∗ ( k [ η − ] (2) ) ≃ W( k ) (2) ∗ = 0coker(8 n : W( k ) (2) → W( k ) (2) ) ∗ = 4 n − > n : W (2) → W (2) ) ∗ = 4 n > else(2) π ∗ MSp[ η − ] ≃ W( k )[ y , y , . . . ] , where | y i | = 2 i (3) π ∗ MSL[ η − ] ≃ W( k )[ y , y , . . . ] Here
MSp and
MSL denote the algebraic symplectic and special linear cobordism spectra [PW10a] , re-spectively.
Result (1) above (i.e. the computation of π ∗ ( [ η − ])) has attracted substantial attention before; see § Corollary 1.4 (see Proposition 8.12) . Let k have exponential characteristic e = 2 . Then the spectra ko[1 /e ] , H ˜ Z [1 /e ] ∈ SH ( k ) are cellular. See § (2) -cooperations and kw (2) -operations.1.3. Proof sketch.
To orient the reader, we provide a rapid sketch of our proof of Theorem 1.1. Grantingthe construction of ϕ , we proceed as follows. Write F for the fiber of ϕ . By a connectivity argument,it suffices to show that the canonical map HW (2) α −→ F ∧ HW is an equivalence, where HW = [ η − ] ≤ is the η -periodic Eilenberg-MacLane spectrum. By base change, it suffices to prove the result for primefields; in particular we may assume that vcd ( k ) < ∞ . It suffices to show that α [1 /
2] and α/ < ∞ and characteristic = 2) itsuffices to check that we have an isomorphism on homotopy groups. The homotopy groups of kw aregiven by W( k )[ β ], where W( k ) is the Witt ring and β ∈ π kw is the Bott element.For α [1 /
2] we are dealing with a rational problem; in particular HW ⊗ Q ≃ [ η − ] ⊗ Q and so π ∗ (HW ∧ kw) ⊗ Q ≃ W( k )[ β ] ⊗ Q . It follows that for n > ϕ ( β n ) = a n β n − for some a n ∈ W( k ) ⊗ Q , which we need to show is a unit. One of the basic properties of the construction of ϕ (related to (1.2) below) is that ϕ ( β n ) = (9 n − β n − , so α [1 /
2] is indeed an equivalence.
TOM BACHMANN AND MICHAEL J. HOPKINS
The case of α/ α ∧ instead.Note that under our assumption that vcd ( k ) < ∞ we have W( k ) ∧ ≃ W( k ) ∧ I , where I ⊂ W( k ) is thefundamental ideal. Our major intermediate result is as follows. Lemma 1.5 (see Theorem 6.1) . We have π ∗ ((kw ∧ HW) ∧ ) ≃ ( W( k ) ∧ I ≤ ∗ ≡ else ; moreover the generators x i ∈ π i ((kw ∧ HW) ∧ ) are compatible with base change. Consequently for n > ϕ ( x n ) = b n x n − for some b n ∈ W( k ) ∧ I , which we need to showis a unit. Since W( k ) ∧ I is a local ring with residue field F independent of k , and the generators arecompatible with base change, we may extend k arbitrarily. We may thus assume that k is quadraticallyclosed, so that W( k ) = F .We now employ the motivic special linear cobordism spectrum MSL. Since kw is SL-oriented and η -periodic, by work of Ananyveskiy [Ana15] (see also §
4) we have π ∗ (kw ∧ MSL) ≃ kw ∗ [ p , p , . . . ] , with | p i | = 4 i . In the case when W( k ) = F , we have partial information on the action of ϕ on kw ∗ MSL.
Lemma 1.6 (see Lemma 7.6) . Suppose that W( k ) = F . Then ϕ ◦ i ( p i ) = 1 . Now consider the canonical ring map γ : π ∗ (kw ∧ MSL) → π ∗ ((kw ∧ HW) ∧ ). We get ϕ ◦ i γ ( p i ) = γ ( ϕ ◦ i ( p i )) = γ (1) = 1 = 0 . This implies that ϕ ( γ ( p i )) = 0, and so b i = 0 as needed.1.4. Organization.
We begin in § §§ § ψ n ( β ) = n n ǫ β. We construct the Adams operations as E ∞ -ring maps ψ n : KO[1 /n ] → KO[1 /n ] ∈ SH ( S ) , for n odd. Our construction begins by making the Adams operations on KGL into C -equivariant mapsof E ∞ -rings, by using the Gepner–Snaith theorem [GS09]. Then we take 2-adically completed homotopyfixed points which, by the homotopy fixed point theorem for Hermitian K -theory, gives us operations onKO ∧ . By means of a fracture square, we combine this with a more naive operation on KO[1 / n ] to yieldthe Adams operation on KO[1 /n ]. Since our definition is rather indirect, establishing (1.2) is a fairlydelicate problem. Our proof eventually boils down to discreteness of the space of E ∞ -endomorphismsof the classical spectrum ku, which is a well-known consequence of Goerss–Hopkins obstruction theory[GH04].In § A -cohomology of MSL and MSp if A is SL-oriented and η -periodic,to obtain the A -homology of MSL and MSp. Secondly, under the additional assumption that W( k ) = F ,we obtain some information about the action of ψ on kw ∗ MSL and kw ∗ MSp.In § ( k ) < ∞ and E ∈ SH ( k ) veff , thenwe show that E ∧ is η -complete. This is deduced from an improved version of Levine’s slice convergencetheorem, which was recently established [BEØ20, § § π ∗ ((kw ∧ HW) ∧ ), i.e. prove Lemma 1.5. This employs the equiva-lences (kw ∧ HW) ∧ ≃ (ko ∧ K W )[ η − ] ∧ ≃ (ko ∧ K W ) ∧ [ η − ] ∧ ≃ (ko ∧ K W ) ∧ η, [ η − ] ∧ . At least under some finiteness assumptions, which we ignore for the purposes of this introduction. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 5
Here the most important (and nontrivial) step is the last one, which uses our new completeness result(and the fact that Σ , K W ∈ SH ( k ) veff , which was essentially established in [Bac17]). Using that K W /η ≃ k M ≃ (H Z / /τ, we compute π ∗∗ (ko ∧ K W ) ∧ η by employing a Bockstein spectral sequence, together with the knowncomputation of H Z / ∗∗ ko [ARØ17].With all this preparation out of the way, in § § § A we provide an alternative proof of the homotopy fixed point theorem for Hermitian K -theory. It utilizes the improved version of Levine’s slice convergence theorem mentioned above, togetherwith the computation by R¨ondigs–Østvær of the slice spectral sequence for KW [RØ16].1.5. Acknowledgements.
It is our pleasure to thank Robert Burklund and Zhouli Xu for extensivediscussions about the homotopy groups of MSL[ η − ] over C . We would further like to thank AlexeyAnanyevskiy, Marc Hoyois, Dan Isaksen, Tyler Lawson, Denis Nardin, Oliver R¨ondigs and Dylan Wilsonfor helpful comments.1.6. Conventions.
All our statements directly or indirectly involving hermitian K -theory require theassumption that 2 is invertible in the base scheme. We may omit specifying this explicitly to avoidtedious repetition.Given a map of spectra α : E → F , we denote by α also the induced map π ∗ E → π ∗ F .We denote the category of motivic spectra over a scheme S by SH ( S ), and assume basic familiaritywith its construction and properties. See e.g. [BH17, §§ ∞ -category [Lur16, Lur09], and we assume some familiarity with the theory of suchcategories (in particular in § ν ( k ) 2-´etale cohomological dimension of k [ √− A [ p n ] p n -torsion in abelian group AA ∧ p classical p -completion of abelian group L ∧ p A derived p -completion of abelian group § E ∧ p p -completion of spectrum § E [1 /n ] , E ( p ) localization of spectrum § motivic sphere spectrumKO , KGL hermitian and algebraic K -theory motivic spectra § , kw , ko variants of the above § top classical orthogonal K -theory spectrumK ◦ rank 0 summand of K -theory space β, β KGL
Bott element in KO , KGL § ψ n Adams operation § ψ n GS , ψ nh GS Adams operations § ϕ modified Adams operation § K M homotopy module of Milnor K -theory k M homotopy module of mod 2 Milnor K -theory K W homotopy module of Witt K -theory § Z motivic cohomology spectrum § τ Bott element in H Z / η -periodic Witt cohomology spectrum § , MSp algebraic SL-, Sp-cobordism motivic spectra § HP ∞ infinite quaternionic projective spaceHGr , SGr , Gr quaternionic, special linear, and ordinary Grassmannians § ⊞ external product of vector bundles (on a product of varieties)Th( V ) Thom space of a vector bundle, V /V \ η motivic Hopf map § ρ standard endomorphisms of motivic sphere spectrum § n ǫ Milnor-Witt integer 1 + h− i + · · · + h± i W , I Witt ring and fundamental ideal § W , I n sheaf of Witt rings and fundamental ideals E [1 / + , E [1 / − plus and minus part of a 2-periodic spectrum § TOM BACHMANN AND MICHAEL J. HOPKINS E + , E − generalized plus and minus part § π i,j ( E ) , π i ( E ) j bigraded homotopy groups § π i,j ( E ) , π i ( E ) j bigraded homotopy sheaves § E ≥ , E ≤ truncations in the homotopy t -structure § S p,q motivic sphere S p − q ∧ G ∧ qm Σ p,q bigraded suspension (smashing with S p,q ) SH ( S ) category of motivic spectra over S SH ( k ) veff category of very effective spectra § S pc( S ) category of motivic spaces over Sf i effective cover functor [Voe02] s i slice functor [Voe02]˜ f i very effective cover functor [Bac17]˜ s i generalized slice functor [Bac17] r R real realization functor § P ( C ) category of presheaves of spaces on C Map( − , − ) mapping space in an ∞ -categorymap( − , − ) mapping spectrum in a stable ∞ -category[ − , − ] homotopy classes of maps, i.e. π Map( − , − ) S pc category of spaces SH category of spectra G rpd category of ∞ -groupoids (so G rpd ≃ S pc)Sch , Sch S category of qcqs schemes (over S )Sm S category of smooth, qcqs schemes over S Vect S S Perf S category of perfect complexes on S Table of notation. Preliminaries and Recollections
In this section we collect various well-known results which will be used throughout the sequel. Abouthalf of them are specific to motivic homotopy theory (bigraded homotopy sheaves § § § ρ and real realization § § § § Filtered modules. § G we mean a family of subgroups G ⊃ · · · ⊃ F s G ⊃ F s +1 G ⊃ · · · . We denote by gr n ( G ) = F n G/F n +1 G the associated graded. Definition 2.1 ([Boa99], Proposition 2.2) . Let F • G be a filtered abelian group.(1) The filtration is exhaustive if G = colim s F s G .(2) The filtration is Hausdorff if 0 = lim s F s G .(3) The filtration is complete if 0 = lim s F s G .More generally, we may be working with ((bi)graded) filtered modules over a ((bi)graded) filteredring. Since the forgetful functor from modules to abelian groups preserves limits and colimits, there isno ambiguity in the definition of exhaustive/Hausdorff/complete. When working with graded objectsthis is no longer the case, and the definitions have to be applied degreewise. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 7 Lemma 2.2 ([Boa99], Theorem 2.6) . Let α : G → G ′ be a morphism of filtered groups. Assume thatboth filtrations are Hausdorff and exhaustive, and that the filtration on G is complete. Then α is anisomorphism of filtered groups (i.e. α induces G ≃ G ′ and F s G ≃ F s G ′ ) if and only if gr • ( α ) anisomorphism. Corollary 2.3.
Let R be a graded filtered ring. Assume that R is concentrated in non-negative degreesand that the filtration is exhaustive, Hausdorff and complete.(1) Let M be a graded filtered R -module, such that the filtration is exhaustive and Hausdorff. If gr • ( M ) is a free (bigraded) gr • ( R ) -module with only finitely many generators in external degrees(i.e. degree coming from the grading on M ) ≤ n for every n , then M is a free R -module oncorresponding generators.(2) Let A be a graded filtered R -algebra, such that the filtration is exhaustive and Hausdorff. If gr • ( A ) is a polynomial gr • ( R ) -algebra on generators in positive external degrees, only finitely many ofwhich lie in any degree, then A is a polynomial R -algebra on corresponding generators.Proof. (1) Choose generators { ¯ x i } i ∈ I of gr • ( M ) in bidegree ( s i , t i ), where t i corresponds to the originalgrading on M and s i to the filtration. Lift them to x i ∈ F s i M t i . Consider M ′ = M i R [ t i ] { s i } , where R [ t i ] { s i } denotes a free R -module on a generator in degree t i and filtration s i . There is thusa canonical map M ′ → M inducing an isomorphism on gr • . It remains to show that M ′ is complete,exhaustive and Hausdorff. By assumption, for each n , M ′ n = M i : t i ≤ n R n − t i { s i } is a finite sum of complete exhaustive and Hausdorff filtered abelian groups, and so has the same prop-erties.(2) Choose polynomial generators { ¯ y i } i ∈ I of gr • ( A ) in bidegree ( s i , t i ). Lift them to y i ∈ F s i A t i .Then monomials in the ¯ y i are module generators of gr • ( A ) and our assumption implies that there areonly finitely many monomials in degrees ≤ n for any n . Thus (1) applies and A is the free R -algebra onmonomials in the y i . Let A ′ be the polynomial R -algebra on generators { y i } . There is a canonical ringmap A ′ → A , and it is an isomorphism since the underlying module map is. (cid:3) Lemma 2.4.
Let α : G → G ′ be a morphism of filtered (possibly graded) groups. Suppose that thefiltration on G ′ is exhaustive and Hausdorff, and the filtration on G is complete. Suppose furthermorethat gr • ( α ) is surjective. Then α is surjective, and also each F p α is surjective.Moreover in this situation gr • ker( α ) ≃ ker(gr • ( α )) . Proof.
By applying the argument degreewise, we may assume that we are in the ungraded situation.First observe that if x ∈ F N G ′ then there exists y ∈ F N G with α ( y ) ∈ x + F N +1 G ; this is justsurjectivity of gr N ( α ). Applying this to x − α ( y ) ∈ F N +1 G ′ we obtain y ∈ F N +2 G with α ( y + y ) ∈ x + F N +2 G ′ . Iterating we find y i ∈ F N + i G (for all i ≥
0) with α ( y + · · · + y n ) ∈ x + F N + n +1 G ′ (for any n ≥ G , the series P i y i converges to (a possibly non-unique element) y ∈ F N G with α ( y ) − x ∈ F N + n +1 G ′ for every n ; hence by Hausdorffness of G ′ we get α ( y ) = x . Thus F N α issurjective.Since F • G ′ is exhaustive, every x ∈ G ′ satisfies x ∈ F N G ′ for some N ; hence the above argumentshows that x is in the image of α and hence α is surjective.For the claim about kernels, apply the snake lemma [Wei95, Lemma 1.3.2] to the diagram of exactsequences 0 −−−−→ F p +1 ker( α ) −−−−→ F p +1 G −−−−→ F p +1 G ′ −−−−→ y y y −−−−→ F p ker( α ) −−−−→ F p G −−−−→ F p G ′ −−−−→ F p +1 G ′ → F p G ′ ) → gr p ker( α ) → gr p G gr p α −−−→ gr p G ′ → . This is the desired result. (cid:3)
TOM BACHMANN AND MICHAEL J. HOPKINS
Lemma 2.5.
Let α : G → G ′ be a morphisms of filtered (possibly graded) groups. Assume that G isHausdorff and exhaustive and gr • ( α ) is injective. Then α is injective.Proof. By applying the argument degreewise, we may assume that we are in the ungraded situation.Let 0 = x ∈ G . By Hausdorffness and exhaustiveness there exists n with x ∈ F n G \ F n +1 G , whence0 = [ x ] ∈ gr n G . It follows that α ( x ) ∈ F n G ′ and 0 = [ α ( x )] = α ([ x ]) ∈ gr n G ′ , and so α ( x ) = 0. (cid:3) R -modules M, M ′ , we can put a filtration on M ⊗ M ′ by F i ( M ⊗ M ′ ) = X a + b = i ( F a M )( F b M ′ ) ⊂ M ⊗ M ′ . Lemma 2.6.
Let R = k be a field. Then gr • ( M ⊗ M ′ ) ≃ gr • ( M ) ⊗ gr • ( M ′ ) . Proof.
We may assume that
M, N are exhaustively filtered. If M = k ( i ) and M ′ = k ( j ) (i.e. F a M = k if a ≤ i and F a M = 0 else), then gr • ( M ) = k [ i ] and M ⊗ M ′ = k ( i + j ); the result in this case follows. If M , N are finite dimensional we can write M = L α ∈ I k ( i α ) (e.g. use Corollary 2.3(1)) and similarly for M ′ . The result follows since ⊗ and gr p commute with sums. In general write M, N as filtered colimits oftheir finite dimensional subspaces; the result follows since ⊗ and gr • commute with filtered colimits. (cid:3) R and (homogeneous) ideal J , we can give R the filtration by powersof J , i.e. F n R = J n . In particular if S is a (possibly infinite) set of variables (possibly with someassigned degrees) and A is a base ring, we can consider the polynomial ring R = A [ S ] or the exterioralgebra R = Λ A [ S ] (which is graded if the variables are), and filter it by powers of the augmentationideal ker( R → A ). In this case gr ( R ) is called the indecomposable quotient . Lemma 2.7. (1) Let
S, A as above. The natural maps A → gr and (gr ) ⊗ n → gr n induce canonicalisomorphisms gr n ( A [ S ]) ≃ Sym nA (gr ( S )) and gr n (Λ A [ S ]) ≃ Λ nA (gr ( S )) . Here gr ( A [ S ]) is a free A -module on a basis in bijection with S .(2) Let α : R → R be a morphism of graded A -algebras, where R i are either both polynomial or bothexterior algebras, on generators in positive degrees. If gr ( α ) : gr ( R ) → gr ( R ) is surjective(respectively split injective, e.g. injective and A a field, respectively an isomorphism) then so is α .Proof. (1) Let R = A [ S ] or R = Λ A [ S ], respectively. We can give R the grading where all variables havedegree 1; then J n = R ≥ n . This implies that gr n ( R ) ≃ R n . All claims are checked easily.(2) If β : M → N is a surjective (respectively split injective, respectively bijective) morphism of A -modules, then so are Sym n ( β ) and Λ n ( β ). Hence under our assumption gr • ( α ) is surjective (respectivelyinjective, respectively an isomorphism) by (1). The filtrations are complete, exhaustive and Hausdorfffor degree reasons. The claim thus follows from Lemmas 2.5 and 2.4. (cid:3) Derived completion of abelian groups.
For an abelian group A , we write L ∧ p A = lim n cof( A p n −→ A ) ∈ D (Ab)for the derived p -completion. Then there is a short exact sequence0 → lim A [ p n ] → π L ∧ p A ≃ Ext( Z /p ∞ , A ) → A ∧ p := lim n A/p n → . Moreover π L ∧ p A ≃ lim A [ p n ] and π i L ∧ p A = 0 for i = 0 , . See [Sta18, Tag 0BKG].
Lemma 2.8.
If the p -torsion in A is of bounded order, then A ∧ p ≃ L ∧ p A (so in particular π L ∧ p A = 0 ).Proof. We need to prove that lim A [ p n ] = 0 = lim A [ p n ]. By assumption the sequence of sets A [ p n ] iseventually stationary, and a sufficiently high composite of the transition maps is zero (since the transitionmaps are given by multiplication by p ). The result follows since (derived) limits can be computed byrestriction to final subcategories of the indexing category. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 9 Witt groups.
For a ring A , we denote by GW( A ) (respectively W( A )) its Grothendieck–Witt ring(respectively its Witt ring) [Kne77, §§ I.4, I.5]. We write I( A ) ⊂ W( A ) for the fundamental ideal, i.e. thekernel of the rank homomorphism [Kne77, § I.7]
Lemma 2.9.
Let A be a Dedekind domain with P ic ( A ) = 0 . Write K for the fraction field of A . Thenin the following commutative diagram, all maps are injective GW( A ) −−−−→ GW( K ) y y W( A ) × Z −−−−→ W( K ) × Z . Proof.
Since W( A ) ֒ → W( K ) [MH73, Corollary IV.3.3], it suffices to show that GW( A ) ֒ → W( A ) × Z . By[Kne77, § I.5, Proposition] the kernel of GW( A ) → W( A ) is generated by “metabolic spaces” of the form H ( V ) for V a vector bundle on A . By [Kne77, § I.4, Proposition 2] the map H factors through K ( A ),and so it suffices to show that K ( A ) −−→ Z is injective. But K ( A ) ≃ Z ⊕ P ic ( A ) [Mil71, Corollary1.11], whence the claim. (cid:3) Lemma 2.10.
Let A be a Dedekind domain with vcd ( F rac ( A )) < ∞ and P ic ( A ) = 0 . Then all torsionin GW( A ) and W( A ) is -primary, and of bounded order.Proof. It follows from Lemma 2.9 that it suffices to establish the claims about W( k ), where k is a field.For this case see e.g. [Bac18e, Lemma 29]. (cid:3) Corollary 2.11.
Assumptions as in Lemma 2.10. The map
GW( A ) → GW( A ) ∧ is injective, andsimilarly for W( A ) .Proof. For any abelian group G , elements in the kernel of G → G ∧ must be in the kernel of G → G/ n for all n , and hence be infinitely 2-divisible. We show the only such element in GW( A ) , W( A ) is 0. Since Z has no infinitely 2-divisible elements (other than 0), it suffices to prove the claim about W( A ). Asabove we may assume that A = k is a field. Since 2 ∈ I( k ), any infinitely 2-divisible element lies in ∩ n I( k ) n . This group is zero [MH73, Theorem III.5.1], whence the result. (cid:3) Lemma 2.12.
Let vcd ( k ) < ∞ . Then the I -adic and -adic filtrations on W( k ) induce the sametopology. In particular W( k ) ∧ ≃ W( k ) ∧ I .Proof. We have 2W( k ) ⊂ I( k ) and I( k ) vcd ( k )+1 ⊂ k ) [EL99, last Theorem], which implies theclaim. (cid:3) Lemma 2.13.
Let k be a field. Then W( k ) (2) is a local ring with maximal ideal I . In particular x ∈ W( k ) (2) is a unit if and only if rk( x ) = 0 .Proof. There is a certain (possibly empty, possibly infinite) set Ω and a homomorphism σ : W( k ) → Z Ω inducing Spec(W( k )) ≃ (Ω × Spec( Z )) / Ω × { (2) } [MH73, Remark after Lemma III.3.5]. SinceSpec(W( k ) (2) ) identifies with the subspace of prime ideals not containing an odd integer it is given by(Ω × Spec( Z (2) )) / Ω × { (2) } . This has [Ω × { (2) } ] as unique maximal element, whence the result. (cid:3) The homotopy t -structure. SH ( k ) admits a t -structure with non-negative part generated [Lur16, Proposition1.4.4.11] by spectra of the form Σ ∞ + X ∧ G ∧ nm , for X ∈ Sm k and n ∈ Z . We denote by E E ≥ n and E E ≤ n the truncation functors.2.4.2. For E ∈ SH ( k ), denote by π i ( E ) j the Nisnevich sheaf on Sm k associated with the presheaf X [Σ i Σ ∞ + X, E ∧ G ∧ jm ] . One may prove that [Hoy15, Theorem 2.3] SH ( k ) ≥ = { E ∈ SH ( k ) | π i ( E ) j = 0 for all i < , j ∈ Z } , and similarly SH ( k ) ≤ = { E ∈ SH ( k ) | π i ( E ) j = 0 for all i > , j ∈ Z } . t -structure is non-degenerate [Hoy15, Corollary 2.4, Remark 2.5], i.e. ∩ i SH ( k ) ≥ i = 0 = ∩ i SH ( k ) ≤ i . π i ( E ) j =: F is an unramified sheaf [Mor05b, Lemma 6.4.4]. This means in particular that F = 0 if and only if for every finitely generated separable field extension K/k , which we may view as thefraction field of a smooth k -variety, the generic stalk F ( K ) is zero.2.4.5. The heart SH ( k ) ♥ can be identified with the category of homotopy modules [Mor03, Theorem5.2.6]. Namely for E ∈ SH ( k ) ♥ there are canonical isomorphism π ( E ) j − ≃ ( π ( E ) j ) − , where on the right hand side the ( − ) − means Voevodsky’s contraction operation [Mor03, Definition4.3.10]. Moreover, any sequence of Nisnevich sheaves F i together with isomorphisms F i − ≃ ( F i ) − suchthat the cohomology of each F i is homotopy invariant (i.e. F i is “strictly homotopy invariant”) gives riseto an object of SH ( k ) ♥ , and this is an equivalence of categories.2.4.6. It turns out that [Mor04a, Theorem 6.2.1] π ( ) ∗ ≃ K MW ∗ . This implies that [Mor12, Lemma 3.10] π ( [ η − ]) ∗ ≃ W [ η ± ] . Here K MW ∗ is the homotopy module of Milnor-Witt K -theory [Mor12, § W is the sheaf ofunramified Witt rings, i.e. the Zariski sheafification of X W( X ).2.4.7. Another common indexing convention is π i,j ( E ) = π i − j ( E ) − j . We also use the common abbreviations π i ( E ) j = π i ( E ) j ( k ) and π i,j ( E ) = π i,j ( E )( k ) . Note that π i,j ( E ) = [ S i,j , E ] . If the base S is not the spectrum of a field, we will only employ the notation π i,j ( E ), with the abovemeaning.2.5. Completion and localization. S (or more generally S ⊂ π ∗∗ ( )) and E ∈ SH ( S ) there exists an initial S -periodic spectrum E ′ under E . In other words for every x ∈ S the endomorphism of E ′ induced by x isan equivalence. See e.g. [BH17, § S = { x } and (given a prime p ) S = { n | ( n, p ) = 1 } we respectively denote E ′ by E [1 /x ] and E ( p ) . These localizations may be computed as the mappingtelescope of appropriate endomorphisms [BH17, Lemma 12.1], i.e. “in the expected way”. We write E ⊗ Q for the result of inverting all primes (i.e. S = Z \ SH ( S )[ x − ] ⊂ SH ( S )for the category of x -periodic spectra.2.5.2. Given a prime p and E ∈ SH ( S ), there exists an initial p -complete spectrum E ∧ p under E ; inother words whenever F is p -periodic (i.e. F ≃ F [1 /p ]) then [ F, E ∧ p ] = 0. Moreover E ∧ p ≃ lim n E/p n , where E/p n denotes the cofiber of the endomorphism of multiplication by p n . See e.g. [Bac18d, § ; e.g. E ∧ η ≃ lim n E/η n . -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 11 p -completion of abeliangroups L ∧ p A ∈ SH from § Lemma 2.14. (1) Let E ∈ SH ( k ) . Then π i ( E ( p ) ) j ≃ ( π i ( E ) j ) ( p ) .(2) Let E ∈ SH ( S ) . There is a split short exact sequence → π L ∧ p π i ( E ) j → π i ( E ∧ p ) j → π L ∧ p π i − ( E ) j → . The map π i ( E ) j → π i ( E ∧ p ) j factors as π i ( E ) j → π L ∧ p π i ( E ) j → π i ( E ∧ p ) j , all maps being the canonical ones. Beware the absence of underlines in (2)!
Proof. (1) Immediate.(2) Since map(Σ i,j , − ) preserves p -completions, this follows from the analogous statement for ordi-nary spectra (see e.g. [BK87, Proposition VI.5.1]). For last assertion, apply the result to E ≥ i → E . (cid:3) Corollary 2.15.
Let E ∈ SH ( S ) . If π i,j ( E ) → π i,j ( E ) ∧ p is injective then so is π i,j ( E ) → π i,j ( E ∧ p ) .Proof. By Lemma 2.14 the map π i,j ( E ) → π i,j ( E ∧ p ) factors as π i,j ( E ) α −→ π L ∧ p π i,j ( E ) ֒ → π i,j ( E ∧ p ) soit suffices to show that α is injective. As reviewed in § π i,j ( E ) ֒ → π i,j ( E ) ∧ p factors as π i,j ( E ) α −→ π L ∧ p π i,j ( E ) → π i,j ( E ) ∧ p . The result follows. (cid:3) Lemma 2.16.
Let C be a presentable stable ∞ -category, E ∈ C and p an integer. Then E ≃ if andonly if E [1 /p ] ≃ and E ∧ p ≃ (equivalently, E/p ≃ ; a fortiori this holds if E ( p ) ≃ ).Similarly if C is presentably symmetric monoidal, L ∈ C is invertible and x ∈ π ( L ) then E ≃ if andonly if ≃ E ∧ x and ≃ E [1 /x ] .Proof. We give the proof of the second statement, the proof of the first one is obtained by replacing x by p . Note that by the definition of E ∧ x as a localization, we have E ∧ x ≃ E/x ≃
0, i.e. x : E → E ∧ L is an equivalence, i.e. E ≃ E [1 /x ]. We thus need to show that E ≃ E ≃ E [1 /x ] and E [1 /x ] ≃ (cid:3) Very effective spectra.
We write SH ( k ) veff ⊂ SH ( k )for the subcategory generated under colimits (equivalently, colimits and extensions [Bac17, Remark afterProposition 4]) by spectra of the form Σ ∞ + X , with X ∈ Sm k . This is the non-negative part of a t -structure [Lur16, Proposition 1.4.4.11]. If E ∈ SH ( k ) eff , then E ∈ SH ( k ) eff ≥ = SH ( k ) veff (respectively E ∈ SH ( k ) eff ≤ ) if and only if π i ( E ) = 0 for i < i >
0) [Bac17, § Inverting ρ . ρ : → G m the map corresponding to − ∈ O × . Then there is a canonical equivalence [Bac18a] SH ( S )[ ρ − ] ≃ SH ( RS ) , where RS denotes the real space associated with S [Sch94, (0.4.2)] and SH ( RS ) means sheaves of spectraon the topological space RS . One puts Sper( A ) := R Spec( A ). Corollary 2.17.
Let { k α /k } be the set of real closures of k . Then the canonical functor SH ( k )[ ρ − ] → Y α SH ( k α )[ ρ − ] ≃ Y α SH is conservative.Proof. Immediate from knowing the stalks of the small real ´etale site of k [Sch94, Proposition 3.7.2].The last equivalence comes from Sper( k ) ≃ {∗} for k real closed. (cid:3) Given topological spaces
X, Y , write C ( X, Y ) for the set of continuous maps from X to Y . Corollary 2.18.
For a local ring k , we have π ∗ k [ ρ − ] ≃ C (Sper( k ) , π s ∗ ) ≃ C (Sper( k ) , Z ) ⊗ π s ∗ . Here Z and π s ∗ are given the discrete topologies.Proof. We need to compute [Σ ∗ , ] SH (Sper( k )) . We can do this using the descent spectral sequence H pr ´ et ( k, π sq ) ⇒ [Σ q − p , ] SH (Sper( k )) . Since the real ´etale cohomological dimension of local rings is 0 [Sch94, Theorem 7.6], the spectral sequencecollapses and yields [Σ ∗ , ] SH (Sper( k )) ≃ H r ´ et ( k, π s ∗ ) ≃ C (Sper( k ) , π s ∗ ) . This proves the first equivalence. For the second, it suffices to prove that if X is a topological spaceand A is a finitely generated abelian group, then C ( X, A ) ≃ C ( X, Z ) ⊗ A . For this it is enough to showthat the functor C ( X, − ) is exact. Since finite sums of abelian groups are products, their preservation isclear. Given an exact sequence 0 → A → B → C → → C ( X, A ) → C ( X, B ) → C ( X, C ) → C ( X, − ) preserves surjections. This is true sinceif α : A → B is a surjection of abelian groups, then there exists a set-theoretic (and so continuous) section s : B → A , and then C ( X, s ) is a set-theoretic section of C ( X, α ), whence the latter is surjective. (cid:3) RS = {∗} (such as if S = Spec ( R ), with R = Z , Z [1 / , Q , R , and so on) then SH ( S )[ ρ − ] ≃SH . In this situation we denote by r R the composite r R : SH ( S ) → SH ( S )[ ρ − ] ≃ SH ( RS ) ≃ SH . By construction, the equivalence is given by taking global sections. If S = Spec( R ), then r R is equivalentto the functor induced from the one sending a smooth variety X/ R to its topological space of real points[Bac18a, § E ∈ SH ( k ) there is a functorial splitting E [1 / ≃ E [1 / , /η ] ∨ E [1 / ∧ η =: E [1 / − ∨ E [1 / + . Equivalently, the ring GW( k )[1 / ≃ [ [1 / , [1 / SH ( k ) splits asGW( k )[1 / ≃ W( k )[1 / × Z [1 / . We obtain a splitting at the level of categories SH ( k )[1 / ≃ SH ( k )[1 / − × SH ( k )[1 / + . For any E ∈ SH ( k ), η acts as an isomorphism on E [1 / − and as zero on E [1 / + . On the other hand(2.1) ηρ = − SH ( k )[ η − ] , and ρ is nilpotent on E [1 / + . See [Bac18a, Lemma 39] for all of this. We deduce the following. Corollary 2.19.
Let k be uniquely orderable. Then SH ( k )[1 / , /η ] r R −→ SH [1 / is an equivalence.For any field k , with set of real closures { k α } , the functor SH ( k )[1 / , /η ] ( r α R ) α −−−−→ Y α SH [1 / is conservative.Proof. The first claim is clear from the above discussion. The second is an immediate consequence ofCorollary 2.17. (cid:3)
One easily deduces the motivic Serre finiteness theorem [ALP17](2.2) π ∗ ( [ η − ]) ⊗ Q ≃ W ⊗ Q . Remark . Since η is the geometric Hopf map, r R ( η ) comes from the Hopf map S → P ( R ) ≃ S .This is just the squaring map, and so r R ( η ) = 2. This observation is closely related to (2.1). -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 13 Adams operations for the motivic spectrum
KOThroughout this section we will work with base schemes S such that 1 / ∈ S .3.1. Summary.
See § , KGL and the Bott element β (andthe table of notation in § n ǫ ). Theorem 3.1.
Let n be odd. For every scheme with / ∈ S we construct a map ψ n : KO S [1 /n ] → KO S [1 /n ] ∈ SH ( S ) . These maps satisfy the following properties.(1) They are compatible with base change (up to homotopy).(2) We have ψ n ( β ) = n · n ǫ · β .(3) ψ n is a morphism of E ∞ -ring spectra.(4) The following diagram commutes (up to homotopy) KO[1 /n ] −−−−→ KGL[1 /n ] ψ n y ψ n y KO[1 /n ] −−−−→ KGL[1 /n ] , where ψ n : KGL[1 /n ] → KGL[1 /n ] is the usual Adams operation; see e.g. [Rio10, Definition5.3.2 and sentences thereafter] .Remark . Shortly after this article was written, Fasel–Haution supplied a much simpler constructionof operations ψ n FH : KO[1 /n ] → KO[1 /n ] [FH20]. These operations satisfy (1) and (4) by construction,and (2) is an immediate consequence of the construction (see Lemma 3.36 for details). Their operationsare only constructed as homotopy ring maps [FH20, Theorem 5.2.4] rather than E ∞ -ring maps, so theydo not (on the nose) “satisfy” (3). However in the sequel we never use that our operations are E ∞ (rather than just homotopy ring maps), so the article [FH20] can be used as a drop-in replacement forthe entirety of this section. Remark . For a scheme X , exterior powers of vector bundles induce a special λ -ring structure onGW( X ) [Zib18, FH20]. As in any special lambda ring, there are thus induced Adams operations. Fromour construction, it is unclear if these “geometric” operations coincide with the ones induced by thespectrum maps ψ n . On the other hand the spectrum maps ψ n FH are by construction closely related tothe geometric operations (see Lemma 3.36). Remark . We show in § ψ n ≃ ψ n FH (see Proposition 3.38). One can view this as eitheridentifying ψ n with the geometric operation, or lifting ψ n FH to an E ∞ -operation. Remark . One may show that for any n (even or odd), ψ n ( β ) = n · n ǫ · β , where by ψ n we mean the“geometric” operation mentioned above. This implies that any spectrum map realizing this geometricoperation must invert n and n ǫ (since β is a unit). If n is odd, then n ǫ maps to a unit in Z [1 /n ] (namely n ) and also in W( k )[1 /n ] (namely 1), whence is a unit in GW( k )[1 /n ] and so KO[1 / ( n · n ǫ )] ≃ KO[1 /n ].One the other hand if n is even then the image of n ǫ in W( k ) is 0 and soKO[1 / ( n · n ǫ )] ≃ KO[1 / + [1 / ( n · n ǫ )] ∨ KO[1 / − [1 / ( n · n ǫ )] ≃ KO[1 /n ] + . An Adams operation ψ n : KO[1 /n ] + → KO[1 /n ] + for n even satisfying all expected properties exists butis not very interesting; see Remark 3.26. Remark . The endofunctors of SH ( k ) given by E E ≥ and E E [ η − ] are respectively lax andstrong symmetric monoidal (the former being the composite of the strong symmetric monoidal functor SH ( k ) ≥ ֒ → SH ( k ) and its hence lax symmetric monoidal right adjoint, and the latter being a smashinglocalization). It follows that the functor SH ( k ) → SH ( k ) , E E ≥ [ η − ]is lax symmetric monoidal. Applying it to the homotopy ring map ψ n : KO[1 /n ] → KO[1 /n ] we obtain ψ n : kw[1 /n ] → kw[1 /n ] ∈ CAlg(h SH ( k )) . Example . We will use many times the following fact: if E → F ∈ SH ( S ) is any morphism, thenthe induced map π ∗∗ ( E ) → π ∗∗ ( F ) is a π ∗∗ ( )-module morphism. For example, suppose that KO ( S ) isgenerated by elements of the form h a i , for a ∈ O ( S ) × (e.g. S the spectrum of a field or Z [1 /d !] [BW20,Lemma 5.5]). Then π , ( ) → π , (KO) is surjective, and hence any ring map ψ : KO → KO actstrivially on π , (KO). More generally, ψ acts trivially on the image of π ∗∗ ( ) → π ∗∗ (KO), e.g. on η . Here is a sketch of our construction. We view KGL as a motivic spectrum with (homotopy coher-ent) C -action coming from passage to duals. Using the Gepner–Snaith theorem, we construct a C -equivariant E ∞ -endomorphism ψ n GS of KGL[1 /n ]. By the homotopy fixed point theorem, over sufficientlynice base schemes, like Z [1 / hC is 2-adically equivalent to KO. We obtain for n odd an E ∞ -endomorphism ψ nh GS on KO + := KGL hC . There is a fracture square for KO involving KO + , KO[1 / − and KO ∧ [1 / − . To build our Adams operation on KO[1 /n ] we will choose a compatible operation ψ n − onKO[1 / − . We have SH ( Z [1 / / − ≃ SH [1 /
2] (see § / − corresponds to the topologi-cal orthogonal K -theory spectrum KO top [1 /
2] under this equivalence. It thus seems natural to let ψ n − bethe topological Adams operation on KO top [1 / n ]. Since KO ∧ [1 / − ∈ SH ( Z [1 / ⊗ Q − ≃ D ( Q ), compat-ibility of ψ nh GS and ψ n − is purely a question about homotopy groups. The operation ψ n − on KO top [1 / n ]acts on a generator of π by multiplication by n . We can write ψ nh GS ( β ) = aβ , for some a ∈ GW( Z [1 / ∧ ,and the compatibility is equivalent to requiring that the image of a in W( Z [1 / ∧ [1 / ≃ Q be n . Ourproof of this ultimately relies on the fact that the space of E ∞ -endomorphisms of KU top is discrete, i.e.Goerss–Hopkins obstruction theory. Remark . This proof seems very unsatisfying to the authors. We believe that the Grothendieck–Wittspace of any scheme should admit E ∞ Adams operations after inverting the relevant integer, functoriallyin the scheme. Presumably this should be related to the spectral λ -ring theory of Barwick–Glasman–Mathew–Nikolaus. This operation would coincide with Zibrowius’ one essentially by construction, andthe fact that ψ n ( β ) = n · n ǫ · β would be a fairly straightforward computation. Taking suspensionspectra and inverting the Bott element (see § E ∞ Adams operationon KO[1 /n ].Unfortunately we have been unable to implement this idea, forcing us to perform the contortionssketched above instead.Along the way, we also determine the real realization of KO. Recall the real realization functor r R from § β : S , → KO gives a periodicity of degree 4 in r R (KO S ), sothis spectrum cannot possibly be the topological KO top . Lemma 3.9.
Let S = Spec( R ) .(1) r R (KGL) = 0 (2) The map r R (KO) → r R (KO ∧ ) identifies with KO top [1 / → (KO top ) ∧ [1 / . Motivic ring spectra KO and KGL . BC is the ordinary 1-category with one object C , and space of endomorphismsgiven by the group C of order 2. The category BC ⊳ is obtained by adding an initial object ∗ ; in otherwords BC ⊳ is equivalent to the opposite of the ordinary 1-category of C -orbits (i.e. the subcategory ofFin C on the two objects C and ∗ ). For a category C , Fun( BC , C ) is the category of objects in C witha homotopy coherent C -action. Given X ∈ Fun( BC , C ) we write X hC ∈ C for its limit. The functorFun( BC ⊳ , C ) → Fun( BC , C ) given by evaluation at C admits a (partially defined) right adjoint (theright Kan extension) sending X ∈ Fun( BC , C ) to the diagram X hC → X [Lur09, Proposition 4.3.2.17and Definition 4.3.2.2]. It follows that given any object ( X → Y ) ∈ Fun( BC ⊳ , C ) there is a functoriallyinduced morphism X → Y hC ∈ C (provided that Y hC exists).3.2.2. The group completion or (direct sum) K -theory functorCMon( G rpd) → CMon( G rpd) gp → S pc ∗ (with the second functor being the forgetful one, using that G rpd ≃ S pc) is lax symmetric monoidal(e.g. use [GGN16, Theorem 5.1] and the fact that right adjoints of symmetric monoidal functors are laxsymmetric monoidal) and hence induces K ⊕ : CAlg(CMon( G rpd)) → CAlg( S pc ∗ ) . Here we use the convention that for a symmetric monoidal category C ⊗ , CAlg( C ) = CAlg( C ⊗ ) denotesthe category of E ∞ -algebras, and for any category D (symmetric monoidal or not) with finite productsCMon( D ) := CAlg( D × ). The symmetric monoidal structure on CMon( S pc) is given by tensor product,and the one on S pc ∗ by smash product. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 15 S be a scheme. Write Vect( S ) for the ordinary 1-groupoid of vector bundles on S . Thiscarries the following structures. • For
V, W ∈ Vect( S ), there is the direct sum V ⊕ W ∈ Vect( S ); this way Vect( S ) becomes anobject in CMon( G rpd). • For
V, W ∈ Vect( S ), there is their tensor product V ⊗ W ∈ Vect( S ). This operation distributesover the sum, promoting Vect( S ) to CAlg(CMon( G rpd)). • For V ∈ Vect( S ), there is the dual V ∗ = Hom( V, O S ). This operation commutes with sum andtensor product, and there is a functorial isomorphism ( V ∗ ) ∗ ≃ V . Hence Vect( S ) is promotedto Fun( BC , CAlg(CMon( G rpd))). • For f : X → Y ∈ Sch, there is a pullback operation f ∗ : Vect( Y ) → Vect( X ). This is compatiblewith composition in f , and also with all the previous structures.All in all we obtain Vect : BC × Sch op → CAlg(CMon( G rpd)) . Applying K ⊕ we get K ⊕ Vect : BC × Sch op → CAlg( S pc ∗ ), or equivalently K ⊕ Vect ∈ Fun( BC , CAlg( P (Sch) ∗ )) . If X ∈ Sch is affine, then K ⊕ Vect( X ) is the K -theory space of X [TT90, Theorem 7.6], but in generalthis is not correct. We can fix this issue by passing to Zariski sheaves:K := L Zar K ⊕ Vect ∈ Fun( BC , CAlg( P (Sch) ∗ )) . Then for every X ∈ Sch the space K( X ) is the (Thomason–Trobaugh) K -theory space of X [TT90,Theorem 8.1].3.2.4. Consider the homotopy fixed points Vect( S ) hC . This is an ordinary 1-category, which can bedescribed as follows: an object consists of a vector bundle V together with isomorphism α : V ≃ V ∗ ,such that α ∨ corresponds to α under the double dual identification. In other words, Vect( S ) hC is thecategory of non-degenerate symmetric bilinear forms (see e.g. [Sch10a, Definition 2.4]) on S , which wealso denote by Bil( S ).Using § BC ⊳ to obtain(Bil → Vect) : BC ⊳ × Sch op → G rpd . Using that CAlg(CMon( G rpd)) → G rpd preserves limits (being a right adjoint), this further upgrades to(Bil → Vect) ∈ Fun( BC ⊳ , CAlg(CMon( G rpd)) . Applying as before direct sum K -theory and sheafification, we obtain the presheafGW := L Zar K ⊕ Bil ∈ P (Sch Z [1 / ) ∗ . We are restricting the base schemes here to 1 / ∈ S because then this definition of GW given us the correct Grothendieck-Witt space [Sch10a, Remark 4.13] [Sch10b, Corollary 8.5]. All in all we have thusbuilt (GW → K) ∈ Fun( BC ⊳ , CAlg( P (Sch Z [1 / ) ∗ )) . P (Sch) ∗ to P (Sm S ) ∗ and motivically localizing, we obtain L mot K ∈ CAlg( S pc( S ) ∗ )and then Σ ∞ L mot K ∈ CAlg( SH ( S )). We have the Bott element β KGL ∈ e K ( P ) ≃ [ S , , K] and henceobtain β ′ KGL ∈ π , Σ ∞ L mot K. By definition we haveKGL S := (Σ ∞ L mot K)[ β ′− ] ∈ CAlg( SH ( S )) . See [Hoy16a, Proposition 3.2, Lemma 3.3 and Theorem 3.8] for details on periodic E ∞ -ring spectra; theupshot is that KGL S is represented by the prespectrum ( L mot K , L mot K , . . . ) with bonding maps givenby multiplication by β . It follows from [Hoy16a, Example 3.4] and [TT90, Proposition 6.8 and Theorem10.8] that if S is Noetherian and regular, then the canonical map K | Sm S → Ω ∞ KGL S is an equivalence;in other words KGL represents algebraic K -theory.There is a similar Bott element β ∈ g GW ( HP ∧ HP ) ≃ [ S , , GW] [PW10b, Definition 5.3, Theorem5.1], and inverting it in the suspension spectrum we obtainKO S := (Σ ∞ L mot K)[ β ′− ] ∈ CAlg( SH ( S )) . To be precise, the generator of C acts via Vect( S ) → Vect( S ), V V ∗ , α : V ≃ −→ W ( α ∗ ) − : V ∗ → W ∗ . Wemust pass to inverses in order to obtain a functor Vect( S ) → Vect( S ) instead of Vect( S ) op → Vect( S ). Arguing as above, using [Sch17, Theorems 9.6, 9.8 and 9.10] we see that if S is Noetherian, regular, and1 / ∈ S , then the canonical map GW | Sm S → Ω ∞ KO S is an equivalence; in other words KO representshermitian K -theory. In fact (in this situation) [Sch17, Theorem 9.10] implies that(3.1) Ω ∞ Σ n,n KO ≃ GW [ n ] , where GW [ n ] is the presheaf of n -shifted Grothendieck–Witt spaces [Sch17, Definition 9.1].One may show that the image ¯ β of β under the map GW → K is β (see e.g. [RØ16, Proposition3.3]). It follows that we could equivalently define KGL as (Σ ∞ L mot K)[ ¯ β ′− ], and in particular we obtaina morphism of E ∞ -rings KO → KGL. We now make this C -equivariant.3.2.6. Consider the functor F : CAlg( SH ( S )) → C at , E π ∗∗ E ;in other words F ( E ) is the set of subsets of π ∗∗ E . We view this power set as a category (in fact poset)by partially ordering it by inclusion. The functor F classifies a fibration (by [Lur09, Theorem 3.2.0.1])CAlg( SH ( S )) mrk → CAlg( SH ( S )) . Thus the objects of CAlg( SH ( S )) mrk are pairs ( E, X ) with E ∈ CAlg( SH ( S )) and X ⊂ π ∗∗ ( E ), andthe morphisms ( E, X ) → ( E ′ , X ′ ) are morphisms E α −→ E ′ such that α ∗ ( X ) ⊂ X ′ . In particular given afunctor b : C →
CAlg( SH ( S )), a lift of b to CAlg( SH ( S )) mrk is a section of F , or in other words a choicefor every object c ∈ C of X c ⊂ π ∗∗ ( b ( c )) subject to the condition that for every map α : c ′ → c ∈ C wehave b ( α ) ∗ ( X c ′ ) ⊂ X c .The functor CAlg( SH ( S )) → CAlg( SH ( S )) mrk , E ( E, π ∗∗ ( E ) × ) has a left adjoint which sends( E, X ) to the initial E -algebra in which all elements of X become invertible.3.2.7. We have the functor(Σ ∞ L mot GW → Σ ∞ L mot K) : BC ⊳ → CAlg( SH ( S )) . We lift this to a functor BC ⊳ → CAlg( SH ( S )) mrk by choosing the sets { β ′ } and { ¯ β ′ } respectively.Here ¯ β ′ is the image β ′ of β ′ in π ∗∗ Σ ∞ L mot K) and so fixed by the C -action. Composing with thelocalization functor CAlg( SH ( S )) mrk → CAlg( SH ( S )) we obtain(KO S → KGL S ) ∈ Fun( BC ⊳ , CAlg( SH ( S )) . S as well.3.2.9. It can be shown that the space L mot K (respectively L mot GW) is motivically equivalent to theproduct of Z and the infinite Grassmannian (respectively the infinite orthogonal Grassmannian) [ST15,Proposition 8.1 and Theorem 8.2] [Hoy16b, Corollary 2.10]. Since Grassmannians are stable under basechange, so are the motivic spaces L mot K and L mot GW, and hence so are the spectra KGL and KO.3.2.10. Recall the real realization functor r R from § r R (KO) (and r R (KGL)). Proof of Lemma 3.9. (1) Since KGL ≃ Σ ∞ ( Z × Gr)[ β − ] we get r R (KGL) ≃ Σ ∞ ( Z × Gr( R ))[ r R ( β KGL ) − ].It suffices to show that r R ( β KGL ) ∈ π ( Z × Gr( R )) is nilpotent. Since Gr( R ) ≃ BO , by Bott periodicity[Kar05, § π ( Z × Gr( R )) ≃ π ( O ) ≃
0, whence the result.(2) The Wood cofiber sequence [RØ16, Theorem 3.4] Σ , KO η −→ KO → KGL together with r R ( η ) = 2(Remark 2.20) and (1) implies that r R (KO) is 2-periodic. We deduce that r R (KO) ≃ r R (KO[1 / − ) ≃ r R (KW[1 / /
2] in [R¨on16, Theorem 4.4]. It also follows that r R (KO ∧ )is rational, and to conclude it suffices to show that π ∗ r R (KO ∧ ) ≃ Q [ β ± ]. We have (see Corollary 2.19for the first step) π ∗ r R (KO ∧ ) ≃ π ∗ (KO ∧ [1 / − ) ≃ π ∗ (KO ∧ [1 /η, / ≃ π ∗ (KO ∧ [1 / ( βη ) , / . For n < π n (KO) ≃ W ( R ) ≃ Z if n ≡ βη is an isomorphism [Sch17, Proposition 6.3] [Bal05, Theorem 1.5.22]. This implies that π n (KO ∧ ) ≃ Z ∧ or 0, depending on n as before, the same is true in the colimit along βη . The result follows. (cid:3) The Gepner–Snaith and homotopy fixed point theorems. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 17 G m : X
7→ O ( X ) × defines an abelian group object in P (Sm S ). Sending an elementto its inverse inverse lifts this to G m ∈ Fun( BC , Ab( P (Sm S ) ≤ )). Let L ⊂ Vect denote the subgroupoidspanned sectionwise by the trivial line bundle O . This is closed under tensor products and duals, andhence defines a subfunctor L ֒ → Vect : BC → CMon( P (Sm S )) . By inspection Ω L is discrete, and identifies with G m . Since L is (sectionwise) connected, we thus obtain B G m ≃ L ֒ → Vect. Composing with Vect → Ω ∞ KGL and using the adjunctionΣ ∞ + ⊣ Ω ∞ : CMon( P (Sm S )) ⇆ CAlg( SH ( S ))we obtain a map Σ ∞ + B G m → KGL ∈ Fun( BC , CAlg( SH ( S ))) . We also have the map ˜ β KGL : S , ≃ Σ ∞ P → Σ ∞ P ∞ ≃ Σ ∞ B G m → Σ ∞ + B G m , employing the stable splitting X + ≃ X ∨ S . The image of ˜ β KGL in π ∗∗ KGL is the Bott element β KGL [GS09, Proposition 4.2]. We may thus lift the map Σ ∞ + B G m → KGL to Fun( BC , CAlg( SH ( S )) mrk )by choosing the sets { ˜ β KGL , σ ˜ β KGL } and { β KGL , − β KGL } respectively above Σ ∞ + B G m and KGL (here σ denotes the action by C , so that in particular σβ KGL = − β KGL ). Inverting the marked elements andnoting that β KGL , − β KGL are units in KGL we obtain the
Gepner–Snaith map Σ ∞ + B G m [ ˜ β − , σ ˜ β − ] → KGL ∈ Fun( BC , CAlg( SH ( S ))) . Lemma 3.10.
The Gepner–Snaith map is an equivalence.Proof.
We have Σ ∞ + B G m [ ˜ β − ] ≃ KGL by [GS09, Theorem 4.17]. The image of σ ˜ β KGL in this ring is aunit (e.g. because it corresponds to − β KGL ), so the further inversion does nothing. (cid:3) G m is a discrete abelian group object, we have the endomorphisms ψ n : G m → G m , x x n for all n ∈ Z , and they all commute. In particular we obtain ψ n : G m → G m ∈ Fun( BC , Ab( P (Sm S ) ≤ )) . This deloops to ψ n : B G m → B G m ∈ Fun( BC , CMon( P (Sm S ))) . Lemma 3.11.
The composite S , β KGL −−−→ Σ ∞ + B G m Σ ∞ + Bψ n −−−−−→ Σ ∞ + B G m → KGL is homotopic to nβ KGL .Proof.
The stable splitting P → P induces the mapK ( P ) ≃ K ( ∗ )[ O (1)] / ( O (1) − → K ( P ) ≃ K ( ∗ ) a + b O (1) b. The element P → B G m + Bψ −−→ B G m + → KGL ∈ K ( P )corresponds to O (1) ⊗ n = ([ O (1) −
1] + 1) n = 1 + n [ O (1) − nβ KGL as desired. (cid:3)
We can thus form the following composite in Fun( BC , CAlg( SH ( S )) mrk )(Σ ∞ + B G m , { ˜ β KGL , σ ˜ β KGL } ) ψ n −−→ (Σ ∞ + B G m , { ψ n ( ˜ β KGL ) , σψ n ( ˜ β KGL ) } ) → (KGL , { nβ KGL , − nβ KGL } ) . Inverting the marked elements we obtainKGL ≃ Σ ∞ + B G m [ ˜ β − , σ ˜ β − ] → KGL[ nβ − ] ≃ KGL[1 /n ] . Further inverting n in the source, we finally arrive at ψ n GS : KGL[1 /n ] → KGL[1 /n ] ∈ Fun( BC , CAlg( SH ( S ))) . Proposition 3.12.
The underlying map ψ n GS : KGL[1 /n ] → KGL[1 /n ] coincides (up to homotopy) withthe map ψ n Riou constructed by Riou [Rio10, Definition 5.3.2 and sentences thereafter] . Note that if C ⇆ D is an adjunction then so is Fun( A , C ) ⇆ Fun( A , D ), e.g. by [BH17, Lemmas D.3 and D.6]. Proof.
Since both operations are stable under base change, we may assume that S = Spec ( Z ). By [Rio10,Remark 5.2.9] the map [KGL , KGL] → [Σ ∞ + P ∞ , KGL] ≃ Z J U K is injective. Here U = O (1) − ψ n Riou is(1 + U ) n [Rio10, Definition 5.3.2]. It suffices to verify that the same holds for ψ n GS . The image in thiscase is given by Σ ∞ + P ∞ ≃ Σ ∞ + B G m Bψ n −−−→ Σ ∞ + B G m → KGL . This corresponds to O (1) ⊗ n = (1 + U ) n by construction. (cid:3) Remark . In light of [Rio10, § ψ n GS act on (higher) algebraic K -groups in thesame way as any of the other standard constructions.3.3.3. The map KO → KGL ∈ Fun( BC ⊳ , CAlg( SH ( S )))induces by definition a map KO → KGL hC . Proposition 3.14.
Let S be Noetherian, regular, / ∈ S and vcd ( s ) < ∞ for all s ∈ S . Then themap KO → KGL hC is a -adic equivalence.Proof. It suffices to show that for every (absolutely) affine, smooth S -scheme X the morphismmap(Σ n,n X + , KO) / → map(Σ n,n X + , KGL hC ) / n ≥
0, and then since X isarbitrary we may assume that n = 0. Since X is a QL-scheme in the sense of [BKSØ15, Definition2.1], the claim follows from [BKSØ15, Theorem 2.4 and Corollary 2.6]. To be precise, their definition of(Hermitian) K -theory (and the involution on K -theory) uses perfect complexes, but the evident functorVect → Perf is duality preserving (when using the 0-shifted duality on Perf, which is why we reduced to n = 0) and induces an equivalence on (Hermitian) K -theory spaces, as we have seen in § (cid:3) Definition 3.15.
Let n be odd, whence invertible in Z ∧ . We denote by ψ nh GS : KO ∧ → KO ∧ the map induced from ψ n GS by completing at 2 and taking homotopy fixed points. (If the base schemedoes not satisfy the assumptions of Proposition 3.14, pull back from Z [1 / Action on the Bott element.
For an algebraically closed field ¯ K of characteristic zero, the groupK ( ¯ K ) is uniquely divisible, and K ( ¯ K ) = ¯ K × ≃ Q / Z ⊕ D where Q / Z corresponds to the roots of unityand D is uniquely divisible [Wei13, Theorem VI.1.6]. This implies (using e.g. Lemma 2.14(2)) that π (KGL( ¯ K ) ∧ ) ≃ Z ∧ . Lemma 3.16.
The action of ψ n on π (KGL( ¯ K ) ∧ ) is given by multiplication by n .Proof. We have π (KGL( ¯ K ) ∧ ) ≃ π L ∧ K ( ¯ K ), compatibly with the Adams action. It thus suffices toshow that the action on K ( ¯ K ) is multiplication by n , i.e. ψ n ([ a ]) = [ a n ]. This is [Wei13, ExampleIV.5.4.1]. (cid:3) Viewing GL( K ) as a discrete topological group, apply the topological +-construction to obtain aspace B GL( K ) + × Z with π i ( B GL( K ) + × Z ) = K i ( K ) for i ≥
0. In this model, the C -action onK( K ) ≃ B GL( K ) + × Z is induced from the automorphism of GL( K ) given by A A − T (and theidentity on Z ). If K = R or K = C , we can give GL( K ) its usual topology instead; denote the result byGL( K top ). Functoriality of the +-construction yieldsK( K ) → B GL( K top ) + × Z ∈ Fun( BC , SH ≥ ) . These maps are in fact p -adic equivalences for all p [Sus84, Corollary 4.7]. Lemma 3.17.
Let E ∈ Fun( BC × BC ′ , CAlg( SH )) denote ku ∧ with its usual action by complex con-jugation and passage to dual bundles. Suppose given a map ψ : E → E such that the induced map on π is given by multiplication by the odd integer n . Then the induced endomorphism of (( E hC ) ≥ ) hC ′ acts by multiplication by n − on π − . -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 19 Proof.
By Goerss–Hopkins obstruction theory, the space Map
CAlg( SH ) (KU ∧ , KU ∧ ) is discrete and iso-morphic to Hom CAlg ♥ ( π ∗ KU ∧ , π ∗ KU ∧ ) ≃ ( Z ∧ ) × via evaluation at the Bott element [GH04, Corollary7.7]. K (1)-localization and connective cover provide inverse equivalences Map CAlg( SH ) (KU ∧ , KU ∧ ) ≃ Map
CAlg( SH ) (ku ∧ , ku ∧ ). It follows thatMap Fun( BC × BC ′ , CAlg( SH )) ( E, E ) ≃ Map
CAlg( SH ) ( E, E ) hC × C ′ is also discrete. The upshot of all this is that ψ is just given by the ordinary ψ n , made equivariant withrespect to C × C ′ in the usual way.We have ( E hC ) ≥ ≃ ko ∧ and the induced action by C ′ is trivial (since ko ≃ BO + × Z and transposecoincides with inverse on orthogonal matrices). By the above discussion, the map induced by ψ is theusual Adams operation ψ n on ko ∧ , made compatible with the trivial action in the canonical (trivial)way. We thus need to show that the Adams action on π − (ko ∧ ) hC ′ ≃ (ko ∧ ) RP ∞ + is by multiplicationby n − .We first consider the KO-cohomology. Let us write x ∈ ko and β ∈ ko for the generators, so that x = 4 β . The groups (KO ∧ ) ∗ ( RP ∞ + ) can be read off from [Fuj67, Theorem 1] as follows(KO ∧ ) ∗ RP ∞ + ≃ Z ∧ [ λ, x, β, β − ] / ( λ + 2 λ, x − β );the (KO ∧ ) ∗ ≃ Z ∧ [ x, β, β − ] / ( x − β )-algebra structure (coming from pullback along RP ∞ → ∗ ) is theevident one. Here | λ | = 0 and λ corresponds to O (1) −
1. The algebra structure is compatible withAdams operations (here we are crucially using that ψ is compatible with the trivial action in the trivialway). Note that ψ ( λ ) = λ . For this it suffices to show that ψ ( O (1)) = O (1); but ψ ( O (1)) = O ( n )[Ada62, Theorem 5.1(iii)] and O (2) ≃ O (e.g. since O (2) = (1 + λ ) = 1 since λ = − λ ), so this holdssince n is odd. It follows that the Adams action on (KO ∧ ) ∗ ( RP ∞ + ) is via multiplication by n − ∗ , sincethis holds for the generators 1 ( ψ being a ring map), λ (as seen above), and x, β ± (which can be checkedin KO ∗ , where it e.g. follows from the injection into KU ∗ )It remains to observe that (ko ∧ ) ( RP ∞ + ) → (KO ∧ ) ( RP ∞ + ) is injective. Indeed the obstruction to thisis [Σ − RP ∞ + , (KO ∧ ) < ] = [Σ − RP ∞ + , (KO ∧ ) ≤− ] = 0 . (cid:3) Theorem 3.18.
Let S be a scheme with / ∈ S and n odd. The action of ψ nh GS on the Bott element β is given by multiplication by n · n ǫ .Proof. It suffices to prove the result for S = Spec( Z [1 / ψ nh GS ( β ) = aβ , for some a ∈ GW( Z [1 / ∧ ; we need to show that a = n · n ǫ . Since the map KO ∧ → KGL ∧ is compatible withthe Adams action by construction, it follows (e.g. from Lemma 3.11) that rk( a ) = n . Hence (usingLemma 2.9) it suffices to prove that a has image n in W( Z [1 / S = Spec ( R ) instead. Consider E = KGL ∧ ( C ) := map SH ( R ) (Σ ∞ + Spec( C ) , KGL ∧ ) ∈ Fun( BC × BC ′ , SH );here the first action comes from complex conjugation and the second from the C -action on KGL (i.e.taking duals). We have a map E → K( C top ) ∧ ≃ ku ∧ which is an equivalence by Suslin’s theorem [Sus84,Corollary 4.7]. It is C × C ′ -equivariant for the usual action on ku ∧ . By Lemma 3.16 the action of ψ n on π E is by multiplication by n . We may hence apply Lemma 3.17 to deduce that the Adams actionon π − (( E hC ) ≥ ) hC ′ is by multiplication by n − . By the Quillen–Lichtenbaum conjecture for algebraic K -theory [RØ05, Theorem 2] we have ((KGL( C ) ∧ ) hC ) ≥ ≃ KGL( R ) ∧ . We thus learn that ψ nh GS acts bymultiplication by n − on π − (KO( R ) ∧ ) ≃ W( R ) ∧ { η β − } . The claim (over R ) follows since the Adamsaction on η must be trivial (since it comes from the sphere spectrum; see Example 3.7).Now we go back to S = Spec ( Z [1 / ψ nh GS ( β ) = aβ , for some a ∈ GW( Z ) ∧ ⊂ GW( Z [1 / ∧ . Indeed then we could determine a by comparison with S = Spec( R ).This argument indeed works; see Lemma 3.37. Since the proof of Lemma 3.37 is rather involved (in thatit relies on [CDH + ψ nh GS acts on π − (KO ∧ ) ≃ W( Z [1 / ∧ { β − η } by multiplication by n − . By arguing as in for example[BW20, proof of Theorem 5.8], we find that( ∗ ) W( Z [1 / ≃ Z [ g ] / ( g , g ) , where g := h i − F ) i −→ Spec( Z ) j ←− Spec( Z [1 / . The sequence of functors Perf( F ) i ∗ −→ Perf( Z ) j ∗ −→ Perf( Z [1 / F ) → K( Z ) → K( Z [1 / i ∗ is duality preserving if we give Perf( F ) the duality Hom( − , i ! O );then all the functors become C -equivariant and we get an induced localization sequenceKGL ∧ ( F ) hC → KGL ∧ ( Z ) hC → KGL ∧ ( Z [1 / hC . Since i ! O ≃ Σ O we see that we get the shifted duality on K( F ). We have K( F ) ∧ ≃ H Z ∧ [Wei13,Corollary IV.1.13], and the C -action is given by multiplication by −
1, the duality being shifted. Inparticular (see e.g. [ ˇCad99, Lemma 1]) π ∗ (K( F ) ∧ ) hC = H −∗ ( BC , Z ∧ ) = ( ∗ even Z / ∗ < . We thus get a short exact sequence( ∗∗ ) 0 = π − KGL ∧ ( F ) hC i ∗ −→ π − KGL ∧ ( Z ) hC j ∗ −→ π − KGL ∧ ( Z [1 / hC ∂ −→ π − KGL ∧ ( F ) hC = Z / . We now determine the image of the injection j ∗ . By the homotopy fixed point theorem and ( ∗ ) wehave π − KGL ∧ ( Z [1 / hC ≃ W( Z [1 / ∧ ≃ Z ∧ [ g ] / ( g , g ) ≃ Z ∧ { } ⊕ Z / { g } . The filtration induced by the homotopy fixed point spectral sequence is the I-adic one, so the first twosubquotients aregr ( Z [1 / Z [1 / / I ≃ F { } and gr ( Z [1 / Z [1 / / I ≃ F {h− i , h i} , which must be a subquotient of the appropriate group e ( i )2 ( Z [1 / H i ( C , K i ( Z [1 / ∧ ) on the E page. By [Mil71, Corollary 16.3] K ( Z [1 / ∧ ≃ ( Z [1 / × ) ∧ ≃ Z ∧ ⊕ Z / , and also K ( Z [1 / ∧ ≃ Z ∧ . We deduce that e ( i )2 ( Z [1 / i = 0, and at most 2 if i = 1; since the subquotient gr i ( Z [1 / e ( i )2 ( Z [1 / i ( Z [1 / e (0)2 ( Z ) → e (0)2 ( Z [1 / ( Z ) → gr ( Z [1 / K ( Z ) ∧ ≃ Z × , which implies that the map e (1)2 ( Z ) → e (1)2 ( Z [1 / ≃ F {h− i , h i} is an injection onto F {h− i} . These facts together imply that j ∗ : π − KGL ∧ ( Z ) hC → π − KGL ∧ ( Z [1 / hC does not hit the element g . Consequently ∂ ( g ) = 1 and ∂ (1 + ǫg ) = 0 for some ǫ ∈ { , } . Let us put α := 1 + ǫg . It follows from exactness of ( ∗∗ ) that π − KGL ∧ ( Z ) hC ≃ Z ∧ { α } ⊂ π − KGL ∧ ( Z [1 / hC . Note that we have an action of ψ n := ψ n GS also on KGL( Z ), and hence on KGL ∧ ( Z ) hC . It followsthat ψ n ( α ) = pα for some p ∈ Z . We know (by Example 3.7) that ψ acts on π − KGL ∧ ( Z [1 / hC bymultiplication by some element a + ǫ ′ g ∈ W( Z [1 / a ∈ Z and ǫ ′ ∈ Z /
2. Comparison with thecase S = Spec( R ) shows that a = n − . We thus get pj ∗ α = j ∗ ( pα ) = j ∗ ( ψ ( α )) = ψ ( j ∗ ( α )) = ( n − + ǫ ′ g ) j ∗ α and so p (1 + ǫg ) = ( n − + ǫ ′ g )(1 + ǫg )= n − + ( n − ǫ + ǫ ′ ) g. Comparing coefficients of 1 yields p = n − , and then comparing coefficients of g yields ǫ ′ = 0. This wasto be shown. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 21 The -adic fracture square. Recall the following well-known fact.
Lemma 3.19.
Let C be a stable, presentable, compactly generated ∞ -category and n ∈ Z . Then forevery E ∈ C the natural commutative square E −−−−→ E [1 /n ] y y E ∧ n −−−−→ E ∧ n [1 /n ] is cartesian.Proof. Let X ∈ C be compact. Functors of the form map( X, − ) preserve limits and colimits, so comple-tion and localization, and form a conservative collection. This reduces the result to C = SH where it iswell-known; see e.g. [Bau11, Proposition 2.2]. (cid:3) We can apply this with C = SH ( S ) and n = 2 to obtain the fracture square(3.2) E −−−−→ E [1 / ≃ E [1 / − ∨ E [1 / + y y E ∧ −−−−→ E ∧ [1 / ≃ E ∧ [1 / − ∨ E ∧ [1 / + ;here we have used the decomposition of 2-periodic spectra into + and − parts (see § Definition 3.20.
For E ∈ SH ( S ) we define E ± = E ∧ × E ∧ [1 / ± E [1 / ± . Note that there are maps E → E ± and E ± → E ∧ → E ∧ [1 / ∓ . Lemma 3.21.
Let C be an ∞ -category and X, A , A , B , B ∈ C with maps X → B × B and A i → B i .Then X × B × B ( A × A ) ≃ ( X × B A ) × B A . Proof.
Consider the following commutative diagram( X × B A ) × B A A × A A X × B A A × B B A X B × B B . The right hand squares and horizontal rectangles are cartesian, hence so are the left hand squares, andhence so is the left hand vertical rectangle, by the pasting law [Lur09, Lemma 4.4.2.1]. (cid:3)
Our slightly unconventional Definition 3.20 is partially justified by the following result.
Corollary 3.22.
For E ∈ SH ( S ) we have cartesian squares E −−−−→ E [1 / − y y E + −−−−→ E ∧ [1 / − and E −−−−→ E [1 / + y y E − −−−−→ E ∧ [1 / + Proof.
Immediate from Lemma 3.21, the fracture square (3.2) and the definition of E ± . (cid:3) Remark . Note that if E ∈ CAlg( SH ( S )) then E ∧ , E [1 / + = E [1 / ∧ η and E [1 / − = E [1 / , /η ]are E ∞ -rings, and hence so are E ± (being pullbacks of E ∞ -rings along E ∞ -ring maps). All in all thefracture squares for E from Corollary 3.22 are pullback diagrams in CAlg( SH ( S )). Proof of the main theorem.Lemma 3.24 (Heard) . The diagram ( α : KO → KGL) ∈ Fun( BC ⊳ , SH ( S )) is a universal η -completelimit diagram: if F : SH ( S ) → C is any functor preserving binary products, then F ( α/η )) : F (KO /η ) → ( F (KGL /η )) hC is an equivalence.Proof. This is essentially [Hea17, § /η → KGL /η ) ∈ Fun( BC , SH ( S )) identifieswith (KGL → KGL C ), where by KGL C we mean the product KGL × KGL with its switch action. Thisis clearly a universal limit diagram. In slightly more detail, let i : ∗ → BC and p : BC → ∗ be thecanonical functors. Then X C ≃ i ∗ ( X ) and Y hC ≃ p ∗ ( Y ); thus ( X C ) hC ≃ p ∗ i ∗ X ≃ (id) ∗ X ≃ X . (cid:3) We can use this to identify KO + , in the sense of Definition 3.20. Lemma 3.25.
Let S satisfy the assumptions of Proposition 3.14 (such as Spec( Z [1 / ). Then for n odd we have KO[1 /n ] + ≃ KGL[1 /n ] hC ∈ SH ( S ) . The same holds for n even and any S .Proof. For n even we have KO[1 /n ] + ≃ KO[1 /n ] ∧ η ≃ KGL[1 /n ] hC , by Lemma 3.24. Thus we now consider n odd.Taking homotopy fixed points in the 2-adic fracture square for KGL[1 /n ] we obtain a cartesian squareKGL[1 /n ] hC −−−−→ KGL[1 / n ] hC y y (KGL ∧ ) hC −−−−→ (KGL ∧ [1 / hC , noting that KGL[1 /n ] ∧ ≃ KGL ∧ . By Proposition 3.14 we have (KGL ∧ ) hC ≃ KO ∧ ≃ KO[1 /n ] ∧ . Wealso have KGL[1 / n ] hC ≃ (KGL[1 / n ] hC ) ∧ η ( L. . ) ≃ KO[1 / n ] ∧ η ≃ KO[1 /n ][1 / + ;here we have used that KGL[1 / n ] hC is η -complete since KGL[1 / n ] is ( η acting by 0). The sameargument shows that (KGL ∧ [1 / hC ≃ KO ∧ [1 / + ≃ KO[1 /n ] ∧ [1 / + . Hence the above cartesiansquare identifies with the defining square of KO[1 /n ] + . (cid:3) Remark . Using KO[1 /n ] + ≃ KGL[1 /n ] hC , for any n we can define ( ψ n ) + : KO[1 /n ] + → KO[1 /n ] + as ( ψ n GS ) hC . If n is even this is the best we can do (see Remark 3.5). For n odd we shall see how toextend this to an endomorphism of all of KO[1 /n ]. Definition 3.27.
For n odd and S satisfying the assumptions of Proposition 3.14, we denote by ψ nh GS : KO[1 /n ] + → KO[1 /n ] + the map induced from ψ n GS : KGL[1 /n ] → KGL[1 /n ] (see § /n ] hC ≃ KO[1 /n ] + .For S = Spec( Z [1 / / n ] − ∈ SH ( Z [1 / / − r R ≃ SH [1 / r R (KO) ≃ KO top [1 / ψ n top : KO top [1 /n ] → KO top [1 /n ] we mean the topological Adams operation, forexample obtained by taking homotopy fixed points of the complex realization of the operation on KGL. Lemma 3.28. In CAlg( SH ( Z [1 / there exists a unique (up to homotopy) map ψ n : KO[1 /n ] → KO[1 /n ] such that the two squares KO[1 /n ] −−−−→ KO[1 /n ] + ψ n y ψ nh GS y KO[1 /n ] −−−−→ KO[1 /n ] + and KO[1 /n ] −−−−→ KO[1 / n ] − ≃ KO top [1 / n ] ψ n y ψ n top y KO[1 /n ] −−−−→ KO[1 / n ] − ≃ KO top [1 / n ] commute (in CAlg( SH ( Z [1 / ).Proof. Applying Remark 3.23 with E = KO[1 /n ] (and S = Spec( Z [1 / CAlg( SH ( Z [1 / (KO[1 /n ] , KO[1 /n ]) −−−−→ Map
CAlg( SH ( Z [1 / (KO[1 /n ] , KO[1 /n ] + ) y y Map
CAlg( SH ( Z [1 / (KO[1 /n ] , KO[1 / n ] − ) −−−−→ Map
CAlg( SH ( Z [1 / (KO[1 /n ] , KO ∧ [1 / − ) =: M. We have two maps a, b : KO[1 /n ] → KO ∧ [1 / − ∈ CAlg( SH ( Z [1 / ψ nh GS and ψ n top ,respectively. A choice of E ∞ -ring map ψ n with the desired properties is a path in M from a to b . Thus -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 23 such a map exists if a, b are homotopic, and is unique if π of the corresponding component of M vanishes.Since KO ∧ [1 / − ∈ SH ( Z [1 / / − ⊗ Q r R ≃ D ( Q ), we find that M ≃ Map
CAlg( D ( Q )) ( r R (KO) ⊗ Q , r R (KO ∧ [1 / − )) . We claim that r R (KO) ⊗ Q is the free E ∞ -ring over Q on an invertible generator in degree 4. Indeed ifwe denote that universal object by F (Σ )[ x − ], then the canonical map F (Σ )[ x − ] → r R (KO) ⊗ Q induces an isomorphism on π ∗ by Lemma 3.9 and the well-known computation π ∗ F (Σ n ) ≃ Q [ x ] (with | x | = 2 n ; this follows e.g. from [RS17, Corollary 8.4]). We deduce that M ⊂ Ω ∞ +4 r R (KO ∧ [1 / − )is given by the union of those path components corresponding to invertible elements of π r R (KO ∧ [1 / − ).It thus follows from Lemma 3.9 again that π M ≃ Q × { x } and π M = 0 (for any choice of base point).In other words ψ n exists if and only if a and b act in the same way on β , and if so it is unique. Since bothmaps act by multiplication by n (see Theorem 3.18 for ψ nh GS , and for ψ n top we can use e.g. comparisonwith the action on KU and Lemma 3.11), the result follows. (cid:3) Proof of Theorem 3.1.
Since everything is stable under base change, we may assume that S = Spec( Z [1 / ψ n be the map constructed in Lemma 3.28. Hence ( ∗ ) the induced endomorphism of KO + ≃ KGL hC is given by ψ nh GS ≃ ( ψ n GS ) hC . It follows that ( ∗∗ ) the induced endomorphism of KO ∧ ≃ (KO + ) ∧ ≃ (KGL ∧ ) hC is also given by ψ nh GS .(1,3) Hold by definition.(2) By Corollary 2.15 and Lemma 2.11, the map π , (KO) → π , (KO ∧ ) is injective, and hence itsuffices to prove the claim for ( ψ n ) ∧ . By ( ∗∗ ), this is ψ nh GS , so the claim follows from Theorem 3.18.(4) The map KO → KGL factors as KO → KO + ≃ KGL hC → KGL, so the claim follows from( ∗ ). (cid:3) Remark . The construction of ψ n also implies that if S is a regular QL -scheme, then under theequivalence KO[1 /n ] + ≃ KGL[1 /n ] hC we have ( ψ n ) + ≃ ψ nh GS (see Definition 3.27).3.7. The motivic orthogonal image of j spectrum. In this section we show that our Adams op-eration can be used to construct a “motivic image of orthogonal j ” spectrum. This idea originated indiscussions with JD Quigley and Dominic Culver regarding [CQ19]. None of the results in this sectionare used in the sequel.We begin by determining the lowest two “generalized slices” of the sphere spectrum. Theorem 3.30.
Let k be a field of exponential characteristic e = 2 .(1) The unit map u −→ KO ∈ SH ( k ) induces an equivalence on ˜ s .(2) There is a fibration sequence Σ , H Z / → ˜ s ( )[1 /e ] ˜ s ( u ) −−−→ ˜ s (KO)[1 /e ] ≃ Σ , H Z / . Proof. (1) For E ∈ SH ( k ) there is a functorial cofiber sequence [Bac17, Lemma 11(1)] s ( E ≥ ) → ˜ s E → f π ( E ) ∗ . It thus suffices to show that → KO induces an equivalence on s ( − ≥ ) and π ( − ) . Indeed then it alsoinduces an equivalence on f π ( − ) ∗ , so on s ( π ( − ) ∗ ), and on s ( − ≥ ) (the latter since s is a stablefunctor), and hence on ˜ s ( − ) by the cofiber sequence. It is well-known that the unit map → KOinduces an isomorphism on π , . Similarly we know that s ( ) ≃ H Z and s (KO ≥ ) ≃ H Z (e.g. use[Bac17, Theorem 16]). In the commutative diagramGW( k ) ≃ π , ( ) ≃ −−−−→ π , (KO ≥ ) ≃ GW( k ) y y Z ≃ π , s ( ) α −−−−→ π , s (KO ≥ ) ≃ Z the vertical maps are both the rank map; it follows that α is an isomorphism. Since this holds over anyfield, the map s ( ) → s (KO ≥ ) induces an isomorphism on π , ; since this is the only non-vanishingeffective homotopy sheaf of H Z the map is an equivalence.(2) We invert the exponential characteristic throughout. For E ∈ SH ( k ) there is a functorial cofibersequence [Bac17, Lemmas 11(2) and 8] f Σ π ( E ) ∗ → ˜ s ( E ) β −→ s ( E ≥ ) . Note that s of the first term of the cofiber sequence vanishes, so ( ∗ ) the map β is equivalent to theprojection ˜ s E → s ˜ s E .We shall show that (a) f Σ π ( ) ∗ ≃ Σ , H Z /
24, (b) the unit map induces an isomorphism s ( ≥ ) → s (KO ≥ ) ≃ Σ , H Z /
2, and (c) f Σ π (KO) ∗ = 0. This will imply the result.(a) First note that π ( f KO) − ≃ π (KO) − ≃ π (KO [ − ) − ∗∗ ) ≃ Z − ≃ , where ( ∗∗ ) is obtained from e.g. [Bac17, Table 1]. This implies via [RSØ16b, (1.2)] that π ( ) − ≃ Z / . Since this holds compatibly over any field (see [RSØ16b, Remark 5.8]), Lemma 3.31 below implies theclaim.(b) The claim that s (KO ≥ ) ≃ Σ , H Z / ∗ ) and [Bac17, Theorem 16]. Since → KO induces an isomorphism π ( − ) it also does on f π ( − ) ∗ and hence on s π ( − ) ∗ . Consideringthe cofiber sequence s ( − ≥ ) → s ( − ≥ ) → s ( π ( − ) ∗ ), it thus suffices to show that s ( ) → s (KO ≥ )is an equivalence. Note that s (KO ≥ ) ≃ s f KO ≥ ≃ s ko. The claim thus follows from [ARØ17,Theorem 3.2] and [RSØ16b, Corollary 2.13 and Lemma 2.28] (both spectra are given by Σ , H Z / s KO ≃ Σ , H Z / ∗ ).This concludes the proof. (cid:3) Lemma 3.31.
Let k be a field, H ∈ SH ( k ) veff ♥ . Suppose that (1) π , ( H ) ≃ Z /n for some n ∈ Z , and(2) for every finitely generated separable field extension K/k , the map π , ( H )( k ) → π , ( H )( K ) is anisomorphism. Then H ≃ H Z /n .Proof. Write GW ∈ SH ( k ) veff ♥ for the unit. The isomorphism π , ( H ) ≃ Z /n induces a map α : GW → H . We also have the rank map β : GW → H Z /n . We shall show that α, β are surjectionswith equal kernels; this will prove the result. Since π , ( H )( k ) → π , ( H )( K ) is an isomorphism and GW ( k ) → π , ( H )( k ) is surjective, α is surjective. We claim that α ( K ) is the rank map. Indeed let¯ K/K be a separable closure and consider the commutative diagram GW ( K ) α ( K ) −−−−→ π , ( H )( K ) ≃ Z /n y (cid:13)(cid:13)(cid:13) Z ≃ GW ( ¯ K ) α ( ¯ K ) −−−−→ π , ( H )( ¯ K ) ≃ Z /n. The map α ( ¯ K ) is the unique morphism of abelian groups sending 1 to 1, whence α ( K ) is the rank mapas desired. We deduce that ker( α )( K ) = ker( β )( K ), and hence ker( α ) = ker( β ) by unramifiedness (andsince “taking the underlying sheaf” is an exact conservative functor [Bac17, Proposition 5(3)]). Theresult follows. (cid:3) We put ksp := ˜ f Σ , KO, so that Σ , ksp ≃ ˜ f KO . Corollary 3.32.
For n odd, the Adams operation ψ n − /n ] → ko[1 /n ] lifts to ϕ n : ko[1 /n ] → Σ , ksp[1 /ne ] .Proof. We invert e throughout.Let us write ˜ f ≤ for the cofiber of ˜ f → id. Then we have a cofiber sequence Σ , ksp → ko → C :=˜ f ≤ ko. We need to show that the compositeko[1 /n ] ψ n − −−−−→ ko[1 /n ] → C [1 /n ]is zero. Considering the cofiber sequence → ko → D, it suffices to show that (a) the composite → ko ψ n − −−−−→ ko → C is zero, and (b) any map D → C iszero.(a) It is enough to show that → ko ψ n − −−−−→ ko is zero. This is clear since ψ n (1) = 1.(b) Recall that Σ , SH ( k ) veff defines the non-negative part of a t -structure on SH ( k ) with ˜ f and˜ f ≤ as non-negative and negative truncation, respectively (see e.g. [BH17, § B]). It follows that C isin the negative part of this t -structure, and hence it is enough to show that D is in the non-negative -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 25 part. Consider the following commutative diagram, in which all rows and columns are cofiber sequences(defining E, F ) ˜ f −−−−→ −−−−→ ˜ f ≤ y y y ˜ f ko −−−−→ ko −−−−→ ˜ f ≤ ko y y y E −−−−→ D −−−−→ F. It suffices to show that
E, F ∈ Σ , SH ( k ) veff . This is clear for E , since Σ , SH ( k ) veff is closed undercolimits. It follows from Theorem 3.30 that F ≃ Σ , H Z / ∈ Σ , SH ( k ) veff .This concludes the proof. (cid:3) Definition 3.33.
A motivic orthogonal image of j spectrum is any spectrum j o in a fiber sequencej o → ko (2) ϕ −→ Σ , ksp (2) . Clearly the unit map (2) → ko (2) lifts to j o . Theorem 7.8 states that the lifted unit map (2) → j o isan η -periodic equivalence.3.8. Geometric operations.
In this section we compare our stable Adams operations to the onesconstructed by Fasel–Haution [FH20]. None of the results in this section are used in the sequel.3.8.1. We begin with the following adaptation of [PW10b, Theorem 13.1].
Lemma 3.34.
For n = 0 the map [KO[1 /n ] , KO[1 /n ]] CAlg(h SH ( Z [1 / → KSp[1 /n ] ( HP ∞ ) × GW( Z [1 / /n ] , α ( α ( H (1)) , α ( β ) /β ) is injective. Here H (1) ∈ KSp ( HP ∞ ) corresponds to the tautological bundle.Proof. There is a presentation KO ≃ colim i Σ ∞ +(4 − i, − i ) X i , for a sequence of pointed, smooth, affine Z [1 / X i [PW10b, Theorem 12.3]. By [PW10b, proof of Theorems 13.1, 13.2, 13.3], the inducedmap γ : [KO , KO[1 /n ]] → lim i KO[1 /n ] i − , i − ( X i ) is an injection (in fact, equivalence). Note that γ ( α ) i = α ( γ (id) i ). The isomorphism KO[1 /n ] i − , i − ( X i ) ≃ KSp[1 /n ] ( X i ) { β − i } hence shows that aring map α : KO[1 /n ] → KO[1 /n ] is determined by its effect on KSp[1 /n ] ( X ) for X smooth affine, andon β . By [PW10b, Theorem 8.1], since X is affine the action of α is determined by the action on theclass corresponding to the tautological bundle of HGr( r, ∞ ) (for various r ), and on β . The claim nowfollows by the computation of the cohomology of HGr( r, ∞ ) in terms of HP ∞ [PW10c, Theorems 11.4and 8.1]. (cid:3) Recall that over any base scheme S with 1 / ∈ S we haveKSp ( HP ∞ S ) = KO , ( HP ∞ S ) ≃ M i ≥ KO − i, − i ( S ) { b KO1 ( γ ) i } . Since KO i, i ( S ) = GW( S ) and KO i − , i − ( S ) = Z make sense also for S = Spec( Z ) (with GW( Z ) = Z ⊕ Z {h− i} ), we will by slight abuse of notation putKSp ( HP ∞ Z ) := M i ≥ KO − i, − i ( Z ) { b KO1 ( γ ) i } . Corollary 3.35.
Let n be odd and R ⊂ [KO[1 /n ] , KO[1 /n ]] SH ( Z [1 / denote the set of those homotopyring maps such that α ( β ) /β ∈ GW( Z )[1 /n ] ⊂ GW( Z [1 / /n ] and α ( H (1)) ∈ KSp ( HP ∞ Z )[1 /n ] ⊂ KSp ( HP ∞ Z [1 / )[1 /n ] . Then the map R → GW( Z )[1 /n ] , α α ( β ) /β is an injection.Proof. Consider the commutative diagram R −−−−→ KSp[1 /n ] ( HP ∞ Z ) × GW( Z )[1 /n ] α α ⊗ Q y y [KO ⊗ Q , KO ⊗ Q ] CAlg(h SH ( Z [1 / −−−−→ (KSp ⊗ Q ) ( HP ∞ Z [1 / ) × (GW( Z [1 / ⊗ Q ) . The top horizontal map is injective by Lemma 3.34, and the right hand vertical map is injective sinceGW( Z ) , Z are torsion-free. It follows that the left hand vertical map is injective.Write H − ∈ KO − , − ( ∗ ) for the trivial symplectic bundle. We shall now show that a homotopy ringmap α : KO ⊗ Q → KO ⊗ Q (over Z [1 / β and H − . Since SH ( S ) ⊗ Q ≃ ( SH ( S ) ⊗ Q ) + × ( SH ( S ) ⊗ Q ) − as symmetric monoidal categories, we need only prove the same claim about (KO ⊗ Q ) ± . We haveequivalences of homotopy ring spectra (KO ⊗ Q ) + ≃ H Z [ t, t − ] ⊗ Q and (KO ⊗ Q ) − ≃ ( ⊗ Q ) − [ u, u − ](see e.g. [DF19, Theorem D]); here | t | = (4 ,
2) and | u | = (8 , , Σ n,n ( ⊗ Q ) − ] r R ≃ [ Q , Q [ n ]] D ( Q ) = 0 for n = 0and similarly [H Z , Σ n,n H Z ⊗ Q ] SH ( Z [1 / = 0 for n = 0(see e.g. [Rio10, Remark 5.3.16]), it follows that α is determined by its effect on t, u . Since t can bechosen to be the image of H − , and u the image of β , the claim follows.We deduce the following: given α ∈ R , write α ( β ) = aβ and α ( H − ) = bH − , for some a ∈ GW( Z )[1 /n ], b ∈ Z [1 /n ] (uniquely determined). Then a and b determine α . To conclude the proof, we need to showthat a determines b . Since H − = 2 hβ , we find that b = rk( a ), so that a determines b up to a sign. Itwill thus suffice to prove that b ≡ top (see e.g.[ARØ17, Lemma 2.13]), we may as well show: if α : KO ∧ → KO ∧ ∈ SH is a homotopy unital map,then ( α − H − ) ≡ ∧ , KO ∧ ] SH ≃ Z ∧ J T K , where T = ψ − α = P i ≥ a i T i for certain a i ∈ Z ∧ . Since α is unital wemust have 1 = α (1) = a . Since ψ ( H − ) = 25 H − (e.g. by comparison with the Adams action on KU ∧ )we get T ( H − ) = 24 H − ≡ (cid:3) X over Z [1 / ± ( X ) = KO ( X ) ⊕ KSp ( X ). This is a commutative, Z / λ -ring with λ -operations given by exterior powers of vector bundles [FH20, Theorem 4.2.4]. Wedenote the associated Adams operations by ψ n geo . Recall the stable operations ψ n FH : KO[1 /n ] → KO[1 /n ]from [FH20, § Lemma 3.36.
For n odd, we have π Ω ∞ ( ψ n FH ) = ψ n geo | KO ( − )[1 /n ] and π Ω ∞ +4 , ( ψ n FH ) = h ( − n ( n − / i nn ǫ ψ n geo | KSp ( − )[1 /n ] . Moreover ψ n FH ( β ) = n n ǫ β .Proof. Put α i = π Map(Σ i, i − , ψ n FH ), ω = h ( − n ( n − / i nn ǫ . By construction, for X ∈ Sm S and E ∈ KO ( X ) or E ∈ KSp ( X ) we have(3.3) α i (( H (1) − H − ) ⊠ E ) = ( H (1) − H − ) ⊠ α i +1 ( E ) ∈ GW ± ( HP ∧ X + ) . Furthermore by construction [FH20, p. 16], the map α − is given by ω − ψ n geo . Since ψ n geo ( H (1) − H − ) = ω ( H (1) − H − ) [FH20, Lemma 5.1.4], and ψ n geo preserves products in all of GW ± ( X ), (3.3) inductivelyimplies that α i = ω i ψ n geo . All claims follow. (cid:3) K -theory. For part (2) below we require somedifficult results about Hermitian K -theory over schemes in which 2 is not invertible from [CDH + Lemma 3.37. (1) For any scheme S , the ring spectrum KGL hC ∈ SH ( S ) is SL -oriented.(2) For S = Spec( Z ) we have π ∗ , ∗ ((KGL hC ) ∧ ) ≃ GW( Z ) ∧ [ β, β − , H − ] / (I( Z ) H − , H − − hβ ) . Proof. (1) The construction of the ring map MSL → KO → KGL in [BW20, Corollary B.3] shows thatover any base, MSL → KGL refines to a C -equivariant map (for the trivial action on MSL). The resultfollows.(2) We have map( X, KGL) ≃ K(Perf X ), for any X ∈ Sm Z . The C -action on KGL corresponds tothe action on Perf X by dualization E Hom( E, O X ). The equivalence Ω P K ≃ K is implementedby the pushforward i ∗ : K( X ) → K( X × P ). Since i ∗ : Perf X → Perf X × P is duality preserving ifon Perf X × P we use the n -shifted duality and on Perf X the ( n + 1)-shifted duality, we find that for n ≥ n,n X, KGL) ≃ K(Perf X ) is the n -shifted one. Iterated tensoring with O X [1]induces a C -equivariant automorphism of Perf X , intertwining the 4 n -shifted duality and the usual one, -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 27 or the (4 n + 2)-shifted duality and the negative of the usual one [Sch10b, Proposition 7]. Denote by β ∈ π , (KGL hC ) the element corresponding to 1 ∈ π , (KGL hC ) under the induced equivalence; inother words β is represented by the perfect complex O [2] (with a certain canonical 4-shifted dualitynaively expressed as Hom( O [2] , O [4]) ≃ Hom( O , O [2]) ≃ O [2]). Multiplication by β is an automorphismof KGL hC (indeed this can be checked before taking fixed points, and β corresponds to an automorphismof Perf X , and hence of KGL). It follows that A = π ∗ , ∗ ((KGL hC ) ∧ ) is (8 , A , = GW( Z ) ∧ and A , = Z ∧ , in such a way that the map α : A → π ∗ , ∗ ((KGL hC Z [1 / ) ∧ ) ≃ (KO ∗ , ∗ Z [1 / ) ∧ is the canonical one in degrees (0 ,
0) and (4 , α ( β ) = β by construction, the map α is an injection,and we can check all the desired relations over Z [1 / A , and A , . Since we have determined above the dualities on map( , KGL) ≃ K( Z ) and map( S , , KGL) ≃ K( Z ) to be the usual one and its negative, the desired result is an immediateconsequence of [CDH +
20, Theorem 1]. (cid:3)
Proposition 3.38.
Let S be a scheme with / ∈ S , and n an odd integer. The two maps ψ n , ψ n FH : KO[1 /n ] → KO[1 /n ] ∈ SH ( S ) are homotopic.Proof. By definition, the maps are pulled back from Spec( Z [1 / S = Spec( Z [1 / β by multiplication by n n ǫ . Hence by Corollary 3.35,it suffices to check that both maps send H (1) to an element of KO , ( HP ∞ Z )[1 /n ] ⊂ KO , ( HP ∞ Z [1 / )[1 /n ].First we treat ψ n FH . Lemma 3.36 shows that up to a factor in GW( Z )[1 /n ] × , ψ n FH ( H (1)) = ψ n geo ( H (1)).By definition, this is a linear combination with integer coefficients of exterior powers of H (1), which areclearly already defined over Z .For ψ n , note that it suffices to show that the subring KO ∗ , ∗ ( HP ∞ Z ) ∧ is preserved. Lemma 3.37(together with the homotopy fixed point theorem over Z [1 / ∧ ) hC ) ∗ , ∗ ( HP ∞ Z ) → ((KGL ∧ ) hC ) ∗ , ∗ ( HP ∞ Z [1 / ) ≃ (KO ∧ ) ∗ , ∗ ( HP ∞ Z [1 / ) . The result follows since ( ψ n ) ∧ = ψ nh GS by definition, and the latter is already defined over Z . (cid:3) Cobordism spectra
Summary.
We will be using the algebraic cobordism spectra MSL and MSp [BH17, Example 16.22][PW10a]. They can be constructed explicitly out of the Thom spaces of tautological bundles on speciallinear (respectively quaternionic) Grassmannians; we review this in § − , − Σ ∞ + HP ∞ → Σ − , − Th( γ ) → MSp , where γ denotes the tautological bundle on HP ∞ .There are notions of SL-oriented and Sp-oriented cohomology theories [PW10a, Definitions 5.1 and8.1]. We will only deal with cohomology theories represented by homotopy commutative ring spectra A ∈ CAlg(h SH ( S )). In this situation one way of exhibiting an SL-orientation (respectively Sp-orientation) isto exhibit a homotopy ring map MSL → A (respectively MSp → A ) (see e.g. [BW20, Proposition 4.13]or [PW10a, Theorems 5.5 and 13.2]).If A is Sp-oriented and V is a symplectic vector bundle bundle on X ∈ Sm S , then we obtain the Borelclasses b i ( V ) ∈ A i, i ( X ) [PW10a, Definition 11.5]. One has [PW10a, Theorem 8.2] A ∗∗ ( HP ∞ ) ≃ A ∗∗ J b ( γ ) K . Similarly if A is SL-oriented and V is an oriented vector bundle, then we obtain the Pontryagin classes p i ( V ) ∈ A i, i ( X ) [Ana15, Definition 19]. When dealing with η -periodic cohomology theories, we willimplicitly shift everything into weight zero (by multiplying by powers of η ); so then we write b i ( V ) ∈ A i ( X ) and p i ( V ) ∈ A i ( X ) . The cohomology of MSp and MSL have been worked out in [PW10a, Theorem 13.1] [Ana15, Theorem10] . We perform the straightforward dualization: Theorem 4.1.
Let A be a cohomology theory (i.e. A ∈ CAlg(h SH ( S )) ).(1) Suppose that A is Sp -oriented. The Kronecker pairing A ∗∗ ( HP n ) ⊗ A ∗∗ ( HP n ) → A ∗∗ is perfect.Denote by { β i } ni =0 ∈ A ∗∗ ( HP n ) the dual basis to { b ( γ ) i } ni =0 ∈ A ∗∗ ( HP n ) . The β i are compatiblewith HP n ֒ → HP n +1 ; denote by β i ∈ A ∗∗ ( HP ∞ ) their common images. Write b i ∈ A i, i (MSp) for the image of β i +1 under the map induced by (4.1) . Then A ∗∗ (MSp) ≃ A ∗∗ [ b , b , b , . . . ] / ( b − . (2) Suppose that A is η -periodic and SL -oriented. The canonical map MSp → MSL annihilates b i for i odd; write p i for the image of b i in A i (MSL) . Then A ∗ MSL ≃ A ∗ [ p , p , . . . ] / ( p − . Example . KO admits a ring map from MSL (see e.g. [BW20, Corollary B.3]), and hence is SL-oriented(whence also Sp-oriented). Since MSL is very effective, ko = ˜ f KO is also SL-oriented. Similarly so arekw = ko[ η − ] and KW = KO[ η − ], since they receive ring maps from ko.If A denotes any of the above theories, then by Remark 3.6 we obtain Adams operations on A -homology. We are particularly interested in the case S = Spec( k ) and A = kw (2) . Note that if W( k ) = F then kw ≃ kw (2) and so all ψ n act on kw ∗ E for any E ∈ SH ( k ) and n odd. Proposition 4.3.
Let S = Spec( k ) , where k is a field (of characteristic = 2 ). Suppose that W( k ) = F .Then ψ ( p i ) ∈ p i + βp i − + β kw ∗ MSL . In fact ψ ( b i ) ∈ b i + β kw ∗ MSp + ( βb i − i even i odd . We also record the following well-known facts.
Lemma 4.4.
We have r R (MSp) ≃ MU and r R (MSL) ≃ MSO . Real realization.
For an algebraic group G , write BG for the stack of G -torsors, viewed as asheaf of groupoids on Sm S . Suppose given furthermore an inclusion G → GL n . Then we obtain amap BG → B GL n → K ◦ corresponding to the tautological virtual bundle on B GL n , and consequentlya Thom spectrum M G ∈ SH ( S ) [BH17, § G -torsor E , write V ( E ) = E × G A n for theassociated vector bundle, where G acts on A n via the embedding into GL n . Lemma 4.5.
There is a cofibration sequence Σ ∞− n,n + ( A n \ hG → Σ ∞− n,n + BG → M G.
Proof.
By definition,
M G ≃ colim E : X → BG Th( V ( E )) , where the colimit is over all smooth schemes with a map to BG , i.e. smooth schemes and G -torsorson them, and Th( V ( E )) is the associated Thom spectrum. In other words there is a cofiber sequenceΣ ∞− n,n + V ( E ) \ → Σ ∞− n,n X + → Th( V ( E )) and hencecolim E : X → BG Σ ∞− n,n + V ( E ) \ → Σ ∞− n,n + BG → M G. In [Ana15], the author works over perfect fields of characteristic = 2 only. However, all results hold over general baseschemes [personal communication]. Indeed the only place where the assumption on the base is used is in Lemma 12. Thisholds over general bases, as can be seen as follows. It suffices to show that the endomorphisms [ x : y ] [ y : x ] and[ x : y ] [ − x : y ] of P are A -homotopic. Since both have determinant −
1, and SL ( Z ) is A -connected (being generatedby elementary matrices), the result follows. There might be some concern here which topology we are using, but we shall only apply the following discussion tospecial groups, where all torsors are Zariski-locally trivial. Again, the G -orbits should be taken as sheaves in some sufficiently strong topology. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 29 By universality of colimits in ∞ -topoi [Lur09, Theorem 6.1.0.6], passage to G -orbits preserves pullbacks.It follows that there is a cartesian square( E × ( A n \ hG ≃ V ( E ) \ −−−−→ ( A n \ hG y p y E hG ≃ X −−−−→ ∗ hG ≃ BG.
Thus colim E : X → BG V ( E ) \ ≃ ( A n \ hG , by universality of colimits again (this time in the presheaf ∞ -topos). The result follows. (cid:3) If G = GL n , the association E V ( E ) induces an equivalence between G -torsors and vector bundlesof rank n . Similarly if G = SL n , we get an equivalence between G -torsors and oriented vector bundles(i.e. carrying a trivialization of the determinant), and if G = Sp n then we obtain an equivalence withsymplectic bundles of rank 2 n (i.e. carrying a non-degenerate, alternating bilinear form). See e.g.[AHW18, §
3] for this. The Grassmannian variety Gr( n, k ) represents the functor of n -dimensional vectorsubbundles of O k ; similarly SGr( n, k ) represents the functor of n -dimensional vector subbundles of O k together with an orientation, and HGr( n, k ) represents the functor of 2 n -dimensional vector subbundles V of O k such that the restriction of the canonical alternating form on O k to V remains non-degenerate.There are thus canonical GL n , SL n and Sp n torsors on Gr( n, k ) , SGr( n, k ) and HGr( n, k ), respectively,inducing maps(4.2) Gr( n, ∞ ) → B GL n , SGr( n, ∞ ) → B SL n and HGr( n, ∞ ) → B Sp n . Each of these maps is well-known to be a motivic equivalence; see e.g. [MV99, Proposition 2.6], [AHW18,proof of Theorem 4.1.1], [PW10b, proof of Theorem 8.2].We can use this to connect to more standard definitions of Thom spectra.
Lemma 4.6.
We have motivic equivalences M GL n ≃ Σ ∞− n,n Th( γ GL n ) , M SL n ≃ Σ ∞− n,n Th( γ SL n ) and M Sp n ≃ Σ ∞− n, n Th( γ Sp n ) , where γ GL n → Gr( n, ∞ ) (respectively γ SL n → SGr( n, ∞ ) , γ Sp n → HGr( n, ∞ ) ) denotes the tautologicalbundle.Proof. We need to prove that the motivic Thom spectrum functor inverts the motivic equivalence Gr n → B GL n , and similarly for SL n , Sp n . This follows from [BH17, Proposition 16.9 and Remark 16.11]. (cid:3) By definition [BH17, Example 16.22], the ( E ∞ -ring maps of) spectra(4.3) MSp → MSL → MGLare obtained by applying the motivic Thom spectrum formalism to the maps K Sp ◦ → K SL ◦ → K ◦ . SinceK Sp ◦ A ≃ colim n B Sp n (argue as in the discussion just before [BH17, Theorem 16.13]), we find (usingLemma 4.6 and [BH17, Proposition 16.9 and Remark 16.11]) thatMSp ≃ colim n M Sp n ≃ colim n Σ ∞− n, n Th( γ Sp n ) , as expected. SimilarlyMSL ≃ colim n Σ ∞− n,n Th( γ SL n ) and MGL ≃ colim n Σ ∞− n,n Th( γ GL n ) . Corollary 4.7.
We have r R (MSp) ≃ MU and r R (MSL) ≃ MSO . In fact r R (MSp n ) ≃ MU n and r R (MSL n ) ≃ MSO n .Proof. It suffices to show the “in fact” part. By Lemma 4.5, we haveMSL n ≃ Σ ∞− n,n cof( T n → G n ) , where T n = ( A n \ h SL n and G n = ∗ h SL n . Since SL n is a special group [Ser58, § T n ≃ colim m ∈ ∆ op SL × mn × ( A n \ ∈ S pc( S ) ∗ . Since r R preserves colimits and finite products, we find that r R ( T n ) ≃ colim m ∈ ∆ op SL n ( R ) × m × ( R n \ ≃ ( R n \ h SL n ( R ) . The inclusion SO n → SL n ( R ) is a homotopy equivalence [Hal15, § E.5], so that this is the same as( R n \ hSO n . Similarly we find that r R ( G n ) ≃ BSO n , whence r R (MSL n ) ≃ Σ − n cof(( R n \ hSO n → BSO n ) ≃ MSO n , as desired. The argument for MSp is similar, using that Sp n is special [Ser58, § n ( R ) ≃ U n [AG01, § (cid:3) Homology. A ∈ CAlg(h SH ( S )) and X ∈ SH ( S ). Then we have the Kronecker pairing A ∗∗ ( X ) ⊗ A ∗∗ A ∗∗ ( X ) → A ∗∗ ( f : Σ ∗∗ → A ∧ X ) ⊗ ( g : X → Σ ∗∗ A ) (Σ ∗∗ f −→ A ∧ X id ∧ g −−−→ A ∧ A → A ) . This is easily seen to be A ∗∗ -bilinear. Lemma 4.8. If X is cellular and strongly dualizable, and either A ∗∗ X or A ∗∗ X is flat over A , then theKronecker pairing is perfect.Proof. Write DX for the dual and put ⊗ := ⊗ A ∗∗ . Replacing X by DX if necessary, we may assumethat A ∗∗ X is flat. Then for any cellular object Y we get A ∗∗ ( DX ∧ Y ) ≃ A ∗∗ ( DX ) ⊗ A ∗∗ ( Y ); indeedthis holds for spheres by construction and is a natural transformation of homological functors preservingfiltered colimits (here we use that A ∗∗ X is flat).Let u : → DX ∧ X and c : X ∧ DX → be the unit and co-unit of the strong duality between X and DX . Since X is cellular, A ∗∗ ( DX ∧ X ) ≃ A ∗∗ ( DX ) ⊗ A ∗∗ ( X ), and hence A ∗∗ ( u ) , A ∗∗ ( c ) define mapsof the correct shape to exhibit a strong duality between A ∗∗ ( X ) and A ∗∗ ( DX ). Consider the followingdiagram A ∗∗ ( X ) ⊗ A ∗∗ A ∗∗ ( X ) ⊗ A ∗∗ ( DX ) ⊗ A ∗∗ ( X ) A ∗∗ ⊗ A ∗∗ ( X ) A ∗∗ ( X ) ⊗ A ∗∗ ( DX ∧ X ) A ∗∗ ( X ∧ DX ) ⊗ A ∗∗ ( X ) A ∗∗ ( X ∧ ) A ∗∗ ( X ∧ DX ∧ X ) A ∗∗ ( ∧ X ) . ≃ id ⊗ A ∗∗ ( u ) ≃ ≃ ≃ A ∗∗ ( c ) ⊗ id A ∗∗ (id ∧ u ) A ∗∗ ( c ∧ id) The vertical maps are lax monoidal structure maps; the equivalence A ∗∗ ( DX ∧ X ) ≃ A ∗∗ ( DX ) ⊗ A ∗∗ ( X )has already been established. The diagram commutes because A ∗∗ is lax symmetric monoidal. Up tosuppressing tensoring with the unit, the bottom horizontal composite is the identity (by definition of u, c exhibiting a strong duality). It follows that the top horizontal composite (inverting the middle verticalmaps) (id ⊗ A ∗∗ ( c )) ◦ ( A ∗∗ ( u ) ⊗ id) is the identity. A similar diagram shows that ( A ∗∗ ( c ) ⊗ id) ◦ (id ⊗ A ∗∗ ( u ))is the identity. Thus A ∗∗ ( u ) , A ∗∗ ( c ) exhibit a strong duality between A ∗∗ ( X ) and A ∗∗ ( DX ). This wasto be shown. (cid:3) In preparation for later, we also observe the following.
Lemma 4.9.
Let ψ : A → A be a ring automorphism. Then the Kronecker pairing satisfies h ψx, y i = ψ h x, ψ − y i . Proof.
The commutative diagram (in which suspensions have been suppressed) x −−−−→ X ∧ A y ∧ id −−−−→ A ∧ A id ∧ ψ y id ∧ ψ y X ∧ A y ∧ id −−−−→ A ∧ A m −−−−→ A ψ − ∧ ψ − y ψ − y A ∧ A m −−−−→ A shows that ψ − h ψx, y i = h x, ψ − y i . The result follows. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 31 Let X → X → · · · ∈ SH ( S )be a directed system and put X = colim i X i . Suppose that lim i A ∗∗ ( X i ) = 0. Then A ∗∗ ( X ) ≃ lim i A ∗∗ ( X i ) , and we can give this the inverse limit topology (i.e. give each A ∗∗ ( X i ) the discrete topology and takethe limit in bigraded topological abelian groups). Corollary 4.10.
Assume in addition that each X i is strongly dualizable and cellular, and A ∗∗ ( X i ) or A ∗∗ ( X i ) is flat. Then (4.4) A ∗∗ ( X ) ≃ Hom A ∗∗ ( A ∗∗ ( X ) , A ∗∗ ) and A ∗∗ ( X ) ≃ Hom A ∗∗ ,c ( A ∗∗ ( X ) , A ∗∗ ) . Here
Hom A ∗∗ ,c means continuous homomorphisms (for the discrete topology on the target and the inverselimit topology on the source).Proof. For the first claim we compute A ∗∗ ( X ) ≃ lim i A ∗∗ ( X i ) ≃ lim i Hom A ∗∗ ( A ∗∗ ( X i ) , A ∗∗ ) ≃ Hom A ∗∗ (colim i A ∗∗ ( X i ) , A ∗∗ ) ≃ Hom A ∗∗ ( A ∗∗ ( X ) , A ∗∗ ) , using the Milnor exact sequence [GJ09, Proposition VI.2.15], Lemma 4.8, and compactness of the spheres.For the second claim, note that by definition a basis of open neighborhoods of 0 in A ∗∗ ( X ) is given byker( A ∗∗ ( X ) → A ∗∗ ( X i )). Any continuous homomorphism thus factors through A ∗∗ ( X i ) for some i ,yielding the formula Hom A ∗∗ ,c ( A ∗∗ ( X ) , A ∗∗ ) ≃ colim i Hom A ∗∗ ( A ∗∗ ( X i ) , A ∗∗ ) . But also colim i Hom A ∗∗ ( A ∗∗ ( X i ) , A ∗∗ ) ≃ colim i A ∗∗ ( X i ) ≃ A ∗∗ ( X ) , using Lemma 4.8 and compactness of the spheres again. (cid:3) E over S and an algebraic group G → GL n , a G -orientation of E consists ofa choice of Thom class t ( V ) ∈ E n,n (Th( V )) for every G -bundle V on a smooth S -scheme, satisfyingcertain naturality and normalization axioms (see e.g. [Ana19, Definition 3.3]). Lemma 4.11.
Let E be G -oriented and V a Nisnevich locally trivial G -bundle on X ∈ Sm S (e.g. G special and V arbitrary). Then the map E ∧ Th( V ) id E ∧ ∆ −−−−→ E ∧ Th( V ) ∧ X + id E ∧ t ( V ) ∧ id X −−−−−−−−−→ E ∧ Σ n,n E ∧ X + m −→ E ∧ Σ n,n X + is an equivalence. We call this equivalence the (homological) Thom isomorphism ; it induces in particular t : E ∗∗ Th( V ) ≃ E ∗− n, ∗− n ( X ). Proof.
By the smooth projection formula, we may assume that X = S . By Nisnevich separation [Hoy16b,Proposition 6.23] and naturality of the Thom class, we may assume that V is trivial. In this case themap is homotopic to the identity, by definition. (cid:3) r, n ) denote the quaternionic Grassmannian of 2 r -dimensional symplectic subspaces in2 n -dimensional symplectic space (see e.g. [PW10c]). For example HGr(1 , n ) = HP n . Lemma 4.12.
Suppose that A is Sp -oriented. We have A ∗∗ ( HP ∞ ) ≃ A ∗∗ { β , β , . . . } , and the canonicalmap α : ( HP ∞ ) n → HGr( n, ∞ ) induces A ∗∗ (HGr( n, ∞ )) ≃ Sym n ( A ∗∗ ( HP ∞ ) ⊗ n ) . Proof. Σ ∞ + HGr( r, n ) ∈ SH ( S ) is cellular and strongly dualizable [RSØ16a, Proposition 3.1]. By [PW10c,Theorem 11.4] the maps A ∗∗ (HGr( r, n )) ← A ∗∗ (HGr( r, n + 1))are surjective, and(4.5) A ∗∗ (HGr( r, ∞ )) ≃ lim n A ∗∗ (HGr( r, n )) ≃ A ∗∗ J b , . . . , b r K . We deduce (using Corollary 4.10) that A ∗∗ ( HP ∞ ) is the topological dual of A ∗∗ ( HP ∞ ) ≃ A ∗∗ J b K , sothat compatible classes β i exist as claimed. In particular(4.6) A ∗∗ ( HP ∞ ) ≃ A ∗∗ { β , β , . . . } . Since (4.5) holds over any base, we find that A ∗∗ (( HP ∞ ) n ) ≃ A ∗∗ J a , . . . , a n K , where a i = b ( γ i ), γ i being the tautological bundle on the i -th factor HP ∞ . There is a map α :( HP ∞ ) n → HGr( n, ∞ ) such that α ∗ ( γ ) ≃ γ ⊞ γ ⊞ · · · ⊞ γ n =: E. By the Cartan formula [PW10c, Theorem 10.5], the map α ∗ : A ∗∗ (HGr( n, ∞ )) → A ∗∗ (( HP ∞ ) n )is given by b i b i ( E ) = σ i ( a , . . . , a n ) , where σ i is the i -th elementary symmetric polynomial. It is thus a split injection onto ( A ∗∗ (( HP ∞ ) n )) Σ n .The map α ∗ is obtained by passing to continuous duals, and hence as claimed. (cid:3) Remark . The above result can also be deduced from [Ana17, Theorem 5.10].
Lemma 4.14.
Let γ be the tautological bundle on HP ∞ . Then γ \ is (motivically) contractible.Proof. Let γ n be the tautological bundle on HP n . Then γ n \ n /G × Sp n − , where G ⊂ Sp isthe subgroup fixing the first vector. One checks (e.g. using Sp = SL ) that G ≃ G a . Since all of thesegroups are special, the ´etale quotients are homotopy quotients, and we deduce that γ n \ ≃ Sp n / Sp n − .Taking colimits we get γ \ ≃ Sp / Sp ≃ ∗ . (cid:3) Proof of Theorem 4.1(1).
Consider the motivic space BSp A ≃ colim n HGr( n, ∞ ) (see (4.2)). This has an H -space structure coming from addition of vector bundles. The diagram( HP ∞ ) n −−−−→ BSp nα y m y HGr( n, ∞ ) −−−−→ BSpcommutes (essentially by construction); here the horizontal maps classify the tautological bundles. Theinclusion at the base point ( HP ∞ ) n → ( HP ∞ ) n +1 covers the canonical map HGr( n, ∞ ) → HGr( n + 1 , ∞ )and induces in cohomology the quotient map A ∗∗ (( HP ∞ ) n +1 ) ≃ A ∗∗ J b K ˆ ⊗ n +1 → A ∗∗ J b K ˆ ⊗ n ≃ A ∗∗ (( HP ∞ ) n ) . Passing to continuous duals, we deduce that under the identification A ∗∗ (HGr( n, ∞ )) ≃ Sym n ( A ∗∗ HP ∞ ),the map HGr( n, ∞ ) → HGr( n + 1 , ∞ ) corresponds to multiplication by β . This implies that A ∗∗ (BSp) ≃ colim n A ∗∗ (HGr( n, ∞ )) ≃ colim n Sym n ( A ∗∗ HP ∞ ) ≃ A ∗∗ [ β , β , . . . ] / ( β − . The Thom isomorphism (Lemma 4.11) induces an isomorphism of rings A ∗∗ BSp t −→ A ∗∗ MSp. Itremains to identify the classes t ( β i ). Let γ be the tautological bundle on HP ∞ . The defining cofibersequence of pointed spaces γ \ → γ ≃ HP ∞ s −→ Th( γ )has the property that γ \ s : A ∗∗ { β , β , . . . } ≃ e A ∗∗ ( HP ∞ ) ≃ A ∗∗ (Th( γ )) . To conclude the proof, we shall show that s ( β i ) = t ( β i − ).To see this, it suffices to show that the composite˜ A ∗∗ ( HP ∞ ) s ≃ A ∗∗ (Th( γ )) t ≃ A ∗− , ∗− ( HP ∞ )maps β i to β i − . By construction, its dual is A ∗ +4 , ∗ +2 ( HP ∞ ) t ( γ ) · −−−→ A ∗∗ (Th( γ )) s −→ ˜ A ∗∗ ( HP ∞ ) , Recall that HGr( n, k ) represents the functor of 2 n -dimensional symplectic subbundles of the trivial symplectic bundle O k . On HP k we thus have γ ֒ → O k whence γ ⊞ n ֒ → O nk ; this defines a map ( HP k ) n → HGr( n, nk ). Now take colimits. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 33 i.e. multiplication by s ∗ ( t ( γ )). By definition, this is b = b ( γ ) [PW10c, Proposition 7.2], and so the claimfollows by dualization. (cid:3) n, k ) for the special linear Grassmannian varieties [Ana15, Definition 22]. Lemma 4.15. Σ ∞ + SGr( n, k ) ∈ SH ( S ) is cellular and strongly dualizable.Proof. By definition we have SGr( n, k ) = det γ n,k \
0, where γ n,k is the tautological bundle on Gr( n, k ).We thus have a cofiber sequence SGr( n, k ) → Gr( n, k ) → Th(det γ n,k ) . Since Gr( n, k ) and Th(det γ n,k ) ≃ P (det γ n,k ⊕ O ) / P (det γ ) [MV99, Proposition 2.17] are strongly du-alizable [CD09, Proposition 2.4.31] so is SGr( n, k ). For cellularity we use the description SGr( n, k ) =SL n /P ′ k . The group P ′ k is an extension of special groups (SL k , SL n − k and G a ) and thus special. Thusthe ´etale quotient defining SGr( n, k ) is a Zariski quotient, and hence (the action being free) a homotopyquotient: SGr( n, k ) ≃ (SL n ) hP ′ k ≃ colim i ∈ ∆ op P × ik × SL n . It thus suffices to show that SL n and P ′ k are (stably) cellular; this is proved in [Wen10, Proposition4.1]. (cid:3) Essentially by construction, the space SGr( n, k ) represents the functor of n -dimensional subbundles of O k , together with a choice of trivialization of the determinant. If V is a symplectic bundle, then det( V )is trivialized (by the Pfaffian; see e.g. the discussion just before [Ana12, Definition 4.5]), and so V is alsoa special linear bundle. This induces maps HGr( n, k ) → SGr(2 n, k ) and HGr( n, ∞ ) → SGr(2 n, ∞ ). Lemma 4.16.
Suppose that A is SL -oriented and η -periodic. The composite ( HP ∞ ) n → HGr( n, ∞ ) → SGr(2 n, ∞ ) → SGr(2 n + 1 , ∞ ) induces A ∗∗ (SGr(2 n + 1 , ∞ )) ≃ Sym n ( A ∗∗ ( HP ∞ ) / { β , β , . . . } ) . Proof.
Write α for the composite. We first determine the map α ∗ : A ∗∗ (SGr(2 n + 1 , ∞ )) → A ∗∗ (( HP ∞ ) n ) ≃ A ∗∗ J a , . . . , a n K . By [Ana15, Theorem 10] we have A ∗∗ (SGr(2 n + 1 , ∞ )) ≃ A ∗∗ J p , . . . , p n K ;here p i = p i ( γ ) are the Pontryagin classes of the tautological bundle γ on SGr(2 n + 1 , ∞ ). Thus α ∗ ( p i ) = p i ( E ), where E := γ ⊞ γ ⊞ · · · ⊞ γ n is the tautological bundle on ( HP ∞ ) n . We compute using[Ana15, Lemma 12] p t ( E ) = Y i p t ( γ i ) = Y i (1 + a i t ) . In other words α ∗ ( p i ) = σ i ( a , . . . , a n ) . It follows that α ∗ is a split injection onto A ∗∗ J a , a , . . . , a n K Σ n . By Lemma 4.15, Corollary 4.10 and[Ana15, Remark 13] the map α ∗ is the topological dual of α ∗ , which is easily checked to be as claimed. (cid:3) Proof of Theorem 4.1(2).
By the Thom isomorphism (Lemma 4.11) we have A ∗∗ MSL ≃ colim n A ∗∗ SGr(2 n + 1 , ∞ ) , which by the above identifies with A ∗∗ (MSp) / ( b , b , . . . ) . This was to be shown. (cid:3)
Adams action.
In this section we work over a field k of characteristic = 2. Recall that for anySp-oriented cohomology theory A we have A ∗∗ ( HP ∞ ) ≃ A ∗∗ J b K , where b = b ( γ ) [PW10a, Theorem 8.2].Recall also that we put KW = KO[ η − ] and kw = KW ≥ . Lemma 4.17.
Suppose that W( k ) = F . Then ψ ( b ) = b (1 + βb ) ∈ kw ( HP ∞ ) . Proof.
We implicitly invert 3 throughout this proof.Since kw ∗∗ ( HP ∞ ) ֒ → KW ∗∗ ( HP ∞ ), it suffices to prove the claim for KW. Note thatKO , ( HP ∞ ) ≃ M i ≥ KO − i, − i { b KO1 ( γ ) i } ≃ M i ≥ Z { b KO1 ( γ ) i } ֒ → KGL , ( HP ∞ );here we use that KO n, n ≃ GW( k ) = Z by assumption, and KO n +4 , n +2 ≃ Z as always (see e.g.[Bac17, Table 1]). Write α : KO → KGL for the canonical map and H (1) ∈ KO , ( HP ∞ ) ≃ KSp ( HP ∞ )for the class of the tautological bundle. We shall first determine ψ ( H (1)), and to do this we determine α ( ψ ( H (1))) = ψ ( α ( H (1))) (see Theorem 3.1(4)). We have α ( H (1)) = β − γ , where γ ∈ KGL ( HP ∞ )is the tautological bundle. By Remark 3.13, “our” Adams operation on KGL is just the classical one, socan be computed in terms of exterior powers of vector bundles (see e.g. [Wei13, § II.4]). It follows thatfor any rank 2 vector bundle V we have ψ ( V ) = V ⊗ − V ⊗ det V ∈ KGL ( X ) . If the bundle is symplectic, then det V = 1. Hence ψ ( α ( H (1))) = ψ ( β − γ ) = 3 − β − ( γ ⊗ − γ )= 3 − ( β − γ )( β ( β − γ ) −
3) = α (3 − H (1)( βH (1) − . Here we have used that ψ is a ring map and ψ ( β KGL ) = 3 β KGL (by construction). Hence by injectivityof α we get ψ ( H (1)) = 3 − H (1)( βH (1) − . By [Ana17, Theorem 6.10] we have b KO1 ( γ ) = H (1) − H − , where H − ∈ KO , ( ∗ ). Hence also ψ ( H − ) ∈ KO , ( ∗ ) . It follows that ψ ( b ) ∈ KW ( HP ∞ ) is the image of ψ ( H (1)) − ψ ( H − ) ∈ KO , ( HP ∞ ). Onehas KW ∗ ≃ W( k )[ β ± ], with | β | = 4 (see e.g. § = 0, so ψ ( b ) is just the image of ψ ( H (1)). Since 2 = 0 ∈ KW ∗ , the result follows. (cid:3) Remark . Our proof is complicated by the fact that we did not want to use the geometric descriptionof the Adams operations on KO ∗∗ ( HP ∞ ); in particular this made it difficult to determine ψ ( b ) withoutthe assumption that W( k ) = F . If we allow ourselves this description (i.e. Proposition 3.38 and Lemma3.36) we get ψ ( b ( γ )) = ψ ( H (1) − H − ) h− i · · ǫ ∈ KO , ( HP ∞ ) . Using that for a rank 2 symplectic bundle E one has ψ ( E ) = E − E , one easily deduces that ψ ( b ) = b (1 − βb / ∈ kw ( HP ∞ ) , over any field. Lemma 4.19.
For a scheme S and n odd, the maps ψ n : KW[1 /n ] → KW[1 /n ] ∈ SH ( S ) and (provided S is a field) ψ n : kw[1 /n ] → kw[1 /n ] are equivalences.Proof. The second map is obtained from the first by applying the connective cover functor, so the secondclaim follows from the first. The first claim is compatible with base change, so we may check it overSpec( Z [1 / ψ n : π ∗ KW[1 /n ] → π ∗ KW[1 /n ] is an isomorphism; recall that π ∗ KW[1 /n ] = W [1 /n, β, β − ]. Henceby Example 3.7 and Theorem 3.1(2,3) the map on π i is given by multiplication by n i , which is anisomorphism as needed (and all other homotopy sheaves vanish). (cid:3) Proof of Proposition 4.3.
It suffices to prove the “in fact” statement.Denote the inverse of ψ by ψ / . Note that ψ ( β ) = 9 β = β , so that ψ acts by the identity on kw ∗ ,and hence so does ψ / . Applying Lemma 4.9 to ψ / we deduce that h ψ / x, y i = h x, ψ y i , i.e. that theaction of ψ / on kw ∗ HP ∞ is dual to the action of ψ on kw ∗ HP ∞ . On kw ∗ HP ∞ we have ψ ( b n ) = ψ ( b ) n = b n (1 + βb ) n = b n + nβb n +2 + O ( β ) . -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 35 Hence(4.7) ψ / ( β i ) = β i + β ( i − β i − + O ( β ) . It follows that for any x ∈ kw ∗ HP ∞ we have ψ / ( x ) = x + O ( β ). Since ψ is β -linear and inverse to ψ / , we deduce that x = ψ ψ / ( x ) = ψ ( x ) + O ( β ), i.e. ψ ( x ) = x + O ( β ). Using this when applying ψ to (4.7) we find that β i = ψ ( β i ) + β ( i − β i − + O ( β )) + O ( β ) , whence ψ ( β i ) = β i + β ( i − β i − + O ( β ) . Using that the β i ∈ kw ∗ HP ∞ maps to b i − ∈ kw ∗ MSp and 2 = 0 ∈ kw ∗ the result follows. (cid:3) Some completeness results
Summary.Theorem 5.1.
Let k be a field of char( k ) = 2 , and suppose that vcd ( k ) < ∞ . If E ∈ SH ( k ) veff , then E ∧ is η -complete. In particular, the map π ∗∗ E ∧ → π ∗∗ E ∧ ,η is an isomorphism. This is the only form of the result we shall use in the sequel. In § π ∗∗ -isomorphism. The extension to an equivalence of spectra uses a technicalresult established in the remaining subsections; this is not relevant for the sequel and so can be skipped.5.2. Main result.
We first recall and extend the slice completeness result of [BEØ20, § slice tower [Voe02] is a functorial tower f • E → E ∈ SH ( k ); the slice completion of E issc E = lim n cof( f n E → E ) . Proposition 5.2.
Let k be a field of exponential characteristic e , t coprime to e with vcd t ( k ) < ∞ . If E ∈ SH ( k ) ≥ , then E ∧ t,ρ → sc( E ) ∧ t,ρ is an equivalence. Note that the fact that E ∧ t,ρ → sc( E ) ∧ t,ρ is a π ∗∗ -isomorphism was established in [BEØ20, Corollary5.13]. We will give the full proof of Proposition 5.2 in § Example . Let E ∈ SH ( k )[ η − ] ≥ . Since ρ = 2 on SH ( k )[ η − ], we deduce that E/ n is slice complete.It follows that slice spectral sequence techniques as in [OR19] could be used (over fields of characteristic = 2 and vcd < ∞ ) to study the relationship between [ η − ] ∧ and kw ∧ . We pursue a different strategyin the sequel.The following argument is closely related to part of [HKO11a, Lemma 20]. Lemma 5.4.
Let E ∈ SH ( k ) . Then E ∧ is η -complete if and only if E/ (2 , ρ ) is η -complete.Proof. Necessity is clear since η -complete spectra are stable under finite colimits. We thus show suffi-ciency. Consider the fiber sequence F → E/ → ( E/ ∧ η . Then F is η -periodic, 2-complete, and F/ρ ≃ F/ρ ≃ F/
2. We deducethat F ≃ F being 2-complete); in other words E/ η -complete. The result follows since η -completespectra are stable under limits. (cid:3) Proof of Theorem 5.1.
By Proposition 5.2, E/ (2 , ρ ) → sc( E ) / (2 , ρ ) is an equivalence. Since slices are η -complete (being modules over H Z (2) ≃ s ( (2) ); see e.g. [BH17, Theorem B.4]), we deduce that E/ (2 , ρ )is η -complete (being a limit of finite extensions of slices, by the effectivity assumption). We conclude byLemma 5.4. (cid:3) Remaining proofs. We are using here the slightly extended definition of virtual cohomological dimension introduced in [BEØ20, § t ( k ) = max { cd p ( k [ √− | p | t } . Definition 5.5. (1) By a tower in a category C we mean an object of Fun( Z op , C ), i.e. a diagram E • = . . . E → E → E → E − → · · · ∈ C . (2) Suppose that C is pointed. We call a tower E • nilpotent if for every n ∈ Z there exists N > n such that the composite map E N → E n is zero. Lemma 5.6.
Let E • ∈ SH be a tower of spectra such that each tower π i ( E • ) of abelian groups isnilpotent. Then lim i →∞ E i ≃ . Proof.
For fixed i , the tower π i ( E • ) is Mittag–Leffler, and hence lim s π i ( E s ) ≃ s π i E s ≃
0. The claim thus follows from the Milnor exact sequence [GJ09,Proposition VI.2.15]. (cid:3)
Recall that if A • → A is a tower in C /A with C an abelian category, there is a canonical descendingfiltration on A denoted by F k A = im( A k → A ) . C be a triangulated category with a t -structure. Fix X ∈ C . For every E ∈ C , we have anatural tower · · · → E ≥ → E ≥ → E ≥ → · · · → E. Applying [ X, − ] we obtain a tower in abelian groups, and hence a natural descending filtration on [ X, E ]which we denote by G • [ X, E ]. Given F ∈ C ♥ , we put H i ( X, F ) = [ X, Σ i F ]. Lemma 5.7.
Let α : E → F ∈ C induce the zero map H i ( X, π i E ) → H i ( X, π i F ) . Then α ( G i [ X, E ]) ⊂ G i +1 [ X, F ] . Proof.
An element of G i [ X, E ] is represented by a map X → E ≥ i . Its image in [ X, F ] is lands in G i +1 [ X, F ] if and only if the composite X → E ≥ i → E → F factors through F ≥ i +1 → F . For this it isenough that the composite X → E ≥ i → F ≥ i → Σ i π i ( F ) is zero. Since this factors as X → Σ i π i ( E ) → Σ i π i ( F ), the result follows. (cid:3) We now specialise this to C = SH ( k ), X (the suspension spectrum of) a smooth variety. Remark . Let X ∈ Sm k . Then G k [ X, E ] = 0 for k > dim X ; in fact [ X, E ≥ k ] = 0. This followsfrom the fact that the Nisnevich topos of X has homotopy dimension at most dim X (see e.g. [BH17,Proposition A.3(3)]). Corollary 5.9.
Let X ∈ Sm k be connected of dimension ≤ d . Let α : E → F ∈ SH ( k ) induce the zeromap on the generic stalk π i ( − ) ( k ( X )) , for i = 0 , , . . . , d . Then the map α ∗ : [ X, E ] → [ X, F ] increases filtration by (at least) .Proof. We first claim that α : H k ( X, π i ( E ) ) → H k ( X, π i ( F ) ) is the zero map, for k ≥ ≤ i ≤ d .Since Zariski and Nisnevich cohomology of the homotopy sheaves of E agree [Mor05b, Lemma 6.4.7],for this it suffices to show that α : π i ( E ) | X Zar → π i ( F ) | X Zar is the zero map. This follows fromunramifiedness of homotopy sheaves [Mor05b, Lemma 6.4.4] and our assumption on α ( k ( X )). We havethus proved the claim.Applying Lemma 5.7, we deduce that ( ∗ ) α ( G i [ X, E ]) ⊂ G i +1 [ X, F ] for 0 ≤ i ≤ d . Since Σ ∞ + X ∈SH ( k ) ≥ , for i < X, F ] ≃ [ X, F ≥ ] ≃ [ X, F ≥ i +1 ], and so ( ∗ ) still holds. For i > d we have G k [ X, E ] = 0 by Remark 5.8, and so again ( ∗ ) holds.This concludes the proof. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 37 Definition 5.10.
Let E • ∈ SH ( k ) be a tower.(1) We call E • locally nilpotent if for every connected X ∈ Sm k and i, j ∈ Z , the tower π i,j ( E • )( k ( X ))is nilpotent.(2) We call E • sectionwise nilpotent if for every X ∈ Sm k and i, j ∈ Z , the tower[Σ i,j X, E • ]is nilpotent. Proposition 5.11.
Let E • ∈ SH ( k ) be locally nilpotent. Then it is sectionwise nilpotent.Proof. Let X ∈ Sm k . It suffices to show that there exists N >
X, E N ] → [ X, E ] is thezero map. Writing X as the disjoint union of its (finitely many) connected components, and using that[ X ` X , E ] ≃ [ X , E ] ⊕ [ X , E ], we may assume that X is connected, say of dimension d . We claimthat for any i there exists N ( i ) > i such that [ X, E N ( i ) ] → [ X, E i ] increases postnikov filtration by (atleast) 1. Assuming this for now, if N = N ◦ ( d +1) (0), then the composite[ X, E N ] → [ X, E N ◦ d (0) ] → [ X, E N ◦ ( d − (0) ] → · · · → [ X, E ]increases postnikov filtration by d + 1, and hence is the zero map by Remark 5.8.It hence suffices to prove the claim; clearly we may assume that i = 0. The local nilpotence assumptionimplies that there exists N = N (0) > π i ( E N ) ( k ( X )) → π i ( E ) ( k ( X )) are allzero, for 0 ≤ i ≤ d . The claim now follows from Corollary 5.9.This concludes the proof. (cid:3) Corollary 5.12.
Let E • ∈ SH ( k ) be locally nilpotent. Then lim i E i ≃ .Proof. Immediate from Proposition 5.11 (which shows that π i map(Σ j,k X, E • ) is nilpotent) and Lemma5.6 (which implies that map(Σ j,k X, lim i E i ) ≃ lim i map(Σ j,k X, E i ) ≃ (cid:3) Proof of Proposition 5.2.
By [BEØ20, Theorem 5.3(1)] (using [BEØ20, Remark 5.12]) we know that f • ( E ) / ( t n , ρ m ) is locally nilpotent. The claim thus follows from Corollary 5.12. (cid:3) The HW -homology of kw6.1. Summary.
Throughout k is a field with vcd ( k ) < ∞ . Recall (see Lemmas 2.12, 2.10 and 2.8)that then W( k ) ∧ I ≃ W( k ) ∧ ≃ L ∧ W( k ) . See § , kw. The following result will be improved upon inProposition 7.7. Theorem 6.1.
Let vcd ( k ) < ∞ . We have π ∗ ((kw ∧ HW) ∧ ) ≃ ( W( k ) ∧ I ∗ = 4 i ≥ else . There exist generators t i ∈ π · i ((kw ∧ HW) ∧ ) and x i ∈ π i ((kw ∧ HW) ∧ ) such that the following hold.(1) Let k ′ /k be an extension with vcd ( k ′ ) < ∞ . Then x i | k ′ generates π i ((kw k ′ ∧ HW k ′ ) ∧ ) ≃ W( k ′ ) ∧ .(2) For i ≥ we have t i ∈ (2 + I( k ) ) t i +1 .(3) Writing i = P n ǫ n n for the binary expansion of i ≥ , up to a unit (of W( k ) ∧ I ) we have x i = Y n t ǫ n n . (4) x can be chosen to be .Remark . It follows that π ∗ ((kw ∧ HW) ∧ ) is a flat W( k ) ∧ -algebra generated by t , t , . . . , subject tothe relations t i = (2 + r i ) t i +1 for certain r i ∈ I( k ) .6.2. The motivic dual Steenrod algebra. mod motivic cohomology spectrum H Z /
2. It is a consequence of the solution ofthe Beilinson–Lichtenbaum and Milnor conjectures (see e.g. [KRØ18, (7.1)]) that π ∗ (H Z / ∗ ≃ k M ∗ [ τ ] . Here k M ∗ denotes the homotopy module of mod Milnor K -theory [Mor12, Example 3.33], and τ ∈ π (H Z / . In particular we have a cofiber sequence(6.1) Σ , − H Z / τ −→ H Z / → k M . τ i , ξ j ∈ H Z / ∗∗ H Z / , i = 0 , , , . . . , j = 1 , , . . . with | τ i | = (2 i +1 − , i −
1) and | ξ i | = (2 i +1 − , i − . Then, when viewed as a left H Z / (6.2) H Z / ∧ H Z / ≃ H Z / τ , τ , . . . , ξ , ξ , . . . ] / ( τ i − ( τ + ρτ ) ξ i +1 − ρτ i +1 ) . We may occasionally write ξ := 1. The switch map on H Z / ∧ H Z / Z / ∗∗ H Z / antipode which we denote by x x . Lemma 6.3. (1) The action of H Z / ∗∗ H Z / on H Z / ∗∗ is determined by Sq ( τ ) = ρ , Sq i ( x ) = 0 for x = τ and i > , or i ≥ and x = 1 or x ∈ k M ( k ) .(2) The right unit η R : H Z / ∗∗ → H Z / ∗∗ H Z / is given by η R ( τ ) = τ + ρτ and η R ( a ) = a , for a ∈ k M ∗ ( k ) .Proof. The right unit is a special case of a homology coaction, and so dual to the action of H Z / ∗∗ H Z / Z / ∗∗ : η R ( x ) = P I Sq I ( x ) · d Sq I . Thus (1) and (2) are (essentially) equivalent. The right unitis k M ∗ ( k )-linear (see e.g. Example 3.7 and use that π ( ) ∗ ≃ K MW ∗ → π (H Z / ≃ k M ∗ is surjective)and satisfies η R (1) = 1. Thus for x ∈ k M ∗ ( k ) we get η R ( x ) = x . For degree reasons, we must have η R ( τ ) = aτ + bτ + cξ , for some a ∈ F , b ∈ k M ( k ), c ∈ k M ( k ). This translates into Sq ( τ ) = aτ ,Sq ( τ ) = b and Sq ( τ ) = c (using [Voe03b, Lemmas 13.1 and 13.5]). Hence a = 1. We have c = 0 by[Voe03b, Lemma 9.9]; b = ρ is holds essentially by definition (using that Z (1)[1] = G m and consideringthe long exact sequence computing H ∗ ( k, Z / (cid:3) Corollary 6.4.
The monomials Y i,j τ iǫ i ξ jn j form a left H Z / ∗∗ -module basis of H Z / ∗∗ H Z / .Proof. Since the conjugates form a left basis, it is clear that these monomials form a right basis. Itsuffices to prove that elements of the form τ p m (where m is one of the monomials) form a k M ∗ -basis. ByLemma 6.3, elements of the form ( τ + ρτ ) p m form a k M ∗ -basis, where m is one of the monomials in theclaim. Order the monomials in the τ i , ξ i and τ lexicographically, with τ < τ i < ξ i < τ i +1 . The relationfrom (6.2) ensures that τ i is a sum of monomials > τ i ; this implies that if τ p m is any such monomial, τ τ p m is a sum of larger monomials. Since τ = τ (see again [HKØ, Theorem 5.6]), it follows that thematrix expressing the elements { ( τ + ρτ ) p m } p,m in terms of the basis { τ p m } p,m is triangular (with unitdiagonal). The result follows. (cid:3) Z / ∗∗ H Z / ≃ (H Z / ∗∗ H Z / ∗ (see e.g. [Hoy15,Proposition 5.5]). By passing to the dual of the monomial basis of H Z / ∗∗ H Z /
2, any monomial m ∈ H Z / p,q H Z / dual b m ∈ H Z / p,q H Z / Warning . We can extend this map H Z / ∗∗ -linearly, but the extension is not compatible with gradings:if a ∈ H Z / r,s then c am ∈ H Z / p − r,q − s H Z / am ∈ H Z / p + r,q + s H Z / in the naive sense that the right hand side has an H Z / Z / -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 39 Example . Since H Z / pq is concentrated in q >
0, except for H Z / , = Z /
2, we find that H Z / ∗ , H Z / F -vector space by 1 and b τ . In other wordsH Z / , H Z / F , H Z / , = F { b τ } and H Z / p, H Z / . Similarly H Z / ∗ , H Z / τ, k M , b ξ , b τ , τ b τ , k M b τ , d ξ τ , d τ τ , so thatH Z / , H Z / F { τ } , H Z / , H Z / k M ⊕ F { τ b τ } , H Z / , H Z / F { b ξ } ⊕ k M { b τ } , H Z / , H Z / F { b τ } ⊕ F { d τ ξ } , H Z / , H Z / F { d τ τ } , and H Z / p, H Z / . If α ∈ H Z / ∗∗ H Z / α L = α ∧ id , α R = id ∧ α : H Z / ∧ H Z / → H Z / ∧ H Z / α L ∗ , α R ∗ : H Z / ∗∗ H Z / → H Z / ∗∗ H Z / . Lemma 6.7. (1) For any α ∈ H Z / ∗∗ H Z / , the map α L ∗ is right H Z / ∗∗ -linear, and α R ∗ is left H Z / ∗∗ -linear(2) We have the formulas b τ L ∗ ( τ ) = 1 b τ L ∗ ( τ i ) = 0 , i > b τ L ∗ ( ξ i ) = 0 , i ≥ b τ L ∗ ( τ ) = 1 b τ L ∗ ( τ i ) = 0 , i = 1 b τ L ∗ ( ξ i ) = 0 , i ≥ b ξ L ∗ ( τ ) = τ b ξ L ∗ ( τ i ) = 0 , i = 1 b ξ L ∗ ( ξ ) = 1 b ξ L ∗ ( ξ i ) = 0 , i = 1 b τ R ∗ ( τ i ) = ξ i , i ≥ b τ R ∗ ( ξ i ) = 0 , i ≥ , and b ξ R ∗ ( τ i ) = 0 , i ≥ b ξ R ∗ ( ξ i ) = ξ i − , i ≥ . (Here in the last formula ξ − := 0 .)(3) b τ L ∗ is a derivation (i.e. satisfies b τ L ∗ ( ab ) = b τ L ∗ ( a ) b + a b τ L ∗ ( b ) ), b τ L ∗ is a derivation on the right H Z / ∗∗ -subalgebra on all the generators except τ , and b ξ L ∗ is a derivation on the right H Z / ∗∗ -subalgebra on all the generators except τ , τ .Proof. (1) Immediate from the definitions.(2) For E ∈ SH ( k ) we have an actionH Z / ∗∗ H Z / ⊗ H Z / ∗∗ H Z / ∗∗ E → H Z / ∗∗ E, α ⊗ e α ∗ ( e ) . By the eightfold way [Boa82, p. 190], this is obtained from the coaction∆ : H Z / ∗∗ E → H Z / ∗∗ H Z / ⊗ H Z / ∗∗ H Z / ∗∗ E by partial dualization: if ∆( e ) = P i a i ⊗ e i then α ∗ ( e ) = P i h a i , α i e i . Applying this with E = H Z /
2, itfollows that for α ∈ H Z / ∗∗ H Z / x ∈ H Z / ∗∗ H Z / x ) = P x i ⊗ y i , we have α L ∗ ( x ) = X i h x i , α i y i , and similarly α R ∗ ( x ) = X i h y i , α i x i . The formulas now follow from the formulas for the comultiplication in H Z / ∗∗ H Z / Z / g be one of the monomialgenerators of H Z / ∗∗ H Z / a, b ∈ H Z / ∗∗ H Z /
2. Suppose we can write∆( a ) = 1 ⊗ a + g ⊗ a ′ + X a i ⊗ a ′ i , ∆( b ) = 1 ⊗ b + g ⊗ b ′ + X b i ⊗ b ′ i , where the a i , b i are monomials. Suppose further that when expanding a i b j into monomials, g does notappear. Then b g L ∗ ( a ) = a ′ , b g L ∗ ( b ) = b ′ and∆( ab ) = g ⊗ ( ab ′ + a ′ b ) + . . . , where g does not appear on the left in the omitted terms. It follows that b g L ∗ ( ab ) = ab ′ + a ′ b , which iswhat we wanted. ∆( a ) , ∆( b ) can always be written in the desired form, the only problem may occurwhen expanding the product. As long as g itself is not a product, the only way the assumption can failis from an “unexpected” contribution, i.e. coming from τ i = ( τ + ρτ ) ξ i +1 + ρτ i +1 . It thus suffices toensure that ∆( a ) , ∆( b ) do not contain τ i on the left hand side, for 0 ≤ i < N for certain N depending on g . This will happen if τ i for i < N are excluded as generators (considering again the explicit formulasfor the comultiplication). One checks that for g = τ , τ , ξ , respectively N = 0 , , (cid:3) Spectra employed in the proof. f KO for the spectrum of very effective hermitian K -theory . We also put ku = f KGL ≃ ˜ f KGL. Note that we have canonical ring maps ko → ku → s (KGL) ≃ H Z . Proposition 6.8.
The canonical map H Z / ∗∗ ko → H Z / ∗∗ H Z / is injective and hits the elements ξ , ξ , . . . , τ , τ , . . . . The resulting map H Z / ∧ ko ← H Z / ξ , ξ , . . . , τ , τ , . . . ] / ( . . . ) ֒ → H Z / ∧ H Z / is an algebra isomorphism.Similarly H Z / ∗∗ kgl → H Z / ∗∗ H Z / is injective and H Z / ∧ kgl ≃ H Z / ξ , ξ , . . . , τ , τ , . . . ] / ( . . . ) ֒ → H Z / ∧ H Z / . Also H Z / ∧ H Z ≃ H Z / ξ , ξ , . . . , τ , τ , τ , . . . ] / ( . . . ) ֒ → H Z / ∧ H Z / . Proof.
Once we have produced the maps in question, to show they are equivalences it suffices to showisomorphisms on π ∗∗ over any field extension. Since all the maps and spectra are stable under basechange, we thus need only prove the assertions on the level of π ∗∗ .This is essentially contained in [ARØ17]. We first explain the case of H Z . We have the cofiber sequenceH Z −→ H Z p −→ H Z / ∂ −→ ΣH Z . The map id H Z / ∧ H Z is given by 2 = 0 (since 2 ∈ π ∗∗ ( )), hence after smashing with H Z / Z / ∗∗ H Z / ≃ H Z / ∗∗ H Z ⊕ C. Putting δ = p ◦ ∂ , we find that H Z / ∗∗ H Z = ker( δ R ∗ ). One knows that δ = b τ corresponds to theBockstein Sq . Using Lemma 6.7 we find that δ L ∗ vanishes on the right H Z / ∗∗ -algebra M generated by τ , τ , . . . , ξ , ξ , . . . and does not vanish in τ M . Dualizing, we find that δ R ∗ vanishes on M and does notvanish on τ M (these are now left H Z / ∗∗ -modules). By Lemma 6.4 we have H Z / ∗∗ H Z / M ⊕ τ M .It follows that ker( δ R ∗ ) = M , as desired.The argument kgl is essentially the same, using Σ , kgl β KGL −−−→ kgl → H Z . It is not immediatelyapparent that β KGL : H Z / ∧ kgl → H Z / ∧ kgl is the zero map, since β KGL π ∗∗ ( ). We can argue asfollows. Over the ring π ∗∗ (H Z / ∧ kgl) the formal group laws x + y + β KGL xy and x + y are isomorphic;in particular the former must have infinite height [Rav86, Lemma A2.2.9]. Thus 0 = [2] kgl ( x ) = β KGL x ,whence β KGL is zero in π ∗∗ (H Z / ∧ kgl). Continuing with the above argument, this time it turns outthat δ = b τ [ARØ17, Lemma 2.9]; the rest of the argument goes through as before.For ko the same argument works, using Σ , ko η −→ ko → kgl. Since η ∈ π ∗∗ ( ) it is immediate that itacts by 0 on H Z / ∧ ko. The boundary map δ turns out to be b ξ [ARØ17, Lemma 2.12] [Voe03b, Lemma13.1]. The rest goes through as before. (cid:3) η − ] the Witt theory spectrum . As the name suggest, it representsBalmer–Witt theory [Hor05]. We put kw = KW ≥ . The image of the Bott element β ∈ π , KO yieldsan element β ∈ π KW = π kw and we have [Bal05, Theorem 1.5.22] π ∗ (KW) ≃ W [ β ± ] and π ∗ (kw) ≃ W [ β ] . Lemma 6.9.
The canonical map ko → KO → KW induces an equivalence ko[ η − ] ≃ kw .Proof. Since ko ∈ SH ( k ) ≥ , the map ko → KW indeed factors through kw. We have π ∗ (ko[ η − ]) ≃ colim h π ∗ (ko) η −→ π ∗ (ko) − η −→ . . . i . Since π ∗ (ko) − n ≃ π ∗ (KO) − n for ∗ , n ≥ π ∗ (ko) − n = 0 for ∗ <
0, the result follows. (cid:3)
We put HW = [ η − ] ≤ ; in other words this is just the homotopy module W [ η ± ]. The unit mapinduces(6.3) HW ≃ kw ≤ ≃ kw /β. Warning . In other works HW would perhaps have been denoted K W [ η − ] or W [ η ± ] (and HW wouldhave denoted something else). Since this object is so central for our work, we reserve the prominentnotation. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 41 f HW. One may show (e.g. see [Bac17, Theorem 17]) that π ( f HW) ∗ ≃ K W ∗ is the homotopy module of Witt K -theory ; in other words K W ∗ = I ∗ (and multiplication by η inducesthe inclusion I ∗ +1 ֒ → I ∗ ; for ∗ < I ∗ = W ). There is a canonical map K W ∗ → k M [Mor04b]; bythe resolution of the Milnor conjecture [Voe03a, OVV07, Mor05a] this induces a cofiber sequence(6.4) Σ , K W η −→ K W → k M . Taking effective covers, we obtain a map f HW → f k M ≃ H Z / f HW −−−−→ K W y y H Z / −−−−→ k M is cartesian [Bac17, Theorem 17]. Lemma 6.11.
The map τ : Σ , − H Z / → H Z / lifts uniquely (up to homotopy) to a map ˜ τ : Σ , − H Z / → f HW , and cof(˜ τ ) ≃ K W . Proof.
Since Σ , − H Z / ≃ Σ G ∧− m ∧ H Z / ∈ SH ( k ) ≥ and K W , k M ∈ SH ( k ) ≤ , the long exact sequencefor [Σ , − H Z / , − ] shows that there is a unique lift as claimed. We can view this as a morphism fromthe bicartesian square Σ , − H Z / −−−−→ id y y Σ , − H Z / −−−−→ τ ) −−−−→ K W y y cof( τ ) −−−−→ k M . The bottom horizontal map is an equivalence by (6.1), hence so is the top one. This was to be shown. (cid:3)
Corollary 6.12.
We have K W ∈ Σ , − SH ( k ) veff .Proof. Since SH ( k ) veff is closed under colimits (and H Z / , f HW ∈ SH ( k ) veff ), this is immediate fromLemma 6.11. (cid:3) Determination of π ∗ (kw ∧ HW) ∧ . E ∈ SH ( k ), consider the η -multiplication tower(6.6) · · · → Σ , E η −→ Σ , E η −→ E. Functorially associated with this is a spectral sequence (the η -Bockstein spectral sequence) with [Lur16,beginning of § E p,q,w = π p + q (cof( η : Σ − p +1 , − p +1 E → Σ − p, − p E )) w , p ≤ d r : E p,q,wr → E p − r,q + r − ,wr +1 π p + q ( E ) w . Here by the last line we mean that the E ∞ -page in position ( p, q, w ) is is related to π p + q ( E ) w (but weare not claiming any kind of convergence). Remark . Since the tower (6.6) is compatible with base change, so is the associated spectral sequence(6.7).The boundary map in the cofiber sequenceΣ , E η −→ E p −→ E/η ∂ −→ Σ , E induces the Bockstein δ = p∂ : E/η → Σ , E/η.
By construction, δ = 0 and so δ ∗ gives π ∗ ( E/η ) ∗ the structure of a chain complex. We write itshomology (respectively cycles, respectively the entire complex) in spot corresponding to π a ( E/η ) b as H a ( π ∗ ( E/η ) ∗ , δ ∗ ) b (respectively Z a ( π ∗ ( E/η ) ∗ , δ ∗ ) b , C a ( π ∗ ( E/η ) ∗ , δ ∗ ) b ). Recall the notion of conditionalconvergence from [Boa99, Definition 5.10]. Lemma 6.14.
By suitably re-indexing the spectral sequence (6.7) we obtain a conditionally convergentspectral sequence E s,f,w = π s ( E/η ) w + f ⇒ π s ( E ∧ η ) w d r : E s,f,wr → E s − ,f + r,wr . Here E s,f,w = 0 for f < . We have E s, ,w = Z s ( π ∗ ( E/η ) ∗ , δ ∗ ) w E s,f,w = H s ( π ∗ ( E/η ) ∗ , δ ∗ ) f + w , f > . Proof.
We have cof( η : Σ − p +1 , − p +1 E → Σ − p, − p E ) ≃ Σ − p, − p E/η and hence E p,q,w ≃ π p + q ( E/η ) w − p .The re-indexing is obtained by putting s = p + q and f = − p .By Lemma 6.16 below, the spectral sequence converges conditionally tocof(lim p Σ p,p E → E ) ≃ lim p cof(Σ p,p E η p −→ E ) ≃ lim p E/η p ≃ E ∧ η . By construction, the d -differentials are induced by δ ∗ , whence the identification of the E -page. (cid:3) Remark . If E → E/η is an E ∞ -ring map, then the spectral sequence above can be identified withthe descent spectral sequence for this map. In particular, it is multiplicative. Furthermore F is an E -module, then the spectral sequence for F is a module over the one for E .In the proof of Lemma 6.14, we have made use of the following well-known fact. Let E • : Z op → SH be a tower of spectra. Then as above there is an associated spectral sequence E p,qn ( E • ). Lemma 6.16.
The spectral sequence E p,qn ( E • ) converges conditionally to cof(lim E • → colim E • ) .Proof. Let E ′ p = cof(lim E • → E p ). Then there is a morphism of towers E • → E ′• inducing a morphismof spectral sequences E p,qn ( E • ) → E p,qn ( E ′• ). By construction, this induces an isomorphism on the E -page, and hence on all following pages. Noting that lim E ′• ≃ E ′• ≃ cof(lim E • → colim E • ),we may replace E • by E ′• , and so assume that lim E • ≃
0. Conditional convergence to the colimit meansby definition [Boa99, Definition 5.10] thatlim π i ( E • ) ≃ ≃ lim π i ( E • ) , for i ∈ Z . By the Milnor exact sequence [GJ09, Proposition VI.2.15], this follows from (and is in fact equivalentto) lim E • ≃ (cid:3) K W → K W /η ≃ k M and G = E ∧ K W , we obtain the spectral sequence(6.8) E ( G ) ∗ , ∗ , ∗ = C ( π ∗ ( E ∧ k M ) ∗ , δ ∗ )[ h ] ⇒ π ∗ (( E ∧ K W ) ∧ η ) ∗ E ( G ) ∗ , ∗ , ∗ = Z ( π ∗ ( E ∧ k M ) ∗ , δ ∗ )[ h ] /h · im( δ ∗ ) . Here h = 1 ∈ E ( K W ) , , − ≃ k M ( k ) , and we have used the module structure to act with h ∈ E ( K W ) on E ( G ). Example . Taking E = , we get E ( G ) s,f,w = 0 unless s = 0, so the spectral sequence collapsesat E . The spectral sequence converges to π (( K W ) ∧ η ) ∗ = ( K W ( k ) ∗ ) ∧ I ; on E = E ∞ we see the I-adicfiltration (as we must) with subquotients given by k M ( k ) ∗ . The element h detects η ∈ K W ( k ) − . Inparticular h is a permanent cycle.Via the spectral sequence, we obtain a filtration F • π ∗ ( G ∧ η ) ∗ on the bigraded group π ∗ ( G ∧ η ) ∗ . Multi-plication by η induces a map π ∗ ( G ∧ η ) ∗ → π ∗ ( G ∧ η ) ∗− which maps F • to F • +1 and on associated gradedcorresponds to multiplication by h . Taking the colimit we obtain a filtration on π ∗ ( G ∧ η ) ∗ [ η − ] = colim i π ∗ ( G ∧ η ) ∗− i ≃ π ∗ ( G ∧ η [ η − ]) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 43 with(6.9) F • π ∗ ( G ∧ η [ η − ]) ∗ = colim i F • + i π ∗ ( G ∧ η ) ∗− i and associated graded gr • π ∗ ( G ∧ η [ η − ]) ∗ ≃ E ∞ ( G ) ∗ , • , ∗ [ h − ] . Note that even though the filtration F • π ∗ ( G ∧ η ) ∗ terminates at F (i.e. F π ∗ ( G ∧ η ) ∗ = π ∗ ( G ∧ η ) ∗ ) this neednot be the case for the induced filtration on π ∗ ( G ∧ η [ η − ]) ∗ : we could well have F ( F − ( F − ( . . . . Note also that we are not making any claim about the completeness etc. of the filtrations.6.4.3. We may wish to apply spectral sequence (6.8) with E = H Z /
2. To begin with, using that(H Z / /τ ≃ k M (see (6.1)), the form of H Z / ∗∗ H Z / η R ( τ ) = τ + ρτ (Lemma 6.3) we find that(6.10) π ∗∗ (H Z / ∧ k M ) ≃ k M ∗ ( k )[ τ , τ , . . . , ξ , ξ , . . . ] / ( τ i − ρτ i +1 ) . Now we determine the differential.
Lemma 6.18.
The action of δ ∗ on π ∗∗ (H Z / ∧ k M ) satisfies δ ∗ ( τ i ) = 0 and δ ∗ ( ξ i ) = ξ i − Proof.
We claim that there exists a commutative square k M δ −−−−→ Σ , k M x x H Z / ˜ δ −−−−→ Σ , H Z / , where the vertical maps are the canonical projections. Indeed using the cofiber sequenceΣ H Z / τ −→ Σ , H Z / → Σ , k M → Σ H Z / Z / , Σ H Z /
2] = 0, which follows from the form of the motivic Steenrod algebra(see Example 6.6). We have (again by Example 6.6)[H Z / , Σ , H Z / ≃ F { b ξ } ⊕ k M ( k ) { b τ } , so that ˜ δ = a b ξ + b b τ , for some a ∈ F and b ∈ k M ( k ). Comparison with Lemma 6.7 (and noting thatwe are looking at ˜ δ R ∗ ) yields δ ∗ ( τ i ) = bξ i and δ ∗ ( ξ i ) = aξ i − . Since η = 0 on H Z /
2, smashing the cofiber sequence for K W /η ≃ k M with H Z / π ∗∗ (H Z / ∧ k M ) ≃ π ∗∗ (H Z / ∧ K W ) ⊕ π ∗∗ (Σ , H Z / ∧ K W ) , with the first summand given by ker( δ ∗ ). Since H Z / ∧ k M = 0 (e.g. π ( − ) ∗ = k M ∗ ) we find that neitherof the (isomorphic) summands can be trivial, and so δ ∗ = 0. After base change to an algebraically closedfield we get k M (¯ k ) = 0, so that the only way to get δ ∗ = 0 is a = 1. Now we compute δ ∗ ( δ ∗ ( τ )) = δ ∗ ( bξ ) = b. But δ = 0, so that b = 0. The result follows. (cid:3) E = ko. Recall that we have determined π ∗∗ (H Z / ∧ k M )in (6.10). Lemma 6.19.
The map ko → H Z / induces an isomorphism π ∗∗ (ko ∧ k M ) ≃ k M ∗ ( k )[ ξ , ξ , . . . , τ , τ , . . . ] / ( τ i − ρτ i +1 ) ֒ → π ∗∗ (H Z / ∧ k M ) . The homology of δ ∗ acting on this is given by k M ∗ ( k )[ τ , τ , . . . ] / ( τ i − ρτ i +1 ) . Proof.
Applying antipodes in Lemma 6.8, we find that π ∗∗ (ko ∧ H Z / ֒ → π ∗∗ (H Z / ∧ H Z /
2) is the right H Z / ∗∗ -algebra generated by ξ , ξ , . . . , τ , τ , . . . . Since these generators form part of a right H Z / ∗∗ -basis (see Corollary 6.4), multiplication by η R ( τ ) is injective and we obtain π ∗∗ (ko ∧ k M ) as the quotient.This proves the first claim.It follows from Remark 6.15 that δ ∗ is a derivation, which by Lemma 6.18 satisfies δ ∗ ( τ i ) = 0, δ ∗ ( ξ i ) = ξ i − (and also δ ∗ ( k M ∗ ( k )) = 0, since these elements come from the sphere). This implies thatthe homology of δ ∗ is given by k M ∗ ( k )[ τ , τ , . . . ] / ( τ i − ρτ i +1 ) ⊗ H ′ , where H ′ is the homology of δ ∗ restricted to the subring F [ ξ , ξ , . . . ] . We can determine this as follows, adapting [Ada95, Proof of § F i ⊂ F [ ξ , ξ , . . . ] bethe F -vector space with basis { ξ ni , ξ ni ξ i +1 } n ≥ . Then δ ∗ ( F i ) ⊂ F i and H ∗ ( F i , δ ) = F { } . Since weare working over a field, H ∗ O i F i ! ≃ O i H ∗ ( F i ) ≃ F { } . It thus remains to observe that N i F i = F [ ξ , ξ , . . . ]; equivalently every monomial (in which ξ occursto even power) can be written uniquely as a product Q i m i , with m i one of the basis elements of F i .This is easily checked. (cid:3) Lemma 6.20.
The spectral sequence E ∗ (ko ∧ K W ) ∗ , ∗ , ∗ collapses at E . In other words, there are no further differentials, and E = E ∞ . In particular the spectral sequenceconverges strongly [Boa99, Theorem 7.1]. Proof.
We have E (ko ∧ k W ) = Z ( δ ∗ )[ h ] /h · im( δ ∗ ) and so in particular E (ko ∧ K W ) ∗ ,f> , ∗ = h · H ( δ ∗ )[ h ].Recall that the generator τ i has bidegree (2 i +1 − , i − τ i ∈ π i (H Z / ∧ H Z / − i . Thus τ i defines an element with s = 2 i and w = 1 − i (and f = 0) in our spectral sequence. Similarly k M ∗ ( k )yields elements with s = 0 , w = ∗ , f = 0. Since i ≥
2, it follows that E (ko ∧ K W ) ∗ ,f> , ∗ is concentratedin stems s ≡ s by 1, we find that any differential emanating frompositive filtration vanishes.Now we prove by induction on r that all differentials vanish on E r , starting with r = 2. By theinduction hypothesis we have E r = E . Let d r ( x ) = y be any differential. Then d r ( hx ) = hd r ( x ) = hy is another differential, h being a permanent cycle (see Example 6.17). Since hx has positive filtration f (since f ( x ) ≥ f ( h ) = 1), by the above we have hy = 0. But multiplication by h is injective on E ∗ ,f> , ∗ = E ∗ ,f> , ∗ r and f ( y ) >
0, so y = 0. This was to be shown. (cid:3) Now we invert η to obtain a new filtration, as in (6.9). Lemma 6.21.
The filtration F • π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ is complete, Hausdorff and exhaustive.Proof. By strong convergence of the spectral sequence, the filtration F • π ∗ (ko ∧ K W ) ∧ η ) ∗ is complete,Hausdorff and exhaustive. It is clear that exhaustive filtrations are stable under colimits. We thus needto show completeness and Hausdorffness, or in other words that R lim i F i π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ ≃ . We claim that for i > η : F i π ∗ ((ko ∧ K W ) η ) ∗ → F i +1 π ∗ ((ko ∧ K W ) η ) ∗− is an isomorphism. Indeed giving F i π ∗ ((ko ∧ K W ) η ) ∗ the obvious induced filtration, this becomes amorphism of complete, Hausdorff, exhaustively filtered groups inducing an isomorphism on associatedgradeds (given by h : h i H ( δ ∗ )[ h ] → h i +1 H ( δ ∗ )[ h ]), so this follows from Lemma 2.2. Hence for i > F i π ∗ ((ko ∧ K W ) η ) ∗ → F i π ∗ ((ko ∧ K W ) η [ η − ]) ∗ is an isomorphism Thus0 ≃ R lim i F i π ∗ ((ko ∧ K W ) ∧ η ) ∗ ≃ −→ R lim i F i π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ , and the result follows. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 45 Proof of Theorem 6.1.
We first compute π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ . This is a filtered graded moduleover π ∗ (( K W ) ∧ η [ η − ]) ∗ ≃ W( k ) ∧ I [ η ± ] . Here a priori W( k ) ∧ I must mean the derived I-adic completion, but since vcd ( k ) < ∞ the I-adic and 2-adic completion agree (by Lemma 2.12), and the ordinary completion agrees with the derived completion(by Lemmas 2.10 and 2.8). Note that gr • π ∗ (( K W ) ∧ η [ η − ]) ∗ = k M ∗ ( k )[ h ± ]. Put u i − = h − i τ i ; we have( s, f, w )( u i ) = 4(2 i , − i , n = P i ǫ i i put y n = Y i u ǫ i i . Observe thatgr • π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ ≃ k M ∗ ( k )[ h ± , τ , τ , . . . ] / ( τ i − ρτ i +1 ) ≃ k M ∗ ( k )[ h ± ] { y , y , . . . } . This is a free module on k M ∗ [ h ± ] with at most one generator in every degree, and all our filtrations arecomplete, Hausdorff and exhaustive, so by Corollary 2.3 we get π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ∗ ≃ W( k ) ∧ I [ η ± ] { x , x , . . . } , where x i is a lift of y i .It follows from Corollary 6.12 and Theorem 5.1 that (ko ∧ K W ) ∧ ≃ (ko ∧ K W ) ∧ ,η and hence((ko ∧ K W ) ∧ η [ η − ]) ∧ ≃ ((ko ∧ K W ) ∧ ,η [ η − ]) ∧ ≃ ((ko ∧ K W ) ∧ [ η − ]) ∧ ≃ (ko ∧ K W [ η − ]) ∧ . Since ko[ η − ] ≃ kw (Lemma 6.9) and K W [ η − ] ≃ HW, we deduce that(kw ∧ HW) ∧ ≃ ((ko ∧ K W ) ∧ η [ η − ]) ∧ . Note that the W( k ) ∧ I is derived 2-complete, and hence all of π ∗ ((ko ∧ K W ) ∧ η [ η − ]) is. In other words π ∗ ((ko ∧ K W ) ∧ η [ η − ]) ≃ π ∗ (((ko ∧ K W ) ∧ η [ η − ]) ∧ )(by Lemma 2.14). We have thus computed π ∗ ((HW ∧ kw) ∧ ) ≃ W( k ) ∧ I { x , x , . . . } . (1) Since our spectral sequence as well as the motivic dual Steenrod algebra are stable under basechange (Remark 6.13), so are the y i . Corollary 2.3 shows that any set of lifts will generate.(2) Let t i be a lift of u i . Then t i is detected by u i = u i +1 hρ . Since hρ = − η [ −
1] = − ( h− i −
1) = − ∈ W( k )we have t i = − t i +1 + (higher filtration) = 2 t i +1 + (higher filtration) , whence the claim.(3) True by construction.(5) Clearly 1 lifts y = 1, whence the claim.This concludes the proof. 7. Main theorem
Lemma 7.1.
The unit map u : [ η − ] → kw is -connected: cof( u ) ∈ SH ( k ) ≥ .Proof. Immediate from examination of the homotopy sheaves (see § § (cid:3) Corollary 7.2. (1) Let ψ : kw (2) → kw (2) be any map such that ψ (1) = 1 . Then in the followingcommutative diagram, the dotted arrow can be filled uniquely up to homotopy. Σ kw (2) kw (2) kw (2) βψ − ϕ (2) Let ϕ : kw (2) → Σ kw (2) be any map. Then in the following commutative diagram, the dottedarrow can be filled uniquely up to homotopy. [ η − ] (2) fib( ϕ ) kw (2) Σ kw (2) u ϕ Proof.
Put F = fib( ϕ ). Note that ( ∗ ) by Lemma 7.1, if E ∈ SH ( k ) ≤ then [kw , E ] ≃ [ , E ]. We havecof( β : Σ kw (2) → kw (2) ) ≃ HW (2) (see (6.3)); hence ϕ exists if (a) kw ψ − −−−→ kw (2) → HW (2) is zero,and is unique if (b) [kw , Σ − HW (2) ] = 0. The factorization → F exists if (c) [ , Σ kw (2) ] = 0 and isunique if (d) [ , Σ kw (2) ] = 0.We have HW (2) ∈ SH ( k ) ≤ , whence by ( ∗ ) for (a) it suffices to show that −→ kw ψ − −−−→ kw (2) → HW (2) is zero, which holds since ψ (1) = 1 by assumption. For (b), again by ( ∗ ) it is enough to show that[ , Σ − HW (2) ] = 0. This is clear since Σ − HW (2) ∈ SH ( k ) < . (c) and (d) are immediate. (cid:3) We apply Corollary 7.2 to the map ψ = ψ constructed in Remark 3.6 to obtain ϕ : kw (2) → Σ kw (2) .From now on, this is the only map we will denote by ϕ . Remark . The defining property of ϕ implies that for E ∈ SH ( k ) (2) and a ∈ kw ∗ ( E ) we get βϕ ( a ) = ψ ( a ) − a . In particular, if kw ∗ E is β -torsion free, then(7.1) ϕ ( a ) = ( ψ ( a ) − a ) /β. Specializing even further, assume that 2 = 0 ∈ W( k ). Then ψ ( β ) = 9 β = β (by Theorem 3.1(2)) andhence(7.2) ϕ ( βx ) = βϕ ( x ) . Remark . Arguing as in Example 3.7, we see that ϕ : kw ∗ E → kw ∗− E is W( k ) (2) -linear. Example . Since kw ∗ ≃ W( k )[ β ] is β -torsion free, we deduce from Theorem 3.1(2) that ϕ ( β n ) = ( ψ ( β n ) − β n ) /β = ( ψ ( β ) n − β n ) /β = (9 n − β n − . Recall from Theorem 4.1(2) (and Example 4.2) thatkw ∗ MSL ≃ kw ∗ [ p , p , p , . . . ] / ( p − . Lemma 7.6.
Suppose that W( k ) ≃ F . Consider the action of ϕ on kw ∗ MSL .(1) We have ϕ ( p i ) ∈ p i − + β kw ∗ MSL .(2) We have ϕ ◦ i ( p i ) = 1 .Proof. Since kw ∗ MSL ≃ kw ∗ [ p , . . . ] is β -torsion free, by (7.1) and Proposition 4.3 we get ϕ ( p i ) ∈ β − ( p i + βp i − + β kw ∗ MSL − p i ) = p i − + β kw ∗ MSL , whence (1). By (7.2), ϕ commutes with β , and hence by iteration we find that ϕ ◦ i ( p i ) = 1 + a i β , forsome a i ∈ kw − MSL. (2) follows since kw − MSL = 0. (cid:3)
Proposition 7.7.
Consider the maps π ∗ (kw ∧ MSL) α −→ π ∗ (kw ∧ kw (2) ) r −→ π ∗ (kw ∧ HW (2) ) . Put ˜ x i = α ( p i ) and x i = r ˜ x i .(1) The canonical maps induce equivalences (of right modules) kw ∧ kw (2) ≃ _ i kw (2) { ˜ x i } and kw ∧ HW (2) ≃ _ i HW (2) { x i } . (2) x ∈ π i (kw ∧ HW (2) ) ≃ W( k ) (2) is a generator if and only if ϕ ◦ i ( x ) generates π (kw ∧ HW (2) ) . -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 47 Proof. (1) We first prove the claim about kw ∧ HW. By stability under base change, we may assumethat vcd ( k ) < ∞ . Since the field is arbitrary, it suffices to show that the map induces an isomorphismon π ∗ . This we can check separately after inverting 2 and after completing at 2 (see Lemma 2.16). Notethat (2.2) implies that [ η − ] ⊗ Q ≃ HW ⊗ Q and hence (kw ∧ HW) ⊗ Q ≃ kw ⊗ Q , and so π ∗ (kw ∧ HW) ⊗ Q ≃ ( W( k ) ⊗ Q ∗ = 4 n ≥
00 else . Similarly by Theorem 6.1 we have π ∗ ((kw ∧ HW) ∧ ) ≃ ( W( k ) ∧ I ∗ = 4 i ≥
00 else . We thus need to show that the image of p i in π i (kw ∧ HW) ⊗ Q ≃ W( k ) ⊗ Q and π i ((kw ∧ HW) ∧ ) ≃ W( k ) ∧ I is a unit. We first deal with the 2-complete case, in which it suffices to show that the image of p i is aunit modulo I [Sta18, Tag 05GI]. Since W( k ) / I = F is independent of k , we may reduce to the casewhere k is quadratically closed and hence W( k ) = F ; it hence suffices to show that the image y i of p i is non-zero. But ϕ ◦ i ( y i ) is the image of ϕ ◦ i ( p i ) = 1 (by Lemma 7.6). Since 1 = 0 ∈ π (kw ∧ HW) ∧ wehave ϕ ◦ i ( y i ) = 0 and so y i = 0.In the rational case we are dealing with ρ -periodic spectra, so we may reduce to the case where k isreal closed and hence W( k ) = Z (see § p i is non-zero inW( k ) ⊗ Q ≃ Q . Consider the commutative diagram (“fracture square”)(kw ∧ HW) (2) −−−−→ (kw ∧ HW) ⊗ Q y y (kw ∧ HW) ∧ −−−−→ ((kw ∧ HW) ∧ ) ⊗ Q . On applying π i , we obtain a commutative diagram π i (kw ∧ HW) (2) −−−−→ π i (kw ∧ HW) ⊗ Q y y Z ∧ ι −−−−→ Z ∧ [1 / . Since Z ∧ is torsion-free the map ι is injective. p i maps to a generator in the lower left hand corner, hencehas non-zero image in the lower right hand corner. It follows that it must also have non-zero image inthe upper right hand corner. This completes the proof for kw ∧ HW.To treat kw ∧ kw, we note that we have built a map γ : W n ≥ Σ n kw (2) → kw ∧ kw (2) of connectiveobjects in kw (2) - M od. The extension of scalars functor kw (2) - M od → HW (2) - M od is conservative onbounded below objects (e.g. by [Bac18b, Lemma 29] and [Bac18c, Corollary 4]). But γ ⊗ kw (2) HW (2) : _ n ≥ Σ n HW (2) → kw ∧ HW (2) is just the equivalence constructed above. The result follows.(2) We have x = ax i for some a ∈ W( k ) (2) . We need to show that a is a unit if and only if ϕ ◦ i ( x ) isa generator. Since W( k ) (2) is a local ring (Lemma 2.13), we may check this modulo I, and hence basechange to a quadratic closure. We may thus assume that W( k ) = F . Now by construction (i.e. Lemma7.6(2)) we have ϕ ◦ i ( x i ) = 1, and hence ϕ ◦ i ( x ) = a generates π if and only if a is a unit, if and only if x generates π i . (cid:3) Theorem 7.8.
Let k be any field of characteristic = 2 . Denote by ϕ the map obtained via Corollary7.2(1) from the map ψ = ψ constructed in Remark 3.6. Then the canonical map [ η − ] (2) → fib( ϕ ) obtained via Corollary 7.2(2) is an equivalence. In other words, there is fiber sequence [ η − ] (2) → kw (2) → Σ kw (2) . Proof.
Write F = fib( ϕ ). The “extension of scalars” functor SH ( k )[ η − ] (2) → HW (2) - M od is conserva-tive on bounded below objects (e.g. by [Bac18b, Lemma 29] and [Bac18c, Corollary 4]). It consequentlysuffices to show that HW (2) → F ∧ HW is an equivalence. Since the field is arbitrary, it is enough toshow that we have an isomorphism on π ∗ . Considering the long exact sequence for π ∗ F and Proposition7.7, it suffices to show that ϕ : π i (kw ∧ HW (2) ) → π i − (kw ∧ HW (2) ) is an isomorphism for i >
0. Inother words we need to show that ϕ ( x i ) generates π i − as a W( k ) (2) -module. By Proposition 7.7(2) this holds if and only if ϕ ◦ ( i − ( ϕ ( x i )) generates π . Since x i generates π i , this is indeed the case, againby Proposition 7.7(2). (cid:3) Applications
Homotopy groups of the η -periodic sphere. Determination of the groups π ∗ ( [ η − ])( k ) (orsome completed or localized variants), for various fields k , has been pursued by various authors over theyears. The case k = C was first to be approached, by Guillou–Isaksen [GI15]. They resolved it up toa conjecture about the classical Adams–Novikov spectral sequence, which was subsequently proved byAndrews–Miller [AM17]. The cases k = R and k = Q (both up to 2-adic completion) were done byGuillou–Isaksen [GI16] and Wilson [Wil18], respectively. All of these authors use the motivic Adamsspectral sequence. In contrast, Ormsby–R¨ondigs [OR19] use the slice spectral sequence; this allows themto treat all fields of (characteristic = 2 and) finite cohomological dimension in which − = 2. Theorem 8.1.
Let k be a field, char( k ) = 2 . We have π ∗ ( k [ 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. The cases ∗ ≤ π ∗ [ η − ] ⊗ Q ≃ W( k ) ⊗ Q (see (2.2)), and hence π ∗ [ η − ] is torsion for ∗ >
0. Thus (for ∗ > π ∗ [ η − ] ≃ π ∗ [1 /η, / ⊕ π ∗ [ η − ] (2) . We first show that π ∗ [1 /η, / ≃ W [1 / ⊗ π s ∗ . By Corollary 2.18 and § π ∗ ( [1 /η − , / K ) ≃ C (Sper( K ) , Z ) ⊗ π s ∗ ⊗ Z [1 / . The case ∗ = 0 yields C (Sper( K ) , Z ) ⊗ Z [1 / ≃ π ( [1 /η, / K ) ≃ W( K )[1 / . The claim follows.It remains to determine π ∗ ( [ η − ]) (2) . This is an immediate application of Theorem 7.8, by takingthe associated long exact sequence of homotopy sheaves of the fibration sequence [ η − ] (2) → kw (2) ϕ −→ Σ kw (2) . Since π ∗ kw ≃ W [ β ] (see § ∗ 6∈ { n, n − | n > } , whereasin these cases we get the kernel and cokernel of ϕ : W (2) ≃ π ∗ kw (2) → π ∗− kw (2) ≃ W .
By Example 7.5, this is multiplication by 9 n −
1. Since we are working 2-locally, by Lemma 8.2 below,up to a unit this is the same as 8 n . (cid:3) Lemma 8.2.
For n ≥ we have ν (9 n −
1) = ν (8 n ) . Proof.
Writing 9 n − n − n + X p ≥ (cid:18) np (cid:19) p , it suffices to show that ν ( (cid:0) np (cid:1) p ) > ν (8 n ). Since (cid:0) np (cid:1) = n ( n − ... ( n − p +1) p ! , it is enough to show that( ∗ ) ν ( p !) < ν (8 p − ) = 3( p − ν ( p !) ≤ p , whence ( ∗ ) holds for p ≥ (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 49 Homotopy groups of
MSp[ η − ] . Next we compute the homotopy sheaves of MSp[ η − ]. This willrequire some slightly involved algebraic manipulations. Lemma 8.3.
Suppose that E ∈ SH ( k ) (2) such that kw ∗ E is β -torsion free. Then for a, b ∈ kw ∗ E wehave ψ ( a ) ϕ ( b ) + ϕ ( a ) b = ϕ ( ab ) . Proof.
Via (7.1) this is a direct computation: ψ ( a ) ϕ ( b ) + ϕ ( a ) b = β − [ ψ ( a )( ψ ( b ) − b ) + ( ψ ( a ) − a ) b ]= β − [ ψ ( a ) ψ ( b ) − ab ] = β − [ ψ ( ab ) − ab ] = ϕ ( ab ) . (cid:3) Lemma 8.4.
Suppose that W( k ) = F . Then the operation ϕ : kw ∗ MSp → kw ∗− MSp is surjective,with kernel F [ y , y , . . . ] ⊂ kw ∗ MSp ≃ F [ β, b , b , . . . ] where deg y i = 2 i and y = β .Proof. Let R = F [ β ], which we view as a filtered ring via the filtration by powers of β . We also viewkw ∗ MSp as filtered by powers of β . Note thatgr • R ≃ F [ β ′ ] and gr • kw ∗ MSp ≃ F [ β ′ , b ′ , b ′ , . . . ];here β ′ ∈ gr and b ′ i ∈ gr are the images of β ∈ F and b i ∈ F . By (7.2) we have ϕ ( βx ) = βϕ ( x ); inother words ϕ is a filtered morphism. Note also that all our filtrations are degreewise finite, and hencecomplete, Hausdorff and exhaustive. By Lemma 2.4 and Corollary 2.3(2), it thus suffices to show thatgr • ( ϕ ) is surjective with kernel F [ y ′ , y ′ , . . . ], such that β lifts y ′ . Since ϕ ( β ) = 0 this makes sense, andsince β lifts β ′ it suffices to show the claim with y ′ = β ′ . By Proposition 4.3 and (7.1) we have ϕ ( b i ) ∈ β kw ∗ MSp + ( b i − i even0 i odd . This implies that(8.1) ϕ ( b ′ i ) = ( b ′ i − i even0 i odd . Also by Proposition 4.3 we have ψ ( b i ) ≡ b i (mod β )and hence, since ψ is a ring map and kw ∗ MSp is generated by the b i (over kw ∗ ), we get ψ ( x ) ≡ x (mod β ) for all x ∈ kw ∗ MSp . Via Lemma 8.3 this implies that ϕ ( ab ) ≡ aϕ ( b ) + ϕ ( a ) b (mod β )for all a, b ∈ kw ∗ MSp and hence(8.2) ϕ ( ab ) ≡ aϕ ( b ) + ϕ ( a ) b for all a, b ∈ gr • kw ∗ MSp . We also have ϕ ( β ) = 8 β = 0 and hence(8.3) ϕ ( β ′ ) = 0We have the decompositiongr • kw ∗ MSp ≃ F [ β ′ , b ′ , b ′ , . . . ] ≃ F [ b ′ , b ′ , . . . ] ⊗ F F [ β ′ , b ′ , b ′ , . . . ] =: A ⊗ F B. Using that ϕ is a derivation (i.e. (8.2)) and the action on the generators (i.e. (8.1), (8.3)) we see that ϕ = ϕ | A ⊗ F id B . Since ⊗ F B is an exact functor, it is thus sufficient to prove that ϕ | A is surjective withkernel F [ y , y , . . . ]. This is established in Lemma 8.5 below (put x i = b ′ i ). (cid:3) Lemma 8.5.
Consider the graded ring A = F [ x , x , . . . ] , where | x i | = i . Give it the derivation ϕ with ϕ ( x i ) = x i − (and ϕ ( x ) = 1 ). Then ϕ is surjective withkernel a polynomial ring F [ y , y , . . . ] where | y i | = i . Proof.
We begin by establishing surjectivity of ϕ . Define(8.4) I : A → A, f X n ≥ x n +1 ϕ ( n ) ( f ) , where ϕ ( n ) means the n -fold iteration of ϕ . Since ϕ lowers degrees, ϕ ( n ) ( f ) = 0 for n sufficiently large,so the sum is finite. Direct computation shows that ϕ ◦ I = id; hence ϕ is surjective as desired.Let J ⊂ A be the ideal generated by x , x , . . . , and give A the filtration by powers of J . Thengr • ( A ) ≃ Λ[ x , x , . . . ] ⊗ F [ t , t , . . . ]where Λ[ . . . ] denotes an exterior algebra in internal degree (i.e. coming from the filtration) zero, and t i in internal degree 1 corresponds to x i . Indeed using Lemma 2.6 it suffices to consider the case A ′ = F [ x ]and J ′ = ( x ), which is easily verified by hand. Since we are in characteristic 2, ϕ annihilates squaresand so is a J -filtered homomorphism; thus it descends to gr • A .We first study ϕ | Λ[ x ,x ,... ] . Suppose that n is not a power of two. We claim that ϕ : Λ[ x , x , . . . , x n − ] n → Λ[ x , x , . . . , x n − ] n − is surjective. If f ∈ Λ[ x , x , . . . , x n − ] n − then ϕ ( n − ( f ) ∈ Λ[ x , x , . . . , x n − ] = F . If ϕ ( n − ( f ) = 0then I ( f ) ∈ Λ[ x , x , . . . , x n − ] and hence f is in the image of the relevant restriction of ϕ . It is thusenough (in order to prove the claim) to show that there exists g ∈ Λ[ x , x , . . . , x n − ] n with ϕ ( n ) ( g ) = 1.Indeed then given f ∈ Λ[ x , x , . . . , x n − ] n − with ϕ ( n − ( f ) = 0 we conclude that f + ϕ ( g ) is in theimage of ϕ , and hence so is f . Let n = i + · · · + i k . Then since ϕ is a derivation we find ϕ ( n ) ( x i . . . x i k ) = X α (cid:18) nα (cid:19) ϕ ( α ) ( x i ) · · · ϕ ( α k ) ( x k ) = (cid:18) ni , . . . , i k (cid:19) ;here the sum is over multi-indices α of length k and sum n , and the coefficients are multinomial coeffi-cients. We need to find i + · · · + i k = n with 1 ≤ i r < n such that the multinomial coefficient is odd. ByKummer’s theorem for multinomial coefficients (see e.g. [How74, Lemma 2.2]), this is possible because n is not a power of 2: write n = P kj =1 a j for a finite strictly increasing sequence { a j } and let i j = 2 a j ;then ν (cid:18)(cid:18) ni , . . . , i k (cid:19)(cid:19) = X j S ( i j ) − S ( n ) = 0 , where S ( i ) is the sum of the base 2 digits of i . This concludes the proof of the claim.We deduce the following: if n is not a power of 2, there exists(8.5) y n ∈ x n + Λ[ x , . . . , x n − ] n with ϕ ( y n ) = 0. This yields a map α : Λ[ y n | n = 2 k ] → ker( ϕ | gr ( A ))which we shall show is an isomorphism; here the source denotes an exterior algebra on generators indegrees different from powers of 2. First note that α is injective, since the compositeΛ[ y n | n = 2 k ] α −→ ker( ϕ | gr ( A )) ֒ → gr ( A ) ≃ Λ[ x n | n ≥ α is surjective.This is a dimension counting argument; it suffices to show that ker( ϕ | gr ( A )) and Λ[ y n | n = 2 k ] have thesame Poincar´e series P t . Surjectivity of ϕ implies surjectivity of the quotient gr ( ϕ ); this (together withthe rank-nullity theorem) implies that P t (ker( ϕ | gr ( A ))) = (1 − t ) P t ( ϕ | gr ( A )) . Since P t ( B ⊗ C ) = P t ( B ) P t ( C ) and P t (Λ[ x n ]) = 1 + t n we deduce that P t ( ϕ | gr ( A )) = (1 − t ) Y n ≥ (1 + t n ) = Y n =2 k (1 + t n ) = P t (Λ[ y n | n = 2 k ]) . We have thus proved that ker( ϕ | gr ( A )) = Λ[ y n | n = 2 k ] . Consider the polynomial ring S = F [˜ y n | n = 2 k ][ x k | k ≥
0] = F [ x , ˜ y , x , ˜ y , ˜ y , ˜ y , x , . . . ] . Note that gr • ( ϕ ) is acts trivially on gr > and is a derivation. This implies that(8.6) ker(gr • ( ϕ )) = ker(gr ( ϕ )) ⊗ F F [ t , t , . . . ] , -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 51 and also that gr • ( ϕ ) is surjective (since we have shown that it is so on gr ). Hence by Lemma 2.4 wededuce that gr • (ker( ϕ )) ≃ ker(gr • ( ϕ )) . It follows that each of the elements y i ∈ ker(gr ( ϕ )) lifts to ˜ y i ∈ ker( ϕ ); since also x i ∈ ker( ϕ ) we obtaina map γ : S → ker( ϕ ) which we shall show is an isomorphism. We give S the filtration by powers ofthe ideal (˜ y n , x k | n = 2 k ); then γ is a filtered morphism (since y i = 0 and hence ˜ y i ∈ F ). For degreereasons, S and ker( ϕ ) are complete, Hausdorff and exhaustive. Hence by Lemma 2.2 it suffices to showthat gr • ( γ ) is an isomorphism. Note that (e.g. by Lemma 2.6) we havegr • ( S ) ≃ Λ[˜ y n | n = 2 k ] ⊗ F [˜ y n , x k | n = 2 k ] . Comparing with (8.6), we see that gr • ( γ ) = γ ⊗ γ , where γ is an isomorphism by construction. It isthus enough to show that γ : F [˜ y n | n = 2 k ][ x k | k ≥ → F [ t , t , . . . ]is an isomorphism. Note that by (8.5) we have γ (˜ y n ) ∈ x n + ( x , . . . , x n − ) A + F and hence (since we are in characteristic 2) γ (˜ y n ) ∈ x n + ( x , . . . , x n − ) A + F . It follows that the map induced by γ on indecomposables is unitriangular in the natural bases, so anisomorphism; hence so is γ by Lemma 2.7(2).This concludes the proof. (cid:3) Corollary 8.6.
Let k be any field of characteristic = 2 . Then π ∗ MSp (2) [ η − ] ≃ W (2) [ y , y , . . . ] . Proof.
To ease notation, we implicitly invert η throughout this proof.We first show the claim about π ∗ instead of π ∗ . Note that W( k ) (2) is a local ring (Lemma 2.13). Weneed to show that ϕ : kw ∗ MSp (2) → kw ∗− MSp (2) is surjective with kernel as indicated. The map ϕ is a map of W( k ) (2) -modules which are degreewisefinitely generated free. Note that kw ∗ MSp (2) / I is independent of k and hence the same holds for ϕ/ I.Thus by Lemma 8.4 the claim holds modulo I. It follows (using Nakayama’s lemma) that ϕ is splitsurjective: we may choose C ⊂ kw ∗ MSp (2) such that kw ∗ MSp (2) = ker( ϕ ) ⊕ C and ϕ : C → kw ∗− MSp (2) is an isomorphism. Thus ker( ϕ/ I) ≃ ker( ϕ ) / I. This implies in particular that any element in ker( ϕ/ I)lifts to ker( ϕ ), and that any family of elements of ker( ϕ ) which form a basis of ker( ϕ/ I) form a basis ofker( ϕ ) (again using Nakayama’s lemma). Lifting the polynomial generators ¯ y i ∈ ker( ϕ/ I) arbitrarily to y i ∈ ker( ϕ ) we deduce that monomials in the y i form a basis of ker( ϕ ); the result about π ∗ follows.Since π ∗ MSp (2) is a W (2) -algebra, we obtain a map W (2) [ y , y , . . . ] → π ∗ MSp (2) . We shall showthis is an isomorphism. To do so, we need to see that the map induces an isomorphism on fields, orequivalently that our generators y i are stable under base change. The above proof shows that a family { y i } will generate (over some field K ) if and only if it generates modulo I. Since W( K ) / I is independentof K , the result follows. (cid:3) Theorem 8.7.
Let k be any field of characteristic = 2 . Then π ∗ MSp[ η − ] ≃ W [ y , y , . . . ] , where | y i | = 2 i .Proof. To ease notation, we implicitly invert η throughout this proof.If char( k ) > k ) = W( k ) (2) [MH73, Theorem III.3.6], and hence the result follows fromCorollary 8.6. We may thus assume that char( k ) = 0, and by essentially smooth base change [BH17,Lemma B.1] that k = Q . Write J for the augmentation ideal ker( π ∗ MSp → π ∗ HW). Let n > M = ( J/J ) n . We claim that M ≃ W ( k ). Assuming this for now, let z n be a generator of M ,and y n ∈ π n MSp a lift of z n . We obtain a map α : W [ y , y , . . . ] → π ∗ MSp, which we shall show is anisomorphism. It suffices to show that α (2) and α [1 /
2] are isomorphisms (see e.g. Lemma 2.16). It followsfrom Corollary 8.6 that M (2) ≃ W( k ) (2) { y ′ n } , where the y ′ n form a family of polynomial generators of π ∗ MSp (2) . Thus α (2) is an isomorphism (Lemma2.7(2)). To show that α [1 /
2] is an isomorphism, since we are dealing with ρ -periodic spectra, it suffices toshow that there is an isomorphism on global sections (see § k = Q ). Moreover,since r R (MSp) ≃ MU (see Lemma 4.4), we know that [Rav86, Theorems 4.1.6 and A2.1.10] π ∗ MSp[1 / ≃ π ∗ MU[1 / ≃ Z [1 / , t , t , . . . ]with | t i | = 2 i . This implies that M [1 / ≃ Z [1 / { t n } and so α [1 /
2] is an isomorphism (Lemma 2.7(2) again).It remains to prove the claim that M ≃ W( Q ). As we have seen, M (2) ≃ W( Q ) (2) and M [1 / ≃ Z [1 / ≃ W( Q )[1 /
2] (see e.g. [MH73, Theorem III.3.10] for the latter isomorphism). We first show that M is finitely generated as a W( Q )-module. Indeed let x m ∈ M [1 /
2] and ya ∈ M (2) (with a ∈ Z oddand x, y ∈ M ) generate M [1 /
2] and M (2) respectively; then x and y generate M . By [Bou98, § II.5.2,Theorem 1], a finitely generated module is invertible (by which we mean locally free of rank 1) if andonly if it is stalkwise invertible. These properties hold for M , so it is an invertible W( Q )-module. Theresult will follow if we show that Pic(W( Q )) is trivial. It follows from idempotent lifting [Wei13, ExerciseI.2.2] that for any ring R we have Pic( R ) ≃ Pic( R red ). It thus suffices to show that W( Q ) red ≃ Z . Since Q is uniquely orderable, this follows from [MH73, Theorem III.3.8]. (cid:3) Homotopy groups of
MSL[ η − ] . We can easily adapt the arguments to MSL as well.
Theorem 8.8.
The canonical map
MSp → MSL induces π ∗ MSL[ η − ] ≃ W [ y , y , . . . ] . Proof.
We have a map α : W [ y , y , . . . ] → π ∗ MSL[ η − ] which we need to show is an equivalence; itsuffices to do this for α [1 /
2] and α (2) . For α [1 /
2] the claim reduces to the analogous result in topology(using Lemma 4.4 and § / → MSO[1 /
2] induces π ∗ MSO[1 / ≃ Z [1 / , t , t , . . . ][Sto15, § IX, Proposition p. 178, Theorem p. 180]. For α (2) we use the resolution η − MSL (2) → kw ∧ MSL (2) ϕ −→ Σ kw ∧ MSL (2) . It suffices to show that π ∗ α (2) is an isomorphism (since the base field is arbitrary and the formation of α is compatible with base change). We hence need to show that ϕ is surjective with kernel as indicated.Using that W( k ) (2) is a local ring (Lemma 2.13), this reduces to checking modulo I, i.e. we may basechange to a field with W( k ) = F . Examining the proof of Lemma 8.4, this is easily seen to hold. (cid:3) , MSL and kw . The map MSp[ η − ] → MSL[ η − ] is an E ∞ -ring map (see (4.3)) which annihi-lates y i for i odd, for degree reasons. It hence induces for i odd an MSp[ η − ]-module map MSp[ η − ] /y i → MSL[ η − ]. Put MSp[ η − ] / ( y , y , . . . , y n +1 ) = n O i =0 MSp[ η − ] /y i +1 ∈ MSp[ η − ]- M od . The canonical map MSp[ η − ] → MSp[ η − ] /y n +3 inducesMSp[ η − ] / ( y , y , . . . , y n +1 ) → MSp[ η − ] / ( y , y , . . . , y n +3 ) , and we put MSp[ η − ] / ( y , y , . . . ) = colim n MSp[ η − ] / ( y , . . . , y n +1 ) . Corollary 8.9.
There is a canonical equivalence
MSp[ η − ] / ( y , y , . . . ) ≃ MSL[ η − ] . Proof.
We implicitly invert η throughout this proof. Since y i maps to 0 in MSL (for i odd), the mapMSp → MSL factors over MSp /y i . We can thus form the compositeMSp / ( y , . . . , y n +1 ) ≃ n O i =0 MSp /y i +1 → MSL ⊗ n → MSL , with the last map being multiplication. Taking the colimit we obtain MSp / ( y , y , . . . ) → MSL. To seethat this is an equivalence we can compute the effect on homotopy sheaves. Since ( y , y , . . . ) is a regularsequence in π ∗ MSp ≃ W [ y , y , . . . ] we find π ∗ (MSp / ( y , y , . . . )) ≃ W [ y , y , . . . ];the result thus follows from Theorem 8.8. (cid:3) -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 53 There is an E ∞ -map MSL → kw [BW20, Corollary B.3]. Lemma 8.10.
The induced map π (MSL[ η − ]) → π (kw) is surjective.Proof. It suffices to show surjectivity after tensoring with Z [1 /
2] and Z (2) .For Z [1 /
2] this reduces (via Lemmas 3.9 and 4.4) to the topological result that the map MSO[1 / α top −−−→ ko[1 /
2] (obtained by applying r R to MSL[ η − ] → kw) is surjective on π . We shall make use of someof the theory of complex orientations and formal group laws, see e.g. [Rav86, §§ F induced by MU[1 / → MSO[1 / α top −−−→ ko[1 / → ku[1 / /
2] is the usual one, the associated formal group is the multiplicative one, and hencethe formal group law must be isomorphic to the multiplicative one. Thus there exists a formal powerseries f ( x ) = b x + b x + . . . such that F ( x, y ) = f − ( f ( x ) + f ( y ) + tf ( x ) f ( y )), where t ∈ π ku[1 /
2] isthe generator. Since MU ∗ is generated by the coefficients of the universal group law it carries, it sufficesto show that t is among the coefficients of F . Applying the isomorphism x ux does not change thisproperty, so we may assume that b = 1. Noting that the coefficients of F lie in π ∗ ko[1 /
2] = Z [1 / , t ],one finds that b = t/
2. A tedious but straightforward verification shows that the coefficient of xy is − t ; hence the desired result.For Z (2) , since W( k ) (2) is a local ring we may base change to a field with W( k ) = F . Consider thecommutative diagram π MSL (2) [ η − ] −−−−→ π (MSL ∧ kw) (2) y y F ≃ π kw (2) −−−−→ π (kw ∧ kw) (2) . We wish to show that the left hand vertical map is non-zero. By Lemma 8.4, the image of the top mapis spanned by β , which is mapped to β R (the image of β under the right unit u R : kw → kw ∧ kw) inthe bottom right hand corner. Since kw ∧ kw is a ring, the right unit kw → kw ∧ kw has a retractionand so induces an injection on homotopy groups; in particular β R = 0. The result follows. (cid:3) Corollary 8.11.
There exist generators y , y , y , · · · ∈ π ∗ MSL[ η − ] such that MSL[ η − ] / ( y , y , . . . ) ≃ kw . Proof.
Lemma 8.10 shows that we may choose y such that α ( y ) = β , where α : MSL → kw is thecanonical E ∞ -map. Now let n >
1. We have α ( y n ) = aβ n for some a ∈ W( k ); replacing y n by y n − ay n ensures that α ( y n ) = 0. Arguing as in the proof of Corollary 8.9 can thus form a mapMSL[ η − ] / ( y , y , . . . ) → kw ∈ MSL[ η − ]- M odwhich induces an isomorphism on π ∗ . This concludes the proof. (cid:3) Cellularity results.
It is well-known that the spectra KO and KW are cellular (i.e. in the subcat-egory of SH ( k ) generated under colimits and desuspensions by S p,q ) [RSØ16a]. Unfortunately in generalif E is cellular there is little reason to believe that truncations like E ≥ or π E are cellular. Our maintheorem allows us to make some deductions of this form. Proposition 8.12.
Let k have exponential characteristic e = 2 . The spectra kw , HW , ko[1 /e ] , kgl[1 /e ] , H ˜ Z [1 /e ] , f ( K W ) ∈ SH ( k ) are cellular.Proof. By Lemma 8.13 below, to prove that E is cellular, it suffices to show that E [1 /η ] and E/η arecellular. Since ko /η ≃ kgl [ARØ17, Proposition 2.11] and ko[1 /η ] ≃ kw, we may remove ko from the list.The argument in [SØ12, Proposition 5.12] (employing [LYZ13, Proposition B.1]) shows that kgl[1 /e ] ≃ MGL[1 /e ] / ( x , x , . . . ) is cellular (since MGL is). Put H W Z := f ( K W ). By [Bac17, Proposition 23] wehave H W Z /η ≃ H Z / ∨ Σ H Z /
2, which is cellular, and by [Bac17, Theorem 17] we have H W Z [1 /η ] ≃ HW;hence we may remove H W Z from the list. Again by [Bac17, Proposition 23] we have a cofiber sequenceΣ , H W Z → H ˜ Z → H Z ∨ Σ H Z /
2. Since H Z [1 /e ] is cellular, we may also remove H ˜ Z from the list.It remains to deal with kw and HW. Since MSp is cellular (see [RSØ16a, Proposition 3.1]), cellularityof kw is immediate from Corollaries 8.9 and 8.11. Finally HW is cellular since HW ≃ kw /β . (cid:3) Lemma 8.13.
Let E ∈ SH ( S ) and x ∈ π ∗∗ ( ) . Then E is cellular if and only if both E/x and E [1 /x ] are. Proof.
Since cellular spectra are closed under colimits, necessity is clear. We show sufficiency. Let
C ⊂ SH ( S ) denote the subcategory of cellular spectra. Since C is closed under colimits, the inclusionhas a right adjoint r , since C is generated by a set of compact objects from SH ( S ), r preserves colimits,and since C is stable under desuspension the functor r is stable. Moreover we have r (Σ p,q E ) ≃ Σ p,q r ( E ),and r ( E x −→ Σ ∗∗ E ) ≃ ( r ( E ) x −→ Σ ∗∗ r ( E )). It follows ( ∗ ) that r commutes with formation of ( − ) /x and( − )[1 /x ].We seek to show that r ( E ) → E is an equivalence. By Lemma 2.16 it suffices to show that r ( E ) /x → E/x and r ( E )[1 /x ] → E [1 /x ] are equivalences. Since r ( E ) /x ≃ r ( E/x ) and r ( E )[1 /x ] ≃ r ( E [1 /x ]) by( ∗ ), the result follows. (cid:3) ∧ HW (2) .Lemma 8.14. Let h : kw (2) → (kw ∧ HW) (2) be the Hurewicz map. Then h ( β ) = 8 ux , for some unit u ∈ W( k ) (2) .Proof. We have h ( β ) = ax and ϕ ( β ) = 8. By Proposition 7.7(2), ϕ ( x ) = v for some unit v . Hence8 = h ( ϕ ( β )) = ϕ ( h ( β )) = ϕ ( ax ) = av. The result follows. (cid:3)
Lemma 8.15.
Put f = 1 if k contains a subfield of vcd ≤ (e.g. char( k ) > ) and f = 2 otherwise.There exist generators x i ∈ π i kw ∧ HW (2) such that f x m x n = f (cid:18) m + nn (cid:19) x m + n . In other words the x i satisfy the identities of a divided power algebra, up to possibly a factor of 2. Proof.
The claim being stable under base change, we may assume that vcd ( k ) ≤ k = Q , in whichcase vcd ( k ) = 2. Let t i generate π · i . We can write t i = b i t i +1 . By Theorem 6.1(2,3) we have b i ∈ I( k )and b i ≡ u (mod I ) for some u ∈ (W( k ) ∧ I ) × . Note that 1 − u ∈ I and 2(1 − u ) ∈ I , so that b i ≡ ). We may thus write b i = 2 + c i for some c i ∈ I . If vcd ( k ) ≤ = 2I [EL99, lastTheorem], whereas if k = Q we have I = 8 Z h i [MH73, III (5.9)]. Hence in either case f c i = 2 f d i forsome d i ∈ I( k ). It follows that f t i = f · w i · t i +1 , where w i = 1 + d i ∈ W( k ) (2) is a unit (Lemma 2.13).Now we inductively define s i generating π · i such that(8.7) f s i = f · (cid:18) i +1 i (cid:19) s i +1 . Indeed we put s = t and assuming that s n has been chosen we get s n = a n t n for some unit a n , hence f s n = f ( a n t n ) = f a n · w n t n +1 = f · (2 a n w n t n +1 ) . Noting that (cid:0) n +1 n (cid:1) = 2 v n where v n is odd by Kummer’s theorem [Mol12, Theorem 2.6.7], we may put s n +1 = ( v − n a n w n ) t n +1 .Let y i generate π i . For n = P i ǫ i i put δ n = 1 /n ! Y i (2 ǫ i i !);then ν ( δ n ) = 0 by Legendre’s formula [Mol12, Theorem 2.6.4]. We can write x n := δ n Y i s ǫ i i = e n y n . By Theorem 6.1(3) e n is a unit modulo I, and hence a unit in W( k ) (2) by Lemma 2.13. Hence the x i aregenerators.We can verify the divided power relations as follows. Write n = P i ǫ i ( n )2 i . Then f · x n x m = f · ( n + m )! n ! m ! · Q i ǫ i ( n ) i !2 ǫ i ( m ) i ! s ǫ i ( n )+ ǫ i ( m ) i ( n + m )! ;these expressions make sense because we are working in a Z (2) -algebra. The product on the right handside consists of factors of the form 2 r ! s r , possibly repeated. We can simplify it by repeatedly applying To be precise, the denominator ( n + m )! has to be cancelled with the various factors of 2 r !, which works by Kummer’stheorem. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 55 the relation f (2 r ! s r ) = f r +1 ! s r +1 coming from (8.7). In the end we will thus have transformed it intoa product of the same form, but with no repeated factors. The s i occurring correspond to sum of thebinary expansions of n and m , i.e. the binary expansion of n + m . Hence the right hand side is f (cid:18) n + mm (cid:19) x n + m , as desired. (cid:3) Corollary 8.16.
We have HW ∧ HW (2) ≃ _ n ≥ Σ n HW (2) / n. Proof.
Since HW ≃ kw /β we have HW ∧ HW (2) ≃ (kw ∧ HW (2) ) /h ( β ). By Lemma 8.14, up to a unitmultiple we have h ( β ) = 8 x . Thus by Proposition 7.7(1), our result will hold if we show that the mapW( k ) (2) ≃ π n − kw ∧ HW (2) × x −−−→ π n kw ∧ HW (2) ≃ W( k ) (2) is (up to a unit) given by multiplication by 8 n . This follows from Lemma 8.15. (cid:3) Remark . Consider the commutative diagramkw (2) ϕ −−−−→ Σ kw (2) β −−−−→ kw (2) y y y kw ∧ HW (2) ϕ −−−−→ Σ kw ∧ HW (2) β −−−−→ kw ∧ HW (2) . Theorem 7.8 implies that the lower map denoted ϕ induces an isomorphism on positive homotopy groups.Thus in order to determine the effect (up to unit) of the lower map β on homotopy groups (in positivedegrees), it suffices to determine the effect of the lower composite. If W( k ) has no torsion (e.g. k = R ),then it suffices to determine this effect rationally. But rationally the vertical maps are isomorphisms, soit suffices to determine the effect of the top composite on homotopy. This is given on π n by multiplicationby 9 n − n (see Lemma 8.2). This providesan alternative proof of Corollary 8.16 over such fields.8.7. kw ∗ kw (2) .Lemma 8.18. Write ϕ n for the n -fold iteration of ϕ . The map kw ∧ kw (2) id ∧ ϕ • −−−−→ Y n ≥ kw ∧ Σ n kw (2) ≃ M n ≥ Σ n kw ∧ kw (2) → M n ≥ Σ n kw (2) is an equivalence of left kw -modules (where the last map is multiplication).Proof. We have Q ≃ L for connectivity reasons. Since the base field is arbitrary and the map iscanonical, it suffices to show that we have an isomorphism on π ∗ . By Proposition 7.7, this is a mapof degreewise finite free left kw ∗ (2) -modules. We may thus show that there is an isomorphism moduloI, i.e. we may assume that W( k ) ≃ F , and it suffices to show that the (left module) generators arepreserved. By Proposition 7.7, generators of the source are obtained as the images of the p i underMSL ∧ kw → kw ∧ kw. It hence suffices to show that ϕ n ( p n ) = 1. This is Lemma 7.6(2). (cid:3) Corollary 8.19.
We have kw ∗ (2) kw ≃ kw ∗ (2) J ′ ϕ K , in the sense that the underlying kw ∗ (2) -module is kw ∗ (2) J ϕ K but the composition product is determined by ϕβ = 9 βϕ + 8 .Proof. By adjunction, we have[kw , Σ ∗ kw (2) ] ≃ [kw ∧ kw (2) , Σ ∗ kw (2) ] kw (2) , where [ − , − ] kw (2) denotes homotopy classes of maps of (strong) kw (2) -modules. Now kw ∧ kw (2) ≃ L n Σ n kw (2) as kw (2) -modules, so that[kw ∧ kw (2) , Σ ∗ kw (2) ] kw (2) ≃ Y n kw ∗ (2) { q n } , where q n : kw ∧ kw (2) → Σ n kw (2) is the projection. By Lemma 8.18, q n is (or rather may be chosen tobe) adjoint to ϕ n . The additive structure follows. Since ϕ commutes with multiplication by kw , the multiplicative structure is determined once weknow ϕβ . Since kw ∗ kw (2) is β -torsion free, it suffices to know βϕβ . We compute βϕβ = ( ψ − β = 9 βψ − β = 9 β ( βϕ + 1) − β = β · (9 βϕ + 8) . The result follows. (cid:3)
Appendix A. The homotopy fixed point theorem
We shall supply an alternative proof of the homotopy fixed point theorem (also known as homotopylimit problem) for hermitian K -theory of (certain) fields. The original reference is [HKO11b] and uses adelicate analysis of some problems in equivariant motivic stable homotopy theory. We shall instead usethe improved version of Levine’s slice converges theorem from [BEØ20, §
5] and the computation of theslice spectral sequence of KW [RØ16] (see also Remark A.2). We fix throughout a base field k .Let E ∈ SH ( k ). We denote by E → ( f • E = · · · → f E → f E → . . . )the slice tower ; here f n ( E ) := cof( f n E → E ). We have colim n f • E ≃ n f n E = sc E . Lemma A.1.
Let char( k ) = 2 , vcd ( k ) < ∞ .(1) lim n map( , f n KW / ≃ map( , KW / (2) lim n map( , f n (KO) / (2 , ρ )) ≃ map( , KO / (2 , ρ )) Proof. (1) By [RØ16, Theorem 6.12] the map map( , KW) → lim n map( , f n KW) induces an isomor-phism on π i for i k ) → W( k ) ∧ I for i ≡ ( k ) < ∞ , this is thus a 2-adic equivalence (see e.g.Lemmas 2.12 (showing that W( k ) ∧ I ≃ W( k ) ∧ ), 2.10 and 2.8 (showing that W( k ) ∧ ≃ L ∧ W( k )) and 2.14(allowing us to compute the homotopy groups of map( , KW) ∧ )), whence the claim.(2) Consider the cofiber sequence ko → KO → E . By [BEØ20, Corollary 5.13] we havelim n map( , f n (ko) / (2 , ρ )) ≃ map( , ko / (2 , ρ ));hence it suffices to prove the analogous claim about E . By [BEØ20, Corollary 5.13] again we havelim n map( , f n (kw) / (2 , ρ )) ≃ map( , kw / (2 , ρ )) . Since ρ = − η -periodic spectra, (1) implies that alsolim n map( , f n (KW) / (2 , ρ )) ≃ map( , KW / (2 , ρ )) , and hence ( ∗ ) the same holds for E [ η − ] (note that kw = ko[ η − ] and KW = KO[ η − ]). We know thehomotopy sheaves π i (KW) (given by W in degrees divisible by 4, else 0) and also π i (KO) for i < ∗∗ ) the map E → E [ η − ]induces an isomorphism on π i ( E ) for all i , and hence on f . It hence suffices to prove the followingclaim: for any spectrum F ∈ SH ( k ) , the map f F → F induces equivalences map( G ∧ qm , f n ( f F ) / (2 , ρ )) ( a ) ≃ map( G ∧ qm , f n ( F ) / (2 , ρ )) and map( G ∧ qm , f ( F ) / (2 , ρ )) ( b ) ≃ map( G ∧ qm , F/ (2 , ρ )) , -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 57 for any q ≥ , n ∈ Z . Indeed then we findlim n map( , f n ( E ) / (2 , ρ )) ( a ) ≃ lim n map( , f n f ( E ) / (2 , ρ )) ( ∗∗ ) ≃ lim n map( , f n f ( E [ η − ]) / (2 , ρ )) ( a ) ≃ lim n map( , f n ( E [ η − ]) / (2 , ρ )) ( ∗ ) ≃ map( , E [ η − ] / (2 , ρ )) ( b ) ≃ map( , f ( E [ η − ]) / (2 , ρ )) ( ∗∗ ) ≃ map( , f ( E ) / (2 , ρ )) ( b ) ≃ map( , E/ (2 , ρ )) . Since all functors in sight commute with taking the cofiber of 2, we may replace F by F/ G ∧ qm , G/ρ ) ≃ cof(map( G ∧ qm , G ∧− m ∧ G ) ρ −→ map( G ∧ qm , G )) ≃ cof(map( G ∧ q +1 m , G ) ρ −→ map( G ∧ qm , G )) . Thus we may ignore taking the cofiber of ρ as well. Then G ∧ qm ∈ SH ( k ) eff implies that the ( b ) holds,and that ( a ) holds for n < n ≥ f f n ≃ f n f (since f f n ≃ f n ≃ f n f ). (cid:3) Remark
A.2 . In the above proof, we have referred to the paper [RØ16] for a certain spectral sequencecomputation. This paper references the homotopy fixed point theorem, which may seem to lead tocircular reasoning in the proof of Theorem A.3 below. However, the only reason why [RØ16] uses thehomotopy fixed point theorem is to determine the slices of KO. The paper [ARØ17] determines s ∗ (ko)independently, and from this we can deduce the slices of s ∗ (KO) via s ∗ (KO) ≃ s ∗ (ko)[ β − ]. Thus thereis no circularity. Theorem A.3 (homotopy fixed point theorem) . Let char( k ) = 2 , vcd ( k ) < ∞ . The canonical map KO / → KGL hC / ∈ SH ( k ) induces an isomorphism on π ∗∗ .Proof. By [Hea17, Corollary 3.9] (see also Lemma 3.24), the map KO → KGL hC is an η -equivalence.Noting that KGL is η -complete and hence so is KGL hC , it hence suffices to show that π ∗∗ (KO / ≃ π ∗∗ (KO ∧ η / F → KO → KO ∧ η ; we need to show that π ∗∗ ( F/
2) = 0. Since F is η -periodic, ρ = − F and we may as well show that π ∗∗ ( F/ (2 , ρ )) = 0. Again by η -periodicity,it suffices to show that map( , F/ (2 , ρ )) = 0. We have F ≃ lim h . . . η −→ Σ , KO η −→ Σ , KO η −→ KO η −→ . . . i . Passing to the final subsystem of multiplication by η , and using the β -periodicity KO ≃ Σ , KO, wecan rewrite this as F ≃ lim (cid:20) . . . η β − −−−−→ Σ − KO η β − −−−−→ Σ − KO η β − −−−−→ KO η β − −−−−→ . . . (cid:21) . We deduce that map( , F/ (2 , ρ )) ≃ lim t map( , Σ − t KO / (2 , ρ )) L. A. (2) ≃ lim t lim n map( , Σ − t f n (KO) / (2 , ρ )) ≃ lim n lim t map( , Σ − t f n (KO) / (2 , ρ ))=: lim n F n . Noting that F − = 0, it suffices to show that S n := fib( F n → F n − ) ≃ n . Unwinding thedefinitions, we have S n = lim t map( , Σ − t s n (KO) / (2 , ρ )) . It follows from [ARØ17, p.9] (see also [RØ16, Theorems 4.18 and 4.27]) that s n (KO) / (2 , ρ ) is a sum ofspectra of the form Σ n − i Σ n,n H Z / (2 , ρ ) , for various i ≥
0. Since (see e.g. § π ∗ , Σ n − i Σ n,n H Z / ≃ ( k M n − i −∗ ( k ) ∗ ≥ n − i , s n (KO) / (2 , ρ )) ∈ SH ≤ n +3 . This implies that S n = lim t Σ − t map( , s n (KO) / (2 , ρ )) ∈ SH ≤ n +3 − t , for any t , and thus S n = 0. This concludes the proof. (cid:3) References [Ada62] J Frank Adams. Vector fields on spheres.
Annals of Mathematics , pages 603–632, 1962.[Ada95] John Frank Adams.
Stable homotopy and generalised homology . University of Chicago press, 1995.[AG01] V. I. Arnol’d and A. B. Givental’.
Symplectic Geometry , pages 1–138. Springer Berlin Heidelberg, Berlin,Heidelberg, 2001.[AHW18] Aravind Asok, Marc Hoyois, and Matthias Wendt. Affine representability results in A –homotopy theory, ii:Principal bundles and homogeneous spaces. Geometry & Topology , 22(2):1181–1225, 2018.[ALP17] Alexey Ananyevskiy, Marc Levine, and Ivan Panin. Witt sheaves and the η -inverted sphere spectrum. Journalof Topology , 10(2):370–385, 2017.[AM17] Michael Andrews and Haynes Miller. Inverting the hopf map.
Journal of Topology , 10(4):1145–1168, 2017.[Ana12] Alexey Ananyevskiy. On the relation of special linear algebraic cobordism to witt groups. arXiv preprintarXiv:1212.5780 , 2012.[Ana15] Alexey Ananyevskiy. The special linear version of the projective bundle theorem.
Compositio Mathematica ,151(3):461–501, 2015.[Ana17] Alexey Ananyevskiy. Stable operations and cooperations in derived witt theory with rational coefficients.
Annalsof K-theory , 2(4):517–560, 2017.[Ana19] Alexey Ananyevskiy. Sl-oriented cohomology theories. arXiv preprint arXiv:1901.01597 , 2019.[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.
Compositio Mathematica , 154(5):883–917,2018. arXiv:1608.08855.[Bac18b] Tom Bachmann. Motivic tambara functors. 2018. arXiv:1807.02981.[Bac18c] Tom Bachmann. On the conservativity of the functor assigning to a motivic spectrum its motive.
Duke Math.J. , 167(8):1525–1571, 06 2018. arXiv:1506.07375.[Bac18d] Tom Bachmann. Rigidity in ´etale motivic stable homotopy theory. 2018. arXiv:1810.08028.[Bac18e] Tom Bachmann. Some remarks on units in grothendieck–witt rings.
Journal of Algebra , 499:229 – 271, 2018.arXiv:1707.08087.[Bal05] P Balmer. Witt groups. handbook of k-theory, 539-576, 2005.[Bau11] Tilman Bauer. Bousfield localization and the hasse square, 2011.[BEØ20] Tom Bachmann, Elden Elmanto, and Paul Arne Østvær. Stable motivic invariants are eventually ´etale local.arXiv:2003.04006, 2020.[BH17] Tom Bachmann and Marc Hoyois. Norms in motivic homotopy theory. 2017. arXiv:1711.03061.[BK87] Aldridge Knight Bousfield and Daniel Marinus Kan.
Homotopy limits, completions and localizations , volume304. Springer Science & Business Media, 1987.[BKSØ15] AJ Berrick, M Karoubi, M Schlichting, and PA Østvær. The homotopy fixed point theorem and the quillen–lichtenbaum conjecture in hermitian k-theory.
Advances in Mathematics , 278:34–55, 2015.[Boa82] J Michael Boardman. The eightfold way to bp-operations.
Current trends in algebraic topology , 2(Part 1):187–226, 1982.[Boa99] J Michael Boardman. Conditionally convergent spectral sequences.
Contemporary Mathematics , 239:49–84,1999.[Bou98] N. Bourbaki.
Commutative Algebra: Chapters 1-7 , volume 1. Springer Science & Business Media, 1998.[BW20] Tom Bachmann and Kirsten Wickelgren. A -euler classes: six functors formalisms, dualities, integrality andlinear subspaces of complete intersections. arXiv:2002.01848, 2020.[ ˇCad99] Martin ˇCadek. The cohomology of bo ( n ) with twisted integer coefficients. Journal of Mathematics of KyotoUniversity , 39(2):277–286, 1999.[CD09] Denis-Charles Cisinski and Fr´ed´eric D´eglise. Triangulated categories of mixed motives. arXiv preprintarXiv:0912.2110 , 2009.[CDH +
20] Baptiste Calm`es, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, De-nis Nardin, Thomas Nikolaus, and Wolfgang Steimle. Hermitian k -theory for stable infinity categories iii.grothendieck–witt groups of rings, 2020. in preparation.[CQ19] Dominic Leon Culver and JD Quigley. Complex motivic kq -resolutions. arXiv preprint arXiv:1905.11952 , 2019.[DF19] Fr´ed´eric D´eglise and Jean Fasel. The borel character. arXiv preprint arXiv:1903.11679 , 2019. -PERIODIC MOTIVIC STABLE HOMOTOPY THEORY OVER FIELDS 59 [Eis13] David Eisenbud. Commutative Algebra: with a view toward algebraic geometry , volume 150. Springer Science& Business Media, 2013.[EL99] Richard Elman and Christopher Lum. On the cohomological 2-dimension of fields.
Communications in Algebra ,27(2):615–620, 1999.[FH20] Jean Fasel and Olivier Haution. The stable adams operations on hermitian k-theory, 2020.[Fuj67] Michikazu Fujii. k o -groups of projective spaces. Osaka Journal of Mathematics , 4(1):141–149, 1967.[GGN16] David Gepner, Moritz Groth, and Thomas Nikolaus. Universality of multiplicative infinite loop space machines.
Algebraic & Geometric Topology , 15(6):3107–3153, 2016.[GH04] Paul G Goerss and Michael J Hopkins. Moduli spaces of commutative ring spectra.
Structured ring spectra ,315(151-200):22, 2004.[GI15] Bertrand J Guillou and Daniel C Isaksen. The η -local motivic sphere. Journal of Pure and Applied Algebra ,219(10):4728–4756, 2015.[GI16] Bertrand Guillou and Daniel Isaksen. The η –inverted R –motivic sphere. Algebraic & Geometric Topology ,16(5):3005–3027, 2016.[GJ09] Paul G Goerss and John F Jardine.
Simplicial homotopy theory . Springer Science & Business Media, 2009.[GS09] David Gepner and Victor Snaith. On the motivic spectra representing algebraic cobordism and algebraic k-theory.
Doc. Math , 14:359–396, 2009.[Hal15] Brian Hall.
Lie groups, Lie algebras, and representations: an elementary introduction , volume 222. Springer,2015.[Hea17] Drew Heard. The homotopy limit problem and the cellular picard group of hermitian k -theory. arXiv preprintarXiv:1705.02810 , 2017.[HKØ] Marc Hoyois, Shane Kelly, and Paul Arne Østvær. The motivic steenrod algebra in positive characteristic. toappear in J.Eur.Math.Soc. [HKO11a] Po Hu, Igor Kriz, and Kyle Ormsby. Convergence of the motivic adams spectral sequence. Journal of K-theory:K-theory and its Applications to Algebra, Geometry, and Topology , 7(03):573–596, 2011.[HKO11b] Po Hu, Igor Kriz, and Kyle Ormsby. The homotopy limit problem for hermitian k-theory, equivariant motivichomotopy theory and motivic real cobordism.
Advances in Mathematics , 228(1):434–480, 2011.[HM07] Rebekah Hahn and Stephen Mitchell. Iwasawa theory for k (1)-local spectra. Transactions of the AmericanMathematical Society , 359(11):5207–5238, 2007.[Hop98] Michael J Hopkins. k (1)-local e ∞ -ring spectra. Topological Modular Forms, in: Math. Surveys Monogr , 201:287–302, 1998.[Hor05] Jens Hornbostel. a -representability of hermitian k-theory and witt groups. Topology , 44(3):661–687, 2005.[Hor18] Jens Hornbostel. Some comments on motivic nilpotence.
Transactions of the American Mathematical Society ,370(4):3001–3015, 2018.[How74] Frederic Howard. The number of multinomial coefficients divisible by a fixed power of a prime.
Pacific Journalof Mathematics , 50(1):99–108, 1974.[Hoy15] Marc Hoyois. From algebraic cobordism to motivic cohomology.
Journal f¨ur die reine und angewandte Mathe-matik (Crelles Journal) , 2015(702):173–226, 2015.[Hoy16a] Marc Hoyois. Cdh descent in equivariant homotopy k-theory. arXiv preprint arXiv:1604.06410 , 2016.[Hoy16b] Marc Hoyois. Equivariant classifying spaces and cdh descent for the homotopy k-theory of tame stacks. arXivpreprint arXiv:1604.06410 , 2016.[Kar05] Max Karoubi. Bott periodicity in topological, algebraic and hermitian k-theory.
Handbook of , pages 111–137,2005.[Kne77] Manfred Knebusch. Symmetric bilinear forms over algebraic varieties. In G. Orzech, editor,
Conference on qua-dratic forms , volume 46 of
Queen’s papers in pure and applied mathematics , pages 103–283. Queens University,Kingston, Ontario, 1977.[KRØ18] Jonas Irgens Kylling, Oliver R¨ondigs, and Paul Arne Østvær. Hermitian k -theory, dedekind ζ -functions, andquadratic forms over rings of integers in number fields. arXiv preprint arXiv:1811.03940 , 2018.[Lur09] Jacob Lurie. Higher topos theory . Number 170. Princeton University Press, 2009.[Lur16] Jacob Lurie. Higher algebra, May 2016.[LYZ13] Marc Levine, Yaping Yang, and Gufang Zhao. Algebraic elliptic cohomology theory and flops, i. arXiv preprintarXiv:1311.2159 , 2013.[MH73] John Willard Milnor and Dale Husemoller.
Symmetric bilinear forms , volume 60. Springer, 1973.[Mil71] John Milnor.
Introduction to algebraic K-theory . Number 72. Princeton University Press, 1971.[Mol12] Victor H. Moll.
Numbers and Functions: From a Classical-Experimental Mathematician’s Point of View . Stu-dent Mathematical Library 065. American Mathematical Society, 2012.[Mor03] Fabien Morel. An introduction to A -homotopy theory. ICTP Trieste Lecture Note Ser. 15 , pages 357–441,2003.[Mor04a] Fabien Morel. On the motivic π of the sphere spectrum. In Axiomatic, enriched and motivic homotopy theory ,pages 219–260. Springer, 2004.[Mor04b] Fabien Morel. Sur les puissances de l’id´eal fondamental de l’anneau de witt.
Commentarii Mathematici Helvetici ,79(4):689–703, 2004.[Mor05a] Fabien Morel. Milnor’s conjecture on quadratic forms and mod 2 motivic complexes.
Rendiconti del SeminarioMatematico della Universita di Padova , 114:63–101, 2005.[Mor05b] Fabien Morel. The stable A -connectivity theorems. K-theory , 35(1):1–68, 2005.[Mor12] Fabien Morel. A -Algebraic Topology over a Field . Lecture Notes in Mathematics. Springer Berlin Heidelberg,2012.[MV99] Fabien Morel and Vladimir Voevodsky. A -homotopy theory of schemes. Publications Math´ematiques del’Institut des Hautes ´Etudes Scientifiques , 90(1):45–143, 1999. [OR19] Kyle Ormsby and Oliver R¨ondigs. The homotopy groups of the η -periodic motivic sphere spectrum. arXivpreprint arXiv:1906.11670 , 2019.[OVV07] Dmitri Orlov, Alexander Vishik, and Vladimir Voevodsky. An exact sequence for with applications to quadraticforms. Annals of mathematics , pages 1–13, 2007.[PW10a] Ivan Panin and Charles Walter. On the algebraic cobordism spectra msl and msp. arXiv preprintarXiv:1011.0651 , 2010.[PW10b] Ivan Panin and Charles Walter. On the motivic commutative ring spectrum bo. arXiv preprint arXiv:1011.0650 ,2010.[PW10c] Ivan Panin and Charles Walter. Quaternionic grassmannians and pontryagin classes in algebraic geometry. arXiv preprint arXiv:1011.0649 , 2010.[Rav86] Douglas C Ravenel.
Complex cobordism and stable homotopy groups of spheres , volume 121. Academic pressNew York, 1986.[Rio10] Jo¨el Riou. Algebraic k-theory, A -homotopy and riemann–roch theorems. Journal of Topology , 3(2):229–264,2010.[RØ05] Andreas Rosenschon and Paul Arne Østvær. The homotopy limit problem for two-primary algebraic k-theory.
Topology , 44(6):1159–1179, 2005.[RØ16] Oliver R¨ondigs and Paul Østvær. Slices of hermitian k–theory and milnor’s conjecture on quadratic forms.
Geometry & Topology , 20(2):1157–1212, 2016.[R¨on16] Oliver R¨ondigs. On the η -inverted sphere. to appear in the TIFR Proceedings of the International Colloquiumon K-theory , 2016.[RS17] Birgit Richter and Brooke Shipley. An algebraic model for commutative h Z–algebras.
Algebraic & GeometricTopology , 17(4):2013–2038, 2017.[RSØ16a] Oliver R¨ondigs, Markus Spitzweck, and Paul Arne Østvær. Cellularity of hermitian k-theory and witt theory. arXiv preprint arXiv:1603.05139 , 2016.[RSØ16b] Oliver R¨ondigs, Markus Spitzweck, and Paul Arne Østvær. The first stable homotopy groups of motivic spheres. arXiv preprint arXiv:1604.00365 , 2016.[RSØ18] O. R¨ondigs, M. Spitzweck, and P. A. Østvær. The motivic Hopf map solves the homotopy limit problem for K -theory. Doc. Math. , 23:1405–1424, 2018.[Sch94] Claus Scheiderer.
Real and Etale Cohomology , volume 1588 of
Lecture Notes in Mathematics . Springer, Berlin,1994.[Sch10a] Marco Schlichting. Hermitian k-theory of exact categories.
Journal of K-theory , 5(1):105–165, 2010.[Sch10b] Marco Schlichting. The mayer-vietoris principle for grothendieck-witt groups of schemes.
Inventiones mathe-maticae , 179(2):349–433, 2010.[Sch17] Marco Schlichting. Hermitian k-theory, derived equivalences and karoubi’s fundamental theorem.
Journal ofPure and Applied Algebra , 221(7):1729–1844, 2017.[Ser58] J-P Serre. Espaces fibr´es alg´ebriques.
S´eminaire Claude Chevalley , 3:1–37, 1958.[SØ12] Markus Spitzweck and Paul Arne Østvær. Motivic twisted k–theory.
Algebraic & Geometric Topology ,12(1):565–599, 2012.[ST15] Marco Schlichting and Girja S Tripathi. Geometric models for higher grothendieck–witt groups in A -homotopytheory. Mathematische Annalen , 362(3-4):1143–1167, 2015.[Sta18] The Stacks Project Authors.
Stacks Project . http://stacks.math.columbia.edu , 2018.[Sto15] Robert E Stong. Notes on cobordism theory . Princeton University Press, 2015.[Sus84] Andrei A Suslin. On the k-theory of local fields.
Journal of pure and applied algebra , 34(2-3):301–318, 1984.[TT90] R. W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In
TheGrothendieck Festschrift III , volume 88 of
Progress in Mathematics , pages 247–435. Birkh¨auser, 1990.[Voe02] V. Voevodsky. Open problems in the motivic stable homotopy theory , i. In
International Press Conference onMotives, Polylogarithms and Hodge Theory . International Press, 2002.[Voe03a] Vladimir Voevodsky. Motivic cohomology with Z / Publications Math´ematiques de l’Institut desHautes ´Etudes Scientifiques , 98:59–104, 2003.[Voe03b] Vladimir Voevodsky. Reduced power operations in motivic cohomology.
Publications Math´ematiques de l’Institutdes Hautes ´Etudes Scientifiques , 98:1–57, 2003.[Wei95] Charles A Weibel.
An introduction to homological algebra . Number 38. Cambridge university press, 1995.[Wei13] Charles A Weibel.
The K -book: An Introduction to Algebraic K -theory , volume 145. American MathematicalSoc., 2013.[Wen10] Matthias Wendt. More examples of motivic cell structures. arXiv preprint arXiv:1012.0454 , 2010.[Wil18] Glen Wilson. The eta-inverted sphere over the rationals. Algebraic & Geometric Topology , 18(3):1857–1881,2018.[Zib18] Marcus Zibrowius. The γ -filtration on the witt ring of a scheme. Quarterly Journal of Mathematics , 69(2):549–583, 2018.
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