Topological models for stable motivic invariants of regular number rings
aa r X i v : . [ m a t h . K T ] F e b TOPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULARNUMBER RINGS
TOM BACHMANN AND PAUL ARNE ØSTVÆR
Abstract.
For an infinity of number rings we express stable motivic invariants in terms of topologicaldata determined by the complex numbers, the real numbers, and finite fields. We use this to extendMorel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck-Witt ringof quadratic forms to deeper base schemes.
Contents
1. Introduction 12. Nilpotent completions 33. Rigidity for stable motivic homotopy of henselian local schemes 74. Topological models for stable motivic homotopy of regular number rings 95. Applications to slice completeness and universal motivic invariants 17References 201.
Introduction
The mathematical framework for motivic homotopy theory has been established over the last twenty-five years [Lev18]. An interesting aspect witnessed by the complex and real numbers, C , R , is thatBetti realization functors provide mutual beneficial connections between the motivic theory and thecorresponding classical and C -equivariant stable homotopy theories [Lev14], [GWX18], [HO18], [BS19],[ES19], [Isa19], [IØ20]. We amplify this philosophy by extending it to deeper base schemes of arithmeticinterest. This allows us to understand the fabric of the cellular part of the stable motivic homotopycategory of Z [1 / in terms of C , R , and F — the field with three elements. If ℓ is a regular prime, anumber theoretic notion introduced by Kummer in 1850 to prove certain cases of Fermat’s Last Theorem[Was82], we show an analogous result for the ring Z [1 /ℓ ] .For context, recall that a scheme X , e.g., an affine scheme Spec( A ) , has an associated pro-space X ´ et ,denoted by A ´ et in the affine case, called the étale homotopy type of X representing the étale cohomologyof X with coefficients in local systems; see [AM69] and [Fri82] for original accounts and [Hoy18, §5]for a modern definition. For specific schemes, X ´ et admits an explicit description after some furtherlocalization, see the work of Dwyer–Friedlander in [DF83, DF94]. For example, they established thepushout square(1) C ∧ ´ et −−−−→ R ∧ ´ et y y ( F ) ∧ ´ et −−−−→ Z [1 / ∧ ´ et Here the completion ( − ) ∧ takes into account the cohomology of the local coefficient systems Z / n ( m ) . Remark . If k is a field, then k ´ et is a pro-space of type K ( π, , where π is the Galois group over k ofthe separable closure of k . If S is a henselian local ring with residue class field k , then k ´ et → S ´ et is anequivalence (by the affine analog of proper base change [Gab94]). For instance, C ´ et ≃ ∗ is contractible, R ´ et ≃ RP ∞ is equivalent to the classifying space of the group C of order two, and ( F p ) ´ et ≃ ( Z p ) ´ et isequivalent to the profinite completion of a circle. That is, up to completion, (1) can be expressed moresuggestively as Z [1 / ´ et ≃ S ∨ RP ∞ . For our generalization to stable motivic homotopy invariants, itwill be essential to keep track of the fields and not just their étale homotopy types. Date : February 3, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Motivic homotopy theory, ℓ -regular number fields, quadratic forms. The presentation of Z [1 / ∧ ´ et has powerful consequences; for example, taking the -adic étale K -theoryof (1) yields a pullback square. Combined with the Quillen–Lichtenbaum conjecture for the two-primaryalgebraic K -theory of Z [1 / , see [Bök84], [Wei97], [Øst03], [HØ03], one obtains the pullback square(2) K ( Z [1 / ∧ −−−−→ K ( R ) ∧ y y K ( F ) ∧ −−−−→ K ( C ) ∧ We show that replacing algebraic K -theory in (2) by an arbitrary cellular motivic spectrum over Z [1 / still yields a pullback square. Let SH ( X ) denote the motivic stable homotopy category of X , see [Jar00],[DRØ03], [Mor03, §5], [BH20b, §4.1]. We write SH ( X ) cell ⊂ SH ( X ) for the full subcategory of cellularmotivic spectra [DI05], i.e., the localizing subcategory generated by the bigraded spheres S p,q for allintegers p, q ∈ Z . For simplicity we state a special case of Theorem 4.13, see Example 4.16. Theorem 1.2.
For every
E ∈ SH ( Z [1 / cell there is a pullback square (3) E ( Z [1 / ∧ −−−−→ E ( R ) ∧ y y E ( F ) ∧ −−−−→ E ( C ) ∧ Here, for X ∈ Sch Z [1 / , we denote by E ( X ) the (ordinary) spectrum of maps from X to p ∗ E in SH ( X ) ,where X ∈ SH ( X ) denotes the unit object and p : X → Z [1 / is the structure map.Example . The motivic spectra representing algebraic K -theory, KGL , hermitian K -theory, KO ,Witt-theory, KW , motivic cohomology or higher Chow groups, H Z , and algebraic cobordism, MGL ,are cellular (at least after localization at ) by respectively [DI05, Theorem 6.2], [RSØ18a, Theorem 1],[RSØ18a, Theorem 1], [Hoy15a, Proposition 8.1] and [Spi18, Corollary 10.4], [DI05, Theorem 6.4]. Werefer to [BH20a, Proposition 8.12] for cellularity of the corresponding (very effective or connective) covers kgl , ko , kw , in the sense of [SØ12], and Milnor-Witt motivic cohomology H e Z , in the sense of [BCD + E = KGL , Theorem 1.2 recovers the stable version of [HØ03, Theorem 1.1], and for E = KO it recovers [BKØ11, Theorem 1.1] (in fact, we extend these results to arbitrary -regular numberfields, not necessarily totally real). The squares for KW , H Z , H e Z , MGL , kgl , ko , kw appear to be new.A striking application of Theorem 1.2 is that it relates the universal motivic invariants over Z [1 / tothe same invariants over C , R , and F . That is, applying (3) to the motivic sphere E = Z [1 / enablescomputations of the stable motivic homotopy groups of Z [1 / . We identify, up to odd-primary torsion,the endomorphism ring of Z [1 / with the Grothendieck-Witt ring of quadratic forms of the Dedekinddomain Z [1 / defined in [MH73, Chapter IV, §3]. This extends Morel’s fundamental computation of π , ( ) over fields [Mor03, §6] to an arithmetic situation. Theorem 1.4.
The unit map Z [1 / → KO Z [1 / induces an isomorphism π , ( Z [1 / ) ⊗ Z [1 / ∼ = GW( Z [1 / ⊗ Z [1 / Remark . The étale homotopy types of various other rings and applications to algebraic K -theoryand group homology of general linear groups were worked out in [DF83], [DF94], [Øst00], [HØ03]. Weshow similar generalizations of (3) with Z [1 / replaced by O F [1 / , for F any -regular number field, orby Z [1 /ℓ ] , Z [1 /ℓ, ζ ℓ ] , where ℓ is an odd regular prime and ζ ℓ is a primitive ℓ -th root of unity; to achievethis we slightly alter the other terms in (3). See Theorems 4.13, 4.17, 4.20, 5.3 for precise statements.Another application, which will be explored elsewhere, is the spherical Quillen–Lichtenbaum propertysaying the canonical map from stable motivic homotopy groups to stable étale motivic homotopy groupsis an isomorphism in certain degrees. Slice completeness is an essential input for showing the sphericalproperty; we deduce this for base schemes such as Z [1 / in Proposition 5.1.As a final comment, we expect that most of the applications we establish hold over more general baseschemes, where convenient reductions to small fields are not possible. The proofs will require significantlydifferent ideas. OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 3
Organization.
In §2 we give proofs for some more or less standard facts about nilpotent completions instable ∞ -categories with t -structures. While these results are relatively straightforward generalizationsof Bousfield’s pioneering work [Bou79], we could not locate a reference in the required generality. Thesenilpotent completions will be our primary tool throughout the rest of the article. In §3 we prove a variantof Gabber rigidity. We show that, for example, if E ∈ SH ( X ) cell where X is essentially smooth over aDedekind scheme, then E ( X hx ) ∧ ℓ ≃ E ( x ) ∧ ℓ for any point x ∈ X such that ℓ is invertible in k ( x ) . Here X hx denotes the henselization of X along x . Our principal results are shown in §4. We establish a generalmethod for exhibiting squares as above and provide a criterion for cartesianess in terms of étale and realétale cohomology, see Proposition 4.6. Next we verify this criterion for regular number rings, reducingessentially to global class field theory — which is also how Dwyer–Friedlander established (1). In §5 wediscuss some applications, including a proof of Theorem 1.4. Notation and conventions.
We freely use the language of (stable) infinity categories, as set out in[Lur09, Lur16]. Given a (stable) ∞ -category C and objects c, d ∈ C , we denote by Map( c, d ) = Map C ( c, d ) (respectively map( c, d ) = map C ( c, d ) ) the mapping space (respectively mapping spectrum). Given asymmetric monoidal category C , we denote the unit object by = C . We assume familiarity with themotivic stable category SH ( S ) ; see e.g., [BH20b, §4.1]. We write Σ p,q = Σ p − q ∧ G ∧ qm for the bigradedsuspension functor and S p,q = Σ p,q for the bigraded spheres. Acknowledgements.
We acknowledge the support of the Centre for Advanced Study at the NorwegianAcademy of Science and Letters in Oslo, Norway, which funded and hosted our research project “MotivicGeometry" during the 2020/21 academic year, and the RCN Frontier Research Group Project no. 250399“Motivic Hopf Equations." 2.
Nilpotent completions
We axiomatize some well-known facts about nilpotent completions in presentably symmetric monoidalstable ∞ -categories with a t -structure. Our arguments are straightforward generalizations of [Bou79] and[Man18]. Theorems 2.1 and 2.2 are the main results in this section.2.1. Overview.
Throughout we let C be a presentably symmetric monoidal ∞ -category (i.e., the tensorproduct preserves colimits in each variable separately) provided with a t -structure which is compatiblewith the symmetric monoidal structure (i.e., C ≥ ⊗ C ≥ ⊂ C ≥ ) and left complete (i.e., for X ∈ C wehave X ≃ lim n X ≤ n ). Given E ∈ CAlg( C ) and X ∈ C recall [MNN17, Construction 2.7] the standardcosimplicial resolution (or cobar construction ) ∆ + → C , [ n ] X ⊗ E ⊗ n +1 whose limit is (for us by definition) the E -nilpotent completion X ∧ E .We call X ∈ C connective if X ∈ ∪ n C ≥ n . Recall that R ∈ CAlg( C ♥ ) is called idempotent if themultiplication map R ⊗ ♥ R → R ∈ C ♥ is an equivalence. Theorem 2.1.
Let C be left complete, E ∈ CAlg( C ≥ ) and X ∈ C . Suppose that π E ∈ CAlg( C ♥ ) isidempotent and X is connective. Then the canonical map X ∧ E → X ∧ π E is an equivalence. One way of producing idempotent algebras is by taking quotients of the unit. Given L , . . . , L n ∈ C ≥ and maps x i : L i → , we set X/ ( x m , x m , . . . , x m n n ) = X ⊗ cof( x ⊗ m n n : L ⊗ m n n → ⊗ · · · ⊗ cof( x ⊗ m : L ⊗ m → The object π ( / ( x , . . . , x n )) ∈ CAlg( C ♥ ) is idempotent. For varying m , the /x mi ’s form an inversesystem indexed on N in an evident way; by taking tensor products, the objects X/ ( x m , . . . , x m n n ) forman N n -indexed inverse system. We define the x -completion of X as the limit X ∧ x ,...,x n := lim m ,...,m n X/ ( x m , . . . , x m n n ) Theorem 2.2.
Suppose each L i ∈ C ≥ is strongly dualizable with dual DL i ∈ C ≥ . If X ∈ C is connectiveand C is left complete, then there is a canonical equivalence X ∧ π ( / ( x ,...,x n )) ≃ X ∧ x ,...,x n To apply Theorem 2.2 in motivic stable homotopy theory we consider, for a scheme S , the homotopy t -structure on SH ( S ) ; see e.g., [BH20b, §B], [SS18, §1]. TOM BACHMANN AND PAUL ARNE ØSTVÆR
Theorem 2.3.
Let S be a quasi-compact quasi-separated scheme of finite Krull dimension and suppose X ∈ SH ( S ) is connective.(1) There is an equivalence X ∧ MGL ≃ X ∧ η .(2) If /ℓ ∈ S then there is an equivalence X ∧ H F ℓ ≃ X ∧ η,ℓ .Proof. The homotopy t -structure is left complete by [SS18, Corollary 3.8]. (1) Owing to [Hoy15a, Theorem 3.8, Corollary 3.9] we have MGL ∈ SH ( S ) ≥ and π (MGL) ≃ π ( /η ) .(2) We need to prove that H F ℓ ∈ SH ( S ) ≥ and π (H F ℓ ) ≃ π ( / ( η, ℓ )) . Since x i ∈ π i,i MGL and Σ i,i MGL = Σ i G ∧ im ∧ MGL ∈ SH ( S ) ≥ i ⊂ SH ( S ) > , both of these claims follow from the Hopkins–Morelisomorphism H F ℓ ≃ MGL / ( ℓ, x , x , . . . ) shown in [Spi18, Theorem 10.3]. (cid:3) Remark . Theorem 2.3 implies that a map α : E → F ∈ SH ( S ) ≥ is an ( η, ℓ ) -adic equivalence if andonly if α ∧ H F ℓ is an equivalence, which is also easily seen by considering homotopy objects. This weakerstatement, however, cannot be used as a replacement for Theorem 2.3 in this work.2.2. Proofs.
Recall that C is a presentably symmetric monoidal ∞ -category equipped with a compatible t -structure. Definition 2.5. (1) Let E ∈ CAlg( C ) . Then X ∈ C is E -nilpotent if it lies in the thick subcategorygenerated by objects of the form E ⊗ Y for Y ∈ C .(2) Let R ∈ CAlg( C ♥ ) be idempotent. Then F ∈ C ♥ is strongly R -nilpotent if F admits a finitefiltration whose subquotients are R -modules. Moreover, X ∈ C is strongly R -nilpotent if it isbounded in the t -structure and all homotopy objects are strongly R -nilpotent. Example . If X ∈ C is an E -module in the homotopy category, then it is a summand of X ⊗ E , andthus X is E -nilpotent. Lemma 2.7.
Suppose R ∈ CAlg( C ♥ ) is idempotent.(1) Let A → B → C → D → E ∈ C ♥ be an exact sequence. If A, B, D, E are strongly R -nilpotent, then so is C .(2) An object X ∈ C is strongly R -nilpotent if and only if it is R -nilpotent and bounded in the t -structure.Proof. (1) The proofs of [Man18, Lemmas 7.2.7–7.2.9] apply unchanged. (2) Example 2.6 implies thatstrongly R -nilpotent objects are R -nilpotent, being finite extensions of homotopy R -modules. It thussuffices to show that if X is R -nilpotent, then its homotopy objects π C i ( X ) ∈ C ♥ are strongly R -nilpotent.This is clear for free R -modules and the property is preserved by taking summands and shifts and cofibersby (1). The result follows. (cid:3) Definition 2.8. (1) If E ∈ CAlg( C ) , X ∈ C , a tower of the form X → · · · → X → X → X is called an E -nilpotent resolution if each X i is E -nilpotent and for every E -nilpotent Y ∈ C , wehave colim n [ X n , Y ] ≃ −→ [ X, Y ] (2) If R ∈ CAlg( C ♥ ) is idempotent and X ∈ C , a tower of the form X → · · · → X → X → X is called a strongly R -nilpotent resolution if each X i is strongly R -nilpotent and for every strongly R -nilpotent Y ∈ C , we have colim n [ X n , Y ] ≃ −→ [ X, Y ] This reference assumes S noetherian, but this is only used to obtain finite homotopy dimension of the Nisnevich topoi,which holds in the stated generality by [CM19, Theorem 3.17] — see also [RØ06, Theorem 4.1]. This reference assumes S noetherian, but since the equivalence exists over Z [1 /ℓ ] it persists after pullback to S . Note that R being idempotent is a property, not additional data. OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 5
Proposition 2.9.
For
X, Y ∈ C and X • , Y • E -nilpotent (respectively strongly R -nilpotent) resolutions,we have Map
Pro( C ) ( X • , Y • ) ≃ lim n Map(
X, Y • ) Thus any map X → Y induces a canonical morphism of towers X • → Y • . In particular, if X ≃ Y , then X • ≃ Y • ∈ Pro( C ) and lim n X n ≃ lim n Y n .Proof. Essentially by definition we have
Map( X • , Y • ) ≃ lim n colim m Map( X m , Y n ) The colimit is equivalent to
Map(
X, Y n ) by the definition of a resolution. (cid:3) Lemma 2.10.
Let E ∈ CAlg( C ) and X ∈ C .(1) The tower of partial totalizations of the standard cosimplicial objects X ⊗ E ⊗• is an E -nilpotentresolution of X .(2) Suppose that E ∈ C ≥ and π E is idempotent. Then if X → X • is any E -nilpotent resolutionby connective objects (e.g., if X is connective, the one arising from (1)), then X → τ ≤• X • is astrongly π ( E ) -nilpotent resolution.Proof. (1) Since partial totalizations are finite limits they commute with ⊗ X , by stability, and are thusgiven by X i = X ⊗ cof( I ⊗ i → ) , where I = fib( → E ) , see [MNN17, Proposition 2.14]. In the notationof loc. cit. we get cof( X i → X i − ) ≃ Σcof( T i ( E, X ) → T i − ( E, X )) and X = 0 . This implies X i is E -nilpotent by [MNN17, Proposition 2.5(1)]. To conclude, it suffices to prove that if Y is E -nilpotent,then colim i map( X i , Y ) ≃ map( X, Y ) . The class of objects Y satisfying the latter equivalence is thick,so we may assume that Y is an E -module. We are reduced to proving that colim i map( I ⊗ i ⊗ X, Y ) = 0 .But this is a summand of colim i map( I ⊗ i ⊗ X ⊗ E, Y ) , Y being an E -module, and the transition maps I ⊗ i +1 ⊗ E → I ⊗ i ⊗ E are null by [MNN17, Proposition 2.5(2)], so the colimit vanishes as desired.(2) We first show that each τ ≤ n X n is strongly R -nilpotent, and more generally, that if Y is E -nilpotentthen each π i ( Y ) is strongly R -nilpotent. By Lemma 2.7(1) we may assume Y is a (free) E -module; in thiscase, each π i ( Y ) is a π ( E ) -module. Suppose Y ∈ C is strongly π ( E ) -nilpotent. Then Y is E -nilpotentsince any π ( E ) -module is an E -module. Finally, we have colim n [ τ ≤ n X n , Y ] ≃ colim n [ X n , Y ] ≃ [ X, Y ] Here the first equivalence holds since Y is bounded above and the second because Y is E -nilpotent. (cid:3) Next we prove that the E -nilpotent completion only depends on π ( E ) . Proof of Theorem 2.1.
For E ∈ CAlg( C ≥ ) and X ∈ C , denote by R n ( E, X ) the n -th partial totalizationof X ⊗ E ⊗• , so that X → R • ( E, X ) is a tower with limit X → X ∧ E . By left completeness and cofinalitywe have X ∧ E ≃ lim m,n τ ≤ m R n ( X, E ) ≃ lim n τ ≤ n R n ( X, E ) By Lemma 2.10, the right hand side is the limit of a strongly π ( E ) -nilpotent resolution, which byProposition 2.9 only depends on X and π ( E ) . (cid:3) Remark . The proof also verifies that any strongly π ( E ) -nilpotent resolution of X has limit X ∧ π E .We now turn to the study of x -completions. Lemma 2.12.
Let L , . . . , L n ∈ C be strongly dualizable and x i : L i → . Let Y ∈ C and suppose that,for every i , the map Y ⊗ L i x i −→ Y is null. Then there is an equivalence colim m ,...,m n map( X/ ( x m , . . . , x m n n ) , Y ) ≃ map( X, Y ) Proof.
As a first observation, note that the maps Y ⊗ L i x i −→ Y and Y Dx i −−→ Y ⊗ DL i correspond underthe equivalence Map( Y ⊗ L i , Y ) ≃ Map(
Y, Y ⊗ D ( L i )) . It follows that Dx i is null.First consider the case n = 1 . By definition we have fib( X → X/x m ) ≃ X ⊗ L ⊗ m . Hence it sufficesto prove colim m map( X ⊗ L ⊗ m , Y ) = 0 . This term can be identified with colim m map( X, ( DL ) ⊗ m ⊗ Y ) ,and the transition maps in this system are null by our first observation. In the general case, we note theequivalence X/ ( x m , . . . , x m n n ) ≃ ( X/x m ) / ( x m , . . . , x m n n ) TOM BACHMANN AND PAUL ARNE ØSTVÆR
Hence we get colim m ,...,m n map( X/ ( x m , . . . , x m n n ) , Y ) ≃ colim m colim m ,...,m n map(( X/x m ) / ( x m , . . . , x m n n ) , Y ) ≃ colim m map( X/x m , Y ) ≃ map( X, Y ) The first equivalence holds since colimits commute, and the other two hold by induction. (cid:3)
Lemma 2.13.
Suppose L ∈ C ≥ is strongly dualizable with strong dual DL ∈ C ≥ . Then, for all X ∈ C ,there are equivalences π i ( X ⊗ L ) ≃ π i ( X ) ⊗ L ≃ π i ( X ) ⊗ ♥ π ( L ) Proof.
By assumption we have C ≥ ⊗ L ⊂ C ≥ . The same holds for DL , which implies C ≤ ⊗ L ⊂ C ≤ .In other words, ⊗ L : C → C is t -exact, and hence π i ( X ⊗ L ) ≃ π i ( X ) ⊗ L . Being in the heart C ♥ , thelatter tensor product is equivalent to π i ( X ) ⊗ ♥ π ( L ) . (cid:3) Let us quickly verify that π ( / ( x , . . . , x n )) is indeed an idempotent algebra in C ♥ . Lemma 2.14.
Let L , . . . , L n ∈ C ≥ and x i : L i → . Then R = π ( / ( x , . . . , x n )) defines anidempotent object of CAlg( C ♥ ) and the multiplication maps π ( L i ) ⊗ ♥ R x i −→ R are null.Proof. Recall that idempotent commutative algebras in C ♥ are the same as maps π ( ) → A ∈ C ♥ suchthat the induced map A → A ⊗ ♥ A is an isomorphism [Lur16, Proposition 4.8.2.9]. Note that π ( / ( x , . . . , x n )) ≃ π ( π ( / ( x , . . . , x n − )) /x n ) More generally, let us prove that if π ( ) → A ∈ C ♥ is an idempotent algebra and L ∈ C ≥ , x : L → , then π ( A/x ) is also an idempotent algebra on which multiplication by x is null. Consider thecommutative diagram of cofiber sequences L ⊗ A ⊗ A/x e −−−−→ A ⊗ A/x u −−−−→ A/x ⊗ A/x d x b x x L ⊗ A ⊗ A c −−−−→ A ⊗ A a −−−−→ A/x ⊗ A Here c and e “multiply L into the left factor A ”, and all the other maps are the canonical projections.Since A is idempotent, π ( A ⊗ A ) ≃ A and π ( A ⊗ A/x ) ≃ π ( A/x ) ≃ π ( A/x ⊗ A ) . Under theseidentifications we have π ( a ) = π ( b ) and so π ( ed ) = π ( bc ) = π ( ac ) = 0 . Since π ( d ) is an epiwe deduce π ( e ) = 0 , and hence π ( u ) is an isomorphism. This concludes the proof since, under ouridentifications, π ( e ) is multiplication by x on π ( A/x ) and π ( u ) is π ( A/x ) → π ( A/x ) ⊗ ♥ π ( A/x ) . (cid:3) We can now identify x -completions as E -nilpotent completions for an appropriate E . Proof of Theorem 2.2.
Lemma 2.14 shows R n = π ( / ( x , . . . , x n )) is idempotent. Step 1 : The map R n ⊗ L i x i −→ R n is null. Indeed, by Lemma 2.13, we have R n ⊗ L i ≃ R n ⊗ ♥ π ( L i ) ,and so this follows from Lemma 2.14. Step 2 : We show the homotopy objects of X/ ( x e , . . . , x e n n ) are strongly R n -nilpotent for all e i ≥ .By an induction argument, using the octahedral axiom, X/x m is a finite extension of copies of X/x .Hence each X/ ( x e , . . . , x e n n ) is a finite extension of copies of X/ ( x , . . . , x n ) ; thus we may assume e i = 1 .By induction on n and Lemma 2.13, together with Lemma 2.7(1), it suffices to show that if M ∈ C ♥ is R i -nilpotent, then both the kernel and cokernel of M ⊗ ♥ π ( L i +1 ) x i +1 −−−→ M are R i +1 -nilpotent. The proof given in [Man18, Lemma 7.2.10] goes through unchanged in our setting. Step 3 : We show that { τ ≤ m X/ ( x e , . . . , x e n n ) } e ,...,e n ; m is a strongly R n -nilpotent resolution of X . Since we assume X is connected, step 2 shows τ ≤ m X/ ( x e , . . . , x e n n ) is bounded with strongly R n -nilpotent homotopy objects. Owing to Lemma 2.7(2) it is in fact strongly R n -nilpotent. We thus need to show that if Y is strongly R n -nilpotent, then colim map( τ ≤ m X/ ( x e , . . . , x e n n ) , Y ) ≃ map( X, Y ) OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 7
Since Y is bounded above, we may remove τ ≤ m in the above expression without changing the colimit.We may assume that Y is an R n -module in C ♥ . By step 1 the map L i ⊗ Y → Y is null, and so the claimfollows from Lemma 2.12. Conclusion of proof : By left completeness we have X ∧ x ,...,x n ≃ lim e ,...,e n ; m τ ≤ m X/ ( x e , . . . , x e n n ) According to step 3, this is the limit of a strongly R n -nilpotent resolution of X , which coincides with X ∧ R n by Remark 2.11. (cid:3) Rigidity for stable motivic homotopy of henselian local schemes
Given a presentably symmetric monoidal stable ∞ -category C and a morphism a : L → with L strongly dualizable, we denote by C ∧ a the a -completion; that is, the localization at maps which becomean equivalence after ⊗ cof( a ) . We refer to [Bac20b, §2.1], [BH20a, §2.5] for more details; in particular,the a -completion of X is given by the object X ∧ a from the previous section.Given a family of objects G ⊂ C (which for us will always be bigraded spheres Σ ∗∗ ), we write C cell for the localizing subcategory generated by G . Noting that C ∧ a is equivalent to the localizing tensor idealgenerated by cof( a ) , by e.g., [Bac20b, Example 2.3], we see that if L ∈ G then these two operationscommute, and so we shall write C ∧ cell a := ( C ∧ a ) cell ≃ ( C cell ) ∧ a Recall the element h := 1 + h− i ∈ π , ( ) , where −h− i is the switch map on G m ∧ G m , and theelement ρ := [ − ∈ π − , − ( ) corresponding to − ∈ O × . Proposition 3.1.
Suppose X is a henselian local scheme and essentially smooth over a Dedekind scheme.Write i : x → X for the inclusion of the closed point and let n ∈ Z .(1) If /n ∈ X then i ∗ : SH ( X ) ∧ cell n → SH ( x ) ∧ cell n is an equivalence.(2) If / n ∈ X then i ∗ : SH ( X ) ∧ cell nh → SH ( x ) ∧ cell nh is an equivalence.(3) i ∗ : SH ( X )[ ρ − ] cell → SH ( x )[ ρ − ] cell is an equivalence. Many proofs in the sequel will follow the pattern of this one. We spell out many details here, whichare suppressed in the following proofs.
Proof. If S is a quasi-compact quasi-separated scheme, e.g., affine, the category SH ( S ) is compactlygenerated by suspension spectra of finitely presented smooth S -schemes [Hoy15b, Proposition C.12].Thus SH ( S ) cell is compactly generated by the spheres, and for every a ∈ π ∗∗ ( S ) , the category SH ( S ) ∧ cell a is compactly generated by Σ ∗∗ /a . Now let f : S ′ → S be a morphism, where S ′ is also quasi-compactquasi-separated. We use f ∗ to transport elements of π ∗∗ ( S ) to π ∗∗ ( S ′ ) , and when no confusion canarise, we denote them by the same letter. Thus, for example, we set SH ( S ′ ) ∧ a := SH ( S ′ ) ∧ f ∗ a The functor f ∗ : SH ( S ) ∧ cell a → SH ( S ′ ) ∧ cell a preserves colimits and the compact generator. Therefore itadmits a right adjoint f ∗ preserving colimits. This implies that f ∗ is fully faithful if and only if the map → f ∗ f ∗ ∈ SH ( S ) ∧ cell a is an equivalence, see e.g., [Bac18b, Lemma 22]; in this case, the functor is anequivalence since its essential image will be a localizing subcategory containing the generator.We can simplify this condition further. By a -completeness and Lemma 3.2 below, it follows that → f ∗ f ∗ is an equivalence if and only if /a → f ∗ f ∗ ( /a ) is an equivalence, i.e., if and only if π ∗∗ ( S /a ) ≃ π ∗∗ ( S ′ /a ) If b ∈ π ∗∗ ( ) , then in our compactly generated situations the b -periodization E [ b − ] is given by thecolimit E [ b − ] = colim (cid:16) E b −→ Σ ∗∗ E b −→ . . . (cid:17) Since f ∗ preserves colimits it commutes with b -periodization by Lemma 3.2. We shall make use of thefact that a map is an equivalence if and only if it is an equivalence after b -periodization and b -completion,see e.g., [BH20a, Lemma 2.16]. Thus to prove fully faithfulness it would also be sufficient, as well asnecessary, to prove π ∗∗ ( S / ( a, b )) ≃ π ∗∗ ( S ′ / ( a, b )) and π ∗∗ ( S [ b − ] /a ) ≃ π ∗∗ ( S ′ [ b − ] /a ) We will use many different variants of these observations in the sequel.(0) We claim the functor SH ( X )[ η − ] → SH ( x )[ η − ] TOM BACHMANN AND PAUL ARNE ØSTVÆR is an equivalence provided / ∈ X , and that SH ( X )[ η − , / → SH ( x )[ η − , / is an equivalence without any assumptions on X . For the first claim, by the above remarks it suffices toprove that π ∗∗ ( [ η − ]) satisfies the required rigidity, which via [Bac20a, Proposition 5.2] reduces to thesame statement for the Witt ring W ( − ) . This is true by [Jac18, Lemma 4.1]. Since SH ( S )[ η − , / ≃SH ( S )[ ρ − , / , see Lemma 3.3, the second claim reduces to (3).(1) It suffices to establish an isomorphism on η -periodization and η -completion. We first treat the η -complete case, i.e., we need to show that → i ∗ i ∗ ∈ SH ∧ cell n,η is an equivalence. By Theorem 2.3(2)with ℓ = n , we have E ∧ n,η ≃ lim ∆ E ∧ H Z /n ∧• +1 for any connective E in SH ( S ) . The cellularization functor SH ( S ) → SH ( S ) cell preserves limits andhence ( n, η ) -completions. Moreover, H Z /n ∈ SH ( S ) cell if /n ∈ S by [Spi18, Corollary 10.4]. Hencethe above formula for E ∧ n,η also makes sense, and is true, in SH ( S ) cell . Thus we need to show themap H Z / ∧ t → i ∗ (H Z / ∧ t ) ∈ SH ( X ) cell is an equivalence, for t ≥ . Lemma 3.2 below implies that i ∗ ( E ∧ i ∗ F ) ≃ i ∗ ( E ) ∧ F , for any E ∈ SH ( x ) , F ∈ SH ( X ) cell . In this way we reduce to t = 1 ; i.e., itsuffices to show π ∗∗ (H Z /n X ) ≃ π ∗∗ (H Z /n x ) Owing to [Spi18, Theorem 3.9], π ∗∗ (H Z /n S ) is given by the Zariski cohomology of S with coefficientsin a truncation of the étale cohomology of µ ⊗− n . When S = X or S = x the scheme S is Zariski local,so π ∗∗ (H Z /n S ) is simply given by certain étale cohomology groups of S with coefficients in µ ⊗− n . Therigidity result follows now from [Gab94, Theorem 1].Next we treat the η -periodic case. If n is even then / ∈ X and so the result follows from (0). If n is odd then n -complete objects are -periodic, and the result also follows from (0).(2) Again it suffices to prove that we have an isomorphism after η -completion and η -periodization;(0) handles the η -periodic case. For the η -complete case, we use that π ( / ( nh, η )) ≃ π (1 / (2 n, η )) , seeLemma 3.3 below, whence ∧ nh,η ≃ ∧ n,η by Theorem 2.2; this reduces to (1).(3) By [Bac18a, Theorem 35] we have SH ( S )[ ρ − ] ≃ SH ( S r ´ et ) , where the right hand side denoteshypersheaves on the small real étale site of S . In this situation we have a natural t -structure, see e.g.,[Bac20b, §2.2], such that the map r ´ et → H r ´ et Z is a morphism of connective ring spectra inducing anisomorphism on π — where by H r ´ et Z we mean the constant sheaf of spectra. Hence, applying Theorem2.1 in this situation, and repeating the above discussion using that H r ´ et Z is cellular and stable underbase change, essentially by definition, we find that in order to prove → i ∗ i ∗ ∈ SH ( S )[ ρ − ] cell is anequivalence it suffices to prove H r ´ et Z → i ∗ H r ´ et Z is an equivalence. In other words, we need to show H ∗ r ´ et ( X, Z ) ≃ H ∗ r ´ et ( x, Z ) Since the real étale and Zariski cohomological dimension coincide [Sch94, Theorem 7.6], we are reducedto H r ´ et , which follows from [ABR12, Propositions II.2.2, II.2.4]. (cid:3) Lemma 3.2.
Let F : C → D be a symmetric monoidal functor between symmetric monoidal categoriesadmitting a right adjoint G , and let x : A → be a morphism in C with A strongly dualizable. Then for X ∈ D , there is a natural equivalence G ( X ⊗ F A ) ≃ G ( X ) ⊗ A , and under this equivalence the map G ( X ⊗ F A ) G (id ⊗ F x ) −−−−−−−→ G ( X ) corresponds to GX ⊗ A x −→ GX Suppose that C , D are presentably symmetric monoidal stable ∞ -categories and G preserves colimits.Write C ′ for the localizing subcategory of C generated by strongly dualizable objects. Then the above resultalso holds for any A ∈ C ′ .Proof. Since F symmetric monoidal, G is lax symmetric monoidal, and there is a canonical map GX ⊗ GF A → G ( X ⊗ F A ) . Composing with the unit A → GF A , we obtain a natural map GX ⊗ A → G ( X ⊗ F A ) , which is an equivalence by the Yoneda lemma. Since this equivalence is natural in A aswell, the claim about x also follows.For the second statement, the subcategory comprised of A ∈ C for which the natural transformation GX ⊗ A → G ( X ⊗ F A ) is an equivalence for all X ∈ D is localizing since G preserves colimits and itcontains all strongly dualizable objects by the first part, hence all of C ′ . (cid:3) OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 9
Lemma 3.3. In π ∗ , ∗ ( ) we have the relations ηh = 0 , h = 2 + ηρ, hρ = 0 It follows that SH ( S )[1 / , /η ] ≃ SH ( S )[1 / , /ρ ] Proof.
By [Dru18, Theorem 1.2] all the Milnor–Witt relations hold in π ∗ , ∗ ( ) , including ηh = 0 . Ourdefinition of h agrees with Druzhinin’s by [Dru18, Lemma 3.10]. We now compute hρ = (2 + η [ − − −
1] = 2[ − −
1] + ([1] − [ − − [ − −
1] = 0 using the logarithm relation [ ab ] = [ a ] + [ b ] + η [ a ][ b ] as well as [1] = 0 which holds by definition.For the last part, note that inverting either η or ρ kills h (by the first or third relation), and hencemakes η and ρ inverses of each other up to a factor of − / (by the second relation). (cid:3) Example . Suppose that / ∈ X , where X is henselian local over a Dedekind scheme. ApplyingProposition 3.1(2) with n = 1 we learn that SH ( X ) ∧ cell h → SH ( x ) ∧ cell h is an equivalence. By Lemma 3.3both the η -periodic and ρ -periodic objects are h -torsion. We conclude SH ( X ) ∧ h [ η − ] ≃ SH ( X )[ η − ] andsimilarly for ρ . Thus there is an equivalence SH ( X )[ η − ] cell ≃ SH ( x )[ η − ] cell A similar equivalence holds for ρ . With reference to Proposition 3.1, this shows (2) implies (3). Example . We have ( E ∧ ab ) ∧ a ≃ E ∧ a since ab -periodic objects are a -periodic. Hence, in Proposition 3.1,(2) implies (1). Remark . Proposition 3.1 remains true with the pair ( X, x ) replaced by any henselian pair ( X, Z ) ,where both X and Z are essentially smooth over not necessarily the same Dedekind scheme. To provethis, since i ∗ commutes with essentially smooth base change, one may assume X henselian local, say withclosed point x . Then Z is also henselian local [Sta18, Tag 0DYD], and the result follows by applyingProposition 3.1 to both X and Z .4. Topological models for stable motivic homotopy of regular number rings
We shall exhibit pullback squares describing SH ( O F [1 /ℓ ]) ∧ cell ℓ for suitable number fields F and primenumbers ℓ in terms of SH ( k ) ∧ cell ℓ for fields of the form k = C , R , F q . To facilitate comparison with thework of Dwyer–Friedlander [DF94] we formally dualize our terminology and exhibit pushout squares inthe opposite category.4.1. Setup.
Let ℓ be a prime (or more generally any integer, but we do not need or use this extragenerality). We shall use the notation ℓ ′ = ℓ if ℓ is odd, and ℓ ′ = ℓh if ℓ = 2 . Definition 4.1. (1) We write CM S ⊂ (CAlg( P r L ) op ) / SH ( S ) cell for the full subcategory comprised of functors F : SH ( S ) cell → C , where C is generated undercolimits by F ( SH ( S ) cell ) (or equivalently by F ( S p,q ) for p, q ∈ Z ).(2) We denote by M ℓ ′ the functor Sch Z [1 /ℓ ] → CM Z [1 /ℓ ] , X
7→ SH ( X ) ∧ cell ℓ ′ , ( f : X → Y ) ( f ∗ : SH ( Y ) ∧ cell ℓ ′ → SH ( X ) ∧ cell ℓ ′ ) op We also put CM = CM Z and, by abuse of notation, M ( X ) := M ( X ) = SH ( X ) cell ∈ CM . Note that CM S = CM /M ( S ) and M ℓ ′ ( X ) = M ( X ) ∧ ℓ ′ . Next we clarify the meaning of colimits in CM S . Lemma 4.2.
Let F : I → CM S be a diagram and write F ′ : I op → C at ∞ for the underlying diagramof categories. Then lim I op F ′ ∈ C at ∞ is presentably symmetric monoidal and admits a natural functorfrom SH ( S ) cell . Let C denote its subcategory generated under colimits by the image of SH ( S ) cell . Thenthere is an equivalence colim I F ≃ C .Proof. The forgetful functor
CAlg( P r L ) SH ( S ) cell / → C at ∞ preserves limits [Lur09, Propositions 5.5.3.13, 1.2.13.8], [Lur16, Corollary 3.2.2.5], and hence the limitadmits a canonical functor from SH ( S ) cell . For D ∈ CM we have
Map CM ( C , D ) ≃ Map
CAlg( P r L ) SH ( S ) cell / ( D , C ) ⊂ Map
CAlg( P r L ) SH ( S ) cell / ( D , lim I op F ′ ) ≃ lim I op Map
CAlg( P r L ) SH ( S ) cell / ( F ( − ) , D ) ≃ lim I op Map CM ( F ( − ) , D ) It remains to show the inclusion is an equivalence, i.e., every map
D → lim I op F ′ in CAlg( P r L ) SH ( S ) cell / factors through C . This holds for the generators, by assumption, so we are done. (cid:3) Next we reformulate and slightly extend our rigidity results from §3.
Lemma 4.3.
Let ¯ x be the spectrum of a separably closed field, X ∈ Sch Z [1 /ℓ ] an essentially smooth overa Dedekind domain, ¯ x → X a map, and y ∈ X a specialization of the image of x . In CM Z [1 /ℓ ] there is acommutative diagram M (¯ x ) M ( y ) M ( X ) s Here the unlabelled maps are the canonical ones. In fact, there is a family of such commutative diagrams,parametrized by the (non-empty) set X hy × X ¯ x .Proof. Let X ′ be the henselization of X along y . By [Sta18, Tags 03HV, 07QM(1)] the map X ′ → X hits the image of ¯ x , and hence there exists a lift s ′ in the commutative diagram ¯ x X ′ X s ′ Applying M and using that M ( y ) → M ( X ′ ) is an equivalence by Proposition 3.1, the result follows. (cid:3) Corollary 4.4.
The following hold under the assumptions in Lemma 4.3.(1) If y ∈ X is separably closed, then s is an equivalence.(2) If ¯ x, ¯ y ∈ Sch Z [1 /ℓ ] are separably closed fields there is a (non-unique) equivalence M (¯ x ) ≃ M (¯ y ) .Proof. (1) We have constructed a symmetric monoidal cocontinuous functor F : SH (¯ y ) ∧ cell ℓ ′ → SH (¯ x ) ∧ cell ℓ ′ under SH ( Z [1 /ℓ ]) cell . Denote its right adjoint by G . Arguing as in the proof of Proposition 3.1, itsuffices to show E → GF E is an equivalence. That is, E → GF E induces an isomorphism on π ∗∗ for E = H F ℓ , E = H r ´ et Z and, if ℓ is even, E = [ η − ] . For any separably closed field of characteristic = ℓ we have π ∗∗ (H F ℓ ) ≃ F ℓ [ τ ] , see e.g., [BKWX20, Corollary C.2(2)], [IØ20, Theorem 18.2.7], W = Z / ,and H r ´ et Z = 0 (the real spectrum being empty). Moreover, all of the maps are algebra maps over thecorresponding algebra for Z [1 /ℓ ] . Thus the map for π ∗∗ H r ´ et Z is trivially an isomorphism, and the onefor π ∗∗ [ η − ] is an isomorphism because as an algebra it is determined by W according to [Bac20a,Proposition 5.2]. The isomorphism for H F ℓ will hold if and only if F ( τ ) = τ , which holds provided F ( τ n ) = τ n for some n ≥ . But, for n ≫ , τ n exists over Z [1 /ℓ ] (if ℓ = 2 this holds with n = 1 , andfor ℓ odd see e.g., [BEØ20, §4.5(2)]).(2) Let x, y ∈ Spec( Z [1 /ℓ ]) be the images of ¯ x, ¯ y . We may assume y is a specialization of x . Let X be the strict henselization of Spec( Z [1 /ℓ ]) along y , with closed point y ′ . By (1) applied with X = X ′ wehave M ( y ′ ) ≃ M (¯ x ) , and by applying it with ( X, ¯ x, y ) = ( { y ′ } , ¯ y, y ′ ) we get M ( y ′ ) ≃ M (¯ y ) . (cid:3) Remark . This common category M (¯ x ) ≃ M (¯ y ) is known as ℓ -complete MU -based (even) syntheticspectra [Pst18].4.2. Criterion.
Recall that for X ∈ Sch Z [1 /ℓ ] the objects H F ℓ , H r ´ et Z ∈ SH ( X ) are cellular and stableunder base change. For H F ℓ this is [Spi18, Corollary 10.4, Theorem 8.22]. For H r ´ et Z this follows from theexpression H r ´ et Z ≃ o (H Z )[1 /ρ ] [Bac18a], where o : SH → SH ( X ) is the unique cocontinuous symmetricmonoidal functor. In particular, any morphism between M ( X ) and M ( Y ) in CM Z [1 /ℓ ] preserves theseobjects. OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 11
Proposition 4.6.
Let X , X , X , X ∈ Sch Z [1 /ℓ ] be essentially smooth over Dedekind schemes andconsider a commutative square (4) M ( X ) −−−−→ M ( X ) y y M ( X ) −−−−→ M ( X ) in CM Z [1 /ℓ ] . In order for (4) to be cocartesian, it suffices that the following conditions hold:(1) For each m , the square map SH ( X ) (Σ , ∗ , H F ℓ ) −−−−→ map SH ( X ) (Σ , ∗ , H F ℓ ) y y map SH ( X ) (Σ , ∗ , H F ℓ ) −−−−→ map SH ( X ) (Σ , ∗ , H F ℓ ) is cartesian.(2) The square Γ r ´ et ( X , F ′ ℓ ) −−−−→ Γ r ´ et ( X , F ′ ℓ ) y y Γ r ´ et ( X , F ′ ℓ ) −−−−→ Γ r ´ et ( X , F ′ ℓ ) is cartesian. Here F ′ ℓ equals F ℓ if ℓ = ℓ ′ , and Z if ℓ ′ = ℓh (i.e., when ℓ even).(3) If | ℓ , then vcd ( K ( X i )) < ∞ .If X contains a primitive ℓ -th root of unity, then condition (1) can be replaced by(1’) For each m , the square Γ Zar ( X , R m ǫ ∗ F ℓ ) −−−−→ Γ Zar ( X , R m ǫ ∗ F ℓ ) y y Γ Zar ( X , R m ǫ ∗ F ℓ ) −−−−→ Γ Zar ( X , R m ǫ ∗ F ℓ ) is cartesian.Proof. To conclude that the square is cocartesian it suffices, by Lemma 4.2, to prove the functor SH ( X ) ∧ cell ℓ ′ → SH ( X ) ∧ cell ℓ ′ × SH ( X ) ∧ cell ℓ ′ SH ( X ) ∧ cell ℓ ′ is fully faithful. Let us denote by p ∗ : SH ( X ) ∧ cell ℓ ′ → SH ( X ) ∧ cell ℓ ′ the right adjoint of the functorcorresponding to M ( X ) → M ( X ) , and similarly for p ∗ , p ∗ . We need to prove that π ∗∗ ( ∧ ℓ ′ ) ≃ π ∗∗ ( p ∗ ( ∧ ℓ ′ ) × p ∗ ∧ ℓ ′ p ∗ ( ∧ ℓ ′ )) Note that each of the left adjoints preserves the compact generators, which is true for any morphismin CM , and hence p i ∗ preserves colimits and therefore it commutes with periodization. Moreover, p i ∗ commutes with ∧E for every E ∈ SH ( X ) ∧ cell ℓ ′ , and with completion at homotopy elements, by Lemma3.2. We may check the desired equivalence after completing at η and after inverting η , and similarly forother homotopy elements. For the η -periodic statement, we further invert respectively complete at .In the -complete (still η -periodic) case, either we have ∤ ℓ and the statement is vacuous, or / ∈ X i and using the fundamental fiber sequence [Bac20a, Corollary 1.2, Proposition 5.7], it suffices to establishthe analogous equivalence for kw ∧ ,ℓ ′ . Recall that kw ∧ ,ℓ ′ is in fact cellular [Bac20a, Proposition 5.7]. Inthe -periodic (still η -periodic) case, arguing as in the proof of Proposition 3.1, it suffices to establish theanalogous equivalence for H r ´ et F ′ ℓ . For the η -complete statement, arguing as in the proof of Proposition3.1, we have ∧ η,ℓ ′ ≃ ∧ H F ℓ and we see that it suffices to establish the analogous equivalence for H F ℓ . Insummary, we need to prove the commutative square of ordinary spectra map SH ( X ) (Σ , ∗ , E ) −−−−→ map SH ( X ) (Σ , ∗ , E ) y y map SH ( X ) (Σ , ∗ , E ) −−−−→ map SH ( X ) (Σ , ∗ , E ) is cartesian for all ∗ ∈ Z and E one of kw ∧ ,ℓ ′ , H F ℓ , H r ´ et F ′ ℓ . Before we start proving this, we need to make another preliminary remark. Suppose that E (0) • −−−−→ E (1) • y y E (2) • −−−−→ E (3) • is a commutative diagram of filtered spectra such that E ( i ) n = 0 for n sufficiently small and the induceddiagrams of associated graded objects gr i E (0) −−−−→ gr i E (1) y y gr i E (2) −−−−→ gr i E (3) are pullbacks for each i . Then the square lim i E (0) i −−−−→ lim i E (1) i y y lim i E (2) i −−−−→ lim i E (3) i is a pullback; indeed, an induction argument implies E (0) i −−−−→ E (1) i y y E (2) i −−−−→ E (3) i is a pullback for every i .Next we show how the conditions (1)–(3) imply that the squares are cartesian. The pullback squarefor H r ´ et F ′ ℓ is precisely condition (2), and the one for H F ℓ is precisely condition (1). The conditioninvolving kw ∧ ,ℓ ′ is only non-vacuous if | ℓ , whence / ∈ X i and kw ∧ ≃ kw ∧ ,ℓ ′ . Consider the filtrationof kw by powers of β , pulled back to X i . The Postnikov filtration gives rise to the said filtration,and so it is complete, and H W gives all subquotients [Bac20a, Theorem 4.4, Lemma 4.3]. Since kw isconnective, the preliminary remark allows us to replace kw ∧ by H W ∧ , which on mapping spectra yields Γ( − , W ∧ ) , where Γ denotes global sections of a Nisnevich sheaf of spectra. On mapping spectra thecellular motivic spectrum K W [Bac20a, Proposition 5.7, Theorem 4.4] yields compatible filtrations of Γ( − , W ) by Γ( − , I n ) , see [Bac20a, Definition 2.6], where I is the fundamental ideal of even dimensionalquadratic forms. Condition (3) together with [Bac20a, Proposition 2.3] implies lim n Γ( − , I n / ≃ .Thus the filtration Γ( − , ( W /I n ) ∧ ) of Γ( − , W ∧ ) is exhaustive. Using the preliminary remark we mayreplace Γ( − , W ∧ ) by Γ( − , I ∗ /I ∗ +1 ) , which coincides with map( G ∧∗ m , (H Z / /τ ) according to [Bac20a,Theorem 2.1, Lemma 2.7]. For this, we may establish the pullback square for H F ℓ , which implies thepullback square for H Z ∧ ℓ and hence for H Z ∧ ℓ / ≃ H Z / since | ℓ .Finally, suppose that ζ ℓ ∈ X . This yields τ ∈ π , − (H F ℓ )( X ) given by the Bockstein on [ ζ ℓ ] . Thecofibers of τ -powers yields a filtration of H F ℓ which pulls back to compatible filtrations on the X i ’s.The explicit construction of the motivic complexes [Spi18, Theorem 3.9] shows that these filtrationsare bounded, separated and exhaustive, and have subquotients Γ Zar ( X i , R m ǫ ∗ F ℓ ) . Via the preliminaryremark, the desired cartesian square thus reduces to condition (1’). (cid:3) Remark . • In all our examples, the chain complexes in conditions (1’) and (2) will be concentrated in asingle degree. • If ¯ x is the spectrum of a separably closed field, then Γ Zar (¯ x, R m ǫ ∗ F ℓ ) = 0 for m > , and similarly Γ r ´ et (¯ x, Z ) = 0 . • If the square (4) in Proposition 4.6 is cocartesian, then conditions (1) and (2) hold, and (1’) holdswhenever ζ ℓ ∈ X . Condition (3) is not necessary in general for the square to be cocartesian(consider for example any square comprised of identity maps).4.3. Models for stable motivic homotopy types.
OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 13
Arithmetic preliminaries.
Lemma 4.8.
Suppose K is a global field with ring of integers O K and put U = Spec( O K [1 /ℓ ]) . If ǫ : U ´ et → U Zar is the change of topology functor, then R i ǫ ∗ µ ℓ ≃ µ ℓ i = 0 a Zar O × /ℓ i = 1 a r ´ et Z / (2 , ℓ ) ⊕ R i = 2 a r ´ et Z / (2 , ℓ ) i > The sheaf R is determined by the exact sequence → R → M x ∈ Spec( O K ) (1) F ℓ → F ℓ ⊕ M x ∈ U (1) i x ∗ F ℓ → Here the middle term is a constant sheaf whereas the right-hand term is a sum of a constant sheaf andskyscraper sheaves, and the map is given by addition in the first component and restriction in the others.Proof.
From [Mil06, Remark II.2.2] we can read off the isomorphisms R ǫ ∗ G m ≃ G m , R i ǫ ∗ G m ≃ for i odd , R i ǫ ∗ G m ≃ a r ´ et R × for i ≥ evenand the short exact sequence → R ǫ ∗ G m → a r ´ et Z / ⊕ M x ∈ Spec( O K ) (1) Q / Z → Q / Z ⊕ M x ∈ U (1) i x ∗ Q / Z → For the exact sequence, recall Br ( K v ) = Z / if v is a real place, = 0 if v is a complex place, and = Q / Z is v is a non-archimedean place [Ser13, p. 163, 193]. Moreover, the kernel of the restriction map isprecisely the sum over the non-archimedean places missing in U . The Kummer short exact sequence → µ ℓ → G m ℓ −→ G m → on U ´ et yields a long exact sequence for R i ǫ ∗ . Since R i ǫ ∗ G m vanishes in odd degrees, R i ǫ ∗ µ ℓ is given bythe kernel or cokernel of multiplication by ℓ . This immediately yields the desired results for i = 2 , , andthe snake lemma produces an exact sequence → R ǫ ∗ µ ℓ → a r ´ et Z / (2 , ℓ ) ⊕ M x ∈ Spec( O K ) (1) F ℓ b −→ F ℓ ⊕ M x ∈ U (1) i x ∗ F ℓ → R ǫ ∗ µ ℓ → a r ´ et Z / (2 , ℓ ) → Since b is a surjection of Zariski sheaves, the result follows. (cid:3) Corollary 4.9.
Suppose
P ic ( U ) is uniquely ℓ -divisible and k ( U ) has a unique place of characteristic ℓ .Then H j ( U, R i ǫ ∗ µ ℓ ) = 0 for j > and H ( U, R i ǫ ∗ µ ℓ ) ≃ µ ℓ ( U ) i = 0 O × ( U ) /ℓ i = 1( Z / (2 , ℓ )) Sper( K ) i > Proof.
Since µ ℓ | U Zar is constant and constant sheaves are flasque, the claims for i = 0 are clear. The claimsabout a r ´ et Z / (2 , ℓ ) follow because R ( U ) ≃ Sper( k ) is discrete. Since the Zariski cohomological dimensionof U is , it remains to show that H ( U, G m /ℓ ) ≃ O × ( U ) /ℓ , H ( U, G m /ℓ ) = 0 , and H ∗ Zar ( U, R ) = 0 for ∗ = 0 , . Using the short exact sequences → µ ℓ → G m → ℓ G m → and → ℓ G m → G m → G m /ℓ → ,the first two claims are equivalent to unique ℓ -divisibility of P ic ( U ) . The exact sequence defining R is,in fact, a flasque resolution, so its H and H are given by the kernel and cokernel of the induced mapon global sections. This induced map is an isomorphism as needed if and only if Spec( O K ) \ U consistsof precisely one point, which holds by assumption. (cid:3) Lemma 4.10.
Let ℓ be prime, K a global field and U ⊂ Spec ( O K ) open. Let H ⊂ O × ( U ) /ℓ be anarbitrary subgroup. There exist x , . . . , x n ∈ U (1) such that the restriction H ⊂ O × ( U ) /ℓ → Y i k ( x ) × /ℓ is an isomorphism. If H is non-trivial, there exist infinitely many such choices. Proof.
First recall the following fact (see e.g., [Neu13, Exercise VI.1.2]): If a ∈ O ( U ) is an ℓ -th power in k ( x ) for all but finitely many x ∈ U (1) , then a is an ℓ -th power.If H is non-trivial, pick = a ∈ H . Since H is a Z /ℓ -vector space, we may write H = h a i × H ′ ,where h a i ≃ Z /ℓ is the subgroup generated by a . By the above fact, there exists x ∈ U (1) such thatthe image of a in k ( x ) × /ℓ is non-zero, and in fact infinitely many choices of x . Since k ( x ) is finite, k ( x ) × /ℓ ≃ Z /ℓ ≃ h a i . We are thus reduced to proving the result for H ′ , and conclude by induction since O × ( U ) /ℓ is finite according to Dirichlet’s unit theorem [Neu13, Corollary 11.7]. (cid:3) In [Gra86], Gras introduced the narrow tame kernel K +2 ( O F ) as the subgroup of K ( O F ) where theregular symbols on all the real embeddings of F vanish, i.e., there is an exact sequence → K +2 ( O F ) → K ( O F ) → r M Z / → We refer to [Gra03, Definition 7.8.1] for the arithmetic notion of ℓ -regular number fields. Definition 4.11.
Let ℓ be a prime number. A number field F is called ℓ -regular if the ℓ -Sylow subgroupof the narrow tame kernel K +2 ( O F ) is trivial.See [Gra86], [GJ89], [RØ00], [BKØ11] for complementary results about these families of number fields.For example, the field of rational numbers Q is ℓ -regular for every prime ℓ , and Q ( ζ ℓ ) is ℓ -regular if ℓ is aregular prime number in the sense of Kummer [Was82]. In [Sie64], Siegel conjectured there are infinitelymany regular prime numbers.We have the following explicit characterization of ℓ -regular number fields. Proposition 4.12.
Let F be a number field. We write O ′ F for the ring of ℓ -integers O F [1 /ℓ ] .(1) F is -regular if and only if the prime ideal (2) does not split in F/ Q and the narrow Picardgroup P ic + ( O ′ F ) has odd order.(2) Let ℓ be an odd prime number and assume µ ℓ ⊂ F . Then F is ℓ -regular if and only if the primeideal ( ℓ ) does not split in F/ Q and the ℓ -Sylow subgroup of the Picard group P ic ( O ′ F ) is trivial.(3) Let ℓ be an odd prime number. Assume µ ℓ F and F contains the maximal real subfield of Q ( ζ ℓ ) .Then F is ℓ -regular if and only if the prime ideals above ( ℓ ) in F do not split in the quadraticextension F ( ζ ℓ ) /F and the ℓ -Sylow subgroups of the Picard groups P ic ( O F ) and P ic ( O F ( ζ ℓ ) ) areisomorphic.Proof. This is a reformulation of [Gra86, Corollary on pp. 328-329]. See also [RØ00, Proposition 2.2]when ℓ = 2 . (cid:3) For further reference we recall that a commutative square of abelian groups A i −−−−→ A i y y p A p −−−−→ A is called bicartesian if it is a pullback when viewed as a commutative square of spectra.4.3.2. Stable motivic homotopy types of -regular number fields. Theorem 4.13.
Suppose F is a -regular number field with r real and c pairs of complex embeddings.Let x, y , . . . , y c ∈ Spec( O ′ F ) be closed points.(1) There is a canonical commutative square in CM M h ( C c + r ) −−−−→ M h ( R r ) ∐ ` i M h ( y i ) y y M h ( x ) −−−−→ M h ( O ′ F ) (2) The square in (1) is a pushout if and only if there is a naturally induced isomorphism ( O ′ F ) × / ≃ ( R × / r × k ( x ) × / × Y i k ( y i ) × / ≃ ( Z / r + c ) (3) There exist infinitely many choices of x, y , . . . , y c such that the map in (2) is an isomorphism. OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 15
Proof.
To simplify notation, throughout this proof we put M := M h .(1) For z ∈ Spec( O ′ F ) and α : K ֒ → C , Lemma 4.3 furnishes a map f z,α : M ( C ) → M ( z ) and ahomotopy between M ( C ) → M ( z ) → M ( O ′ F ) and the map M ( C ) → M ( O ′ F ) induced by α . In (1), thebottom and right-hand maps are the canonical ones. Write α , ¯ α , . . . , α c , ¯ α c , β , . . . , β r for the complexand real embeddings. Let α c + i = ι ◦ β i , where ι : R → C is the canonical embedding. The left-handmap is f x,α i on component i . The top map is f y i ,α i on the i -th component if i ≤ c , and induced by ι onthe remaining components. In all cases, the induced composite map M ( C ) → M ( O ′ F ) is either equal orhomotopic to the map induced by α i . Thus the square commutes.(2) We use the criterion from Proposition 4.6. Condition (3) holds since the fields are finitely generated.For condition (2), the square ( Z / r −−−−→ y y ( Z / r −−−−→ is clearly bicartesian because the map O ′ F → R r induces an isomorphism on real spectra.Next we check condition (1’). Owing to [BKØ11, Proposition 2.1(5)] the -regularity assumptionimplies P ic ( O ′ F ) has odd order, so it is uniquely -divisible, and F has only one place of characteristic .Thus Corollary 4.9 applies and it remains to check the bicartesianess of three squares. The first one is Z / −−−−→ ( Z / r + c y y Z / −−−−→ ( Z / r + c Observe that if
X, Y are connected schemes and f : M ( X ) → M ( Y ) is any map in CM Z [1 / , then f ∗ : H ( Y, Z / → H ( X, Z / is an isomorphism. Indeed this reduces to the case of the structure map M ( X ) → M ( Z [1 / , where it is obvious. Thus the square for m = 0 is bicartesian because the verticalmaps are isomorphisms. When m = 2 the square is the same as in condition (2) above, and hence it isbicartesian. The remaining square for m = 1 takes the form ( O ′ F ) × / −−−−→ k ( x ) × / y y ( R × / r × Q i k ( y i ) × / −−−−→ ( C × / r + c = 0 Since the inclusion of abelian groups into spectra preserves finite products, this square is bicartesian ifand only if the stated condition holds.(3) Dirichlet’s unit theorem [Neu13, Corollary 11.7] implies ( O ′ F ) × ≃ µ ( O ′ F ) × Z r + c ; here µ ( O ′ F ) isthe finite abelian group of roots of unity in O ′ F . It is cyclic, being a finite multiplicative subgroup of afield, and since {± } ∈ µ ( O ′ F ) the group has even order. It follows that ( O ′ F ) × / ≃ Z / × ( Z / r + c .Moreover, -regularity implies the naturally induced map ( O ′ F ) × / → ( R × / r ≃ ( Z / r is surjective[BKØ11, Proposition 2.1(5)]; we write U + ≃ ( Z / c for its kernel. The condition in part (2) holds ifand only if the induced map U + → k ( x ) × / × Q i k ( y i ) × / is an isomorphism. Lemma 4.10 implies thelatter is true for infinitely many choices of x, y i . (cid:3) Remark . As in Examples 3.4 and 3.5, Theorem 4.13(1) implies similar pushout squares with respectto completions at , h , and with respect to periodizations at ρ , η . For example, we have a pushout squarein CM M ( C c + r )[ η − ] −−−−→ M ( R r )[ η − ] ∐ ` i M ( y i )[ η − ] y y M ( x )[ η − ] −−−−→ M ( O ′ F )[ η − ] Remark . The various embeddings α i : K → C differ by automorphisms of C . It follows that onemay choose the maps f x,α i to be of the form σ i ◦ f x,α . Thus, applying an automorphism of M h ( C r + s ) inthe square of Theorem 4.13, we may assume that all the left hand vertical maps M h ( C ) → M h ( x ) arethe same. The square being a pushout now is equivalent to saying that there are lifts of M h ( x ) , M h ( y i ) to CM M h ( C ) / , and an equivalence M h ( O ′ F ) ≃ r _ M h ( R ) ∨ M h ( x ) ∨ c _ M h ( y i ) Here ∨ denotes the coproduct in CM M ( C ) / . Example . When F = Q we consider Z [1 / × ≃ {± } × { (1 / n } and Z [1 / × / ≃ Z / {− , } . Here Z / { } is the kernel of the surjection Z [1 / × / → R × / . We need to find a closed point x ∈ Spec( Z [1 / such that is not a square in k ( x ) . This holds when k ( x ) = Spec( F q ) , where q ≡ ± . In particularthe canonical map SH ( Z [1 / ∧ cell → SH ( R ) ∧ cell × SH ( C ) ∧ cell SH ( F ) ∧ cell is fully faithful. To deduce Theorem 1.2 from the introduction, let E ∈ SH ( Z [1 / ∧ cell and compute map( , E ) using the above square.4.3.3. Stable motivic homotopy types of ℓ -regular number fields. Theorem 4.17.
Let F be a number field with c pairs of complex embeddings and ℓ be an odd primenumber. Suppose F is ℓ -regular and µ ℓ ⊂ F . Let x, y , . . . , y c ∈ Spec( O ′ F ) be closed points.(1) There is a canonical commutative square in CM M ℓ ( C c ) −−−−→ ` c M ℓ ( y i ) y y M ℓ ( x ) −−−−→ M ℓ ( O ′ F ) (2) The square is a pushout if and only if there is a naturally induced isomorphism ( O ′ F ) × /ℓ ≃ k ( x ) × /ℓ × Y i k ( y i ) × /ℓ ( ≃ ( F ℓ ) c ) (3) There exist infinitely many choices of x, y , . . . , y c satisfying (2).Proof. The proof is essentially the same as that of Theorem 4.13. The maps x, y i → Spec( O ′ F ) togetherwith choices of embeddings of K into C induce, via Lemma 4.3, the maps M ℓ ( C ) → M ℓ ( x ) , M ℓ ( y i ) in the commutative square. One verifies, using Corollary 4.9 and Z / (2 , ℓ ) = 0 , that condition (1’) ofProposition 4.6 reduces to the condition stated in (2). The other conditions hold trivially; since K contains a primitive ℓ -th root of unity, the real spectrum Sper( O ′ F ) = ∅ . The existence of infinitely manychoices in (3) follows from Lemma 4.10. (cid:3) Remark . Arguing as in Remark 4.15, we find that there are lifts of M ℓ ( x ) , M ℓ ( y i ) to CM M ℓ ( C ) / , andan equivalence M ℓ ( O ′ F ) ≃ c _ M ℓ ( y i ) ∨ M ℓ ( x ) Example . Theorem 4.17 applies to F = Q ( ζ ℓ ) if ℓ is regular — we note that ( ℓ ) is totally ramifiedin F and K ( Z [ ζ ℓ ]) /ℓ ≡ µ ℓ ⊗ P ic ( Z [ ζ ℓ ]) . In this case, O ′ F = Z [1 /ℓ, ζ ℓ ] and k ( x ) = F p , where p is a primenumber which is congruent to ℓ but is not congruent to ℓ by [DF94, Example 1.9]. Theorem 4.20.
Let ℓ an odd regular prime and p = ℓ a prime number. There is a commutative squarein CM Z [1 /ℓ ] M ℓ ( C ) −−−−→ M ℓ ( R ) y y M ℓ ( F p ) −−−−→ M ℓ ( Z [1 /ℓ ]) The square is a pushout if p generates the multiplicative group of units ( Z /ℓ ) × .Proof. We get the square from Lemma 4.3 and proceed by verifying the conditions in Proposition 4.6.Since Z [1 /ℓ ] has a unique real embedding, condition (2) holds. Condition (3) is vacuous. Next we verifycondition (1). Let us write Γ( X, F ℓ ( i )) for the motivic complex and Γ ´ et ( X, F ℓ ( i )) ≃ Γ ´ et ( X, µ ⊗ iℓ ) for itsétale version. If A = Z [1 /ℓ, ζ ℓ ] , then H et ( A, F ℓ ) = F ℓ , H et ( A, F ℓ ) = A × /ℓ and H ∗ ´ et ( A, F ℓ ) = 0 else, see[Mil06, Remark II.2.2]. Corollary 4.9 implies that Γ( A, F ℓ ( i )) ≃ Γ ´ et ( A, F ℓ ( i )) ≥− i . A transfer argumentshows Γ( Z [1 /ℓ ] , F ℓ ( i )) is a summand of Γ( A, F ℓ ( i )) , and similarly for Γ ´ et . We deduce the equivalence Γ( Z [1 /ℓ ] , F ℓ ( i )) ≃ Γ ´ et ( Z [1 /ℓ ] , F ℓ ( i )) ≥− i The same is true for C , R , F p since they are Nisnevich local. Consequently condition (1) will hold if thesquare Γ ´ et ( Z [1 /ℓ ] , F ℓ ( i )) −−−−→ Γ ´ et ( F p , F ℓ ( i )) y y Γ ´ et ( R , F ℓ ( i )) −−−−→ Γ ´ et ( C , F ℓ ( i )) OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 17 is cartesian and the maps H i ´ et ( R , F ℓ ( i )) ⊕ H i ´ et ( F p , F ℓ ( i )) → H i ´ et ( C , F ℓ ( i )) are surjective for every i . The first condition holds by [DF94, Theorem 2.1]. The second condition isvacuous when i > and easily verified for i = 0 . (cid:3) Remark . By adjoining an ℓ -th root of unity, one obtains the commutative square M ℓ ( C ⊗ Z [ ζ ℓ ]) −−−−→ M ℓ ( R ⊗ Z [ ζ ℓ ]) y y M ℓ ( F p ⊗ Z [ ζ ℓ ]) −−−−→ M ℓ ( Z [1 /ℓ, ζ ℓ ]) This induces a cartesian square in étale cohomology, but not in motivic cohomology (since e.g., the group H , ( Z [1 /ℓ, ζ ℓ ] , F ℓ ) = 0 but the corresponding map on H , is not surjective).4.3.4. Relation to étale homotopy types.
Corresponding to the squares in Theorems 4.13, 4.17 and 4.20,there are analogous squares of étale homotopy types ; in fact, Lemma 4.3, the only non-formal input inthe construction of the said squares, also holds for étale homotopy types. Due to the equivalence H ´ et Z /ℓ n ≃ H Z /ℓ n [( τ ) − ] ∈ SH ( S ) ∧ cell ℓ from e.g., [BEØ20, Theorem 7.4], our cocartesian squares in CM Z [1 /ℓ ] induce cartesian squares in étalecohomology with Z /ℓ n ( i ) -coefficients. By Dwyer–Friedlander [DF94, Theorem 2.1] [DF83, pp. 144–145],the resulting squares of étale homotopy types become pushouts after appropriate homological localization.By analyzing the proof of Theorem 4.20, one sees that condition (1) in Proposition 4.6 is satisfied ifthe following hold: • dim X ≤ , dim X n = 0 else. • The induced square of étale cohomology with F ℓ ( i ) -coefficients is cartesian. • The induced square of Zariski cohomology with F ℓ -coefficients is cartesian. • The group H ( X , R i ǫ ∗ F ℓ ( i )) = 0 for i ≥ .5. Applications to slice completeness and universal motivic invariants
We apply the results in Section 4 to show slice completeness and compute the endomorphism ring ofthe motivic sphere over regular number rings. Our completeness result for Voevodsky’s slice filtration[Voe02] is motivated by applications such as motivic generalizations of Thomason’s étale descent theoremfor algebraic K -theory in [ELSØ17] and [BEØ20], convergence of the slice filtration [Lev13], the solutionof Milnor’s conjecture on quadratic forms in [RØ16], computations of universal motivic invariants in[RSØ19] and of hermitian K -groups in [KRØ20].For the standard nomenclature associated with the slice filtration, such as the effective covers f q andthe effective cocovers f q , and the slice completion sc we refer to [RSØ18b, §3, (3.1), (3.3), (3.10)]. Let SH ( S ) ≥ denote the connective motivic spectra with respect to the homotopy t -structure on SH ( S ) [Hoy15a, §2.1]. The notion of a cell presentation of finite type is defined in [RSØ19, §3.3]. Proposition 5.1.
Suppose F is a -regular number field and set O ′ F := O F [1 / .(1) Let E • ∈ SH ( O ′ F ) ∧ cell be a tower such that lim n p ∗ i ( E n ) ≃ , where p ∗ i denotes the pullback to anyof the fields in Theorem 4.13(1). Then lim n E n ≃ is contractible.(2) If E ∈ SH ( O ′ F ) veff ∩ SH ( O ′ F ) cell is cellular and very effective, then E / is η -complete on homo-topy.(3) Let E ∈ SH ( O ′ F ) ≥ ∩ SH ( O ′ F ) cell and assume the slices of E are cellular and stable under basechange. Then there is an isomorphism π ∗ , ∗ (lim n f n ( E ) / (2 , ρ )) ≃ π ∗ , ∗ ( E / (2 , ρ )) (4) Let E ∈ SH ( O ′ F ) eff ∩SH ( O ′ F ) cell be cellular and effective. Assume E / has a Z (2) -cell presentationof finite type and its slices are cellular and stable under base change. Then E / (2 , η ) is slicecomplete on homotopy and π ∗ , ∗ (sc( E ) ∧ ) ≃ π ∗ , ∗ ( E ∧ ,η ) In particular, there is an isomorphism π ∗ , ∗ (sc( ) ∧ ) ≃ π ∗ , ∗ ( ∧ ) Proof. (1) Let I = {∗ → ∗ ← ∗} be the category so that lim I means pullback. For all X ∈ SH ( O ′ F ) ∧ cell we compute Map( X, lim n E n ) ≃ lim n Map( X, E n ) ≃ lim n lim i ∈ I Map( p ∗ i ( X ) , p ∗ i ( E n )) ≃ lim i lim n Map( p ∗ i ( X ) , p ∗ i ( E n )) ≃ lim i Map( p i ( X ) , lim n p ∗ i ( E n )) ≃ The result follows.(2) Recall that E is η -complete if and only if lim h · · · η −→ Σ , E η −→ Σ , E η −→ E i ≃ Thus by (1) it suffices to check p ∗ i ( E / is η -complete for each i , which holds by [BH20a, Theorem 5.1].(3) The claim holds if and only if lim n f n ( E ) / (2 , ρ ) ≃ on homotopy groups, or equivalently whencomputed in SH ( O ′ F ) cell . The assumptions imply f n ( E ) ∈ SH ( O ′ F ) cell and p ∗ i f n E ≃ f n p ∗ i E . Hence by(1) it suffices to note that lim n f n ( p ∗ i E ) / (2 , ρ ) ≃ p ∗ i ( E ) / (2 , ρ ) owing to [BH20a, Proposition 5.2].(4) For the first statement we need to prove lim n f n ( E / (2 , η )) ≃ on homotopy groups. As in (3),this reduces to the same statement over fields, which holds by [RSØ19, Proposition 3.49]. For the secondstatement we need to show sc( E / ≃ E ∧ η / , which holds by the proof of [RSØ19, Lemma 3.13]: sc( E / is η -complete since E / is effective, and sc( E / /η ≃ sc( E / (2 , η )) ≃ E / (2 , η ) — the first equivalence holdsby inspection of the slices. The final statement follows since the slices of (2) over O ′ F are known andhave the desired properties by [RSØ19, Remark 2.2, Theorem 2.12]. (cid:3) Remark . We expect that analogs of Proposition 5.1 hold over more general base schemes. Moreover,we expect that these results hold without the qualification “on homotopy.” Both shortcomings are a resultof our specific technique for accessing global sections of cellular spectra over arithmetic base schemes.Recall that any unit a ∈ O ( S ) × gives rise to a map [ a ] : → S , ∈ SH ( S ) , and hence an element h a i := 1 + η [ a ] ∈ π , ( S ) This turns π , ( ) into an Z [ O ( S ) × ] -algebra. We made use of the algebra structure in the formulationof Theorem 1.4 for Z [1 / . The generalization to -regular number rings takes the following form. Theorem 5.3.
Suppose F is a -regular number field with r real embeddings and c pairs of complexembeddings. For the endomorphism ring of the motivic sphere over the base scheme O ′ F := O F [1 / there is an isomorphism of Z [( O ′ F ) × ] -algebras π , ( O ′ F ) ⊗ Z (2) ≃ GW( O ′ F ) ⊗ Z (2) induced by the unit map → KO . Moreover, we have the vanishing result π ∗ , ( O ′ F ) ⊗ Z (2) = 0 for ∗ < Proof.
The presentation of Grothendieck-Witt rings of fields of characteristic = 2 by generators andrelations given in [Lam73, Theorem 4.1] implies there are Z [( O ′ F ) × ] -algebra isomorphisms GW( R ) ≃ Z ⊕ Z {h− i} , GW( C ) ≃ Z , GW( F q ) ≃ Z ⊕ Z / In the isomorphism for
GW( F q ) , the right-hand side has trivial multiplication on the square class group F × q / ( F × q ) ≃ Z / . As such, every n -dimensional form in GW( F q ) can be written as either n h i or ( n − h i⊕h a i , where a is a non-square element in F × q (we may choose a = − if and only if q ≡ ).Moreover, by [BKØ11, Proposition 2.1(7)] and the proof of [BW20, Theorem 5.8] one deduces the Z [( O ′ F ) × ] -algebra isomorphism GW( O ′ F ) ≃ Z r ⊕ ( Z / c Thus, for the closed points x, y , . . . , y c ∈ Spec( O ′ F ) in the notation of Theorem 4.13, there is a pullbacksquare of Z [( O ′ F ) × ] -algebras(5) GW( O ′ F ) −−−−→ L r GW( R ) ⊕ L c GW( y i ) y y GW( x ) −−−−→ L r + c GW( C ) The Grothendieck-Witt rings appearing in (5) are quotients of Z [( O ′ F ) × ] . Thus the maps in (5) areunique as Z [( O ′ F ) × ] -algebra maps. Since -adic completion is exact on finitely generated abelian groups,this square remains cartesian after -adic completion. OPOLOGICAL MODELS FOR STABLE MOTIVIC INVARIANTS OF REGULAR NUMBER RINGS 19
Consider the long exact sequence of homotopy groups associated with the pullback square(6) map( O ′ F , ∧ ) −−−−→ map( L r R ⊕ L c y i , ∧ ) y y a map( x , ∧ ) −−−−→ map( L r + c C , ∧ ) We have π ∗ , ( ∧ )( C ) ≃ ( π s ∗ ) ∧ by [Lev14, Corollary 2]. It follows that the right vertical map in (6)is surjective on homotopy groups. Indeed, recall that SH fin is the initial stable symmetric monoidal ∞ -category according to [BGT13, Theorem 3.1]. Thus for any symmetric monoidal stable ∞ -category C and symmetric monoidal functor F : C → SH ( C ) ∧ , there exists a factorization ( π s ∗ ) ∧ → π ∗ ( c ∧ ) F −→ π ∗ , (( C ) ∧ ) and the composite is surjective by Levine’s result. Thus using [Mor12, Corollary 6.43] we deduce thepullback square of rings(7) π , ( ∧ )( O ′ F ) −−−−→ L r GW( R ) ∧ ⊕ L c GW( y i ) ∧ y y GW( x ) ∧ −−−−→ L r + c GW( C ) ∧ Note that (7) comes from a diagram in CM O ′ F . Hence the maps in (7) are π , ( ∧ )( O ′ F ) -algebra maps,so a fortiori Z [( O ′ F ) × ] -algebra maps. The Grothendieck-Witt rings in (7) are quotients of Z [( O ′ F ) × ] ∧ ;thus the lower horizontal and right-hand vertical maps in (7) are unique Z [( O ′ F ) × ] -algebra maps. Thus(7) is the -adic completion of (6) and there is an isomorphism of Z [( O ′ F ) × ] -algebras π , ( ∧ )( O ′ F ) ≃ GW( O ′ F ) ∧ There is a similar pullback square for π , ( − ) ⊗ Q . Since the vanishing π , ( ∧ )( k ) ⊗ Q = 0 holds for k = R [DI16, Figure 4], k = C [Lev14, Corollary 2], and k = F q [WØ17, Theorem 1.3], we deduce thevanishing π , ( ∧ )( O ′ F ) ⊗ Q = 0 Inserted into the fracture square long exact sequence we get a pullback square of Z [( O ′ F ) × ] -algebras(8) π , ( )( O ′ F ) ⊗ Z (2) −−−−→ π , ( )( O ′ F ) ⊗ Q y y π , ( ∧ )( O ′ F ) −−−−→ π , ( ∧ )( O ′ F ) ⊗ Q (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) GW( O ′ F ) ∧ GW( O ′ F ) ∧ ⊗ Q By inspection there are isomorphisms of Z [( O ′ F ) × ] -algebras π , ( )( O ′ F ) ⊗ Q ≃ H r ´ et ( O ′ F , Q ) × H ( O ′ F , Q ) ≃ Q r × Q ≃ GW( O ′ F ) ⊗ Q We refer to [Bac20b, Theorem 7.2] for a proof of the first isomorphism. Since π , ( )( O ′ F ) ⊗ Q is aquotient of Z [( O ′ F ) × ] ⊗ Q , in (8), the right-hand vertical map π , ( )( O ′ F ) ⊗ Q ≃ GW( O ′ F ) ⊗ Q → π , ( ∧ )( O ′ F ) ⊗ Q ≃ GW( O ′ F ) ∧ ⊗ Q is the unique Z [( O ′ F ) × ] -algebra map. This shows we can identify the square of Z [( O ′ F ) × ] -algebras (8)with the corresponding fracture square for GW( O ′ F ) ⊗ Z (2) . It also follows that the unit map to KO induces an isomorphism, since π , (KO O F ) = GW( O ′ F ) is a quotient of Z [ O ′× F ] .Next we show the vanishing π ∗ , ( O ′ F ) ⊗ Z (2) = 0 for ∗ < . From (6), since a is surjective on π ∗ andall terms except possibly the top left vanish on π ∗ for ∗ < , we deduce that π ∗ , (( O ′ F ) ∧ ) = 0 for ∗ < .Since GW( O ′ F ) is finitely generated, the map GW( O ′ F ) ⊗ Q × GW( O ′ F ) ∧ → GW( O ′ F ) ∧ ⊗ Q is surjective. Considering the fracture square for π ∗ , ( O ′ F ) ⊗ Z (2) it thus remains to prove π ∗ , ( O ′ F ) ⊗ Q =0 for ∗ < . This follows from the identification of these groups with subquotients of the rational gammafiltration and rational real étale cohomology, both of which vanish in these degrees, as above. (cid:3) Applying the same proof method establishes the following odd-primary analog of Theorem 5.3.
Theorem 5.4.
Let ℓ be an odd prime number. Suppose F is ℓ -regular and µ ℓ ⊂ F . For the endomorphismring of O ′ F over the base scheme O ′ F := O F [1 /ℓ ] there is an isomorphism of Z [( O ′ F ) × ] -algebras π , ( O ′ F ) ⊗ Z ( ℓ ) ≃ GW( O ′ F ) ⊗ Z ( ℓ ) induced by the unit map → KO . Moreover, we have the vanishing result π ∗ , ( O ′ F ) ⊗ Z ( ℓ ) = 0 for ∗ < The same results hold for the motivic sphere over the base scheme Z [1 /ℓ ] when ℓ is a regular prime. References [ABR12] Carlos Andradas, Ludwig Bröcker, and Jesús M Ruiz.
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