Slices of hermitian K-theory and Milnor's conjecture on quadratic forms
aa r X i v : . [ m a t h . K T ] M a y Slices of hermitian K -theory andMilnor’s conjecture on quadratic forms Oliver R¨ondigs Paul Arne ØstværOctober 15, 2018
Abstract
We advance the understanding of K -theory of quadratic forms by computing the slices of themotivic spectra representing hermitian K -groups and Witt-groups. By an explicit computationof the slice spectral sequence for higher Witt-theory, we prove Milnor’s conjecture relating Galoiscohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. Ina related computation we express hermitian K -groups in terms of motivic cohomology. Contents K -theory 84 Slices 11 K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Homotopy orbit K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Hermitian K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 The mysterious summand is trivial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Higher Witt-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Hermitian K -theory, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Higher Witt-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Hermitian K -theory, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 K -groups 36A Maps between motivic Eilenberg-MacLane spectra 40 Introduction
Suppose that F is a field of characteristic char( F ) = 2. In [33] the Milnor K -theory of F is definedin terms of generators and relations by K M ∗ ( F ) = T ∗ F × / ( a ⊗ (1 − a )); a = 0 , . Here T ∗ F × is the tensor algebra of the multiplicative group of units F × . In degrees zero, oneand two these groups agree with Quillen’s K -groups, but for higher degrees they differ in general.Milnor [33] proposed two conjectures relating k M ∗ ( F ) = K M ∗ ( F ) / K M ∗ ( F ) to the mod-2 Galoiscohomology ring H ∗ ( F ; Z /
2) and the graded Witt ring GW ∗ ( F ) = ⊕ q ≥ I ( F ) q /I ( F ) q +1 for thefundamental ideal I ( F ) of even dimensional forms, via the two homomorphisms: k M ∗ ( F ) GW ∗ ( F ) ≺ s F ∗ H ∗ ( F ; Z / h F ∗ ≻ The solutions of the Milnor conjectures on Galois cohomology [62] – h F ∗ is an isomorphism– and on quadratic forms [39] – s F ∗ is an isomorphism – are two striking applications of motivichomotopy theory. For background and influence of these conjectures and also for the history oftheir proofs we refer to [4], [12], [22], [27], [34], [35], [43], [54]. In this paper we give an alternateproof of Milnor’s conjecture on quadratic forms by explicitly computing the slice spectral sequencefor the higher Witt-theory spectrum KT . Our method of proof applies also to smooth semilocalrings containing a field of characteristic zero.Let X ∈ Sm F be a smooth scheme of finite type over a field F . We refer to [13] for a surveyof the known constructions of the first quadrant convergent spectral sequence relating motiviccohomology to algebraic K -theory MZ ⋆ ( X ) = ⇒ KGL ∗ ( X ) . (1)From the viewpoint of motivic homotopy theory [60], the problem of constructing (1) reduces toidentifying the slices s q ( KGL ) of the motivic spectrum
KGL representing algebraic K -theory.Voevodsky introduced the slice spectral sequence and conjectured that s q ( KGL ) ∼ = Σ q,q MZ . (2)The formula (2) was proven for fields of characteristic zero by Voevodsky [61], [64], and (invokingdifferent methods) for perfect fields by Levine [28]. By base change the same holds for all fields.When char( F ) = 2 we are interested in the analogues of (1) and (2) for the motivic spectra KQ and KT representing hermitian K -groups and Witt-groups on Sm F , respectively [16]. Theorem4.18 shows the slices of hermitian K -theory are given by infinite wedge product decompositions s q ( KQ ) ∼ = ( Σ q,q MZ ∨ W i< q Σ i + q,q MZ / q ≡ W i< q +12 Σ i + q,q MZ / q ≡ . (3)Moreover, in Theorem 4.28 we compute the slices of higher Witt-theory, namely s q ( KT ) ∼ = _ i ∈ Z Σ i + q,q MZ / . (4)2he summand Σ q,q MZ in (3) is detected by showing that s q ( KGL ) is a retract of s q ( KQ ) if q is even. We deduce that s q ( KQ ) is a wedge sum of Σ q,q MZ and some MZ -module, i.e., amotive, which we identify with the infinite wedge summand in (3). Our first results show there isan additional “mysterious summand” Σ q,q M µ of s q ( KQ ). We show M µ is trivial by using basechange and the solution of the homotopy fixed point problem for hermitian K -theory of the primefields [5], [11], cf. [6], [20]. As conjectured in Hornbostel’s foundational paper [16], Theorem 3.4shows there is a homotopy cofiber sequence relating the algebraic and hermitian K -theories:Σ , KQ η ≻ KQ ≻ KGL (5)The stable Hopf map η is induced by the canonical map A r { } → P . We show this over anyfinite dimensional regular noetherian base scheme S equipped with the trivial involution and with 2invertible in its ring of regular functions, i.e., ∈ Γ( S, O S ). A closely related statement is obtainedin [49]. The sequence (5) is employed in our computation of the slices of KQ .The algebraic K -theory spectrum KGL affords an action by the stable Adams operation Ψ − .For the associated homotopy orbit spectrum KGL hC there is a homotopy cofiber sequence KGL hC ≻ KQ ≻ KT . (6)In (6), KGL hC → KQ is induced by the hyperbolic map KGL → KQ , while KQ → KT is thenatural map from hermitian K -theory into the homotopy colimit of the tower KQ η ≻ Σ − , − KQ Σ − , − η ≻ Σ − , − KQ Σ − , − η ≻ · · · . (7)We use the formulas s (Ψ − ) = id and Σ , Ψ − = − Ψ − to identify the slices s q ( KGL hC ) ∼ = ( Σ q,q MZ ∨ W i ≥ Σ i + q )+1 ,q MZ / q ≡ W i ≥ Σ i + q ) ,q MZ / q ≡ . (8)By combining the slice computations in (3) and (8) with the homotopy cofiber sequence in (6) wededuce the identification of the slices of the Witt-theory spectrum KT in (4). Alternatively, thisfollows from (3), (7), and Spitzweck’s result that slices commutes with homotopy colimits [50].Our next goal is to determine the first differentials in the slice spectral sequences as maps ofmotivic spectra. Because of the special form the slices of KQ and KT have, this involves themotivic Steenrod squares Sq i constructed by Voevodsky [62] and further elaborated on in [19].According to (4) the differential d KT ( q ) : s q ( KT ) ≻ Σ , s q +1 ( KT )is a map of the form _ i ∈ Z Σ i + q,q MZ / ≻ Σ , _ j ∈ Z Σ j + q,q MZ / . Let d KT ( q, i ) denote the restriction of d KT ( q ) to the i th summand Σ i + q,q MZ / s q ( KT ). Bycomparing with motivic cohomology operations of weight one, it suffices to consider d KT ( q, i ) : Σ i + q,q MZ / ≻ Σ i + q +4 ,q +1 MZ / ∨ Σ i + q +2 ,q +1 MZ / ∨ Σ i + q,q +1 MZ / . In Theorem 5.3 we show the closed formula d KT ( q, i ) = ( ( Sq Sq , Sq , i − q ≡ Sq Sq , Sq + ρ Sq , τ ) i − q ≡ . (9)3he classes τ ∈ h , and ρ ∈ h , are represented by − ∈ F ; here h p,q is shorthand for the mod-2motivic cohomology group in degree p and weight q . We denote integral motivic cohomology groupsby H p,q . This sets the stage for our proof of Milnor’s conjecture on quadratic forms formulated in[33, Question 4.3]. For fields of characteristic zero this conjecture was shown by Orlov, Vishik andVoevodsky in [39], and by Morel [36], [37] using different approaches.According to (4) the slice spectral sequence for KT fills out the upper half-plane. A strenuouscomputation using (9), Adem relations, and the action of the Steenrod squares on the mod-2 motiviccohomology ring h ⋆ of F shows that it collapses. We read off the isomorphisms E p,q ( KT ) = E ∞ p,q ( KT ) ∼ = ( h q,q p ≡ . To connect this computation with quadratic form theory, we show the spectral sequence convergesto the filtration of the Witt ring W ( F ) by the powers of the fundamental ideal I ( F ) of evendimensional forms. By identifying motivic cohomology with Galois cohomology for fields we arriveat the following result. Theorem 1.1. If char( F ) = 2 the slice spectral sequence for KT converges and furnishes a completeset of invariants e qF : I ( F ) q /I ( F ) q +1 ∼ = ≻ H q ( F ; Z / for quadratic forms over F with values in the mod- Galois cohomology ring. If X ∈ Sm F is a semilocal scheme and F a field of characteristic zero, our computationsand results extend to the Witt ring W ( X ) with fundamental ideal I ( X ) and the mod-2 motiviccohomology of X . Our reliance on the Milnor conjecture for Galois cohomology [62] can be replacedby Levine’s generalized Milnor conjecture on ´etale cohomology of semilocal rings [26], as shown in[15, § § KQ in low degrees. Weformulate our computation of the second orthogonal K -group, and refer to the main body of thepaper for the more complicated to state results on other hermitian K -groups. Theorem 1.2. If char( F ) = 2 there is a naturally induced isomorphism KO ( F ) ∼ = ≻ ker( τ ◦ pr + Sq : H , ⊕ h , ≻ h , ) . Throughout the paper we employ the following notation. S finite dimensional regular and separated noetherian base schemeSm S smooth schemes of finite type over SS m,n , Ω m,n , Σ m,n motivic ( m, n )-sphere, ( m, n )-loop space, ( m, n )-suspension SH , SH eff the motivic and effective motivic stable homotopy categories E , = S , generic motivic spectrum, the motivic sphere spectrum Acknowledgements:
We gratefully acknowledge both the editor and the referee for the twodetailed and helpful referee reports on this paper.
Let i q : Σ q,q SH eff ⊂ ≻ SH be the full inclusion of the localizing subcategory generated by Σ q,q -suspensions of smooth schemes. We denote by r q the right adjoint of i q and set f q = i q ◦ r q . The4 th slice of E is characterized up to unique isomorphism by the distinguished triangle f q +1 ( E ) ≻ f q ( E ) ≻ s q ( E ) ≻ Σ , f q +1 ( E ) (10)in SH [60]. When it is helpful to emphasize the base scheme S we shall write f Sq ( E ) and s Sq ( E ).Smashing with the motivic sphere Σ , has the following effect on the slice filtration. Lemma 2.1.
For all q ∈ Z there are natural isomorphisms f q (Σ , E ) ∼ = ≻ Σ , f q − ( E ) and s q (Σ , E ) ∼ = ≻ Σ , s q − ( E ) . which are compatible with the natural transformations occurring in (10).Proof. Let m and q be integers. The suspension functor Σ m,m restricts to a functorΣ m,mq − : Σ q,q SH eff ≻ Σ q + m ) ,q + m SH eff satisfying Σ m,m i q ( E ) = i q + m ◦ (Σ m,mq E ) for all E ∈ Σ q,q SH eff . This equality induces a uniquenatural isomorphism on the respective right adjoints: r q (cid:0) Σ − m, − m E (cid:1) ∼ = Σ − m, − mq + m r q + m ( E )In particular, there results a natural isomorphism: f q (Σ , E ) = i q r q (Σ , E ) ∼ = i q Σ , q − r q − ( E ) = Σ , i q − r q − ( E ) = Σ , f q ( E )In order to conclude the same for s q , observe that the inclusion i q +1 factors as i q ◦ i qq +1 , where i qq +1 : Σ q +1) ,q +1 SH eff ≻ Σ q,q SH eff is the natural inclusion. The functor i qq +1 has a right adjoint r qq +1 for the same reason that i q does,and the natural transformation f q +1 ≻ f q is obtained from the counit i qq +1 ◦ r qq +1 ≻ Id. Asbefore, the equality Σ m,mq i qq +1 ( E ) = i q + mq + m +1 ◦ (Σ m,mq +1 E ) induces a unique natural isomorphism onright adjoints, which serves to show that the diagram f q +1 (Σ , E ) ∼ = ≻ Σ , f q ( E ) f q (Σ , E ) g ∼ = ≻ Σ , f q − ( E ) g commutes. The remaining statements follow.Let η : S , ≻ denote the Hopf map induced by the canonical map A r { } ≻ P . Sinceevery motivic spectrum E is a module over the motivic sphere spectrum , multiplication with η defines, by abuse of notation, a map η : Σ , E ≻ E . Lemma 2.2.
For every E and q ∈ Z there is a naturally induced commutative diagram: f q +1 (Σ , E ) ∼ = ≻ Σ , f q ( E ) f q +1 Ef q +1 ( η ) g ≻ f q E η g roof. This follows from Lemma 2.1 by naturality.
Example 2.3.
The Hopf map induces a periodicity isomorphism η : Σ , KT ∼ = ≻ KT , cf. thedefinition (18) . In particular, the vertical map on the left hand side in the diagram of Lemma 2.2is an isomorphism. It also implies the isomorphism s q ( KT ) ∼ = Σ q,q s ( KT ) . The slices s q ( E ) are modules over the motivic ring spectrum s ( ), cf. [41, Theorem 3.6.22] and[14, § S is a perfect field, then s ( ) is the Eilenberg-MacLane spectrum MZ by the works of Levine [28] and Voevodsky [64]. We set D p,q,n = π p,n f q ( E ) and E p,q,n = π p,n s q ( E ).The exact couple D ∗ (0 , − , ≻ D ∗ E ∗ ≺ ( , , ) ≺ ( − , , ) gives rise to the slice spectral sequence E p,q,n = ⇒ π p,n ( E ) . (11)Our notation does not connote any information on the convergence of (11). The d -differential d E ( p, q, n ) : π p,n s q ( E ) ≻ π p − ,n s q +1 ( E )of (11) is induced on homotopy groups π p,n by the composite map d E ( q ) : s q ( E ) ≻ Σ , f q +1 ( E ) ≻ Σ , s q +1 ( E )of motivic spectra. The r th differential has tri-degree ( − , r, q th slice, if n > q the maps in · · · ≻ π p,n f q +1 ( E ) ≻ π p,n f q ( E ) ≻ · · · (12)are isomorphisms, and π p,n s q ( E ) is trivial. Thus only finitely many nontrivial differentials enter eachtri-degree, so that (11) is a half-plane spectral sequence with entering differentials. Let f q π p,n ( E )denote the image of π p,n f q ( E ) in π p,n ( E ). The terms f q π p,n ( E ) form an exhaustive filtration of π p,n ( E ). Moreover, E is called convergent with respect to the slice filtration if \ i ≥ f q + i π p,n f q ( E ) = 0for all p, q, n ∈ Z [60, Definition 7.1]. That is, the filtration { f q π p,n ( E ) } of π p,n ( E ) is Hausdorff.When E is convergent, the spectral sequence (11) converges [60, Lemma 7.2]. Precise convergenceproperties of the slice spectral sequence are unclear in general [29].When comparing slices along field extensions we shall appeal to the following base changeproperty of the slice filtration. Lemma 2.4.
Let α : X ≻ Y be a smooth map. For q ∈ Z there are natural isomorphisms f Xq α ∗ ∼ = ≻ α ∗ f Yq , s Xq α ∗ ∼ = ≻ α ∗ s Yq . roof. Any map α : X ≻ Y between base schemes yields a commutative diagram:Σ q,q SH eff ( Y ) α ∗ ≻ Σ q,q SH eff ( X ) SH ( Y ) i Yq g α ∗ ≻ SH ( X ) i Xq g Let α ∗ be the right adjoint of α ∗ . By uniqueness of adjoints, up to unique isomorphism, there existsa natural isomorphism of triangulated functors r Yq α ∗ ∼ = ≻ α ∗ r Xq . Since α is smooth, the functor α ∗ has a left adjoint α ♯ and there is a commutative diagram:Σ q,q SH eff ( X ) α ♯ ≻ Σ q,q SH eff ( Y ) SH ( X ) i Xq g α ♯ ≻ SH ( Y ) i Yq g By uniqueness of adjoints, there is an isomorphism r Xq α ∗ ∼ = ≻ α ∗ r Yq , and hence f Xq α ∗ = r Xq i Xq α ∗ = r Xq α ∗ i Yq ∼ = α ∗ r Yq i Yq = α ∗ f Yq . (13)The desired isomorphism for slices follows since, by uniqueness of adjoints, (13) is compatible withthe natural transformation f q +1 ≻ f q . Theorem 2.5.
Suppose I is a filtered partially ordered set and D : I op ≻ Sm Y , i ≻ X i is adiagram of Y -schemes with affine bonding maps. Let α : X ≡ lim i ∈I X i ≻ Y be the naturallyinduced morphism. For every q ∈ Z there are natural isomorphisms s Xq α ∗ ∼ = ≻ α ∗ s Yq and f Xq α ∗ ∼ = ≻ α ∗ f Yq . Proof.
Let G be a compact generator of the triangulated category Σ q +2 ,q +1 SH eff ( X ) and E ∈ SH ( Y ). According to [42, Theorem 2.12] it suffices to show [ G , α ∗ s Yq ( E )] is trivial. Note that G isof the form α ( j ) ∗ G j for G j a compact generator of the triangulated category Σ q +2 ,q +1 SH eff ( X j ).We consider the overcategory of an arbitrary element j ∈ I and the composite diagram D ↓ j : I op ↓ j ≻ I op D ≻ Sm Y , ( j → i ) X i . We claim the functor Φ : I op ↓ j ≻ I op yields an isomorphism colim D ↓ j ≻ colim D . For i ∈ I there exists an object i ′ and maps i → i ′ ← j , since I is filtered. Thus Φ ↓ i is nonempty.For zig-zags i → i ′ ← j and i → i ′′ ← j there exist maps i ′ → ℓ ← i ′′ such that the induced mapsfrom i to ℓ coincide, and similarly for j ( I is a partially ordered set.) Hence Φ ↓ i is connected.For j → i in I op ↓ j , let e ( i ) : X i ≻ X j be the structure map of the diagram. Thus the maphocolim j ↓I e ( i ) ∗ e ( i ) ∗ F ≻ α ( j ) ∗ α ( j ) ∗ F
7s an isomorphism in SH ( X j ) for all F ∈ SH ( X j ). Since G j is compact and slices commute withbase change along smooth maps by Lemma 2.4, the group[ G , α ∗ s Yq ( E )] = [ α ( j ) ∗ G j , α ∗ s Yq ( E )] ∼ = [ G j , α ( j ) ∗ α ( j ) ∗ β ( j ) ∗ s Yq ( E )] ∼ = [ G j , hocolim j ↓I e ( i ) ∗ e ( i ) ∗ β ( j ) ∗ s Yq ( E )] ∼ = colim j ↓I [ e ( i ) ∗ G j , e ( i ) ∗ β ( j ) ∗ s Yq ( E )]is trivial. With reference to [42, Remark 2.13] the second isomorphism follows in the same way. Lemma 2.6. If α : A ≻ B is a regular ring map then s Bq α ∗ ∼ = α ∗ s Aq .Proof. By Popescu’s general N´eron desingularization theorem [44] the regularity assumption on α is equivalent to B being the colimit of a filtered diagram of smooth A -algebras of finite type. Theresult follows now from Theorem 2.5. Corollary 2.7. If α : F ≻ E is a separable field extension then s Eq α ∗ ∼ = α ∗ s Fq . K -theory Let
KGL = (
K, . . . ) denote the motivic spectrum representing algebraic K -theory [57, § S . Here, K : Sm S ≻ Spt sends a smooth S -scheme to its algebraic K -theoryspectrum. The structure maps of KGL are given by the Bott periodicity operator β : K ∼ ≻ Ω , K. Suppose 2 is invertible in the ring of regular functions on S . Let KO : Sm S ≻ Spt denote thefunctor sending a smooth S -scheme to the (non-connective) spectrum representing the hermitian K -groups for the trivial involution on S and sign of symmetry ε = +1. Similarly, when ε = −
1, weuse the notation
KSp : Sm S ≻ Spt .There are natural maps f : KO ≻ K and f : KSp ≻ K induced by the forgetful functors.Taking the homotopy fibers of these forgetful maps yields the following homotopy fiber sequences:Ω , K h ′ ≻ V Q can ≻ KO f ≻ K Ω , K h ′ ≻ V Sp can ≻ KSp f ≻ K (14)Moreover, there are natural maps h : K ≻ KO and h : K ≻ KSp induced by the hyperbolic functors. Taking the homotopy fibers of these hyperbolic maps yieldsthe following homotopy fiber sequences:Ω , KO can ≻ U Q f ≻ K h ≻ KO Ω , KSp can ≻ U Sp f ≻ K h ≻ KSp (15)Karoubi’s fundamental theorem in hermitian K -theory [23] can be formulated as follows.8 heorem 3.1 (Karoubi) . There are natural weak equivalences φ : Ω , U Sp ∼≻ V Q and ψ : Ω , U Q ∼≻ V Sp.
In their foundational paper on localization in hermitian K -theory of rings [18, Section 1.8],Hornbostel-Schlichting show the following result. Theorem 3.2 (Hornbostel-Schlichting) . The homotopy cofiber of the map KO ≻ KO ( A r { } × S − ) resp . KSp ≻ KSp ( A r { } × S − ) induced by the map A r { } ≻ S is naturally weakly equivalent to Σ , U Q resp. Σ , U Sp . The homotopy cofiber sequences in Theorem 3.2 split by the unit section 1 ∈ A r { } ( S ). SinceΩ , is the homotopy fiber with respect to the unit section, Theorem 3.2 implies the natural weakequivalencesΩ , KO ∼ Ω , Σ , U Q ≺ ∼ U Q and Ω , KSp ∼ Ω , Σ , U Sp ≺ ∼ U Sp.
In [16] Hornbostel shows that hermitian K -theory is represented by the motivic spectrum KQ = ( KO, U Sp, KSp, U Q, KO, U Sp, . . . ) . Our notation emphasizes the connection between hermitian K -theory and quadratic forms. Thestructure maps of KQ are the adjoints of the weak equivalences: KO ∼≻ Ω , U Sp U Sp ∼≻ Ω , KSpKSp ∼≻ Ω , U Q U Q ∼≻ Ω , KO (16) Proposition 3.3.
There are commutative diagrams:
KO f ≻ K U Sp f ≻ K Ω , U Sp ∼ g Ω , f ≻ Ω , K ∼ β g Ω , KSp ∼ g Ω , f ≻ Ω , K ∼ β g KSp f ≻ K U Q f ≻ K Ω , U Q ∼ g Ω , f ≻ Ω , K ∼ β g Ω , KO ∼ g Ω , f ≻ Ω , K ∼ β g The forgetful map f : KQ ≻ KGL is given by the sequence ( f , f , f , f , f , . . . ) of mapsof pointed motivic spaces displayed in diagrams (14) and (15). Similarly, the hyperbolic map h : KGL ≻ KQ is given by the sequence ( h , h , h , h , h , . . . ). Here h and h have beenintroduced before, and h and h are defined by the weak equivalences from Theorem 3.1 and thecanonical maps h ′ and h ′ introduced with the construction of V Q and
V Sp in diagram (14). Byinspection of the structure maps of KQ determined by (16), devising a map KQ ≻ Ω , KQ ofmotivic spectra is tantamount to giving a compatible sequence of maps between motivic spaces KO ≻ Σ , U Q, U Sp ≻ Σ , KO, KSp ≻ Σ , U Sp, U Q ≻ Σ , KSp, . . . .
Next we show the homotopy cofiber sequence (5) relating algebraic and hermitian K -theory viathe stable Hopf map, as reviewed in the introduction. A closely related statement is obtained in[49, Theorem 6.1]. 9 heorem 3.4. The stable Hopf map and the forgetful map yield a homotopy cofiber sequence Σ , KQ η ≻ KQ f ≻ KGL ≻ Σ , KQ . The connecting map factors as
KGL ∼ = ≻ Σ , KGL Σ , h ≻ Σ , KQ .Proof. We show the map KQ ≻ Ω , KQ induced by η is determined by the canonical maps KO ≻ Σ , U Q U Sp ∼ Σ , V Q ≻ Σ , KOKSp ≻ Σ , U Sp U Q ∼ Σ , V Sp ≻ Σ , KSp.
To that end, it suffices to describe the maps KO ( P ) ≻ KO ( A r { } ) U Sp ( P ) ≻ U Sp ( A r { } ) KSp ( P ) ≻ KSp ( A r { } ) U Q ( P ) ≻ U Q ( A r { } )induced by the unstable Hopf map A r { } ≻ P , or equivalently, see [38, p. 73 in § A r { } ) × S ( A r { } ) ≻ A r { } , ( x, y ) ≻ xy − . We note there is an isomorphism of schemes θ : ( A r { } ) × S ( A r { } ) ≻ ( A r { } ) × S ( A r { } ) , ( x, y ) ≻ ( xy, y ) . Now the composite Υ θ is the projection map on the first factor. Thus for KO , U Sp , KSp and
U Q , the homotopy cofibers of the maps induced by Υ and the projection map coincide up to weakequivalence. The latter homotopy cofibers are given in Theorem 3.2.
Lemma 3.5.
The unit map ≻ KGL factors as ≻ KQ f ≻ KGL .Proof.
The unit map ≻ KGL is given by the trivial line bundle over the base scheme S . Thelatter is obtained by forgetting the standard nondegenerate quadratic form on it, which providesthe factorization.Let ǫ : ≻ be the endomorphism of the sphere spectrum induced by the commutativityisomorphism on the smash product S , ∧ S , . Lemma 3.6.
The composition KQ f ≻ KGL h ≻ KQ coincides with multiplication by − ǫ .Proof. Since 1 − ǫ = 1 + h− i is the hyperbolic plane [38, p. 53] the unit map for KQ induces acommutative diagram: ≻ KQ1 − ǫ g ≻ KQ h ◦ f g By smashing with KQ and employing its multiplicative structure we obtain the diagram: KQ ≻ KQ ∧ KQ ≻ KQKQKQ ∧ (1 − ǫ ) g ≻ KQ ∧ KQ g ≻ KQ h ◦ f g The middle vertical map is KQ ∧ ( h ◦ f ), while the two composite horizontal maps coincide withthe identity on KQ . The right hand square commutes because h ◦ f is a map of KQ -modules.10 Slices
This section contains a determination of the slices of hermitian K -theory and Witt theory over anyfield of characteristic not two. These are the first examples of non-orientable motivic spectra forwhich all slices are explicitly known. Our starting point is the computation of the slices of algebraic K -theory. K -theory We recall and augment previous work on the slices of algebraic K -theory. Theorem 4.1 (Levine, Voevodsky) . The unit map ≻ KGL induces an isomorphism of zeroslices s ( ) ≻ s ( KGL ) . Hence there is an isomorphism s q ( KGL ) ∼ = Σ q,q MZ for all q ∈ Z .Proof. By [28], [61], and [64], and base change to any field as in [19], the unit ≻ KGL inducesa map MZ ∼ = s ( ) ≻ s ( KGL ) ∼ = MZ corresponding to multiplication by an integer i ∈ MZ , ∼ = Z . Now s ( ) ≻ s ( KGL ) is a mapof ring spectra by multiplicativity of the slice filtration [14, Theorem 5.19], [41]. It follows that i = 1. Moreover, the q th power β q : S q,q ≻ KGL of the Bott map induces an isomorphismΣ q,q MZ ∼ = s q ( S q,q ) ≻ s q ( KGL ) . K -theory Let P be pointed at ∞ . The general linear group scheme GL (2 n ) acquires an involution given by (cid:18) A BC D (cid:19) ≻ (cid:18) D t B t C t A t (cid:19) − . Geometrically, this corresponds to a strictification of the pseudo-involution obtained by sending avector bundle of rank n to its dual. These involutions induce the inverse-transpose involution onthe infinite general linear group scheme GL and its classifying space B GL [56]. Letting C operatetrivially on the first factor in Z × B GL , the involution coincides sectionwise with the unstableAdams operation Ψ − on the motivic space representing algebraic K -theory. The stable Adamsoperation Ψ − : KGL ≻ KGL is determined by the structure map K ≻ Ω , K = hofib( K ( − × P ) ∞ ∗ ≻ K ( − )) obtained frommultiplication by the class 1 − [ O ( − ∈ K ( P ) of the hyperplane section. Note thatΨ − (1 − [ O ( − − [ O (1)] = 1 − (1 + (1 − [ O ( − O ( − − − (1 − [ O ( − . Thus, up to homotopy, the stable operation Ψ − : KGL ≻ KGL is given levelwise on motivicspaces by the formula Ψ − ,n = ( Ψ − n ≡ − Ψ − n ≡ . S , shifts motivic spectra by one index we getΣ , Ψ − = − Ψ − . (17)When forming homotopy fixed points and homotopy orbits of KGL we implicitly make use of anaive C -equivariant motivic spectrum, i.e., a non-equivariant motivic spectrum with a C -action(given by Ψ − ) which maps by a levelwise weak equivalence to KGL . Recall the Witt-theoryspectrum KT is the homotopy colimit of the sequential diagram KQ η ≻ Σ − , − KQ Σ − , − η ≻ Σ − , − KQ Σ − , − η ≻ · · · . (18)Note that KT is a motivic ring spectrum equipped with an evident KQ -algebra structure. Ournotation follows [16] and reminds us of the fact that KT is an example of a ”Tate spectrum” ormore precisely ”geometric fixed point spectrum” via the homotopy cofiber sequence relating it to KQ and homotopy orbit algebraic K -theory [25]. Theorem 4.2 (Kobal) . There is a homotopy cofiber sequence
KGL hC ≻ KQ ≻ KT . The involution on
KGL induces an MZ -linear involution on the slices s q ( KGL ) ∼ = Σ q,q MZ .Suspensions of MZ allow only two possible involutions, namely the identity (trivial involution) andthe multiplication by − MZ × , ∼ = {± } . Proposition 4.3.
Let ( KGL , Ψ − ) be the motivic K -theory spectrum with its Adams involution.The induced involution on s q ( KGL ) ∼ = Σ q,q MZ is nontrivial if q is odd and trivial if q is even.Proof. This follows from (17) and the equality s (Ψ − ) = id MZ . For the latter, note that theAdams involution and the unit map of KGL yield commutative diagrams:
KGL s ( KGL ) ι ≻ s ( ) s ( ι ) ≻ KGL Ψ − g ι ≻ s ( KGL ) s (Ψ − ) g s ( ι ) ≻ By reference to Theorem 4.1 it follows that s (Ψ − ) = id MZ .Next we consider the composite of the hyperbolic and forgetful maps f ◦ h = Ψ + Ψ − : KGL ≻ KGL . (19) Proposition 4.4.
The endomorphism s q ( f h ) of s q ( KGL ) ∼ = Σ q,q MZ is multiplication by if q iseven and the trivial map if q is odd.Proof. Apply Proposition 4.3, the additivity of s q and s q (Ψ ) = s q (id KGL ) = id s q ( KGL ) . Lemma 4.5.
The slice functor s q commutes with homotopy colimits for all q ∈ Z .Proof. By a general result for Quillen adjunctions between stable pointed model categories shownby Spitzweck [50, Lemma 4.4], the conclusion follows because s q commutes with sums.12he idea is now to combine Theorem 4.1, Proposition 4.3 and Lemma 4.5 to identify the slicesof homotopy orbit K -theory. To start with, we compute the homotopy orbit spectra of MZ forthe trivial involution and the unique nontrivial involution over a base scheme essentially smoothover a field. These computations are parallel to the corresponding computations for the topologicalintegral Eilenberg-MacLane spectrum H Z , as the proofs suggest. In the case where F is a subfieldof the complex numbers, complex realization – which is compatible with homotopy colimits, andsends MZ to H Z by [30, Lemma 5.6] – maps the motivic computation to the topological one. Lemma 4.6.
With the trivial involution on motivic cohomology there is an isomorphism ( MZ , id) hC ∼ = MZ ∨ ∞ _ i =0 Σ i +1 , MZ / . Proof. If E is equipped with an involution then E hC ∼ = (cid:0) E ∧ ( EC ) + (cid:1) C , where EC is a contractiblesimplicial set with a free C -action considered as a constant motivic space. In the case of ( MZ , id),( MZ , id) hC ∼ = (cid:0) MZ ∧ ( EC ) + (cid:1) C ∼ = MZ ∧ ( EC ) + /C ∼ = MZ ∧ RP ∞ + . Since S n, ≻ RP n ≻ RP n +1 is a homotopy cofiber sequence, so isΣ n, MZ ≻ MZ ∧ RP n ≻ MZ ∧ RP n +1 . (20)Clearly MZ ∧ RP ∼ = Σ , MZ /
2. Proceeding by induction on (20) using Lemma A.3, we obtain MZ ∧ RP n ∼ = (W n − i =0 Σ i +1 , MZ / n ≡ n, MZ ∨ W n − i =0 Σ i +1 , MZ / n ≡ . Lemma 4.7.
With the nontrivial involution σ on motivic cohomology there is an isomorphism ( MZ , σ ) hC ∼ = ∞ _ i =0 Σ i, MZ / . Proof.
The nontrivial involution σ on MZ is determined levelwise by the nontrivial involution on MZ n = K ( Z , n, n ) = Z tr ( S n,n ) . Here, Z tr sends a motivic space X to the motivic space with transfers freely generated by X , cf. [10,Example 3.4]. On the level of motivic spaces the nontrivial involution is induced by a degree − S , . Since Z tr commutes with homotopy colimits [47, §
2] weare reduced to identify hocolim C S n,n . When n = 0 the homotopy colimit is contractible. When n > C S n,n ∼ = hocolim C Σ n − ,n S , ∼ = Σ n − ,n (hocolim C S , ) ∼ = Σ n − ,n RP ∞ . By passing to the sphere spectrum the above yields an isomorphismhocolim C ∼ = Σ − , RP ∞ . hC ( MZ , σ ) ∼ = ∞ _ i =0 Σ i, MZ / . Theorem 4.8.
Suppose F is a field equipped with the trivial involution and char( F ) = 2 . Theslices of homotopy orbit K -theory are given by s q ( KGL hC ) ∼ = Σ q,q MZ ∨ W ∞ i = q Σ q +2 i +1 ,q MZ / q ≡ W ∞ i = q − Σ q +2 i +1 ,q MZ / q ≡ . Proof.
This follows from Theorem 4.1, Proposition 4.3 and Lemmas 4.5, 4.6, 4.7. K -theory Throughout this section F is a field of char( F ) = 2. Corollary 4.9.
There is a splitting of MZ -modules s ( KQ ) ∼ = s ( KGL ) ∨ µ which identifies s ( f ) with the projection map onto s ( KGL ) .Proof. Lemma 3.5 shows the unit of KQ and the forgetful map to KGL furnish a factorization ι ≻ KQ f ≻ KGL of the unit of
KGL . By Theorem 4.1 the composite s ( ) s ( ι ) ≻ s ( KQ ) s ( f ) ≻ s ( KGL )is an isomorphism. The desired splitting follows since retracts are direct summands in SH . Proposition 4.10.
The composite KQ f ≻ KGL h ≻ KQ induces the multiplication by mapon s q ( KQ ) for all q ∈ Z .Proof. The unit of MZ induces an isomorphism on zero slices by Theorem 4.1. Since MZ ∈ SH eff is an effective motivic spectrum, the counit f ( MZ ) ≻ MZ is an isomorphism. Thus there is acanonical isomorphism of ring spectra MZ ∼ = f ( MZ ) ≻ s ( MZ ) ∼ = MZ . It follows that smashing with MZ , i.e., passing to motives [46], [47], induces a canonical ring map[ , ] ≻ [ s ( ) , s ( )] ∼ = [ MZ , MZ ] . Lemma 3.6 shows the composite hf coincides with 1 − ǫ . Its image in motives is multiplication by2 because the twist isomorphism of Z (1) ⊗ Z (1) is the identity map [59, Corollary 2.1.5]. Corollary 4.11.
In the splitting s ( KQ ) ∼ = s ( KGL ) ∨ µ the homotopy groups π s,t µ are modulesover π , MZ / ∼ = F . roof. This follows from Corollary 4.9 and Proposition 4.10.The slice functors are triangulated. Thus the homotopy cofiber sequence in Theorem 3.4 induceshomotopy cofiber sequences of slices.
Lemma 4.12.
There is a distinguished triangle of MZ -modules Σ , MZ Σ , (2 , ≻ Σ , ( MZ ∨ µ ) ≻ s ( KQ ) ≻ Σ , MZ , where ∈ Z ∼ = MZ , and ∈ π , µ .Proof. The distinguished triangle follows from Theorems 3.4, 4.1, Lemma 2.1 and Corollary 4.9.Since induced maps between slices are module maps over s ( ) ∼ = MZ , we haveHom MZ (Σ , MZ , Σ , ( MZ ∨ µ )) ∼ = Hom SH ( , MZ ∨ µ ) ∼ = MZ , ⊕ π , µ. This shows MZ ≻ MZ ∨ µ is of the form ( a, α ) ∈ Z ⊕ π , µ . By the proof of Corollary 4.9, s ( f ) isthe projection map onto the direct summand s ( KGL ). The connecting map in the distinguishedtriangle identifies with the (1 , h . It follows that s − ( h ) = 0. ByProposition 4.10, s ( hf ) is multiplication by 2. Thus a = 2 and π , s ( KGL ) ≻ π , s ( KQ ) isinjective. Consider the short exact sequence0 ≻ π , s ( KGL ) ≻ π , s ( KQ ) ≻ π , cone(2 , α ) ≻ π − , s KGL = 0 . Proposition 4.10 shows that s ( KQ ) ≻ s ( KGL ) ≻ s ( KQ ) is multiplication by 2. Hencethe same holds for the induced map π , cone(2 , α ) ≻ π , Σ , MZ = 0 ≻ π , cone(2 , α ). Itfollows that π , cone(2 , α ) is an F -module. We note that the image of (1 , α ) has order 4 unless α = 0: Write π , cone(2 , α ) as the cokernel of (2 , α ) : Z ≻ Z ⊕ A , A an F -module. Its subgroupgenerated by (1 , α ) is { (1 , α ) , (2 , , (3 , α ) , (4 ,
0) = 2(2 , α ) = 0 } , and if α = 0, then (2 , = 0. Corollary 4.13.
There is an isomorphism s ( KQ ) ∼ = Σ , ( MZ / ∨ µ ) and s ( f ) coincides with thecomposite map s ( KQ ) ∼ = Σ , ( MZ / ∨ µ ) pr ≻ Σ , MZ / δ ≻ Σ , MZ ∼ = s ( KGL ) . Proof.
This follows from Lemma 4.12.
Lemma 4.14.
The map s ( h ) : s ( KGL ) ≻ s KQ is trivial.Proof. The first component of s ( h ) : s ( KGL ) ≻ s ( KQ ) ∼ = Σ , ( MZ / ∨ µ )is trivial for degree reasons by Lemma A.3. Thus s ( h ) corresponds to the transpose of the matrix (cid:0) β (cid:1) . By Corollary 4.13, s ( f ) corresponds to the matrix (cid:0) δ (cid:1) . Proposition 4.10 implies that s ( hf ) = (cid:18) βδ (cid:19) is the multiplication by 2 map, hence trivial on Σ , ( MZ / ∨ µ ). We conclude that βδ = 0. Thus β factors as Σ , MZ ≻ Σ , MZ β ′ ≻ Σ , µ, which coincides with Σ , MZ β ′ ≻ Σ , µ ≻ Σ , µ. Since the multiplication by 2 map on µ is trivial, the result follows.15 orollary 4.15. There is an isomorphism s ( KQ ) ∼ = ≻ Σ , MZ ∨ Σ , ( MZ / ∨ µ ) which identifies s ( f ) with the projection map onto Σ , MZ .Proof. Theorem 4.1 and Corollary 4.13 show that, up to isomorphism, Theorem 3.4 gives rise tothe distinguished triangleΣ , MZ Σ , s ( h ) ≻ Σ , ( MZ / ∨ µ ) ≻ s ( KQ ) s ( f ) ≻ Σ , MZ . Lemma 4.14 implies that s ( h ) is the trivial map. Lemma 4.16.
The map s ( h ) : s ( KGL ) ≻ s ( KQ ) coincides with the composite Σ , MZ (2 , ≻ Σ , MZ ∨ Σ , ( MZ / ∨ µ ) . Proof.
Corollary 4.15 identifies s ( f ) with the projection map onto Σ , MZ . Since s ( f h ) = 2 byProposition 4.4, s ( h ) = (2 , γ , γ ) where( γ , γ ) : Σ , MZ ≻ Σ , ( MZ / ∨ µ ) . Note that γ = 0 by Lemma A.3. Similarly s ( hf ) = 2 by Proposition 4.10. Using the matrix s ( hf ) = γ we conclude that γ = 0. Corollary 4.17.
There is an isomorphism s ( KQ ) ∼ = Σ , MZ / ∨ Σ , ( MZ / ∨ µ ) . Moreover, s ( f ) coincides with the composite map s ( KQ ) ∼ = Σ , MZ / ∨ Σ , ( MZ / ∨ µ ) pr ≻ Σ , MZ / δ ≻ Σ , MZ ∼ = s ( KGL ) . Proof.
Apply s to the Hopf cofiber sequence in Theorem 3.4 and use Lemma 4.16. Theorem 4.18.
The slices of the hermitian K -theory spectrum KQ are given by s q ( KQ ) = ((cid:0) Σ q,q MZ (cid:1) ∨ (cid:0) Σ q,q M µ (cid:1) ∨ W i< q Σ i + q,q MZ / q ≡ (cid:0) Σ q,q M µ (cid:1) ∨ W i< q +12 Σ i + q,q MZ / q ≡ . Here M µ ∼ = Σ , M µ and M µ s,t is an F -module for all integers s and t .Proof. Corollary 4.17 identifies the third slice s ( KQ ) ∼ = Σ , MZ / ∨ Σ , ( MZ / ∨ µ ) . On the other hand, Karoubi periodicity and Corollary 4.9 imply the isomorphism s ( KQ ) ∼ = s (Σ , KQ ) ∼ = Σ , s − ( KQ ) ∼ = Σ , µ. It follows that Σ − , MZ ∨ Σ − , MZ is a direct summand of µ . Iterating the procedure furnishesan isomorphism µ ∼ = M µ ∨ _ i< Σ − i, MZ / , where M µ is simply the complementary summand. The result follows.16n the next section we show the mysterious summand M µ of s ( KQ ) is trivial. The followingsummarizes some of the main observations in this section. Proposition 4.19.
The map s q ( f ) : s q ( KQ ) ≻ s q ( KGL ) is the projection onto Σ q, q MZ , s q +1 ( f ) : s q +1 ( KQ ) ≻ s q +1 ( KGL ) is the projection map onto Σ q +1 , q +1 MZ / composed with δ : Σ q +1 , q +1 MZ / ≻ Σ q +2 , q +1 MZ . The map s i ( h ) : s i ( KGL ) ≻ s i ( KQ ) is multiplicationby composed with the inclusion of Σ q, q MZ if i = 2 q , and trivial if i is odd. To prove M µ is trivial we use the solution of the homotopy limit problem for hermitian K -theoryof prime fields due to Berrick-Karoubi [5] and Friedlander [11]. Their results have been generalizedby Hu-Kriz-Ormsby [20] and Berrick-Karoubi-Schlichting-Østvær [6], where the following is shown.Here vcd ( F ) = cd ( F ( √− F . Theorem 4.20. If vcd ( F ) < ∞ there is a canonical weak equivalence KQ / ≻ KGL hC / . To make use of Theorem 4.20 we need to understand how slices compare with mod-2 reductionsof motivic spectra and formation of homotopy fixed points. The former is straightforward becausethe slice functors are triangulated.
Lemma 4.21.
Let E be a motivic spectrum. There is a canonical isomorphism s q ( E ) / ∼ = s q ( E / . Lemma 4.22.
There is a naturally induced commutative diagram: ( MZ , id) hC ∼ = ≻ MZ ∨ _ i< Σ i, MZ / MZ ≺ p r ≻ Proof.
With the trivial C -action on MZ the homotopy fixed points spectrum is given by( MZ , id) hC = sSet ∗ ( RP ∞ + , MZ ) ∼ = MZ ∨ sSet ∗ ( RP ∞ , MZ ) . The canonical map ( MZ , id) hC ≻ MZ corresponds to the map induced by Spec( F ) + ⊂ ≻ RP ∞ + .Induction on n shows there are isomorphisms sSet ∗ ( RP n , MZ ) ∼ = Σ − n, MZ × Q n − i =1 Σ − i, MZ / n ≡ Q n i =1 Σ − i, MZ / n ≡ . Writing sSet ∗ ( RP ∞ , MZ ) = colim n sSet ∗ ( RP n , MZ ) we deduce sSet ∗ ( RP ∞ , MZ ) ∼ = Y i< Σ i, MZ / . Proposition A.5 identifies the latter with W i< Σ i, MZ / Lemma 4.23.
The nontrivial involution on MZ and the connecting map δ : Σ − , MZ / ≻ MZ (on the zeroth summand below) give rise to the commutative diagram: ( MZ , σ ) hC ∼ = ≻ _ i ≤ Σ i − , MZ / MZ ≺ δ ≻ roof. Let (
G, σ ) be an involutive simplicial abelian group with σ ( g ) = − g and EC a contractiblesimplicial set with a free C -action. Then C acts on the space of fixed points sSet ( EC , G ) C byconjugation, i.e., on the simplicial set of maps from EC to ( G, σ ). Now choose EC such thatits skeletal filtration ( S , g ) ⊆ ( S , g ) ⊆ · · · ⊆ ( S n , g ) ⊆ · · · comprises spheres equipped with theantipodal C -action. Here ( S n +1 , g ) arises from ( S n , g ) by attaching a free C -cell of dimension n + 1 along the C -map C × S n ≻ ( S n , g ) adjoint to the identity on S n . Thus the inclusion( S n , g ) ⊆ ( S n +1 , g ) yields a pullback square of simplicial sets: sSet (cid:0) ( S n +1 , g ) , ( G, σ ) (cid:1) C ≻ sSet (cid:0) ( S n , g ) , ( G, σ ) (cid:1) C sSet ( D n +1 , G ) g ≻ sSet ( S n , G ) g (21)By specializing to n = 0, (21) extends to the commutative diagram: sSet (cid:0) ( S , g ) , ( G, σ ) (cid:1) C ≻ sSet (cid:0) ( S , g ) , ( G, σ ) (cid:1) C ∼ = G ≻ sSet ( D , G ) g φ ≻ sSet ( S , G ) ∼ = G × Gx ( x, − x ) g + ≻ G g φ is a fibrant replacement of the diagonal G ≻ G × G . The right hand square is a pullback, andaddition is a Kan fibration. Hence sSet (cid:0) ( S , g ) , ( G, σ ) (cid:1) C is the homotopy fiber of multiplicationby 2 on G . On the other hand, there is a homotopy cofiber sequence S ≻ S ∼ = RP ≻ RP and sSet (cid:0) ( S , g ) , ( G, σ ) (cid:1) C has a canonical basepoint. Thus there is a homotopy equivalenceΩ sSet (cid:0) ( S , g ) , ( G, σ ) (cid:1) C ≃ sSet ∗ ( RP , G ) . By induction on (21), we find for every n ≥ sSet (cid:0) ( S n , g ) , ( G, σ ) (cid:1) C ≃ sSet ∗ ( RP n +1 , G ) , which implies Ω( G, σ ) hC ≃ sSet ∗ ( RP ∞ , G ) . (22)A levelwise and sectionwise application of (22) yields the weak equivalencesΩ , ( MZ , σ ) hC ∼ = sSet ∗ ( RP ∞ , MZ ) ∼ = Y i< S i, MZ / . (For the second weak equivalence see the proof of Lemma 4.6.) The canonical map from ( MZ , σ ) hC to MZ corresponds to the map induced by RP ⊂ ≻ RP ∞ . Proposition A.5 concludes the proof. Proposition 4.24.
Suppose that vcd ( F ) < ∞ . The canonical map s q ( KGL hC ) ≻ s q ( KGL ) hC is an isomorphism for every integer q . roof. Let F be an arbitrary field of characteristic not 2. Lemma 4.26 shows that KGL hC satisfiesthe properties which were used to determine the slices of KQ . As in the proof of Theorem 4.18,this results in a splitting s q ( KGL hC ) = ((cid:0) Σ q,q MZ (cid:1) ∨ (cid:0) Σ q,q M ν (cid:1) ∨ W i< q Σ i + q,q MZ / q ≡ , (cid:0) Σ q,q M ν (cid:1) ∨ W i< q +12 Σ i + q,q MZ / q ≡ . Here M ν ∼ = Σ , M ν and M ν s,t is an F -module for all integers s and t . Hence the vanishing of M ν can be deduced from a computation of the zero slice of KGL hC / (cid:0) KGL / (cid:1) hC . Moreover, thehomotopy norm cofiber sequence KGL hC ≻ KGL hC ≻ \ KGL C ≻ Σ , KGL hC from [20, Diagram (20)] and Theorem 4.8 imply that M ν is also a direct summand of s (cid:0) \ KGL C (cid:1) .Suppose that vcd ( F ) < ∞ . Theorem 4.20 shows the canonical maps KQ / ≻ KGL hC / KT / ≻ \ KGL / C are equivalences. Thus the Tate motivic spectrum \ KGL / C is cellularin the sense of [9], since KQ is cellular [48]. (The latter is deduced using the model for KQ from[40].) Hence the zero slice of \ KGL / C is the motivic Eilenberg-MacLane spectrum associated toa chain complex of abelian groups. This chain complex does not depend on the base field. Thusit suffices to compute the zero slice of \ KGL / C (or, equivalently, of KT /
2) over an algebraicallyclosed field of characteristic not two. For this one can employ the specific cell presentation givenin [17, Theorem 3.2] (see also [45, Section 4]), which extends to any algebraically closed field ofcharacteristic not two via the base change techniques used in the proof of Lemma 5.1. It showsthat KT may be obtained in two steps from the η -inverted sphere [ η ]. The first step produces acanonical map D ∞ ≻ KT from the colimit D ∞ of the sequential diagram [ η ] = D ≻ D ≻ · · · ≻ D n ≻ · · · where D n ≻ D n +1 is the canonical map to the homotopy cofiber of the unique nontrivial mapΣ n − , [ η ] ≻ D n . The second step provides that D ∞ ≻ KT is obtained by inverting theunique nontrivial element κ ∈ π , D ∞ . The computation of the slices of the sphere spectrum [30, §
8] leads to the computation s ( [ η ]) ∼ = MZ / ∨ _ n ≥ Σ n, MZ / s ( D ∞ ) ∼ = _ n ≥ Σ n, MZ / KT ∼ = D ∞ [ κ ], the zero slice of KT is s ( KT ) ∼ = s (cid:0) D ∞ [ κ ] (cid:1) ∼ = s ( D ∞ )[ s κ ] ∼ = _ n ∈ Z Σ n, MZ / . It follows that M ν is contractible, which proves the statement.19 heorem 4.25. Suppose that vcd ( F ) < ∞ . The slices of homotopy fixed point algebraic K -theory KGL hC are given by s q ( KGL hC ) = Σ q,q (cid:16)(cid:0) Σ q, MZ (cid:1) ∨ W i< q Σ i, MZ / (cid:17) q ≡ q,q W i< q +12 Σ i, MZ / q ≡ . Proof.
This follows from Theorem 4.1, Lemmas 4.22, 4.23, and Proposition 4.24.
Lemma 4.26.
The map KQ ≻ KGL hC induces the projection map on all slices.Proof. This is clear from the following claims.1. The map θ : KQ ≻ KGL hC fits into a commutative diagram of homotopy cofiber se-quences: Σ , KQ η ≻ KQ f ≻ KGL Σ , h ◦ β ≻ Σ , KQ Σ , KGL hC Σ , θ g η ≻ KGL hC θ g f ′ ≻ KGL id g Σ , h ′ ◦ β ≻ Σ , KGL hC Σ , θ g
2. The map s θ : s KQ ≻ s KGL hC is the identity on the summand MZ .3. The map h ′ ◦ f ′ : KGL hC ≻ KGL hC is multiplication with 1 − ǫ .4. The map f ′ ◦ h ′ : KGL ≻ KGL coincides with 1 + Ψ − .To prove the first claim, consider the homotopy fiber of the canonical map θ : KQ ≻ KGL hC .The Tate diagram [20, Diagram (20)] shows that it coincides with the homotopy fiber of thecanonical map KT ≻ \ KGL C . Since η acts invertibly on these two motivic spectra, it actsinvertibly on the homotopy fiber of θ . This implies that the diagramΣ , KQ η ≻ KQ Σ , KGL hC Σ , θ g η ≻ KGL hC θ g is a homotopy pullback square, and hence the first claim. See also [21, Remark 5.9]. The secondclaim follows from the factorization of the unit map for algebraic K -theory unit ≻ KQ θ ≻ KGL hC f ′ ≻ KGL . By the proof of Lemma 3.6 the third claim follows from the commutative diagram: KQ f ≻ KGL h ≻ KQKGL hC θ g f ′ ≻ KGL id g h ′ ≻ KGL hC θ g KGL hC in the same way as for KQ , cf. Section 4.3. That is,one obtains an identification s q ( KGL hC ) = ((cid:0) Σ q,q MZ (cid:1) ∨ (cid:0) Σ q,q M ν (cid:1) ∨ W i< q Σ i + q,q MZ / q ≡ (cid:0) Σ q,q M ν (cid:1) ∨ W i< q +12 Σ i + q,q MZ / q ≡ . Here M ν ∼ = Σ , M ν and M ν s,t is an F -module for all integers s and t . Moreover, it follows fromour first claim above that θ : KQ ≻ KGL hC splits as (id , ζ ), where id is the identity on thenon-mysterious summands of the respective slices, and ζ : M µ ≻ M ν up to suspension with S , .By comparison with the identification of the slices of KGL hC in Theorem 4.25, the summand M ν is contractible, which completes the proof. Theorem 4.27.
The mysterious summand M µ is trivial.Proof. By Corollary 2.7 it suffices to consider prime fields. Theorems 4.20, 4.25 and Lemma 4.26show that the mysterious summand is the homotopy fiber of a weak equivalence.
By combining Theorems 4.2, 4.8, 4.18, and 4.27 we have enough information to identify the slicesof Witt-theory. An alternate proof with no mention of homotopy orbit K -theory follows by Lemma4.5 and the identification of s q ( η ) : s q (Σ , KQ ) ≻ s q ( KQ ) worked out in § Theorem 4.28.
The slices of Witt-theory are given by s q ( KT ) ∼ = Σ q,q _ i ∈ Z Σ i, MZ / . Let u : KQ ≻ KT be the canonical map. The next result follows now by an easy inspection. Proposition 4.29.
Restricting the map s q ( u ) : s q ( KQ ) ≻ s q ( KT ) to the summand Σ q, q MZ yields the projection Σ q, q MZ ≻ Σ q, q MZ / composed with the inclusion into s q ( KT ) . Re-stricting the same map to a suspension of MZ / yields the inclusion into s q ( KT ) . On odd slices, s q +1 ( u ) : s q +1 ( KQ ) ≻ s q +1 ( KT ) is the inclusion. Having determined the slices we now turn to the problem of computing the first differentials in theslice spectral sequences for
KGL hC , KQ and KT . K -theory Theorem 4.1 and Lemma 4.21 show there is an isomorphism s q ( KGL / ∼ = Σ q,q MZ / . Hence the differential d KGL / : s q ( KGL / ≻ Σ , s q +1 ( KGL / ,
1) element in the mod-2 motivic Steenrod algebra. By Bott periodicityit is independent of q . Next we resolve the mod-2 version of a question stated in [58, Remark 3.12].21 emma 5.1. If char( F ) = 2 the differential d KGL / : MZ / ≻ Σ , MZ / equals the Milnor operation Q = Sq + Sq Sq .Proof. By Lemma A.2 and the Adem relation Sq = Sq Sq there exist a, b ∈ Z / d KGL / = a Sq + b Sq Sq . (23)Using the Adem relations Sq Sq = Sq Sq , Sq Sq = Sq + Sq Sq , and Sq Sq = 0 we find( a Sq + b Sq Sq ) = a Sq Sq + ab Sq Sq Sq + ab Sq Sq Sq + b Sq Sq Sq Sq =( a + b ) Sq Sq . Since d KGL / squares to zero this implies a = b . Recall the classes 0 = τ ∈ h , and ρ ∈ h , represented by − ∈ F . Over R , and hence Q , d KGL / ( τ ) = ρ by Suslin’s computation of thealgebraic K -theory of the real numbers [52], while d KGL / ( τ ) = a Sq ( τ ) + b Sq Sq ( τ ) = aρ by (23) and Corollary 6.2. These computations imply that a = b = 1 by base change for all fieldsof characteristic zero. To extend this result to fields of odd characteristic, we consider the followingdiagram of motivic spectra over Spec( Z [ ]). KGL / ≺ MGL / (2 , x , x , . . . ) MGL ≻ MGL / (2 , x , x , . . . ) MGL ≻ MZ / MGL denotes Voevodsky’s algebraic cobordism spectrum, x n denotes the canonical imageof Lazard’s generator in MGL ∗ , ∗ , MZ / F -coefficients [51], and the maps are induced by the respective canonical orientations. All mapsin diagram (24) induce equivalences on zero slices. For the map pointing to the left, this followsfrom the description of KGL as (cid:0) MGL / ( x , x , . . . ) MGL (cid:1) [ x − ] [50, Theorem 5.2]. For the mapin the middle, this follows from its construction. By [51, Theorem 11.3], the rightmost map indiagram (24) is an equivalence, and in particular after applying the zero slice functor. Thereresults a commutative diagram s KGL / d ≻ Σ , s KGL / s MZ / ∼ = g ≻ Σ , s MZ / ∼ = g (25)where the map on the top is defined as over a field. By its construction, the motivic spectrum MZ / f MZ / ∗ , and the aforementioned [51, Theorem 11.3] implies that MZ / Z [ ]). There results a canonical isomorphism s MZ / ∼ = MZ / f : Spec( Q ) ≻ Spec( Z [ ])maps diagram (25) to the following diagram over Spec( Q ), by Lemma 2.6, [51, Lemma 7.5] and theprevious argument. s KGL / d ≻ Σ , s KGL / MZ / ∼ = g Q ≻ Σ , MZ / ∼ = g p is an odd prime, base change via i : Spec( F p ) ⊂ + ≻ Spec( Z [ ]) maps diagram (25) to thefollowing diagram. i ∗ s KGL / i ∗ d ≻ Σ , i ∗ s KGL / i ∗ MZ / ∼ = g ≻ Σ , i ∗ MZ / ∼ = g By [51, Theorem 9.19], i ∗ MZ / F p ,and the bottom map in the last diagram is the first Milnor operation by [51, Theorem 11.24].Thus the E -page of the 0th slice spectral sequence for KGL / h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , The differential h , ≻ h , has a nontrivial group as source and a potentially nontrivialgroup as target, but is always zero by Corollary 6.2. Corollary 5.2. If √− ∈ F then d KGL / : π p,n s q ( KGL / ≻ π p − ,n s q +1 ( KGL / is trivial.Proof. By assumption ρ = 0, so the assertion follows from Lemma 5.1 and Corollary 6.2. K -theory, I In what follows we make of use the identification of d KGL / with the first Milnor operation Q togive formulas for the d -differential in the slice spectral sequence for KQ . To begin with, considerthe commutative diagram: s q ( KQ ) d KQ ( q ) ≻ Σ , s q +1 ( KQ ) s q ( KGL ) g d KGL ( q ) ≻ Σ , s q +1 ( KGL ) g s q ( KGL / g Q ≻ Σ , s q +1 ( KGL / g (26)To proceed from here requires two separate arguments depending on whether the d -differentialexits an even or an odd slice. First we analyze the even slices.23heorems 4.18 and 4.27 show that d KQ (2 q ) is a map from s q ( KQ ) = Σ q, q (cid:0) Σ q, MZ ∨ _ i< Σ q +2 i, MZ / (cid:1) to Σ , s q +1 ( KQ ) = Σ q +1 , q +1 _ i ≤ Σ q +2 i +1 , MZ / . Up to suspension with S q, q , Proposition 4.19 shows s q ( KQ ) ≻ s q ( KGL ) ≻ s q ( KGL / MZ ∨ _ i< Σ i, MZ / ≻ MZ ≻ MZ / . Likewise, up to suspension with S q, q , the composite of the right hand side maps in (26) equalsΣ , _ i< Σ i +1 , MZ / pr ≻ Σ , MZ / δ ≻ Σ , MZ pr ≻ Σ , MZ / . By Lemma A.4, Sq ◦ pr is the only element of [ MZ , Σ , MZ /
2] whose composition with Sq andthe projection MZ ≻ MZ / Q ◦ pr = Sq Sq ◦ pr. When restricting to MZ this shows d KQ (2 q ) = (0 , Sq ◦ pr , a (2 q ) τ ◦ pr); a (2 q ) ∈ h , . On odd slices the first differential d KQ (2 q + 1) is a map from s q +1 ( KQ ) = Σ q +1 , q +1 _ i ≤ Σ q +2 i, MZ / , s q +2 ( KQ ) = Σ q +2 , q +2 (cid:0) Σ q +3 , MZ ∨ _ i ≤ Σ q +2 i +1 , MZ / (cid:1) . Restricting d KQ (2 q + 1) to the top summand MZ / i = 0 in the infinite wedgeproduct yields – up to suspension with S q +1 , q +1 – a map d KQ (2 q + 1 ,
0) : MZ / ≻ Σ , MZ ∨ Σ , MZ / ∨ Σ , MZ / . Lemma A.4 shows δ Sq Sq is the only element of [ MZ / , Σ , MZ ] whose composition with theprojection map pr equals Q Sq = Sq Sq . Thus by commutativity of (26), we obtain d KQ (2 q + 1) = ( δ Sq Sq , b (2 q + 1) Sq + φ (2 q + 1) Sq , a (2 q + 1) τ ) , where a (2 q + 1), b (2 q + 1) ∈ h , and φ (2 q + 1) ∈ h , .In what follows we use the computations in this section to explicitly identify d KT and d KQ interms of motivic cohomology classes and Steenrod operations.24 .3 Higher Witt-theory Up to suspension with S q,q , Theorem 4.5 shows that the differential d KT ( q ) : s q ( KT ) ≻ Σ , s q +1 ( KT )takes the form _ i ∈ Z Σ i, MZ / ≻ Σ , _ j ∈ Z Σ j, MZ / . Lemma 2.1 and (1 , , KT ∼ = KT for KT imply d KT ( q ) = Σ q,q d KT (0). Let d KT ( q, i ) denote the restriction of d KT ( q ) to the summand Σ q +2 i,q MZ / s q ( KT ). The (4 , , KT ∼ = KT shows d KT (0) is determined by its values on the summands MZ / , MZ /
2. That is, d KT ( q, i ) is uniquely determined by d KT (0 ,
0) or d KT (0 , Theorem 5.3.
The d -differential in the slice spectral sequence for KT is given by d KT ( q, i ) = ( ( Sq Sq , Sq , i − q ≡ Sq Sq , Sq + ρ Sq , τ ) i − q ≡ . The motivic cohomology classes = τ ∈ h , and ρ = [ − ∈ h , are represented by − ∈ F .Proof. We have reduced to computing the maps d KT (0 ,
0) : MZ / ≻ Σ , MZ / ∨ Σ , MZ / ∨ Σ , MZ / d KT (0 ,
2) : Σ , MZ / ≻ Σ , MZ / ∨ Σ , MZ / ∨ Σ , MZ / . To proceed we invoke the commutative diagram: s q ( KQ ) ≻ s q ( KT )Σ , s q +1 ( KQ ) d KQ ( q ) g ≻ Σ , s q +1 ( KT ) d KT ( q ) g By combining Proposition 4.29 with the computations in Section 5.2 we find d KT (0 ,
0) = ( Sq Sq , Sq + φ (0) Sq , a (0) τ ) d KT (0 ,
2) = ( Sq Sq , Sq + φ (2) Sq , a (2) τ ) . Here a (0), a (2) ∈ h , and φ (0), φ (2) ∈ h , . Since d KT squares to zero, we extract the equations a (2) a (0) = a (0) φ (2) = a (0)( φ (0) + φ (2) + ρ ) = 0 , a (0) + a (2) = 1 ,a (0) ρ + φ (0) = a (2) φ (0) = a (2)( φ (0) + φ (2) + ρ ) = a (2) ρ + φ (2) = 0 . The proof of Proposition 5.4 discusses the two possible values for a (2) in the case of an algebraicallyclosed field. In particular, having a (2) = 0 would result in the group π , KT being nontrivial, whichwould contradict the vanishing of Balmer’s higher Witt groups of fields in degrees not congruent tozero modulo four [3, Theorem 98]. Hence a (2) = 1 for algebraically closed fields, which extends toall fields by base change to an algebraic closure. This implies a (0) = 0, φ (0) = 0, and φ (2) = ρ .25 roposition 5.4. Suppose F has mod- cohomological dimension zero. Then the slice filtrationfor KT coincides with the fundamental ideal filtration of W ( F ) . Moreover, there are isomorphisms π p, f q ( KT ) ∼ = h ,q p ≡ q mod 4 , q ≥ h , p ≡ , q <
00 otherwise . The first two isomorphisms are induced by the canonical map f q ( KT ) ≻ s q ( KT ) .Proof. By the (1 , , KT and Theorem 4.28, it suffices to consider thefiltration · · · h ,n +1 · · · ≻ π p, − n f ( KT ) g ≻ π p, − n f ( KT ) g ≻ π p, − n f ( KT ) · · · h ,n +2 g g h ,n g for p even, and · · · h ,n · · · ≻ π p, − n f ( KT ) g ≻ π p, − n f ( KT ) g ≻ π p, − n f ( KT ) · · · g h ,n +1 g g for p odd. When n = p = 0, π , f ( KT ) ≻ h , is a ring map [14], [41], hence an isomorphism. Itfollows that π , f ( KT ) = 0, hence f q π , ( KT ) = 0 for all q >
0. In particular, the slice filtration isHausdorff, and it coincides with the (trivial) filtration on the Witt ring given by the fundamentalideal. Theorem 5.3 and Lemma 6.1 leave us with two possibilities:1. The first differential E p,q = h ,q ≻ h ,q +1 = E p − ,q +1 is an isomorphism if q ≡ p + 2 mod 4, and trivial otherwise.2. The first differential E p,q = h ,q ≻ h ,q +1 = E p − ,q +1 is an isomorphism if q ≡ p mod 4, and trivial otherwise.In both cases we find E = E ∞ . Our case distinctions can be recast in the following way.1. E ∞ p,q ∼ = h , if q = 0 and p ≡ E ∞ p,q ∼ = h , if q = 0 and p ≡ π , ( KT ). Computing with the first condition yieldsthe desired result on the filtration. One extends to arbitrary n by using the commutative diagram: π p,n f q + i ( KT ) ≻ π p,n f q ( KT ) π p − n, f q − n + i ( KT ) ∼ = g ≻ π p − n, f q − n ( KT ) ∼ = g The following image depicts the first differential when 2 q ≡ i mod 4 with degrees along thehorizontal axis and weights along the vertical axis. Each dot is a suspension of MZ / qq + 1 q − q + 2 ii − i − i + 2 i + 4 τ Sq Sq + ρ Sq Sq Sq K -theory, II Next we determine d KQ by combining Proposition 4.29 with the formula for d KT in Theorem 5.3.Recall the identifications s q ( KQ ) ∼ = Σ q, q (cid:0) Σ q, MZ ∨ _ i The d -differential in the slice spectral sequence for KQ is given by d KQ ( q, i ) = ( ( Sq Sq , Sq , q − > i ≡ Sq Sq , Sq + ρ Sq , τ ) q − > i ≡ d KQ ( q, q ) = ( (0 , Sq ◦ pr , q ≡ , Sq ◦ pr , τ ◦ pr) q ≡ d KQ ( q, q − 1) = ( ( δ Sq Sq , Sq , q ≡ δ Sq Sq , Sq + ρ Sq , τ ) q ≡ . The following image depicts the first differential for KQ with degrees along the horizontal axisand weights along the vertical axis. Each small dot is a suspension of MZ / MZ . 27 q − q q + 14 q + 24 q + 3 8 q q − q − q + 2 8 q + 4 τ τ ◦ pr Sq Sq ◦ pr Sq + ρ Sq δ ◦ Sq Sq Sq Sq In this section we compute the slice spectral sequence of KT over any field F of char( F ) = 2.Note that KT acquires a ring spectrum structure from hermitian K -theory KQ and the tower (7).Recall from Theorem 4.28, cf. Example 2.3, the identification s ( KT ) ∼ = _ i ∈ Z Σ i, MZ / . By the periodicity Σ , KT ∼ = KT induced by multiplication with the Hopf map, this determinesall slices of KT . Recall that the first differential s q ( KT ) ≻ Σ , s q +1 ( KT ) is determined bythe motivic Steenrod operations in (9), a formula proven in Theorem 5.3. With the elements τ , Sq , Sq + ρ Sq , and Sq Sq corresponding to their given colors, the first differentials can berepresented as follows. The group in bidegree ( p, q ) is a direct sum of mod-2 motivic cohomologygroups positioned on the vertical line above p and inbetween the horizontal lines corresponding toweights q and q + 1. The number of direct summands increases linearly with the weight: The next statement is an immediate consequence of Voevodsky’s proof of Milnor’s conjectureon Galois cohomology [62]. Recall the classes τ ∈ h , and ρ ∈ h , are represented by − ∈ F . Lemma 6.1. For ≤ p ≤ q cup-product with τ yields an isomorphism τ : h p,q ∼ = ≻ h p,q +1 . orollary 6.2. If a ∈ h p,q where ≤ p ≤ q , write a = τ q − p c where c ∈ h p,p and let n = q − p . TheSteenrod squares of weight ≤ act on the mod- motivic cohomology ring h ∗ , ∗ by Sq ( τ n c ) = ( ρτ n − c n ≡ n ≡ Sq ( τ n c ) = ( ρ τ n − c n ≡ , n ≡ , Sq Sq ( τ n c ) = ( ρ τ n − c n ≡ n ≡ , , Sq Sq ( τ n c ) = ( ρ τ n − c n ≡ n ≡ , , . Proof. This follows from Lemma 6.1, the computation of Sq i ( τ n ) for i ∈ { , } and the Cartanformula [63, Proposition 9.6].We consider an element φ ∈ h m,n as a stable motivic cohomology operation of bidegree ( m, n )with the same name via multiplication with φ on the left. Only elements of bidegrees (0 , , , 2) will be relevant here. The Adem relations in weight less than or equal to 2 are given by Sq Sq = 0 , Sq τ = τ Sq + ρ, Sq ρ = ρ Sq , Sq Sq = Sq , Sq Sq = 0 , Sq τ = τ Sq + τ ρ Sq , Sq ρ = ρ Sq , Sq Sq = τ Sq Sq , Sq Sq = Sq + Sq Sq , Sq Sq = ρ Sq Sq , Sq Sq = Sq Sq . This concludes the prerequisites for our proof of the following result. Theorem 6.3. The 0th slice spectral sequence for KT collapses at its E -page, and E ∞ p,q ( KT ) ∼ = ( h q,q p ≡ . Proof. The E -page takes the form E p,q = π p, s q ( KT ) = M i ∈ Z h i +( q − p ) ,q = L ⌊ q ⌋ j =0 h j,q q ≡ p mod 2 L ⌊ q − ⌋ j =0 h j +1 ,q q p mod 2 . The group h i +( q − p ) ,q in the sum L i ∈ Z h i +( q − p ) ,q arises from the 2 i th summand of s q ( KT ). Thevanishing of h p,q for p < p > q shows the sum is finite. Hence for every element a ∈ E p,q there exists a unique collection of elements { a j ∈ h j,q } such that a = ( a q , a q − , . . . , a ) ∈ E p,q q ≡ ≡ p mod 2( a q − , a q − , . . . , a ) ∈ E p,q q ≡ ≡ p mod 2( a q , a q − , . . . , a ) ∈ E p,q q ≡ p mod 2( a q − , a q − , . . . , a ) ∈ E p,q q ≡ p mod 2 . By inspection of (9) the components of d KT are given by d KT ( a ) j = ( Sq Sq a j − + Sq a j − + ρ Sq a j − j ≡ q − p mod 4 Sq Sq a j − + Sq a j − + τ a j j ≡ q − p + 2 mod 4 . (27)Here j is the dimension index of the target group. The formula can be read off from the parity ofthe dimension index of the source group. Note that ρ Sq , Sq and τ shift the dimension by 2, 2, and0, respectively. We compute the E -page by repeatedly using the Steenrod square computationsand Adem relations given in the beginning of Section 6.29 ≡ a = ( a q , a q − , . . . ), Corollary 6.2 implies d KT ( a ) j = ( Sq a j − j ≡ q mod 4 τ a j j ≡ q + 2 mod 4 . If d KT ( a ) = 0 the injectivity part of Lemma 6.1 implies 0 = a q − = a q − = · · · , so thatker( d KT ) = h q,q ⊕ h q − ,q ⊕ · · · . We may assume q > 0. For b ∈ E p +1 ,q − the entering differential is given by d KT ( b ) j = ( Sq Sq b j − + Sq b j − + τ b j j ≡ q − p mod 4 Sq Sq b j − + Sq b j − + ρ Sq b j − j ≡ q − p + 2 mod 4 . Corollary 6.2 simplifies this formula to d KT ( b ) j = ( Sq Sq b j − + τ b j j ≡ q − p mod 40 j ≡ q − p + 2 mod 4 . For example, if p ≡ j ≡ q − p mod 4 implies j ≡ q mod 4. Thus for b j − ∈ h j − ,q − one has q − − ( j − ≡ Sq b j − = 0. It follows that E p,q is thehomology of the complex h q − ,q − ⊕ h q − ,q − ⊕ · · · α ≻ h q,q ⊕ h q − ,q ⊕ · · · ≻ , where α ( b q − , b q − , . . . , b m ) = ( Sq Sq b q − , Sq Sq b q − + τ b q − , . . . , τ b m ). Here m ≡ q mod 4and 0 ≤ m ≤ 3. Lemma 6.1 implies α is split injective by mapping ( a q , a q − , . . . , a m ) to (cid:0) τ − a q − + φ ( a q − + τ φ ( a q − + · · · + τ φ ( a m ))) , . . . , τ − a m +4 + φ ( a m ) , τ − a m (cid:1) , where φ is the composite map h p,q τ − ≻ h p,q − Sq Sq ≻ h p +4 ,q τ − ≻ h p +4 ,q − . It follows that E p,q ∼ = h q,q for all p ≡ p ≡ a = ( a q − , a q − , . . . ), Corollary 6.2 implies d KT ( a ) j = ( j ≡ q − Sq Sq a j − + τ a j j ≡ q + 1 mod 4 . If d KT ( a ) = 0 then 0 = a q − = a q − = · · · by applying inductively the injectivity statementin Lemma 6.1. Thus we have ker( d KT ) = h q − ,q ⊕ h q − ,q ⊕ · · · . For b ∈ E p +1 ,q − the entering differential is given by d KT ( b ) j = ( Sq Sq b j − + Sq b j − + τ b j j ≡ q − p mod 4 Sq Sq b j − + Sq b j − + ρ Sq b j − j ≡ q − p + 2 mod 4 . d KT ( b ) j = ( Sq b j − + τ b j j ≡ q − p mod 40 j ≡ q − p + 2 mod 4 . Thus E p,q is the homology of the complex h q − ,q − ⊕ h q − ,q − ⊕ · · · d KT ≻ h q − ,q ⊕ h q − ,q ⊕ · · · ≻ . Since the restriction of d KT to h q − ,q − ⊕ h q − ,q − ⊕ · · · is surjective by Lemma 6.1, E p,q = 0. p ≡ a = ( a q , a q − , . . . ), Corollary 6.2 implies d KT ( a ) j = ( j ≡ q − Sq a j − + τ a j j ≡ q mod 4 . Hence the subgroup ker( d KT ) can be identified with { ( a q , a q − , . . . ) ∈ h q,q ⊕ h q − ,q ⊕ · · · : τ a j = Sq a j − for all j ≡ q mod 4 , ≤ j ≤ q } . For b ∈ E p +1 ,q − the entering differential is given by d KT ( b ) j = ( Sq Sq b j − + Sq b j − + ρ Sq b j − j ≡ q mod 4 Sq Sq b j − + Sq b j − + τ b j j ≡ q − . Corollary 6.2 simplifies this formula to d KT ( b ) j = ( Sq Sq b j − + ρ Sq b j − j ≡ q mod 4 Sq b j − + τ b j j ≡ q − . If a ∈ ker( d KT ), Lemma 6.1 shows there exists elements b j,q − ∈ h j,q − for all 0 ≤ j < q where j ≡ q − τ b j,q − = a j . For these indices j , the Adem relation Sq τ = τ Sq + τ ρ Sq and d KT ( a ) = 0 imply τ a j +2 = Sq a j = Sq τ b j = τ Sq b j + τ ρ Sq b j = τ ρ Sq b j . Thus ρ Sq b j = a j +2 by Lemma 6.1. It follows that d KT ( b q − , , b q − , . . . ) = ( ρ Sq b q − , τ b q − , ρ Sq b q − , τ b q − ) = ( a q , a q − , a q − , a q − , . . . ) . This shows that E p,q is trivial. p ≡ a = ( a q − , a q − , . . . ) the exiting differential simplifies to d KT ( a ) j = ( Sq Sq a j − + ρ Sq a j − j ≡ q − Sq a j − + τ a j j ≡ q − Sq to Sq a j − + τ a j yields Sq ( Sq a j − + τ a j ) = τ Sq Sq a j − + τ Sq a j + τ ρ Sq a j . d KT ) is comprised of tuples ( a q − , a q − , . . . ) ∈ h q − ,q ⊕ h q − ,q ⊕ · · · for which τ a j = Sq a j − whenever j ≡ q − ≤ j < q . For b ∈ E p +1 ,q − the enteringdifferential simplifies to d KT ( b ) j = ( τ b j j ≡ q − Sq b j − j ≡ q − . If a ∈ ker( d KT ), Lemma 6.1 shows there exist elements b j,q − ∈ h j,q − for all 0 ≤ j < q where j ≡ q − τ b j,q − = a j . For these j , d KT ( a ) = 0 implies τ a j +2 = Sq a j = Sq τ b j = τ Sq b j + τ ρ Sq b j = τ Sq b j . Hence Sq b j = a j +2 by Lemma 6.1. It follows that d KT ( b q − , , b q − , . . . ) = ( Sq b q − , τ b q − , Sq b q − , τ b q − ) = ( a q − , a q − , a q − , a q − , . . . ) . This shows that E p,q is trivial.An inspection of the E -page shows that the 0th slice spectral sequence for KT collapses with E = E ∞ -page: h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , Let I ( F ) denote the fundamental ideal of even dimensional quadratic forms in the Witt ring W ( F ). To conclude our proof of Milnor’s conjecture on quadratic forms, it remains to identify theslice filtration on the Witt ring with the filtration given by the fundamental ideal. Lemma 6.4. The slice filtration of KT induces a commutative diagram: π , f ( KT ) ⊂ ≻ π , f ( KT ) I ( F ) ∼ = g ⊂ ≻ W ( F ) ∼ = g Proof. By [14, Corollary 5.18] the canonical map f ( KT ) ≻ s ( KT ) induces a ring homomor-phism W ( F ) ∼ = π , f ( KT ) ≻ π , s ( KT ) ∼ = h , ∼ = F . 32t is natural with respect to separable field extensions according to Corollary 2.7. Comparing withan algebraic closure of F shows this ring homomorphism is induced by sending a quadratic formto its rank. Since the group π , s ( KT ) = h , is trivial, the map π , f ( KT ) ≻ π , f ( KT )is injective, which proves the result. Corollary 6.5. The identification π , KT ∼ = W ( F ) induces an inclusion I ( F ) q ⊆ f q π , KT .Proof. Since KT is a ring spectrum the claim follows from the multiplicative structure of the slicefiltration [14, Theorem 5.15], [41, Theorem 3.6.9], and Lemma 6.4.Any rational point u ∈ A r { } ( F ) defines a map of motivic spectra [ u ] : ≻ S , . We areinterested in the effect of the map [ u ] on motivic cohomology and KT . Lemma 6.6. The map H , = [ , MZ ] = [ , ∧ MZ ] ([ u ] ∧ MZ ) ∗ ≻ [ , Σ , MZ ] = H , sends to u ∈ A r { } ( F ) = F × ∼ = H , .Proof. Let Z (1) denote the Tate object in the derived category of motives DM F over F . Theassertion follows from the canonically induced diagram:Hom Sm F ( F, A r { } ) ≻ [ , Σ , MZ ]Hom DM F ( Z , O × ) g ≺ ∼ = Hom DM F ( Z , Z (1)[1]) ∼ = g The lower horizontal isomorphism follows from [32, Theorem 4.1]. Corollary 6.7. Suppose u , . . . , u q ∈ A r { } ( F ) are rational points. The map H , ≻ H q,q induced by the smash product [ u ] ∧ · · · ∧ [ u q ] sends to { u , . . . , u q } ∈ K Mq ∼ = H q,q , and likewisefor h , ≻ h q,q . Lemma 6.8. The composition [ u ] ≻ S , η ≻ induces multiplication by h u i − on π , , where h u i is the class of the rank one quadratic form in the Grothendieck-Witt ring defined by u .Proof. This follows from [38, Corollary 1.24].The unit map for KT induces the canonical map from the Grothendieck-Witt ring to the Wittring π , ≻ π , KT . By definition, multiplication by η is an isomorphism on π ∗ , ∗ KT . Thusthe “multiplication by [ u ]” map on π , KT is determined by its effect on the Grothendieck-Wittring π , . Recall that f q π p,n E denotes the image of the canonical map π p,n f q ( E ) ≻ π p,n E . Lemma 6.9. There is a canonically induced short exact sequence ≻ f q +1 π , KT j q ≻ f q π , KT ≻ h q,q ≻ . roof. Theorem 6.3 shows the exact sequence in the proof of [60, Lemma 7.2] takes the form0 ≻ f q π , KT / f q +1 π , KT α ≻ h q,q ≻ \ i ≥ f q + i π − , f q ( KT ) ≻ \ i ≥ f q + i π − , KT ≻ . Moreover, π − , KT is the trivial group. Corollary 6.5 furnishes a map I ( F ) q /I ( F ) q +1 β ≻ f q π , KT / f q +1 π , KT . Combined with the canonical surjective map k Mq γ ≻ I ( F ) q /I ( F ) q +1 from the q -th mod-2 MilnorK-theory group defined in [33], we obtain the composite map α ◦ β ◦ γ : k Mq ≻ h q,q . (28)Lemmas 6.6 and 6.8 show that (28) coincides with Suslin’s isomorphism between Milnor K -theoryand the diagonal of motivic cohomology [32, Lecture 5]. In particular, α is surjective.The above shows that γ is injective, which gives an alternate proof of the main result in [39]. Theorem 6.10. The canonical map k Mq ≻ I ( F ) q /I ( F ) q +1 is an isomorphism for q ≥ . Corollary 6.11. The identification π , KT ∼ = W ( F ) induces an equality I ( F ) q = f q π , KT for q ≥ .Proof. By the definition of β there is a commutative diagram:0 ≻ I ( F ) q +1 ≻ I ( F ) q ≻ I ( F ) q /I ( F ) q +1 ≻ ≻ f q +1 π , KT g j q ≻ f q π , KT g ≻ h q,q β g ≻ β is an isomorphism by Lemma 6.9, Theorem 6.10, and the isomorphism (28). Thus theresult follows by induction using the identification I ( F ) = f π , KT = π , f KT in Lemma 6.4.This finishes our proof of Milnor’s conjecture on quadratic forms. Theorem 6.12. The image of π p + q,q f n ( KT ) in π p + q,q ( KT ) ∼ = W ( F ) coincides with I n − q ( F ) ,where I ( F ) ⊆ W ( F ) is the fundamental ideal. Thus the slice spectral sequence for KT convergesto the filtration of the Witt ring given by the fundamental ideal.Proof. This follows from Corollary 6.11 and the main result in [1], which shows the filtration of W ( F ) by I ( F ) is Hausdorff. For completeness we analyze the map π p, f q +1 ( KT ) ≻ π p, f q ( KT )for columns p ≡ , , Lemma 6.13. Let q ≥ . The canonical map f q +1 ( KT ) ≻ f q ( KT ) induces the trivial map π p, f q +1 ( KT ) ≻ π p, f q ( KT ) for p ≡ , , . roof. The statement is clear for q = 0. Suppose that p ≡ , x ∈ π p, f q +1 ( KT ). Itsimage y ∈ π p, f q ( KT ) lies in the kernel of the map π p, f q ( KT ) ≻ π p, s q ( KT ). By induction, themap π p +1 , s q − ( KT ) ≻ π p, f q ( KT ) is surjective. Hence there is an element z ∈ π p +1 , s q − ( KT )mapping to y . Thus z lies in the kernel of d ( p + 1 , q − d ( p + 1 , q − 1) coincides with the image of d ( p + 2 , q ). In particular, there is an element in π p +2 ,q s q ( KT ) whose image is z , showing that y = 0.Suppose now that p ≡ x ∈ π p, f q +1 ( KT ). Consider its image y ∈ π p, s q +1 ( KT ).Since its image under d ( p, q + 1) is trivial, Theorem 6.3 furnishes an element z ∈ π p +1 , s q +2 ( KT )whose image under d ( p + 1 , q + 2) is precisely y . Consider the difference x − w , where w is theimage of z in π p, f q +1 ( KT ). The image of x − w in π p, f q ( KT ) then coincides with the image of x .Since the image of x − w in π p, s q +1 ( KT ) is zero, there is an element v ∈ π p, f q +2 ( KT ) mappingto x − w . Proceeding inductively yields an element e ∈ \ i ≥ f q + i π p, f q ( KT ) . However, using that π p, KT = 0, this group is trivial by the exact sequence0 ≻ f q π p +1 , KT / f q +1 π p +1 , KT ≻ h q,q ≻ \ i ≥ f q + i π p, f q ( KT ) ≻ \ i ≥ f q + i π p, KT ≻ e = 0, and the image of x in π p, f q ( KT ) is zero. Corollary 6.14. Let q ≥ and p ≡ , . There is a canonically induced split short exactsequence of F -modules ≻ π p, f q ( KT ) ≻ π p, s q ( KT ) ≻ π p − , f q +1 ( KT ) ≻ . Corollary 6.15. For q ≥ there are canonically induced isomorphisms π p, f q ( KT ) ∼ = h q − ,q ⊕ h q − ,q ⊕ · · · p ≡ h q − ,q ⊕ h q − ,q ⊕ · · · p ≡ h q − ,q ⊕ h q − ,q ⊕ · · · p ≡ . Proof. Use Theorem 6.3, Lemma 6.13, and Corollary 6.14. Corollary 6.16. For q ≥ the canonical map Σ , s q ( KT ) ≻ f q +1 ( KT ) induces a split shortexact sequence ≻ h q − ,q ⊕ h q − ,q ⊕ · · · ≻ π , f q +1 ( KT ) ≻ f q +1 π , KT = I ( F ) q +1 ≻ . Moreover, the map π , f q ( KT ) ≻ π , f q − ( KT ) is injective on the image of π , f q +1 ( KT ) .Proof. The latter claim follows by a diagram chase and Theorem 6.3, since E p,q = 0 if p ≡ · · · β ≻ π , s q ( KT ) α ≻ π , f q +1 ( KT ) ≻ π , f q ( KT ) ≻ · · · ≻ π , s q ( KT ) / Ker( α ) ≻ π , f q +1 ( KT ) ≻ f q +1 π , ( KT ) ≻ . Since Ker( α ) = Im( β ) = Im( d ,q − ) = h q − ,q ⊕ h q − ,q ⊕ · · · by Theorem 6.3, the sequence is shortexact. It splits by Lemma 6.1, since the composition of h q − ,q ⊕ h q − ,q ⊕ · · · ≻ π , f q +1 and π , f q +1 ( KT ) ≻ π , s q +1 ( KT ) ∼ = h q +1 ,q +1 ⊕ h q − ,q +1 ⊕ · · · pr ≻ h q − ,q +1 ⊕ h q − ,q +1 ⊕ · · · is given by multiplication with τ ∈ h , .Theorems 6.3 and 6.12 imply Theorem 1.1 stated in the introduction. If X ∈ Sm F is a semilocalscheme and F a field of characteristic zero, our computations and results extend to the Witt ring W ( X ) with fundamental ideal I ( X ) and the mod-2 motivic cohomology of X . Our reliance on theMilnor conjecture for Galois cohomology [62] can be replaced by [15, § X by [24, Theorem 7.6]. The rest of the proof is identical tothe one given for fields. Kerz proved a closely related result in [24, Theorem 7.10]. By periodicityof KT there is an evident variant of Theorem 6.3 for the n th slice spectral sequence of KT forevery n ∈ Z . We note the following result from [2] is transparent from our computation of the slicespectral sequence for KT . Corollary 6.17. If X ∈ Sm F is a semilocal scheme of geometric origin then W ( X ) contains noelements of odd order. If X is not formally real then W ( X ) is a -primary torsion group. K -groups According to Theorem 4.18 the E -page of the 0th slice spectral sequence for KQ takes the form E p,q = π p, s q ( KQ ) = ( H q − p,q ⊕ L i< q h i +( q − p ) ,q q ≡ L i ≤ q − h i +( q − p ) ,q q ≡ . Using the formula in Theorem 5.5 for the first differentials we are ready to perform low-degreecomputations in the slice spectral sequence for KQ . We assume throughout that F is a field ofchar( F ) = 2.The Beilinson-Soule vanishing conjecture predicts the integral motivic cohomology group H q − p,q is trivial if p > q , which holds for instance for finite fields and number fields. (The same groupis uniquely divisible if p ≥ q , except when p = q = 0). We note that H − p, = 0 for p ≥ § E -page of the slice spectral sequence for KQ , the group E p,q inbidegree ( p, q ) is a direct sum of motivic cohomology groups positioned on the vertical line above p and inbetween the horizontal lines corresponding to the weights q and q + 1.36 − − h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , H , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , h , H , h , h , h , h , h , h , h , H , h , h , h , h , h , h , h , h , H , h , h , h , h , h , h , h , h , H , h , h , h , h , h , h , h , h , H , h , h , h , h , h , H , h , H , h , h , h , H , h , H , h , h , H , h , H − , h , h , H , A comparison with KT via Proposition 4.29 implies: Lemma 7.1. Suppose that p ≤ . Then the canonical map induces an identification E p,q ( KQ ) = E p,q ( KT ) for all q ≥ p + 1 if p is even and for all q ≥ p + 2 if p is odd. If F satisfies theBeilinson-Soule vanishing conjecture, the identification holds also for p > . Lemma 7.1 combined with the classical computation of π , KQ as the Grothendieck-Witt groupof F basically determines the 0th and 1st columns of the 0th slice spectral sequence of KQ . In the1st column we note: There are no entering or exiting differentials in weight q ≤ 2. In weight threethe exiting differential has kernel h , by Corollary 6.2. Moreover, the differential E , ( KQ ) = H , ⊕ h , ≻ h , ⊕ h , τ + Sq ≻ h , is surjective by Lemma 6.1 and Corollary 6.2. All d r -differentials exiting E rp,q ( KQ ) for p = 0, p = 1and r ≥ Lemma 7.2. There are isomorphisms E ∞ p,q ( KQ ) ∼ = H , p = q = 0 h q,q p = 0 , q > h , p = q = 1 h , p = 1 , q = 20 p = 1 , q = 1 , . Remark 7.3. There are isomorphisms H , ∼ = Z , h , ∼ = Z / , and h , ∼ = F × / . By computing in the 2nd column we obtain the motivic cohomological description of the secondorthogonal K -group KO ( F ) stated in Theorem 1.2.37 emma 7.4. There are isomorphisms E ∞ ,q ( KQ ) ∼ = ( ker( τ ◦ pr + Sq : H , ⊕ h , ≻ h , ) q = 20 q = 2 . Proof. The claim for q = 2 follows from Theorem 5.5. We note that the kernel of the surjection τ ◦ pr + Sq is the preimage of the subgroup { , ρ } ⊆ h , under pr : H , ≻ h , . If ρ = 0, thisgroup coincides with 2 H , . In weight three, the kernel of τ + Sq : h , ⊕ h , ≻ h , is isomorphic to h , via φ ≻ ( ρ φ, τ φ ). The image of the entering differential correspondsto the image of H , in h , under this isomorphism, hence it coincides with h , . The remainingvanishing follows from Lemma 7.1. Remark 7.5. There are isomorphisms H , ∼ = K ( F ) , h , ∼ = Z / , and h , ∼ = Br ( F ) the -torsionsubgroup of the Brauer group of equivalence classes of central simple F -algebras. Lemma 7.6. There is a short exact sequence → h , → KO ( F ) → H , → and isomorphisms E ∞ ,q ( KQ ) ∼ = H , q = 2 h , q = 30 q = 2 , . Proof. Note that Sq acts trivially on h , by Corollary 6.2. In weight 2 we look at the kernel of τ ◦ pr : H , ≻ h , . By Lemma 6.1 this equals the kernel of pr, i.e., 2 H , ⊆ H , . In weight3, the kernel of the exiting differential is isomorphic to h , via c ≻ ( τ ρ c, τ c ). In weight 4,the kernel of the exiting differential is isomorphic to h , via φ ≻ ( τ ρ φ, τ φ ). The enteringdifferential surjects onto the kernel of the latter map. In weight 5, the group coincides with thecorresponding group for KT , as the change from h , to H , does not affect the spectral sequence,due to the triviality of the differential exiting bidegree (4 , Remark 7.7. There are isomorphisms H , ∼ = K ind ( F ) ; the cokernel of K M ( F ) → K ( F ) , i.e.,the K -group of indecomposable elements, and h , ∼ = Z / . By Lichtenbaum’s weight-two motiviccomplex Γ(2) [31] there is a short exact sequence → h , → H , → H , → . By comparingwith the forgetful map KQ → KGL and identifying h , with h , one concludes that KO ( F ) isisomorphic to K ind ( F ) . Lemma 7.8. There are isomorphisms E ,q ( KQ ) ∼ = q = 2 , H , q = 4 h q,q q ≥ . Proof. As noted above the group H , is trivial. Lemma 6.1 shows the exiting differential in weight3 is injective. In weight 4, the kernel of the exiting differential is H , ⊕ h , . The nontrivial elementin the image of the entering differential is ( δ ( ρ τ ) , τ ). Thus the quotient can be identified with H , . In weight 5, this follows from the case of KT because Sq ◦ pr : H , ≻ h , ≻ h , istrivial. Lemma 7.1 finishes the proof. 38n principle one can continue with a similar analysis of the next columns. To summarize thecomputations above, we note that in low-degrees the E -page takes the form: − − h , h , h , h , h , h , H , h , h , ker( τ ◦ pr + Sq ) h , H , h , h , H , Corollary 7.9. The group KO ( F ) surjects onto H , ∼ = K M . If K M = 0 then KO ( F ) is thetrivial group. The symplectic K -groups KSp ∗ ( F ) of F are the filtered target groups of the second slice spectralsequence for KQ on account of the isomorphism π p, KQ ∼ = KSp p − ( F ). Computations similar tothe above yields the E -page: H , H , h , h , h , H , h , h , / Sq ker Sq , We read off that KSp ( F ) ∼ = 2 H , is infinite cyclic and KSp ( F ) is the trivial group. It followsthat all the classes in the sixth column are infinite cycles and E p,q = E ∞ p,q when ( p, q ) = (6 , , (6 , KSp ( F ) ≻ H , . Its kernel is isomorphic to I ( F ) as shown bySuslin [53, § cd ( F ) ≤ 2, so that h p,q = 0 when p ≥ 3, the seventh column degenerates to ashort exact sequence 0 → h , → KSp ( F ) → H , → . emark 7.10. The computations in this section hold for smooth semilocal rings containing a fieldof characteristic zero, cf. the generalization of Milnor’s conjecture on quadratic forms discussed inthe introduction. A Maps between motivic Eilenberg-MacLane spectra Throughout this section the base scheme is essentially smooth over a field of characteristic unequalto 2 [19, Definition 2.9]. In the following series of results, we identify weight zero and weight oneendomorphisms of motivic Eilenberg-MacLane spectra in SH in terms of Steenrod operations Sq i and the motivic cohomology classes ρ , τ . When the base scheme is a field of characteristic zero,these identifications follow from Voevodsky’s work on the motivic Steenrod algebra [63], while thegeneralization to our set-up relies on [19]. We use square brackets to denote maps in SH . Lemma A.1. [ MZ / , Σ p, MZ / 2] = F p = 0 F { Sq } p = 10 otherwise . Recall that Sq is the canonical composite map MZ / δ ≻ Σ , MZ pr ≻ Σ , MZ / Lemma A.2. [ MZ / , Σ p, MZ / 2] = τ h , p = 0 h , ⊕ h , { τ Sq } p = 1 h , Sq ⊕ h , { Sq } p = 2 h , { Sq Sq } ⊕ h , { Sq Sq } p = 3 h , { Sq Sq Sq } p = 40 otherwise . Lemma A.3. [ MZ , Σ p, MZ / 2] = ( F { pr } p = 00 p = 0 [ MZ / , Σ p, MZ ] = ( F { δ } p = 10 p = 1 . Lemma A.4. [ MZ , Σ p, MZ / 2] = τ h , ◦ pr p = 0 h , ◦ pr p = 1 h , { Sq ◦ pr } p = 2 h , { Sq Sq ◦ pr } p = 30 otherwise[ MZ / , Σ p, MZ ] = δ ◦ τ h , p = 1 δ ◦ h , p = 2 F { δ ◦ Sq } p = 3 F { δ ◦ Sq Sq } p = 40 otherwise . Sq ( τ ) = ρ , Sq ( τ ) = 0 and Sq ( τ ) = τ ρ . It follows that Sq ( τ n ) = ( ρτ n − n ≡ n ≡ Sq ( τ n ) = ( ρ τ n − n ≡ , n ≡ , . Proposition A.5. For every subset A ⊆ Z there is a canonical weak equivalence α : _ i ∈ A Σ i, MZ / ≻ Y i ∈ A Σ i, MZ / . Proof. It suffices to show [Σ p,q X + , α ] is an isomorphism for all p, q ∈ Z and X ∈ Sm F . This is thecanonical map M i ∈ A h i − p, − q ( X + ) ≻ Y i ∈ A h i − p, − q ( X + ) . Work of Suslin-Voevodsky [55] shows the group h i − p,q ( X + ) is nonzero only if 0 ≤ i − p ≤ q , see[19, Corollary 2.14] for base schemes essentially smooth over a field. References [1] J. K. Arason and A. Pfister. Beweis des Krullschen Durchschnittsatzes f¨ur den Wittring. Invent. Math. , 12:173–176, 1971.[2] R. Baeza. ¨Uber die Torsion der Witt-Gruppe W q ( A ) eines semi-lokalen Ringes. Math. Ann. , 207:121–131, 1974.[3] P. Balmer. Witt groups. In Handbook of K -theory. Vol. 1, 2 , pages 539–576. Springer, Berlin, 2005.[4] H. Bass. John Milnor, the algebraist. In Topological methods in modern mathematics (Stony Brook, NY, 1991) ,pages 45–84. Publish or Perish, Houston, TX, 1993.[5] A. J. Berrick and M. Karoubi. Hermitian K -theory of the integers. Amer. J. Math. , 127(4):785–823, 2005.[6] A. J. Berrick, M. Karoubi, M. Schlichting, and P. A. Østvær. The homotopy fixed point theorem and theQuillen-Lichtenbaum conjecture in hermitian K -theory. Adv. Math. , 278:34–55, 2015.[7] S. Bloch and S. Lichtenbaum. A spectral sequence for motivic cohomology. .[8] J. M. Boardman. Conditionally convergent spectral sequences. In Homotopy invariant algebraic structures(Baltimore, MD, 1998) , volume 239 of Contemp. Math. , pages 49–84. Amer. Math. Soc., Providence, RI, 1999.[9] D. Dugger and D. C. Isaksen. Motivic cell structures. Algebr. Geom. Topol. , 5:615–652, 2005.[10] B. I. Dundas, O. R¨ondigs, and P. A. Østvær. Motivic functors. Doc. Math. , 8:489–525 (electronic), 2003.[11] E. M. Friedlander. Computations of K -theories of finite fields. Topology , 15(1):87–109, 1976.[12] E. M. Friedlander. Motivic complexes of Suslin and Voevodsky. Ast´erisque , (245):Exp. No. 833, 5, 355–378,1997. S´eminaire Bourbaki, Vol. 1996/97.[13] D. Grayson. The motivic spectral sequence. In Handbook of K -theory. Vol. 1, 2 , pages 39–69. Springer, Berlin,2005.[14] J. J. Guti´errez, O. R¨ondigs, M. Spitzweck, and P. A. Østvær. Motivic slices and coloured operads. J. Topol. ,5(3):727–755, 2012.[15] R. T. Hoobler. The Merkuriev-Suslin theorem for any semi-local ring. J. Pure Appl. Algebra , 207(3):537–552,2006.[16] J. Hornbostel. A -representability of hermitian K -theory and Witt groups. Topology , 44(3):661–687, 2005.[17] J. Hornbostel. Some comments on motivic nilpotence. arXiv:1511.07292 .[18] J. Hornbostel and M. Schlichting. Localization in hermitian K -theory of rings. J. London Math. Soc. (2) ,70(1):77–124, 2004.[19] M. Hoyois, S. Kelly, and P. A. Østvær. The motivic Steenrod algebra in positive characteristic. arXiv:1305.5690 ,to appear in: JEMS. 20] P. Hu, I. Kriz, and K. Ormsby. The homotopy limit problem for Hermitian K-theory, equivariant motivichomotopy theory and motivic Real cobordism. Adv. Math. , 228(1):434–480, 2011.[21] D. C. Isaksen and A. Shkembi. Motivic connective K -theories and the cohomology of A(1). J. K-Theory ,7(3):619–661, 2011.[22] B. Kahn. La conjecture de Milnor (d’apr`es V. Voevodsky). Ast´erisque , (245):Exp. No. 834, 5, 379–418, 1997.S´eminaire Bourbaki, Vol. 1996/97.[23] M. Karoubi. Le th´eor`eme fondamental de la K -th´eorie hermitienne. Ann. of Math. (2) , 112(2):259–282, 1980.[24] M. Kerz. The Gersten conjecture for Milnor K -theory. Invent. Math. , 175(1):1–33, 2009.[25] D. Kobal. K -theory, Hermitian K -theory and the Karoubi tower. K -Theory , 17(2):113–140, 1999.[26] M. Levine. Relative Milnor K -theory. K -Theory , 6(2):113–175, 1992.[27] M. Levine. Homology of algebraic varieties: an introduction to the works of Suslin and Voevodsky. Bull. Amer.Math. Soc. (N.S.) , 34(3):293–312, 1997.[28] M. Levine. The homotopy coniveau tower. J. Topol. , 1(1):217–267, 2008.[29] M. Levine. Convergence of Voevodsky’s slice tower. Doc. Math. , 18:907–941 (electronic), 2013.[30] M. Levine. A comparison of motivic and classical stable homotopy theories. J. Topol. , 7(2):327–362, 2014.[31] S. Lichtenbaum. The construction of weight-two arithmetic cohomology. Invent. Math. , 88(1):183–215, 1987.[32] C. Mazza, V. Voevodsky, and C. Weibel. Lecture notes on motivic cohomology , volume 2 of Clay MathematicsMonographs . American Mathematical Society, Providence, RI, 2006.[33] J. Milnor. Algebraic K -theory and quadratic forms. Invent. Math. , 9:318–344, 1969/1970.[34] J. Milnor. Collected papers of John Milnor. V. Algebra . American Mathematical Society, Providence, RI, 2010.Edited by H. Bass and T. Y. Lam.[35] F. Morel. Voevodsky’s proof of Milnor’s conjecture. Bull. Amer. Math. Soc. (N.S.) , 35(2):123–143, 1998.[36] F. Morel. Suite spectrale d’Adams et invariants cohomologiques des formes quadratiques. C. R. Acad. Sci. ParisS´er. I Math. , 328(11):963–968, 1999.[37] F. Morel. Milnor’s conjecture on quadratic forms and mod 2 motivic complexes. Rend. Sem. Mat. Univ. Padova ,114:63–101 (2006), 2005.[38] F. Morel. A -algebraic topology over a field , volume 2052 of Lecture Notes in Mathematics . Springer, Heidelberg,2012.[39] D. Orlov, A. Vishik, and V. Voevodsky. An exact sequence for K M ∗ / Ann. of Math. (2) , 165(1):1–13, 2007.[40] I. Panin and C. Walter. On the motivic commutative ring spectrum BO , K-theory Preprint Archives:0978.[41] P. Pelaez. Multiplicative properties of the slice filtration. Ast´erisque , (335):xvi+289, 2011.[42] P. Pelaez. On the functoriality of the slice filtration. J. K-Theory , 11(1):55–71, 2013.[43] A. Pfister. On the Milnor conjectures: history, influence, applications. Jahresber. Deutsch. Math.-Verein. ,102(1):15–41, 2000.[44] D. Popescu. General N´eron desingularization. Nagoya Math. J. , 100:97–126, 1985.[45] O. R¨ondigs. On the η -inverted sphere. arXiv:1602.08798 .[46] O. R¨ondigs and P. A. Østvær. Motives and modules over motivic cohomology. C. R. Math. Acad. Sci. Paris ,342(10):751–754, 2006.[47] O. R¨ondigs and P. A. Østvær. Modules over motivic cohomology. Adv. in Math. , 219(2):689–727, 2008.[48] O. R¨ondigs, M. Spitzweck, and P. A. Østvær. Cellularity of hermitian K -theory and Witt theory. Preprint,2015.[49] M. Schlichting. Hermitian K -theory, derived equivalences, and Karoubi’s fundamental theorem. arXiv:1209.0848 .[50] M. Spitzweck. Relations between slices and quotients of the algebraic cobordism spectrum. Homology, HomotopyAppl. , 12(2):335–351, 2010.[51] M. Spitzweck. A commutative P -spectrum representing motivic cohomology over Dedekind domains. arXiv:1207.4078 . 52] A. A. Suslin. On the K -theory of local fields. In Proceedings of the Luminy conference on algebraic K -theory(Luminy, 1983) , volume 34, pages 301–318, 1984.[53] A. A. Suslin. Torsion in K of fields. K -Theory , 1(1):5–29, 1987.[54] A. A. Suslin. Voevodsky’s proof of the Milnor conjecture. In Current developments in mathematics, 1997(Cambridge, MA) , pages 173–188. Int. Press, Boston, MA, 1999.[55] A. Suslin and V. Voevodsky. Bloch-Kato conjecture and motivic cohomology with finite coefficients. In Thearithmetic and geometry of algebraic cycles (Banff, AB, 1998) , volume 548 of NATO Sci. Ser. C Math. Phys.Sci. , pages 117–189. Kluwer Acad. Publ., Dordrecht, 2000.[56] R. W. Thomason. The homotopy limit problem. In Proceedings of the Northwestern Homotopy Theory Confer-ence (Evanston, Ill., 1982) , volume 19 of Contemp. Math. , pages 407–419, Providence, R.I., 1983. Amer. Math.Soc.[57] V. Voevodsky. A -homotopy theory. In Proceedings of the International Congress of Mathematicians, Vol. I(Berlin, 1998) , volume Extra Vol. I, pages 579–604 (electronic), 1998.[58] V. Voevodsky. Voevodsky’s Seattle lectures: K -theory and motivic cohomology. In Algebraic K -theory (Seattle,WA, 1997) , volume 67 of Proc. Sympos. Pure Math. , pages 283–303. Amer. Math. Soc., Providence, RI, 1999.Notes by C. Weibel.[59] V. Voevodsky. Triangulated categories of motives over a field. In Cycles, transfers, and motivic homologytheories , volume 143 of Ann. of Math. Stud. , pages 188–238. Princeton Univ. Press, Princeton, NJ, 2000.[60] V. Voevodsky. Open problems in the motivic stable homotopy theory. I. In Motives, polylogarithms and Hodgetheory, Part I (Irvine, CA, 1998) , volume 3 of Int. Press Lect. Ser. , pages 3–34. Int. Press, Somerville, MA,2002.[61] V. Voevodsky. A possible new approach to the motivic spectral sequence for algebraic K -theory. In Recentprogress in homotopy theory (Baltimore, MD, 2000) , volume 293 of Contemp. Math. , pages 371–379. Amer.Math. Soc., Providence, RI, 2002.[62] V. Voevodsky. Motivic cohomology with Z / Publ. Math. Inst. Hautes ´Etudes Sci. , (98):59–104,2003.[63] V. Voevodsky. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes ´Etudes Sci. , (98):1–57, 2003.[64] V. Voevodsky. On the zero slice of the sphere spectrum. Tr. Mat. Inst. Steklova , 246(Algebr. Geom. Metody,Svyazi i Prilozh.):106–115, 2004., 246(Algebr. Geom. Metody,Svyazi i Prilozh.):106–115, 2004.