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Slices of hermitian K-theory and Milnor's conjecture on quadratic forms

We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.


Introduction
Voevodsky's solutions of the Milnor conjectures on Galois cohomology [44] and on quadratic forms [29] for fields of characteristic unequal to two are striking applications of motivic homotopy theory. We give an alternate proof of Milnor's conjecture on quadratic forms by explicitly computing the slice spectral sequence for higher Witt-theory KT. Our method of proof applies also to smooth semilocal rings containing a field of characteristic zero.
Let X ∈ Sm F be a smooth scheme of finite type over a perfect field F . We refer to [9] for a survey of the known constructions of the first quadrant convergent spectral sequence relating motivic cohomology to algebraic K-theory MZ ⋆ (X) =⇒ KGL * (X). (1) From the viewpoint of motivic homotopy theory [42], the problem of constructing (1) reduces to identifying the slices s q (KGL) of the motivic spectrum KGL representing algebraic K-theory. Voevodsky introduced the slice spectral sequence and conjectured that The formula (2) was proven for fields of characteristic zero by Voevodsky [43], [46], and (invoking different methods) for perfect fields by Levine [21].
Moreover, the slices of higher Witt-theory factor into infinite wedge product decompositions The summand Σ 2q,q MZ in (3) is detected by showing that s q (KGL) is a retract of s q (KQ). We deduce that s q (KQ) is a wedge sum of Σ 2q,q MZ and some MZ-module, i.e., a motive, which we identify with the infinite wedge summand in (3). Our first results show there is an additional "mysterious summand" Σ 2q,q Mµ of s q (KQ). We show Mµ is trivial by using base change and the solution of the homotopy fixed point problem for hermitian K-theory of the prime fields [3], [8], cf. [4], [15]. In the course of the proof we note the useful result that slices commute with homotopy fixed points for finite groups. As conjectured in Hornbostel's foundational paper [12] there is a homotopy cofiber sequence relating the algebraic and hermitian K-theories The stable Hopf map η is induced by the canonical map A 2 {0} → P 1 . We show this over any finite dimensional regular noetherian base scheme S equipped with the trivial involution and with 2 invertible in its ring of regular functions, i.e., 1 2 ∈ Γ(S, O S ). The sequence (5) is employed in our computation of the slices of KQ.
By combining the slice computations in (3) and (8) with the homotopy cofiber sequence in (6) we deduce the identification of the slices of the Witt-theory spectrum KT in (4). Alternatively, this follows from (3), (7), and Spitzweck's result that slices commutes with homotopy colimits [36].
Our next goal is to determine the d 1 -differentials in the slice spectral sequences. Because the slices of KQ and KT are motives, this involves the motivic Steenrod squares Sq i constructed by Voevodsky [44] and further elaborated on in [14]. According to (4) the differential is a map of the form i∈Z Σ 2i+q,q MZ/2 ≻ P 1 ∧ j∈Z Σ 2j+q,q MZ/2.
Here h p,q is shorthand for the mod-2 motivic cohomology group in degree p and weight q. The classes τ ∈ h 0,1 and ρ ∈ h 1,1 are represented by −1 ∈ F . We denote integral motivic cohomology groups by H p,q . This sets the stage for our proof of Milnor's conjecture on quadratic forms formulated in [25,Question 4.3]. For fields of characteristic zero this conjecture was shown by Orlov, Vishik and Voevodsky in [29], and by Morel [26], [27] using different approaches.
According to (4) the slice spectral sequence for KT fills out the upper half-plane. A strenuous computation using (9), Adem relations, and the action of the Steenrod squares on the mod-2 motivic cohomology ring h ⋆ of F shows that it collapses. We read off the isomorphisms E 2 p,q (KT) = E ∞ p,q (KT) ∼ = h q,q p ≡ 0 mod 4 0 otherwise.
To connect this computation with quadratic form theory, we show the spectral sequence converges to the filtration of the Witt ring W (F ) by the powers of the fundamental ideal I(F ) of even dimensional forms. By identifying motivic cohomology with Galois cohomology for fields we arrive at the following result.
Theorem. Suppose F is a perfect field and char(F ) = 2. Then the slice spectral sequence for KT converges and furnishes a complete set of invariants e q F : I(F ) q /I(F ) q+1 ∼ = ≻ H q (F ; Z/2) for quadratic forms over F with values in the mod-2 Galois cohomology ring.
The same conclusion follows for any field of characteristic unequal to two by comparing with its Cohen ring. A further generalization to smooth semilocal rings containing a field of characteristic zero follows by the same method using Levine's generalized Milnor conjecture onétale cohomology of semilocal rings [20], as shown in [11] and [18], cf. [23].
In Section 7 we perform computations in the slice spectral sequence for KQ in low degrees. We formulate our computation of the second orthogonal K-group, and refer to the main body of the paper for the more complicated to state results on other hermitian K-groups.
Theorem. Suppose F is a perfect field and char(F ) = 2. There is a naturally induced isomorphism Throughout the paper we employ the following notation. S finite dimensional regular and separated noetherian base scheme Sm S smooth schemes of finite type over S S 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, 1 = S 0,0 generic motivic spectrum, the motivic sphere spectrum 2 Slices and the slice spectral sequence Let i q : S 2q,q ∧ SH eff ⊂ ≻ SH be the full inclusion of the localizing subcategory generated by S 2q,q -suspensions of smooth schemes. We denote by r q the right adjoint of i q and set f q = i q • r q . The qth slice of E is characterized up to unique isomorphism by the distinguished triangle in SH [42]. When it is helpful to emphasize the base scheme S we shall write f S q (E) and s S q (E). Smashing with the motivic sphere S 1,1 has the following effect on the slice filtration.
The former isomorphism is compatible with the natural transformation f q+1 ≻ f q .
Let η : S 1,1 ≻ 1 denote the Hopf map induced by the canonical map A 2 {0} ≻ P 1 . Since every motivic spectrum E is a module over the motivic sphere spectrum 1, multiplication with η defines, by abuse of notation, a map η : For every E and q ∈ Z there is a naturally induced commutative diagram: The Hopf map induces a periodicity isomorphism η : S 1,1 ∧ KT ∼ = ≻ KT. In particular, the vertical map on the left hand side in the diagram of Lemma 2.2 is an isomorphism. It also implies the isomorphism s q (KT) ∼ = S q,q ∧ s 0 (KT).
The slices s q (E) are modules over the motivic ring spectrum s 0 (1), cf. [30] and [10]. If the base scheme S is a perfect field, then s 0 (1) is the Eilenberg-MacLane spectrum MZ by the works of Levine [21] and Voevodsky [46]. We set D 1 p,q,n = π p,n f q (E) and E 1 p,q,n = π p,n s q (E). The exact couple Let α * be the right adjoint of α * . By uniqueness of adjoints, up to unique isomorphism, there exists a natural isomorphism of triangulated functors r Y q α * ∼ = ≻ α * r X q . Since α is smooth, the functor α * has a left adjoint α ♯ and there is a commutative diagram: By uniqueness of adjoints, there is an isomorphism The desired isomorphism for slices follows since, by uniqueness of adjoints, (12) is compatible with the natural transformation f q+1 ≻ f q .
Suppose I is a filtered partially ordered set and fix a diagram of Y -schemes with affine bonding maps. There is a naturally induced map α : Theorem 2.5. For α : X ≻ Y as above and q ∈ Z there are natural isomorphisms Proof. Let G be a compact generator of the triangulated category S 2q+2,q+1 ∧ SH eff (X) and E ∈ SH(Y ). According to [31,Theorem 2.12] it suffices to show [G, α * s Y q (E)] is trivial. Note that G is of the form α(j) * G j for G j a compact generator of the triangulated category S 2q+2,q+1 ∧ SH eff (Y j ). Consider the overcategory of j ∈ I and the composite diagram 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 maps from 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 map is an isomorphism in SH(X j ) for all F ∈ SH(X j ). Since G j is compact and slices commute with base change along smooth maps by Lemma 2.4, the group is trivial. With reference to [31,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 B q α * ∼ = α * s A q . Proof. By Popescu's general Néron desingularization theorem [32] the regularity assumption on α is equivalent to B being the colimit of a filtered diagram of smooth A-algebras of finite type. The result follows now from Theorem 2.5.
3 Milnor's conjecture on quadratic forms In this section we compute the slice spectral sequence of KT over a perfect field F of characteristic char(F ) = 2. Note that KT acquires a ring spectrum structure from hermitian K-theory KQ and the tower (7). Recall from (4), cf. Example 2.3, the identification By the periodicity S 1,1 ∧ KT ∼ = KT induced by multiplication with the Hopf map, this determines all slices of KT. Recall that the first differential s q (KT) ≻ S 1,0 ∧ s q+1 (KT) is determined by the motivic Steenrod operations in (9), a formula proven in Theorem 6.3. With the elements τ , Sq 2 , Sq 2 + ρSq 1 , and Sq 3 Sq 1 corresponding to their given colors, the d 1 -differentials can be represented as follows. The group in bidegree (p, q) is a direct sum of mod-2 motivic cohomology groups positioned on the vertical line above p and inbetween the horizontal lines corresponding to weights 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 conjecture on Galois cohomology [44]. Recall the classes τ ∈ h 0,1 and ρ ∈ h 1,1 are represented by −1 ∈ F .
Corollary 3.2. If a ∈ h p,q where 0 ≤ p ≤ q, write a = τ q−p c where c ∈ h p,p and let n = q − p. The Steenrod squares of weight ≤ 1 act on the mod-2 motivic cohomology ring h * , * by Proof. This follows from Lemma 3.1, the computation of Sq i (τ n ) for i ∈ {1, 2} and the Cartan formula [45,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, 1), (1, 1), and (1, 2) will be relevant for us. The Adem relations in weight less than or equal to 2 are given by This concludes the prerequisites for our proof of the following result.
Theorem 3.3. The 0th slice spectral sequence for KT collapses at its E 2 -page, and Proof. The E 1 -page takes the form The group h 2i+(q−p),q in the sum i∈Z h 2i+(q−p),q arises from the 2ith summand of s q (KT). The vanishing of h p,q for p < 0 and p > q shows the sum is finite. Hence for every element a ∈ E 1 p,q there exists a unique collection of elements {a j ∈ h j,q } such that (a q , a q−2 , . . . , a 0 ) ∈ E 1 p,q q ≡ 0 ≡ p mod 2 (a q−1 , a q−3 , . . . , a 0 ) ∈ E 1 p,q q ≡ 1 ≡ p mod 2 (a q , a q−2 , . . . , a 1 ) ∈ E 1 p,q q ≡ 1 ≡ p mod 2 (a q−1 , a q−3 , . . . , a 1 ) ∈ E 1 p,q q ≡ 0 ≡ p mod 2.
By inspection of (9) the components of d KT 1 are given by Here j is the dimension index of the target group. The formula can be read off from the parity of the dimension index of the source group. Note that ρSq 1 , Sq 2 and τ shift the dimension by 2, 2, and 0, respectively. We compute the E 2 -page by repeatedly using the Steenrod square computations and Adem relations given in the beginning of Section 3.
If d KT 1 (a) = 0 then 0 = a q−3 = a q−7 = · · · by applying inductively the injectivity statement in Lemma 3.1. Thus we have For b ∈ E 1 p+1,q−1 the entering differential is given by Corollary 3.2 simplifies this formula to Thus E 2 p,q is the homology of the complex Since the restriction of d KT p ≡ 2 mod 4 For a = (a q , a q−2 , . . . ), Corollary 3.2 implies Hence the subgroup ker(d KT 1 ) can be identified with p+1,q−1 the entering differential is given by Corollary 3.2 simplifies this formula to Thus . . ). This shows that E 2 p,q is trivial. p ≡ 3 mod 4 For a = (a q−1 , a q−3 , . . . ) the exiting differential simplifies to by Corollary 3.2. Applying Sq 2 to Sq 2 a j−2 + τ a j yields p+1,q−1 the entering differential simplifies to Hence Sq 2 b j = a j+2 by Lemma 3.1. It follows that . . ). This shows that E 2 p,q is trivial.
An inspection of the E 2 -page shows that the 0th slice spectral sequence for KT collapses with E 2 = E ∞ -page: 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 the slice filtration on the Witt ring with the filtration given by the fundamental ideal.
Proof. By [10], [30] the canonical map f 0 (KT) ≻ s 0 (KT) induces a ring homomorphism It is natural with respect to separable field extensions according to Corollary 2.7. Comparing with an algebraic closure of F shows this ring homomorphism is induced by sending a quadratic form to its rank. Since the group π 1,0 s 0 (KT) = h 1,0 is trivial, the map is injective, which proves the result.
Proof. Since KT is a ring spectrum the claim follows from the multiplicative structure of the slice filtration [10], [30] (note that KT is not effective), and Lemma 3.4.
Any rational point u ∈ A 1 {0}(F ) defines a map of motivic spectra [u] : 1 ≻ S 1,1 . We are interested in the effect of the map [u] on motivic cohomology and KT. Lemma 3.6. The map Proof. Let Z(1) denote the Tate object in the derived category of motives DM F over F . The assertion follows from the canonically induced diagram: The lower horizontal isomorphism follows from [24,Theorem 4.1].
Lemma 3.8. The composition 1 [u] ≻ S 1,1 η ≻ 1 induces multiplication by u − 1 on π 0,0 1, where u is the class of the rank one quadratic form in the Grothendieck-Witt ring defined by u.
The unit map for KT induces the canonical map from the Grothendieck-Witt ring to the Witt ring π 0,0 1 ≻ π 0,0 KT. By definition, multiplication by η is an isomorphism on π * , * KT. Thus the "multiplication by [u]" map on π 0,0 KT is determined by its effect on the Grothendieck-Witt ring π 0,0 1. 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.
There is a canonically induced short exact sequence Proof. Theorem 3.3 shows the exact sequence in the proof of [42, Lemma 7.2] takes the form Moreover, π −1,0 KT is the trivial group. Corollary 3.5 furnishes a map Combined with the canonical surjective map k M q γ ≻ I(F ) q /I(F ) q+1 from the q-th mod-2 Milnor K-theory group defined in [25], we obtain the composite map Lemmas 3.6 and 3.8 show that (14) coincides with Suslin's isomorphism between Milnor K-theory and the diagonal of motivic cohomology [24,Lecture 5]. In particular, α is surjective.
The above shows that γ is injective, which gives an alternate proof of the main result in [29].
Proof. By the definition of β there is a commutative diagram: Note that β is an isomorphism by Lemma 3.9, Theorem 3.10, and the isomorphism (14). Thus the result follows by induction using the identification This essentially finishes our proof of Milnor's conjecture on quadratic forms.
Thus the slice spectral sequence for KT converges to the filtration of the Witt ring given by the fundamental ideal.
Proof. This follows from Corollary 3.11 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,0 f q+1 (KT) ≻ π p,0 f q (KT) for columns p ≡ 1, 2, 3 mod 4 in Lemma 3.13 below.
Proof. The statement is clear for q = 0. Suppose that p ≡ 1, 2 mod 4 and x ∈ π p,0 f q+1 (KT). Its image y ∈ π p,0 f q (KT) lies in the kernel of the map π p,0 f q (KT) ≻ π p,0 s q (KT). By induction, the map π p+1,0 s q−1 (KT) ≻ π p,0 f q (KT) is surjective. Hence there is an element z ∈ π p+1,0 s q−1 (KT) mapping to y. Thus z lies in the kernel of d p+1,q−1 . Theorem 3.3 shows the kernel of 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 ≡ 3 mod 4 and x ∈ π p,0 f q+1 (KT). Consider its image y ∈ π p,0 s q+1 (KT). Since its image under d p,q+1 is trivial, Theorem 3.3 furnishes an element z ∈ π p+1,0 s q+2 (KT) with d p+1,q+2 (z) = y. Consider the difference x − w, where w is the image of z. The image of x − w in π p,0 f q (KT) then coincides with the image of x. Since the image of x − w in π p,0 s q+1 (KT) is zero, there is an element v ∈ π p,0 f q+2 (KT) mapping to x − w. Proceeding inductively yields an element However, using that π p,0 KT = 0, this group is trivial by the exact sequence from the proof of [42,Lemma 7.2]. Hence e = 0, and the image of x in π p,0 f q (KT) is zero.
Proof. The latter claim follows by a diagram chase and Theorem 3.3, since E 2 p,q = 0 if p ≡ 1 mod 4. Hence the exact sequence induces the short exact sequence Since Ker(α) = Im(β) = Im(d 2,q−1 ) = h q−1,q ⊕ h q−5,q ⊕ · · · by Theorem 3.3, the sequence is short exact. It splits by Lemma 3.1, since the composition of is given by multiplication with τ ∈ h 0,1 . Theorems 3.3 and 3.12 imply the main theorem stated in the introduction. If X ∈ Sm F is a semilocal scheme 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 the Milnor conjecture for Galois cohomology [44] can be replaced by [11, §2.2] or [18,Theorem 7.8], while the isomorphism (14) holds for X by [18,Theorem 7.6]. The rest of the proof is identical to the one given for fields. Kerz proved a closely related result in [18,Theorem 7.10]. By periodicity of KT there is an evident variant of Theorem 3.3 for the nth slice spectral sequence of KT for every n ∈ Z. We note the following result from [2] is transparent from our computation of the slice spectral sequence for KT. Corollary 3.17. If X ∈ Sm F is a semilocal scheme of geometric origin then W (X) contains no elements of odd order. If X is not formally real then W (X) is a 2-primary torsion group.

Algebraic and hermitian K-theory
Let KGL = (K, . . . ) denote the motivic spectrum representing algebraic K-theory [39] over the base scheme S. Here, K : Sm S ≻ Spt sends a smooth S-scheme to its algebraic K-theory spectrum. The structure maps of KGL are given by the Bott periodicity operator Suppose 2 is invertible in the ring of regular functions on S. Let KO : Sm S ≻ Spt denote the functor 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, we use the notation KSp : Sm S ≻ Spt. There are natural maps f 0 : KO ≻ K and f 2 : KSp ≻ K induced by the forgetful functors. Taking the homotopy fibers of these maps, we set Karoubi's fundamental theorem in hermitian K-theory [17] can be formulated as follows.
The homotopy cofiber sequences in Theorem 4.2 split by the unit section 1 ∈ A 1 {0}(S). Since Ω 1,1 is the homotopy fiber with respect to the unit section, Theorem 4.2 implies the natural weak equivalences In [12] Hornbostel shows that hermitian K-theory is represented by the motivic spectrum The structure maps of KQ are the adjoints of the weak equivalences: Proposition 4.3. There are commutative diagrams: The ≻ Ω 1,1 KQ of motivic spectra is tantamount to giving a compatible sequence of maps between motivic spaces Next we show the homotopy cofiber sequence (5) relating algebraic and hermitian K-theory via the stable Hopf map, as reviewed in the introduction.
The connecting map factors as KGL ∼ = ≻ S 2,1 ∧ KGL Proof. We show the map KQ ≻ Ω 1,1 KQ induced by η is determined by the canonical maps To that end, it suffices to describe the maps induced by the unstable Hopf map A 2 {0} ≻ P 1 , or equivalently by We note there is an isomorphism of schemes Now the composite νθ 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 ν and the projection map coincide up to weak equivalence. Proof. The unit map 1 ≻ KGL is given by the trivial line bundle over the base scheme S. The latter is obtained by forgetting the standard nondegenerate quadratic form on it, which provides the factorization.
Let ǫ : 1 ≻ 1 be the endomorphism of the sphere spectrum induced by the commutativity isomorphism on the smash product S 1,1 ∧ S 1,1 . Proof. Since 1 − ǫ = 1 + −1 is the hyperbolic plane [28, p. 53] the unit map for KQ induces a commutative diagram: By smashing with KQ and employing its multiplicative structure we obtain the diagram: The middle vertical map is KQ ∧ (h • f ), while the two composite horizontal maps coincide with the identity on KQ. The right hand square commutes because h • f is a map of KQ-modules.

Algebraic K-theory
We recall and augment previous work on the slices of algebraic K-theory.
Theorem 5.1 (Levine, Voevodsky). Suppose F is a perfect field. The unit map 1 ≻ KGL induces an isomorphism of zero slices Hence there is an isomorphism s q (KGL) ∼ = S 2q,q ∧ MZ for all q ∈ Z.

Homotopy orbit K-theory
Let P 1 be pointed at ∞. The general linear group scheme GL(2n) acquires an involution given by Geometrically, this corresponds to a strictification of the pseudo-involution obtained by sending a vector bundle of rank n to its dual. These involutions induce the inverse-transpose involution on the infinite general linear group scheme GL and its classifying space BGL [38]. Letting C 2 operate trivially on the first factor in Z × BGL, the involution coincides sectionwise with the unstable Adams operation Ψ −1 on the motivic space representing algebraic K-theory. The stable Adams operation is determined by the structure map K ≻ Ω 2,1 K = hofib(K(− × P 1 ) ∞ * ≻ K(−)) obtained from multiplication by the class 1 − [O(−1)] ∈ K 0 (P 1 ) of the hyperplane section. Note that Thus, up to homotopy, the operation Ψ −1 st : KGL ≻ KGL is given levelwise by the formula Since smashing with S 2,1 shifts motivic spectra by one index we get When forming homotopy fixed points and homotopy orbits of KGL we implicitly make use of a naive C 2 -equivariant spectrum, i.e., a (non-equivariant) motivic spectrum with a C 2 -action, whose underlying motivic spectrum maps by a levelwise weak equivalence to KGL. Recall the Witt-theory spectrum KT is the homotopy colimit of the sequential diagram Note that KT is a motivic ring spectrum equipped with an evident KQ-algebra structure. There is a homotopy cofiber sequence relating KQ, KT and homotopy orbit algebraic K-theory [19].

Theorem 5.2 (Kobal).
There is a homotopy cofiber sequence  Proof. This follows from (16) and the equality s 0 (Ψ −1 ) = id MZ . For the latter, note that the Adams involution and the unit map of KGL yield commutative diagrams: By reference to Theorem 5.1 it follows that s 0 (Ψ −1 st ) = id MZ .
Next we consider the composite map Proposition 5.4. The endomorphism s q (f h) of s q (KGL) ∼ = S 2q,q ∧ MZ is multiplication by 2 if q is even and the trivial map if q is odd.
Lemma 5.5. 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 shown by Spitzweck [36,Lemma 4.4], the conclusion follows because s q commutes with sums.
The idea is now to combine Theorem 5.1, Proposition 5.3 and Lemma 5.5 to identify the slices of homotopy orbit K-theory. To start with, we compute the homotopy orbit spectra of MZ for the trivial and nontrivial involutions over a base scheme essentially smooth over a field of characteristic unequal to 2.
Lemma 5.6. With the trivial involution on motivic cohomology there is an isomorphism Proof. If E is equipped with an involution then E hC 2 ∼ = E∧(EC 2 ) + C 2 , where EC 2 is a contractible simplicial set with a free C 2 -action considered as a constant motivic space. In the case of (MZ, id), Since S n,0 ≻ RP n ≻ RP n+1 is a homotopy cofiber sequence, so is Clearly MZ ∧ RP 2 ∼ = S 1,0 ∧ MZ/2. Proceeding by induction on (18) using Lemma A.3, we get Lemma 5.7. With the nontrivial involution on motivic cohomology there is an isomorphism Proof. The nontrivial involution σ on MZ is determined levelwise by the nontrivial involution on MZ n = K(Z, 2n, n) = Z tr (S 2n,n ).
Here, Z tr sends a motivic space X to the motivic space with transfers freely generated by X [7]. On the level of motivic spaces the nontrivial involution is induced by a degree −1 pointed map of the simplicial circle S 1,0 . Since Z tr commutes with homotopy colimits [34] we are reduced to identify hocolim C 2 S 2n,n .
When n = 0 the homotopy colimit is contractible. When n > 0 there are isomorphisms By passing to the sphere spectrum the above yields an isomorphism Finally, applying transfers and arguing as in the proof of Lemma 5.6, we deduce the isomorphism Theorem 5.8. Suppose F is a perfect field with the trivial involution and char(F ) = 2. The slices of homotopy orbit K-theory are given by Proof. This follows from Theorem 5.1 and Lemmas 5.5, 5.6, 5.7.

Hermitian K-theory
Throughout this section F is a perfect field with char(F ) = 2. Proof. The unit of MZ induces an isomorphism on zero slices by Theorem 5.1. Since MZ ∈ SH eff is an effective motivic spectrum, the counit f 0 (MZ) ≻ MZ is an isomorphism. Thus there is a canonical isomorphism of ring spectra It follows that smashing with MZ, i.e., passing to motives [33], [34], induces a canonical ring map Corollary 5.11. In the splitting s 0 (KQ) ∼ = s 0 (KGL) ∨ µ the homotopy groups π s,t µ are modules over π 0,0 MZ/2 ∼ = F 2 .
Proof. This follows from Corollary 5.9 and Proposition 5.10.
The slice functors are triangulated. Thus the homotopy cofiber sequence in Theorem 4.4 induces homotopy cofiber sequences of slices.
Lemma 5.14. The map s 1 (h) : Proof. The first component of is trivial for degree reasons by Lemma A.3. Thus s 1 (h) corresponds to the transpose of the matrix 0 β . By Corollary 5.13, s 1 (f ) corresponds to the matrix δ 0 . Proposition 5.10 implies that is the multiplication by 2 map, hence trivial on S 1,1 ∧ (MZ/2 ∨ µ). We conclude that βδ = 0. Thus β factors as Since the multiplication by 2 map on µ is trivial, the result follows.
Corollary 5.15. There is an isomorphism which identifies s 2 (f ) with the projection map onto S 4,2 ∧ MZ. Proof.
Lemma 5.14 implies that s 1 (h) is the trivial map.

Moreover, s 3 (f ) coincides with the composite map
Proof. Apply s 3 to the Hopf cofiber sequence in Theorem 4.4 and use Lemma 5.16.
Theorem 5.18. The slices of the hermitian K-theory spectrum KQ are given by Here Mµ ∼ = S 4,0 ∧ Mµ and Mµ s,t is an F 2 -module for all integers s and t.
Proof. Corollary 5.17 identifies the third slice On the other hand, Karoubi periodicity and Corollary 5.9 imply the isomorphism It follows that S −2,0 ∧MZ∨S −4,0 ∧MZ is a direct summand of µ. Iterating the procedure furnishes an isomorphism where Mµ is simply the complementary summand. The result follows.
In the next section we show the mysterious summand Mµ of s 0 (KQ) is trivial. The following summarizes some of the main observations in this section.

Higher Witt-theory
By combining Theorems 5.2, 5.8, 5.18, and 5.29 we have enough information to identify the slices of Witt-theory. An alternate proof with no mention of homotopy orbit K-theory follows by Lemma 5.5 and the identification of s q (η) : Theorem 5.20. The slices of Witt-theory are given by Let u : KQ ≻ KT be the canonical map. The next result follows now by an easy inspection.

The mysterious summand is trivial
To prove Mµ is trivial we use the solution of the homotopy limit problem for hermitian K-theory of prime fields due to Berrick-Karoubi [3] and Friedlander [8]. Their results have been generalized by Hu-Kriz-Ormsby [15] and Berrick-Karoubi-Schlichting-Østvaer [4], where the following is shown.
To make use of Theorem 5.22 we need to understand how slices compare with mod-2 reductions of motivic spectra and formation of homotopy fixed points. The former is straightforward.
Lemma 5.24. There is a naturally induced commutative diagram: With the trivial C 2 -action on MZ the homotopy fixed points spectrum is given by The canonical map (MZ, id) hC 2 ≻ MZ corresponds to the map induced by Spec(F ) + ⊂ ≻ RP ∞ + . Induction on n shows there are isomorphisms Writing sSet * (RP ∞ , MZ) = colim n sSet * (RP n , MZ) we deduce Proposition A.5 identifies the latter with i<0 S 2i,0 ∧ MZ/2.
Lemma 5.25. The nontrivial involution on MZ and the connecting map δ : S −1,0 ∧MZ/2 ≻ MZ give rise to the commutative diagram: Proof. Let (G, σ) be an involutive simplicial abelian group with σ(g) = −g and EC 2 a contractible simplicial set with a free C 2 -action. Then C 2 acts on the space of fixed points sSet(EC 2 , G) C 2 by conjugation, i.e., on the simplicial set of maps from EC 2 to (G, σ). Now choose EC 2 such that its skeletal filtration (S 0 , g) ⊆ (S 1 , g) ⊆ · · · ⊆ (S n , g) ⊆ · · · comprises spheres equipped with the antipodal C 2 -action. Here (S n+1 , g) arises from (S n , g) by attaching a free C 2 -cell of dimension n + 1 along the C 2 -map C 2 × 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: By specializing to n = 0, (19) extends to the commutative diagram: The right hand square is a pullback, and addition is a Kan fibration. Hence sSet (S 1 , g), (G, σ) C 2 is the homotopy fiber of multiplication by 2 on G. On the other hand, there is a homotopy cofiber sequence S 1 2 ≻ S 1 ∼ = RP 1 ≻ RP 2 and sSet (S 1 , g), (G, σ) C 2 has a canonical basepoint. Thus there is a homotopy equivalence By induction on (19), we find for every n ≥ 0 a homotopy equivalence A levelwise and sectionwise application of (20) yields the weak equivalences (For the second weak equivalence see the proof of Lemma 5.6.) The canonical map from (MZ, σ) hC 2 to MZ corresponds to the map induced by RP 1 ⊂ ≻ RP ∞ . Proposition A.5 concludes the proof.
Proposition 5.26. Let G be a finite group and E a naive G-equivariant motivic spectrum. There is a natural isomorphism Proof. Let EG be a contractible simplicial set with a free G-action. If E is bifibrant, then E hG is comprised of the fixed points of sSet(EG + , E) equipped with the conjugate action. Define the triangulated functor s <q on SH by taking the homotopy cofiber of the counit f q ≻ id. We claim there is an isomorphism s <q (E hG ) ≻ s <q (E) hG in SH. This implies the same result for the homotopy fiber f q of id ≻ s <q . With this in hand, the assertion follows from the homotopy fiber sequence S −1,0 ∧ s q ≻ f q+1 ≻ f q . In [30] s <q is identified up to natural isomorphism with RF • LU , where F is the identity functor from the stable model category MS of motivic spectra to the left Bousfield localization MS <q of MS, and U is its right adjoint. In order to compute s <q (E) hG we let R <q E be a fibrant replacement in MS <q . Since MS <q is a left Bousfield localization of a cofibrantly generated model category, we can construct R <q by applying Quillen's small object argument to a set of maps. In MS filtered colimits commute with finite limits, in particular fixed points for the action of a finite group, while pushouts with one inclusion map commute with fixed points. Thus R <q commutes with fixed points. It follows that E ≻ R <q E induces a weak equivalence in MS <q on fixed points. Thus E ≻ R <q E is a G-equivariant weak equivalence in MS <q . Since the equivariant model structure is simplicial, sSet(EG + , E) ≻ sSet(EG + , R <q E) is a G-equivariant weak equivalence in MS <q . It follows that is a weak equivalence in MS <q . By the above there is a zig-zag of weak equivalences in MS <q . This shows s <q commutes with homotopy fixed points for finite groups.
Theorem 5.27. The slices of homotopy fixed point algebraic K-theory KGL hC 2 are given by Now since f ′ • θ = f the composition ξ • ω is the identity. Over an algebraically closed field of characteristic unequal to two, the maps ω and ξ are inverse equivalences by [16,Remark 5.9] (in positive characteristic one can employ anétale homotopy realization). In order to conclude the same for any perfect field F of characteristic char(F ) = 2 we consider P . Its slices are determined by Theorem 5.27 up to the action of η on the slices of KGL hC 2 . Now P ≃ KGL after passage to an algebraic closure, so η acts on the q-th slice of KGL hC 2 as follows: If q is even, η induces the inclusion on the summand comprised of suspensions of MZ/2. And if q is odd, η induces the inclusion on the summand comprised of suspensions of MZ/2, together with the canonical map It follows that ω and ξ induce identity maps on all the slices. The slice spectral sequence for KGL converges strongly by [43,Proposition 5.5]. By commuting homotopy limits and using Lemma 5.26 it follows that the slice spectral sequence for KGL hC 2 converges conditionally in the sense that holim q f q (KGL hC 2 ) ≃ * [6]. Conditional convergence is a triangulated property so the slice spectral sequence for P converges conditionally as well. We conclude that [S p,q ∧ X + , ω] and [S p,q ∧ X + , ξ] are inverse homomorphisms for any smooth F -scheme X, which implies our first claim. The second claim follows from the factorization of the unit map for algebraic K-theory By the proof of Lemma 4.6 the third claim follows from the commutative diagram: The previous commutative diagram and (17) imply the fourth and final claim. Using the above one may now compute the slices of KGL hC 2 in the same way as for KQ, cf. Section 5.3. That is, one obtains an identification Here Mν ∼ = S 4,0 ∧ Mν and Mν s,t is an F 2 -module for all integers s and t. Moreover, it follows from our first claim above that θ : KQ ≻ KGL hC 2 splits as (id, ζ), where id is the identity on the non-mysterious summands of the respective slices, and ζ : Mµ ≻ Mν up to S 1,1 -suspension. By comparison with the identification of the slices of KGL hC 2 in Theorem 5.27, the summand Mν is contractible, which completes the proof.
Proof. By Corollary 2.7 it suffices to consider prime fields. Theorems 5.22,5.27 and Lemma 5.28 show that the mysterious summand is the homotopy fiber of a weak equivalence.

Differentials
Having determined the slices we now turn to the problem of computing the d 1 -differentials in the slice spectral sequences for KGL hC 2 , KQ and KT.
Using the Adem relations Sq 3 Sq 3 = Sq 5 Sq 1 , Sq 2 Sq 3 = Sq 5 + Sq 4 Sq 1 , and Sq 1 Sq 1 = 0 we find Thus the E 1 -page of the 0th slice spectral sequence for KGL/2 takes the form: with the first Milnor operation Q 1 to give formulas for the d 1 -differential in the slice spectral sequence for KQ. To begin with, consider the commutative diagram: To proceed from here requires two separate arguments depending on whether the d 1 -differential exits an even or an odd slice. First we analyze the even slices.
In what follows we use the computations in this section to explicitly identify d KT 1 and d KQ 1 in terms of motivic cohomology classes and Steenrod operations.
Proposition 6.4. Suppose F has mod-2 cohomological dimension zero. Then the slice filtration for KT coincides with the fundamental ideal filtration of W (F ). Moreover, there are isomorphisms The first two isomorphisms are induced by the canonical map f q (KT) ≻ s q (KT).
2. The first differential is an isomorphism if q ≡ p mod 4, and trivial otherwise.
In both cases we find E 2 = E ∞ . Our case distinctions can be recast in the following way.
The second case contradicts the vanishing of π 2,0 (KT). Computing with the first condition yields the desired result on the filtration. One extends to arbitrary n by using the commutative diagram: The following image depicts the first differential when 2q ≡ i mod 4 with degrees along the horizontal axis and weights along the vertical axis. Each dot is a suspension of MZ/2.
The following image depicts the first differential for KQ with degrees along the horizontal axis and weights along the vertical axis. Each small dot is a suspension of MZ/2 while a large dot is a suspension of MZ.  (3) the E 1 -page of the 0th slice spectral sequence for KQ takes the form E 1 p,q = π p,0 s q (KQ) = H 2q−p,q ⊕ i< q 2 h 2i+(q−p),q q ≡ 0 mod 2 i≤ q−1 2 h 2i+(q−p),q q ≡ 1 mod 2.
The group E 1 p,q in bidegree (p, q) is a direct sum of motivic cohomology groups positioned on the vertical line above p and inbetween the horizontal lines corresponding to weights q and q + 1. Using the formula in Theorem 6.5 for the d 1 -differentials we are ready to perform low degree computations in the slice spectral sequence for KQ. We assume F is a field of characteristic zero.
The Beilinson-Soule vanishing conjecture predicts that H 2q−p,q is the trivial group if p > 2q, which holds for instance for finite fields and number fields. Recall that H 2q−p,q is uniquely divisible if p ≥ 2q, except when p = q = 0. A comparison with KT via Proposition 5.21 implies: Lemma 7.1. Suppose that p ≤ 3. Then the canonical map induces an identification E 2 p,q (KQ) = E 2 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 the Beilinson-Soule vanishing conjecture, the identification holds also for p > 3. Lemma 7.1 combined with the classical computation of π 0,0 KQ as the Grothendieck-Witt group of F basically determines the 0th and 1st columns of the 0th slice spectral sequence of KQ. In the 1st column we note: There are no entering or exiting differentials in weight q ≤ 2. In weight three the exiting differential has kernel h 2,3 by Corollary 3.2. Moreover, the differential is surjective by Lemma 3.1 and Corollary 3.2. All d r -differentials exiting E r p,q (KQ) for p = 0, p = 1 and r ≥ 2 are trivial.

A Maps between motivic Eilenberg-MacLane spectra
Throughout this section the base scheme is essentially smooth over a field of characteristic unequal to 2 [14]. In the following series of results, we identify weight zero and weight one endomorphisms 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 [45], while the generalization to our set-up relies on [14]. We use square brackets to denote maps in SH. Here Sq 1 (τ ) = ρ, Sq 2 (τ ) = 0 and Sq 2 (τ 2 ) = τ ρ 2 . It follows that Sq 1 (τ n ) = ρτ n−1 n ≡ 1 mod 2 0 n ≡ 0 mod 2 Sq 2 (τ n ) = ρ 2 τ n−1 n ≡ 2, 3 mod 4 0 n ≡ 0, 1 mod 4.
Proposition A.5. For every subset A ⊆ Z there is a canonical weak equivalence α : Proof. It suffices to show [S p,q ∧ X + , α] is an isomorphism for all p, q ∈ Z and X ∈ Sm F . This is the canonical map i∈A h i−p,−q (X + ) ≻ i∈A h i−p,−q (X + ).
Work of Suslin-Voevodsky [37] shows the group h i−p,q (X + ) is nonzero only if 0 ≤ i − p ≤ 2q, see [14,Corollary 2.14] for base schemes essentially smooth over a field.