The very effective covers of KO and KGL over Dedekind schemes

We answer a question of Hoyois--Jelisiejew--Nardin--Yakerson regarding framed models of motivic connective K-theory spectra over Dedekind schemes.

Here KGL is the motivic spectrum representing homotopy algebraic K-theory and KO is the motivic spectrum representing homotopy hermitian K-theory. 1 Again by comparison with the classical situation, this suggests that Σ ∞ fr Vect and Σ ∞ fr Bil should be motivic analogs of connective K-theory spectra.Another way of producing "connective" versions is by passing to (very) effective covers [12,11].It was proved in [8,7] that these two notions of connective motivic K-theory spectra coincide, provided that S is regular over a field.
Our main result is to extend this comparison to more general base schemes.We denote by HZ Spitzweck's motivic cohomology spectrum [11] and by HW the periodic Witt cohomology spectrum [3,Definition 4.6].
Theorem 1.1.Let S be a scheme.

Σ ∞
fr Bil → f0 KO ∈ SH(S) is an equivalence.These assumptions are satisfied if S is essentially smooth over a Dedekind scheme (containing 1/2 in case (2)).
Date: June 6, 2022. 1 As a notational convention for this introduction, whenever we mention KO we shall assume that 1/2 ∈ S.
Remark 1.2.That the assumptions are satisfied for Dedekind schemes is proved in [4,Proposition B.4] for (1) and in [3,Lemma 3.8] for (2).They in fact hold for all schemes; this will be recorded elsewhere.
Notation.We use notation for standard motivic categories and spectra, as in [3] and [8].

Proofs
As a warm-up, we treat the case of KGL.Recall that the functor Σ ∞ fr inverts group-completion.The Bott element lifts to β : (P 1 , ∞) → Vect gp [7, §5].We also have the rank map Vect gp → Z ∈ P Σ (Cor fr (S)).The composite U in an open cover, it holds for S. It follows that we may assume that S is qcqs, e.g.affine.
Since f 1 (HZ) = 0 we find (using Lemma 2.1) that all maps induce an equivalence on f 0 .Since the colimit is KGL, f 0 commutes with colimits (here we use that X is qcqs, via [4, Proposition A.3(2)]) and Σ ∞ fr Vect is effective (like any framed suspension spectrum), the result follows.
The proof for KO is an elaboration on these ideas.From now on we assume that 1/2 ∈ S. Recall from [3, Definition 2.6, Lemma 2.7] the motivic spectrum For the time being, assume S is Dedekind.Taking framed loops we obtain Lemma 2.2.Let S be a Dedekind scheme, 1/2 ∈ S. ( ) is an equivalence.For this and some of the following arguments, it will be helpful to recall that we have an embedding of Spc fr (S) gp into the stable category of spectral presheaves on Cor fr (S).In particular, many fiber sequences in Spc fr (S) are cofiber sequences.
Construction 2.3.The assignment V → (V ⊕ V * , ϕ V ) sending a vector bundle to its associated (hyperbolic) symmetric bilinear bundle upgrades to a morphism where Vect carries the C 2 -action coming from passing to dual bundles, and Bil carries the trivial C 2action.
Proof.Since the presheaves are 1-truncated, all the required coherence data can be written down by hand.
(2) The homotopy orbits spectral sequence yields all as presheaves with framed transfers. Proof.
(2) The homotopy orbit spectral sequence just arises from the Postnikov filtration of Vect gp and the formation of homotopy orbits and hence is compatible with transfers.Its E 2 page takes the form The form of the differentials of the spectral sequence implies that the terms H i (C 2 , a Nis π j Vect gp ) are permanent cycles for i ≤ 1, and survive to E ∞ for (i, j) = (0, 0) and (i, j) = (1, 1).One has a Nis π 0 Vect gp = Z with the trivial action and a Nis π 1 Vect gp = G m [13, Lemma III.1.4] with the inversion action.This already yields the first assertion.A straightforward computation shows that no differential can hit the (i, j) = (0, 1) spot either, yielding the second assertion.Moreover this implies that H 1 (C 2 , G m ) = µ 2 is the bottom of the filtration of π 2 .It follows that there is a map a Nis π 2 (Vect gp ) hC2 → A, where A is a quotient of µ 2 .To prove that A = µ 2 it suffices to check this on sections over a field, in which case we can use the hermitian motivic spectral sequence of [2].
We have a Nis π 0 Bil gp ≃ GW .Thus we can form the following filtration of Bil gp refining the Postnikov filtration Bil gp ← F 1 Bil gp ← F 2 Bil gp ← F 3 Bil gp ← F 4 Bil gp ∈ P Σ (Cor fr (S)) with subquotients given Nisnevich-locally by (2.1) GW , ΣZ/2, Σk M 1 , Σ 2 Z/2.Recall also the framed presheaf Alt ∈ P Σ (Cor fr (S)) sending a scheme to the groupoid of vector bundles with a non-degenerate alternating form.Tensoring with the canonical alternating (virtual) form H(1)−h on HP 1 (where H(1) is the tautological rank 2 alternating form on HP 1 , and h is the standard alternating form on a trivial vector bundle of rank 2) yields maps σ 1 : HP 1 ∧ Alt gp → Bil gp and σ 2 : HP 1 ∧ Bil gp → Alt gp ; by construction we have β = σ 1 σ 2 (recall that HP 1 mot ≃ S 4,2 ).
(1) The composite Proof.(1) Write C for the cofiber computed in the category of spectral presheaves on Cor fr (S).Then C admits a finite filtration, with subquotients corresponding to those in (2.1).Since each of those is the infinite loop space of a motivic spectrum, it follows that C is in fact motivically local.Consequently C corresponds to Bil gp /F 4 Bil gp under the embedding into spectral presheaves.These contortions tell us that there are fiber sequences Hence to prove that the composite is null, it suffices to prove that there are no maps from Σ 4,2 Alt gp into the motivic localizations of the subquotients of the filtration given in (2.1).These motivic localizations are GW , L Nis K(Z/2, 1), L Nis K(k M 1 , 1) and L Nis K(Z/2, 2) (since they are motivically equivalent to the subquotients, and motivically local because they are infinite loop spaces of the motivic spectra H Z, Σk M , Σ 2,1 k M , Σ 2 k M ).It suffices to prove that Ω 4,2 of these subquotients vanishes, which is clear.Next we claim that Σ ∞ fr Bil gp /F 4 Bil gp is stable under base change (among Dedekind schemes containing 1/2).Indeed the defining fiber sequences of F 4 Bil gp are also cofiber sequences, and so Σ ∞ fr Bil gp /F 4 Bil gp is obtained by iterated extension from spectra stable under base change (see Lemma 2.2(2) for k M 1 , [8, proof of Lemma 7.5] for Bil and Alt, and [6, Lemma 16] for Z/2).To prove that the induced map is an equivalence we thus reduce as before to S = Spec(k), k a perfect field of characteristic = 2.In this case the result is a straightforward consequence of the hermitian motivic filtration of [2].
(2) The proof is essentially the same as for (1), but easier.
We now arrive at the main result.
Proof.As before we may assume that S is qcqs.
We know that KO is the colimit of It is hence enough to prove that σ 1 : Σ −8n,−4n Σ ∞ fr Bil → Σ −8n−4,−4n−2 Σ ∞ fr Alt induces an equivalence on f0 for every n ≥ 0, and similarly for σ 2 .(Here we use that S is qcqs, so that f0 preserves filtered colimits.)Given a cofiber sequence A → B → C, in order to prove that f0 A ≃ f0 B, it suffices to show that Map(X, C) = * for every X ∈ SH(S) veff , i.e. that C ∈ SH(S) veff⊥ .
Over Z[1/2], the cofiber of σ 1 has a finite filtration, with subquotients and the cofiber of σ 2 is Σ −4,−2 Σ ∞ fr Z.Using [6, Corollary 22], [8, Theorem 7.3] and Lemma 2.2(3), we can identify the list of cofibers as These spectra are stable under arbitrary base change (essentially by definition), and hence for arbitrary S the cofibers of σ 1 , σ 2 are obtained as finite extensions, with cofibers in the above list.To conclude the proof, it will thus suffice to show that all spectra in the above list are in SH(S) veff⊥ .
Note that if E ∈ SH(S) then E ∈ SH(S) veff⊥ if and only if Ω ∞ E ≃ * .In particular this holds if f 0 E = 0.This holds for Σ m,n HZ as soon as n < 0, by assumption.Hence it also holds for Σ m,n HZ/2 in the same case (f 0 being a stable functor) and for Σ m,n k M ≃ cof(Σ m,n−1 HZ/2 τ − → Σ m,n HZ/2).
[1,otopic after motivic localization, since Z is motivically local and truncated and (P 1 , ∞) Lemma 16], we may assume that S = Spec(Z).Using[4, Proposition B.3]we further reduce to the case where S is the spectrum of a perfect field.In this case Σ ∞ fr Vect ≃ f 0 KGL and so (Σ ∞ fr Vect)/β ≃ s 0 KGL ≃ HZ (see e.g.[1, Proposition 2.7]).Note first that if U ⊂ S is an open subscheme, and any of the assumptions of Theorem 1.1 holds for S, it also holds for U .On the other hand, if one of the conclusions holds for all