η$\eta$ ‐Periodic motivic stable homotopy theory over Dedekind domains

We construct well‐behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer–Witt K$K$ ‐theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence, we lift the fundamental fiber sequence of η$\eta$ ‐periodic motivic stable homotopy theory established in Bachmann and Hopkins (2020) from fields to arbitrary base schemes, and use this to determine (among other things) the η$\eta$ ‐periodized algebraic symplectic and SL${\rm SL}$ ‐cobordism groups of mixed characteristic Dedekind schemes containing 1/2$1/2$ .


INTRODUCTION
Let be a field. We have spectra KO, KW, kw, HW, H ℤ, HZ ∈ ( ) representing interesting cohomology theories for smooth -varieties: KO represents hermitiantheory, KW represents Balmer-Witt -theory, HW represents cohomology with coefficients in the sheaf of Witt groups, HZ represents higher Chow-Witt groups; kw, H ℤ are more technical but have featured prominently in, for example, [2,7]. The first aim of this article is to define extensions of these spectra to other bases. The utility of such extensions is manifold; for example, they can be used in integrality arguments [9]. Thus, let be the prime spectrum of a Dedekind domain, perhaps of mixed characteristic, or a field. Consider the motivic spectrum KO ∈ ( ) (see, for example, [8,Subsection 2.2,4.1] for a definition of the motivic stable category ( )), representing hermitian -theory [18]. From this, we can build the following related spectra Here by ⩽0 , ⩾0 , we mean the truncation in the homotopy -structure on ( ), and by eff ⩽0 we mean the truncation in ( ) eff (see, for example, [8,Section B]). If ∶ ′ → is any morphism, there are natural induced base change maps * (kw ) → kw ′ , and so on. It thus makes sense to ask if the spectra above are stable under base change, that is, if the base change maps are equivalences. This is true for KO (and KW), since this spectrum can be built out of (orthogonal or symplectic) Grassmannians [32,37], which are stable under base change. Our main result is the following. Theorem 1.1 (see Theorem 4.5). All of the above spectra are stable under base change among Dedekind domains or fields, provided that they contain 1∕2.
Over fields, the above definitions of spectra coincide with other definitions that can be found in the literature (see [1,2,11,31]; this is proved in Lemma 4.3). In other words, we construct wellbehaved extensions to motivic stable homotopy theory over Dedekind domains of certain motivic spectra which so far have mainly been useful over fields. In fact, we show that all of the above spectra (which we have built above out of KO by certain universal properties) admit more explicit (and so calculationally useful) descriptions. For example, we show that 0 ( ) * = * , ( ) * = 0 for ≠ 0; here is the Nisnevich sheaf associated with the presheaf of fundamental ideals in the Witt rings.
Remark 1.2. The above description of * * asserts in particular that the sheaf is strictly 1invariant. In fact variants of this property form a starting point of our proofs, and are the reason for assuming that is Dedekind. Unwinding the arguments, one finds that we ultimately rely on the Gersten conjecture (for étale cohomology of essentially smooth schemes over discrete valuation rings) via [16]. Remark 1.3. Using recent results on Gersten resolutions [12], our results may be extended to regular J2 schemes instead of just Dedekind schemes. Alternatively, using the cdh topology instead of the Nisnevich one, they may be extended to all schemes. These facts will be recorded elsewhere.
Our motivating application of these results is as follows. Using Theorem 1.1, together with the fact that equivalences (and connectivity) of motivic spectra over can be checked after pullback to the residue fields of [8,Proposition B.3], one obtains essentially for free the following extension of [7]. Corollary 1.4 (see Theorem 4.12). For as above, there is a fiber sequence (2) → kw (2) → Σ 4 kw (2) ∈ ( ).
Using this, we also extend many of the other results of [7] to Dedekind domains.

Overview
The main observation allowing us to prove the above results is the following. Recall that there is an equivalence ( ) ≃  f r ( ), where the right-hand side means the category of motivic spectra with framed transfers [14,21]. This supplies us with an auxiliary functor ∞ f r ∶  1 f r ( ) →  f r ( ). The Hopf map ∶ → already exists in  1 f r ( ) (see Subsection 3.3). This readily implies that we can make sense of the category  1 f r ( )[ −1 ] of -periodic 1 -spectra with framed transfers, and that there is an equivalence The significance of this is that the left-hand side no longer involves ℙ 1 -stabilization, and hence is much easier to control. In the end, this allows us to relate all our spectra in the list above to a spectrum ko f r which is known to be stable under base change. To do so, we employ (1) work of Jeremy Jacobson [24] on the Gersten conjecture for Witt rings in mixed characteristic, and (2) work of Markus Spitzweck [36] on stability under base change of Hℤ.

Organization
In Section 2, we construct by hand a motivic spectrum with the expected homotopy sheaves. In Section 3, we study some truncations in  1 f r ( )[ −1 ], allowing us among other things to construct a spectrum kw with the expected homotopy sheaves. We prove our main theorems in Section 4. We first give alternative, more explicit definitions of the spectra in our list and deduce stability under base change. Then we show that the spectra we constructed satisfy the expected universal properties. We establish the fundamental fiber sequence of -periodic motivic stable homotopy theory as an easy corollary. Finally, in Section 5 we deduce some applications, mostly in parallel with [7,Section 8].

Notation and terminology
By a Dedekind scheme, we mean a finite disjoint union of spectra of Dedekind domains or fields, that is, a regular Noetherian scheme of Krull dimension ⩽ 1. Given a non-vanishing integer and a scheme , we write 1∕ ∈ to mean that ∈  ( ) × . We denote by pc( ) ⊂ (Sm ) the ∞-category of motivic spaces, that is, the subcategory of motivically local (that is, 1 -invariant and Nisnevich local) presheaves. We write mot for the left adjoint of the inclusion, that is, the motivic localization functor. For a motivic spectrum ∈ ( ), we denote by ( ) the homotopy sheaves (see, for example, [7,Subsection 2.4.2]). Beware that unless the base is a field, these objects are only loosely related to the homotopystructure.
We denote by Nis ,́, and́, respectively, the associated sheaves of sets in the Nisnevich, étale and real étale topologies. We write Nis for the Nisnevich localization of presheaves of spaces or spectra. Unless specified otherwise, all cohomology is with respect to the Nisnevich topology.
All schemes are assumed quasi-compact and quasi-separated. We denote by pc the ∞-category of spaces, and by  the ∞-category of spectra.

2.1
For a scheme (with 1∕2 ∈ ), denote by the Nisnevich sheaf of commutative discrete rings obtained by sheafification from the presheaf of Witt rings [25,Section I.5]. The canonical map

∶ →́≃́ℤ
is the global signature. One may show that ( ) ⊂ 2́ℤ (indeed locally I(−) consists of diagonal forms of even rank [28,Corollary I.3.4], and the signature is thus a sum of an even number of terms ±1) and hence ( ) ⊂ 2́ℤ. Sincéℤ is torsion-free, there are thus induced maps ∕2 ∶ →́ℤ.

2.2
Fix a Dedekind scheme . A -prespectrum over means a sequence of objects ( 1 , 2 , … ) with ∈ Fun(Sm op , ), together with maps → Ω +1 . Such a prespectrum can in particular be exhibited by defining as a presheaf of abelian groups. See, for example, [10, Section 6] for details as well as symmetric (monoidal) variants. A -prespectrum is called a motivic spectrum if each is motivically local, and the structure maps → Ω +1 are equivalences.
From now on, we view the category of Nisnevich sheaves of abelian groups as embedded into Nisnevich sheaves of spectra, and view all sheaves of abelian groups as sheaves of spectra, so that for ∈ Sm we have ( ) ( ) = Nis , and similarly for .
Proof. Let = ( 1 , 2 , … ) be a -prespectrum in Nisnevich sheaves of spectra. If each is connective, we can form a prespectrum Nis ⩽0 ( ) with Nis ⩽0 ( ) ≃ Nis ⩽0 ( ) the truncation in the usual -structure, and bonding maps Even if is a motivic spectrum Nis ⩽0 ( ) need not be; however if it is then it represents the truncation ⩽0 ( ) ∈ ( ) in the homotopy -structure.
In [36, Subsection 4.1.1], there is a construction of a specific -prespectrum such that (1) is a motivic spectrum representing Hℤ∕2 and (2) Nis ⩽0 ( ) ≃ , the equivalence being as -prespectra. The map Hℤ∕2 → (Hℤ∕2)∕ ∈ ( ) corresponds to a map → ′ of -spectra which is immediately seen to be a level wise zero-truncation. It follows that ′ ≃ Nis ⩽0 ( ) ≃ as -prespectra. In particular, Nis ⩽0 ( ) ≃ are motivic spectra, and in fact ≃ ⩽0 (Hℤ∕2) ∈ ( ). Since ( ) ⩾0 is closed under smash products, truncation in the homotopy -structure is lax symmetric monoidal on ( ) ⩾0 and so ⩽0 (Hℤ∕2) admits a canonical ring structure making Hℤ∕2 → (Hℤ∕2) ⩽0 into a commutative ring map. It remains to show that ⩽0 (Hℤ∕2) ≃ is an equivalence of ring spectra. Both of them can be modeled by  ∞ -monoids in the ordinary 1-category of symmetric -prespectra of sheaves of abelian groups on Sm ; that is, just commutative monoids in the usual sense. The isomorphism between them preserves the product structure by inspection. □ The following is the main result of this section.
Since satisfies the analog of ( * ) by Lemma  → Ω Nis ; indeed the cofiber of this map is given by Ω ℤ∕2 ≃ 0 (the vanishing holds, for example, since motivic cohomology vanishes in negative weights). It follows that 0 ( ) * ≃ * and = 0 for ≠ 0; here = for < 0. It also follows that Ω Nis ≃ Nis , and that this spectral sheaf is homotopy invariant.
Recall that a Noetherian valuation ring is a ring which is either a discrete valuation ring or a field [38, Tag 00II]. Lemma 2.10. Let be a Noetherian valuation ring. Then there is a filtered system of Noetherian valuation rings with vcd( ( )) < ∞ (that is, there exists with vcd < for all ) and colim ≃ .
Proof. Let = ( ). Then = colim , where the colimit is over finitely generated subfields ⊂ ; this colimit is filtered. Let = ∩ . We shall show that is a Noetherian valuation ring, and ( ) = . This will imply the result since vcd ( ) < ∞ uniformly in by [ Proof.
(1) holds by definition. (2) The functor Σ ∶  Σ (Cor f r ( )) →  Σ (Sm ) * preserves filtered (in fact sifted) colimits and commutes with Nis [14, Proposition 3.2.14]. Consequently Nis ∶ hv Nis (Cor f r ( )) → hv Nis (Sm ) * also preserves filtered colimits. Being a right adjoint it also preserves limits, and hence commutes with spectrification. Consequently, it suffices to prove the following: given ∈  Σ (Cor f r ( )) and ⩾ 0, we have Σ (Σ ) ∈  Σ (Sm ) ⩾ ; indeed then by what we have already said. Writing Σ as an iterated sifted colimit, using semi-additivity of  Σ (Cor f r ( )) [14, sentence after Lemma 3.2.5] and the fact that Σ commutes with sifted colimits, we find that Σ (Σ ) is given by Σ ( ), that is, the iterated bar construction applied section wise. The required connectivity is well-known; see, for example, [ Remark 3.2. The proof of Lemma 3.1 also shows the following: if ∈ hv Nis (Cor f r ( )) and ∈ Sm , then ( ) ∈ pc is a commutative monoid (Cor f r ( ) being semi-additive) and Σ ∞ ∈ hv Nis  (Cor f r ( )) has underlying sheaf of spectra corresponding to the group completion of .

3.3
The unit ∈ ( 1 ⧵ 0) × defines a framing of the identity on 1 ⧵ 0 and hence a framed correspondence 1 ⧵ 0 ⇝ * . We denote by the corresponding map (obtain by precomposition with ↪ ∨ ≃ 1 ⧵ 0). The following is a key result. It shows that the Hopf map is already accessible in framed 1spectra, which enables all our further results.

3.5
By construction, the forgetful functor Proof. The equivalence  f r ( ) ≃ ( ) restricts to the subcategories of those objects such that (−) = 0 for ≠ 0. These are exactly the objects representable by prespectra valued in sheaves of abelian groups that are motivic spectra when viewed as valued in spectral sheaves. The construction of HW supplies the sheaf with a structure of framed transfers. Then the subsheaf admits at most one compatible structure of framed transfers; the existence of the map → HW together with the above discussion shows that this structure exists. This supplies a description of the (unique) lift of to  f r ( ). From this and Lemma 3.4, it follows that the action by on induces the inclusion * +1 → * on homotopy sheaves. This implies immediately that [ −1 ] → HW induces an isomorphism on homotopy sheaves and hence is an equivalence (see

Remark 3.3). For
∕ the same argument works using Theorem 2.1. The following is one of the main results of this section. is an -periodization. Since this is an equivalence of -periodic objects (see the proof of (1,2)), this is clear.

4.1
Fix a Dedekind scheme with 1∕2 ∈ . In the previous two sections, we have defined ring spectra , and HW (and kw) in ( ).  [11], the two definitions are shown to coincide in [6]), the spectrum ko is defined in [1], and the spectra HW, kw are defined in [7, Subsection 6.3.2].
Proof. We first treat ko. Since very effective covers are stable under pro-smooth base change [8,Lemma B.1], and ko f r is stable under arbitrary base change (see the proof of Theorem 4.5), we may assume that the base field is perfect. In this case pc f r ( ) gp ≃ ( ) veff [14, Theorem 3.5.14(i)] and hence Σ ∞ f r Ω ∞ f r coincides with the very effective cover functor. Thus, ko ≃ ko f r as needed.
The claim for kw follows via [7, Lemma 6.9] and Lemma 3.11 (4). For , , the claim is true essentially by construction. This implies the claim for HW via Lemma 3.9.

)]). For HZ consider the commutative diagram
The left-hand square is defined to be cartesian, so that by [2,Theorem 17], coincides with the usual definition of HZ. The right-hand square is cartesian by [30,Theorem 5.3]. Hence, the outer rectangle is cartesian and ≃ HZ as defined above. This concludes the proof. □ Proof. Immediate from Remarks 3.3 and 3.7. □ The following is one of our main results. Proof. For ko f r this follows from the facts that (1) Ω ∞ KO ∈ pc( ) is motivically equivalent to the orthogonal Grassmannian [37, Theorem 1.1], which is stable under base change, and that (2) the forgetful functor pc f r ( ) → pc( ) commutes with base change [21,Lemma 16]. The case of kw follows from this and Lemma 3.11, which shows that kw ≃ ko f r [ −1 ]. The case of HW now follows from Lemma 4.4.
The spectra Hℤ and Hℤ∕2 are stable under base change [36], and hence so is ≃ (Hℤ∕2)∕ (see Lemma   Proof. We give the proof for ( ), the one for ( ) eff is analogous. Since by definition Σ Σ ∞ + ∧ ∧ ∈ ( ) >0 for > 0, ∈ ℤ, necessity is straightforward. We now show sufficiency. Let  ⊂ ( ) denote the subcategory on those spectra with Map(Σ , ) = * . We need to show that ( ) ⩾0 ⊂ . By definition, for this it is enough to show that (a)  is closed under colimits, (b)  is closed under extensions, and (c) Σ ∞ + ∧ ∧ ∈  for every ∈ Sm and ∈ ℤ. (a) is clear, and (b) follows from the fact that given any fiber sequence of spaces * → → * we must have ≃ * . To prove (c), it is enough to show that Ω ∞ ( ∧ ∧ ) ≃ * ∈ hv Nis pc (Sm ). By [8,Proposition A.3], this can be checked on homotopy sheaves. □ Remark 4.9. In contrast with the case of fields, if has positive dimension, then homotopy sheaves do not characterize ( ) ⩾0 . That is, given ∈ ( ) ⩾0 , it need not be the case that ( ) * = 0 for < 0. † We denote by ⩾0 , ⩽0 (respectively, eff ⩾0 , eff ⩽0 ) the truncation functors for the homotopystructure on ( ) (respectively, ( ) eff ) and 0 for the effective cover functor; see, for example, [8, Section B] for a uniform treatment.  Proof. By [8, Lemma B.1], pro-smooth base change commutes with truncation in the homotopy -structure, in the effective homotopy -structure, and also with effective covers. We may thus assume that = .
Consider the cofiber sequence kw → KW → . To prove that kw ≃ ⩾0 KW, it is enough to show that kw ∈ ( ) ⩾0 and ∈ ( ) <0 . By Remark 4.10, we may check the first claim after base change to fields, and hence by Theorem 4.5 for this part we may assume that is the spectrum of a field, where the claim holds by definition. The second claim follows via Lemma 4.8 from our definition of kw as a connective cover in the Nisnevich topology (see Definition 3.10).
Consider the fiber sequence → [ −1 ] → HW. To prove that ⩽0 [ −1 ] ≃ HW it suffices to show that ∈ ( ) >0 and HW ∈ ( ) ⩽0 . As before the first claim can be checked over fields where it holds by definition, and the second one follows from Lemma 4.8 and the definition of HW (see Definition 3.6). The argument for ⩽0 kw ≃ HW is similar.
(2) The unit map → kw (2) factors uniquely (up to homotopy) through the fiber of .
(3) The resulting sequence is a fiber sequence.
In particular for any scheme with 1∕2 ∈ , there is a canonical fiber sequence

APPLICATIONS
Throughout, we assume that 2 is invertible on all schemes. We shall employ the special linear and symplectic cobordism spectra MSL and MSp; see, for example, [8,Example 16.22] for a definition.

5.1
Recall Here * denotes the classical stable stems.
Proof. We first show that 0 ( [ −1 ]) → 0 (HW) ≃ is an isomorphism. We may do so after ⊗ℤ (2) and ⊗ℤ[1∕2]; the former case is immediate from the fundamental fiber sequence (using Proof. [28,Corollary IV.3.3] shows that for any Dedekind scheme there is a natural exact sequence of abelian groups ( * ) The last map is surjective if has only one point, and hence these sequences constitute a resolution of the presheaf on Nis . The terms of this resolution are acyclic [31,Lemma 5.42] † , and hence this resolution can be used to compute cohomology. To show that * ( , ) = 0 for * > 0 it suffices to show that the right most map of ( * ) is surjective. Clearly, if this is true for then it also holds for any localization of . It hence suffices to prove surjectivity for ℤ; this is [ (2) 2 ) is a polynomial ring on generators̄. Writẽ∈ * MSp (2) for arbitrary lifts of these generators. The proof of [7,Corollary 8.6] shows that if Spec( ) → Spec(ℤ[1∕2]) is an arbitrary field, then * (MSp (2) )( ) ≃ W( ) (2) [̃1,̃2, … ]. We shall show that if is henselian local and essentially smooth over then * (MSp (2) )( ) ≃ * (MSp (2) )( ), where is the closed point; this will imply our claims. Using the fundamental fiber sequence, for this it is enough to show that W( ) ≃ W( ), which holds by [ (2) and ⊗ℤ[1∕2]; the latter cases reduces via real realization to the base field ℝ where we already know it. For the former case, we first verify as above that ∶ kw * MSL (2) → kw * −4 MSL (2) is surjective, and hence * MSL (2) → kw * MSL (2) is injective; the claim follows easily from this. It follows that we may form MSp∕( 1 , 3 , … ) → MSL.
To check that this is an equivalence we may pull back to fields, and so we are reduced to [7,Corollary 8.9]. We have now proved (3), which implies the missing half of (2).
It remains to establish (4). We claim that 4 MSL ℤ[1∕2] → 4 kw ℤ[1∕2] is surjective, and hence an isomorphism since both groups are free W(ℤ[1∕2])-modules of rank 1. This we may check after ⊗ℤ (2) and ⊗ℤ[1∕2]. In the former case, we have a morphism of finitely generated modules over a local ring, so may check ⊗ W(ℤ[1∕2]) (2) 2 , that is, over ℂ, in which case the claim holds by [7,Lemma 8.10]. In the latter case, via real realization we reduce to ℝ, and so again the claim holds by [7,Lemma 8.10]. The upshot is that we may choose 2 in such a way that its image in 4 kw is the generator . Then as in the proof of [7,Corollary 8.11] we modify the other generators to be annihilated in kw to obtain a map MSL∕( 4 , 6 , … ) → kw. This map is an equivalence since it is so after pullback to fields, by [ ′ in the sense that need not be central and so the multiplicative structure is more complicated than a power series ring. We have = 9 + 8.

A C K N O W L E D G E M E N T S
I would like to thank Shane Kelly for help with Lemma 2.10. To the best of my knowledge, the first person suggesting to study ko f r was Marc Hoyois. Open access funding enabled and organized by Projekt DEAL.

J O U R N A L I N F O R M AT I O N
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