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Zariski-Local Framed $\Mathbb {A}^ 1$-Homotopy Theory

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Zariski-Local Framed $\Mathbb {A}^ 1$-Homotopy Theory [email protected] [email protected] ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY ANDREI DRUZHININ AND VLADIMIR SOSNILO Abstract. In this note we construct an equivalence of ∞-categories Hfr,gp S ≃ Hfr,gp S ( ) zf ( ) of group-like framed motivic spaces with respect to the Nisnevich topology and the so-called Zariski fibre topology generated by the Zariski one and the trivial fibre topology introduced by Druzhinin, Kolderup, Østvær, when S is a separated noetherian scheme of finite dimension. In the base field case the Zariski fibre topology equals the Zariski topology. For a non-perfect field k an equivalence of ∞-categories of Voevodsky’s motives DM(k) ≃ DMzar(k) is new already. The base scheme case is deduced from the result over residue fields using the corresponding localisation theorems. Namely, the localisation theorem for Hfr,gp(S) was proved Hfr,gp S by Hoyois, and the localisation theorem for zf ( ) is deduced in the present article from the affine localisation theorem for the trivial fibre topology proved by Druzhinin, Kolderup, Østwær. Introduction It is known that the Nisnevich topology on the category of smooth schemes is a very natural in the context of motivic homotopy theory. In particular, the constructions of Voevodsky’s motives [9, 33, 38, 41], unstable and stable Morel-Voevodsky’s motivic homotopy categories [19, 31, 34, 35], and framed motivic categories [21, 25, 27] are all based on Nisnevich sheaves of spaces. In this note we show that Zariski topology over a field and a modification of it over finite-dimensional separated noetherian schemes called Zariski fibre topology lead to the same ∞-categories of Voevodsky’s motives DM(S) and of the framed motivic spectra SHfr(S) [21]. Consequently this allows us to improve the reconstruction theorem from [30] stating SH(S) ≃ SHfr(S), and write SH(S) ≃ fr SHzf (S). 1. Base field case of DM(k). Consider the ∞-categories DM(k), DMzar(k) of Voevodsky’s motives over a field defined with respect to the Nisnevich and the Zariski topologies. Theorem A. For any field k the obvious fully faithful embedding is an equivalence DM(k) ≃ DMzar(k). (0.1) The equivalence (0.1) is known in the perfect base field case due to results of [39], namely, it is a consequence of the strict homotopy invariance theorem [39, Theorem 5.6], and the injectivity arXiv:2108.08257v1 [math.AG] 18 Aug 2021 on essentially smooth local schemes theorem [39, Cor. 4.18, 4.19]. The equivalence (0.1) formally follows from [41, Corollary 3.2.7], that gives the isomorphism of hom-groups on compact generators, and the proof is contained actually in [41, Proof of Theorem 3.2.6]. The novelty in the article is a general and short proof of (0.1) that deduces it from the ´etale excision isomorphism [39, Proposition 4.12, Corollary 4.13] proved by Voevodsky over any base field. Lemma 1.9 shows the equivalence of topological localisations on any stable ∞-category of presheaves that satisfy the excision axiom (EtEx)´ defined in what follows. Note that the isomor- n n A1 phism Hnis(X, Fnis) ≃ Hzar(X, Fzar) for an -invariant presheaf of abelian groups with transfers proved in [39, Theorem 5.7] is stronger than such a categorical equivalence. The research is supported by the Russian Since Foundation grant 20-41-04401. 1 ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 2 2. Relative case, and framed motivic spectra. We expect that the verbatim version of The- orem A over positive dimensional base schemes does not hold, because DMzar(S) does not satisfy the Localisation Theorem, in the sense of (Loc) in what follows, while DM(S) does, this is proved in [9]. However, we introduce the Zariski fibre topology on SmS which coincides with the Zariski topology over fields and satisfies (Loc) over general bases. Concretely, this topology is generated by the Zariski topology and the trivial fibre topology introduced in the article [16, Definition 3.1] by the first author, Kolderup and Østvær. Assume given a family of preadditive ∞-categories CorrS of correspondences, for noetherian separated base schemes S (see Definition 2.6), we say that CorrS satisfies ´etale excision over fields if ´ gp (EtEx) for any field k, additive presheaf F ∈ PShΣ (Corrk), an essentially smooth local henselian ′ ′ k-scheme U, and an ´etale morphims U → U, such that U ×U Z ≃ Z, for a closed subscheme Z of U, the map ′ ′ F (U) → F (W ) ×F (W ) F (Up−1(x)) ′ ′ is an equivalence, where W = U − Z, W = U ×U W . Given a family of topologies τ on the categories SmS that induces topologies on the categories SmAffS over a of separated noetherian scheme S such that the base change functors are continuous, we say that τ satisfies (LocAff) or (Loc) with respect to Corr, if (LocAff) for an affine scheme S ∈ Sch, a closed immersion i: Z → S, and the open immersion tr j : S − Z → S, for any SHτ (SmAffB ), the canonical sequence ! ∗ i∗i F → F → j∗j F ! ∗ is a fibre sequence, see Section 4.0.1 for the definitions of i∗,i , j∗, j . tr (Loc) the similar property to (LocAff) with S ∈ Sch and F ∈ SHτ (SmS) holds, Let us note that for any topology ν stronger then the Zariski topology, (LocAff) implies (Loc), though it is not used in the article formally. Theorem B. Let CorrS be as above, and τ, and ν be families of topologis on SmAff S as above. Let CorrS,τ,ν satisfy the properties (Embed) in the sense of Definitions 2.8 and 4.21, and ν satisfies (rCovLift), Definition 4.15. Then (LocAff) for τ implies (LocAff) for τ ∪ ν, Definition 4.5. The property (Embed) means in short that for an open immersion U → S there is the fully faithful functor CorrU → CorrS, and the topologies τ,ν on SmAffU are the restrictions of the ones on SmAffS. The property (rCovLift) is a kind of lifting property for ν-coverings with respect to closed immersions Z → S. Applying Lemma 1.9 and Theorem B to the Zariski and Nisnevich topologies on SmS we get the following result. Theorem C. Let CorrS and τ satisfy the properties (FinE) defined in Definition 2.9, Defini- tion 4.25. Suppose that Corrk satisfies property (EtEx)´ over fields k. Then there is a canonical equivalence of categories gp gp Hnis(Corrk) ≃ Hzar(Corrk). Suppose that τ satisfy the property (LocAff) with respect to CorrS , then for a noetherian separated scheme S of finite Krull dimension, gp gp Hnis(CorrS) ≃ Hτ∪zar(CorrS). The properties (FinE) is a kind of continuity with respect to the embeddings of the generic points of the base schemes. Example 0.2. The first part of the above theorem holds for any preadditive ∞-category of cor- respondences equipped with the functor Fr+(k) → Corrk, from the framed category of framed correspondences Fr+(k) defined in [23]. In particular, it holds for K-motives, and GW-motives [14, 26, 42] over fields, Milnor-Witt motives [4, 8, 10, 11] appropriately defined over non-perfect fields, R ∞-categories DMA(S) over base schemes in the sense of [15], and the ∞-categories DM (S) in the sense of [20]. ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 3 Example 0.3. The second equivalence of the above theorem holds for the ∞-categories of framed fr correspondences CorrS = Corr (S) from [21], and τ being the trivial fibre topology [16, Definition 3.1]. See Corollary 1 for details. The latter example leads to the reconstruction of SH(S) as the category of Zariski fibre local fr framed motivic spectra SHzf (S). Informally, this is because the framed transfers are a universal structure on a cohomology theory that ensures the theory is representable by an object of SH(S), while CorS-transfers plays a similar role for DM(S). Framed correspondences in their original form were introduced in the unpublished notes by Voevodsky [40], and the instrument was deeply studied and developed in the Garkusha-Panin theory of framed motives [1, 2, 12, 13, 17, 18, 22, 23, 24, 25, 27, 29, 36]. We use the closely related but different notion of the ∞-category of framed correspondences Corrfr introduced in a joint work of the second author with Elmanto, Hoyois, Khan and Yakerson and studied further in [5, 6, 21]. The property (Loc) for H(S) is proved in Morel-Voevodsky’s and Ayoub’s works [3, 35], and (Loc) for DM is proved in Cisinski-D´eglise’s work [9]. The property (Loc) for Hfr(S)= H(Corrfr(S)) of framed motivic spaces with respect to the Nisnevich topology is proved by Hayois in [30]. [16, §14.2] in an equivalent form recovers (Loc) for H(S) and Hfr(S), and proves the property (Lo- fr cAff) for the categories Htf (S) and Htf (S) of motivic spaces and framed motivic spaces with respect to the trivial fibre topology [16, Definion 3.1], denoted tf and generated by the squares (0.4) in what follows. With the use of the latter result Theorem B, examples 4.11, 4.22 and 4.27, fr,gp fr,gp and lemma 4.19 prove (LocAff) for the category Hzf (S)= Hzf (Corr (S)) of Zariski fibre local group-like framed motivic spaces, where zf = zar ∪ tf. Let us note that the group-completion is used for the simplicity of the argument but is not necessary, and note again that (Loc) follows from (LocAff). Definition. The Zariski fibre topology, denoted zf, on SchS over a scheme S is a completely decomposible topology generated by the Zariski squares and the Nisnevich (pullback) squares of the form ′ / ′ X ×S (S − Z) X (0.4) / X ×S (S − Z) X, where Z is a closed subscheme in S.
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