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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. Such a square is called a trivial fibre square. Remark. It is expected that the Zariski fibre topology is the weakest topology that contains Zariski topology and satisfies the Localisation Theorem for categories DM(S) or SH(S) in the sense of (Loc), or [35, Theorem 2.21]. Let us note also that the Zariski fibre topology is the strongest subtopology of the Nisnevich topology on SchS that equals the Zariski topology over the residue fields. Corollary 1. Let k be a field, and S be a finite-dimensional noetherian separated scheme. Any group-like A1-invariant sheaf in PShfr(S) with respect to the Zariski fibre topology is a Nisnevich sheaf; there are canonical equivalences of categories fr,gp fr,gp fr,gp fr,gp Hnis (k) ≃ Hzar (k), Hnis (S) ≃ Hzf (S), fr fr and similarly for SHS1 (S) and SH (S). Remark. Here we list results that ware known, for a perfect field k, in the following two forms: n n A1 (1) the isomorphism of cohomoligies Hzar(−, Fzar) ≃ Hnis(−, Fnis) for an -invariant presheaf with various type of transfers F , and (2) equivalences of categories of motives with respect to Zariski and Nisnevich topology. Note that (1) is stronger and formally implies (2). The claim in the form (1) for Cor-correspondences is [39, Theorem 5.7], and follows also by the argument presented in [37]. For Chow-Witt corre- spondences the claim in the form (1) was written in [10, Theorem 3.2.9]. Note also that cited ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 4 above categorical results from [41] are formally stronger then (2); and the strict homotopy invari- ance theorem for A1-presheaves with transfers like [39, Theorem 5.6] for a preadditive category of correspondences Corrtr(k) allows to prove the equivalences of endofunctors

LzarLA1 ≃ Lzar,A1 ≃ LnisLA1 ≃ Lnis,A1 (0.5) on PSh(Corrgp(k)). The formulation in terms of (0.5) for the case of framed correspondences appeared in discussions of the first author with the authors of [21], the proof with the use of the strict homotopy invariance proved over perfect fields in [24] and the principle of the arguments cited above for the case of Cor(k) is given in [21, Theorem 3.4.11], or in the appendix of [16]. 3. Structure of the text. Section 1 discusses sheaves that satisfy the excision property with respect to varying topologies (we call them excisive, see Definition 1.6). In Lemma 1.9 we consider a pair of topologies Z ⊂ N, such that Z have enough set of points, and N is completely decomposable. We prove that any Z-sheaf that satisfies the excision property on Z-points with respect to N-squares is an N-sheaf. This is the key lemma for Theorem A, and the base field case of Theorem C (these are Theorem 3.6 and Theorem 4.39 in the text). In Section 2 we introduce a formalism of ∞-categories of correspondences over base schemes S, tr S1,tr S1 tr and define ∞-categories H (S)= H(CorrS), SH (S)= SH (CorrS), SH (S)= SH(CorrS). We show that Lemma 1.9 implies that for a pair of topologies Z and N as above and an ∞-category of correspondences CorrS satisfying the corresponding excision axiom, as in Definition 3.3, there tr tr is an equivalence HZ (S) ≃ HN (S). In the case N = Nis and Z = Zar we obtain our main result over a base field Theorem 3.6. Section 3.2 formulates the used ´etale excision axiom, and Section 3.1 deduces the ´etale excision theorem for framed correspondences in the required form from the ´etale excision theorem provided by [17, 18, 24]. Section 4 extends the result to the arbitrary base scheme case, proving Theorem C and The- S1 S1 orem 4.39. The result claims the equivalence SHzar∪τ (CorrS) → SHnis∪τ (CorrS) under the ad- ditional assumption of the property (LocAff) for a topology τ with respect to CorrS defined in Section 4.0.1. Sections 4.0.2, 4.0.3 and 4.1 contain intermediate steps needed to deduce the property (LocAff) for the topologies zar ∪ τ and nis ∪ τ form the property (LocAff) for τ, and summarised in Section 4.2. The latter deduction additionally uses some specific properties of the Zariski and Nisnevich topologies in comparing with the ones formally used in the base field case of Theorem C. Then in Section 4.3 the claimed equivalence in Theorem 4.39 for the base schemes case is deduced from the result over residue fields. Notation. Throughout the paper we make use of the following conventions and notation.

• SmX is the category of smooth schemes over X. SmAffS is the category of affine schemes over S. • PSh(S) is the ∞-category of presheaves of spaces on the category SmS, PSh(SmAffS ) is the ∞-category of presheaves of spaces on SmAff S. padd add • Cat∞ is the ∞-category of ∞-categories, and Cat∞ , Cat∞ are the ∞-categories of preadditive and additive ∞-categories. The canonical functor padd add gp Cat∞ → Cat∞ ; C 7→ C takes a preadditive ∞-category C to the additive ∞-category Cgp whose mapping spaces are group completions of the corresponding E∞-monoids that are mapping spaces of C. tr • Given an ∞-category denoted by CorrS or Corr(S), we write PSh (S) = PSh(CorrS), and tr tr tr denote by PShΣ (S) the subcategory of radditive presheaves in PSh (S). Denote by hS (X) the presheaf represented by X ∈ CorrS. • Given an essentially surjective functor SmS → CorrS , and a topology ν on SmS, the tr A1 tr subcategory of PShΣ (S) spanned by -invariant objects is denoted by Hτ (S), and the tr subcategory of ν-sheaves is denoted by Shν (S). The latter two subcategories and their intersection are reflective, we denote the corresponding localisation functors by LA1 , Lν, and LA1ν respectively. • We denote by τ ∪ ν the topology generated by topologies τ and ν. ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 5

• Denote by Corr(SmAff S) the subcategory in CorrS spanned by the objects that are in the tr tr image of SmAffS, and PSh (SmAff S) = PSh(Corr(SmAff S)), and similarly for Shτ (S), tr S1,tr tr Hτ (SmAff S), SHτ (SmAff S), SHτ (SmAffS). tr,gp gp • If the ∞-category CorrS as above is preadditive we write PSh (S) = PSh(CorrS ), tr,gp gp tr,gp tr,gp tr tr,gp tr,gp PShΣ (S) = PShΣ(CorrS ), Shτ (S) = PSh (S) ∩ Shτ (S), Shτ (S), H (S) = tr tr,gp tr H (S) ∩ PSh (S). Note that any F ∈ PShΣ (S) admits a canonical structure of an E tr,gp tr,gp ∞-monoid, and PShΣ (S), H (S) are equivalent to the subcategories of group-like objects in PShtr(S), Htr(S) by [28, Lemma 1.6] fr fr • CorrS = Corr (S) is the ∞-category of framed correspondences over S (constructed in fr [21]), and Corr (SmAffS ) is the subcategory spanned by affine S-schemes. We write fr fr for the superscript in the notation of categories for the case CorrS = CorrS . In particular, fr tr fr PSh (S) = PSh (CorrS ). • For any presheaf F ∈ PSh(S), a smooth S-scheme X, and a closed subset Z ⊂ X we −1 denote by F (XZ ) the global sections of the presheaf i F |X where i denotes the embedding Z → X. If W is an essentially smooth scheme obtained as a localisation of a smooth scheme X in a closed subset Z, which are clear from the context, we write F (W ) for F (XZ ).

1. General lemma.

Definition 1.1. For an ∞-category S and an object X ∈ S denote by SX the comma category over X. Define the functor eX : PSh(SX ) → PSh(S) that takes a presheaf F on SX to a presheaf G on S G(W )= F (W → X). a W →X∈SX

Lemma 1.2. For an ∞-category S and X ∈ S the functor eX : PSh(SX ) → PSh(S) is conserva- tive. ′ ′ Proof. Let F → F be a morphism in PSh(SX ). Let w : W → X ∈ SX . If eX (F ) ≃ eX (F ) then ′ the canonical morphism s : W →X∈SX F (s) → s : W →X∈SX F (s). Since the coproduct functor in Spc is conservative the equivalence` F (w) ≃ F ′(`w) follows.  Definition 1.3. Let Z be a Grothendieck topology on a category S. We say that an object F of PSh(S) is a Z-sheaf if for any Z-covering X˜ → X the map F (X) → lim F (CˇX (X˜)) is an ∆ equivalence. For a Grothendieck topology Z on a category S denote by the same symbol the induced topology on SX .

Lemma 1.4. The functor eX from Def. 1.1 preserve Z-sheaves. Proof. The claim follows form the definitions.  For a category S denote by pro−S the category of pro-objects. For any F ∈ PSh(S) we denote by the same symbol the corresponding continuous presheaf on pro−S. Lemma 1.5. Let T be a family of pro-objects in S that define enough set of points for a Grothendieck topology Z on S. Then for a scheme X ∈ S the family TX of pro-objects in SX given by morphisms of the form T → X, T ∈ T , gives enough set of points for the topology Z on SX . Proof. Since a pro-object T ∈ T defines a point of Z on S, for any Z-covering W → W , W ∈ S, the morphism cW : W ×W T → T has a right inverse lW . Since cW ◦ lW = idW , forf any morphism W → X the morpihsmf lW defines the morphism in SX . Then it follows that any pro-object T → X ∈ TX is a point of the topology Z on SX . ′ We are going to show that the set of points TX is enough for the topology Z on SX . Let F → F ′ be a morphism of Z-sheaves on SX such that F (T → X) ≃ F (T → X) for any T → X ∈ TX . ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 6

′ By Lemma 1.4 eX (F ) → eX (F ) is a morphism of Z-sheaves on S. For any T ∈ T and morphism ′ t: T → W , by the above t define a Z-point in SX , hence eX (F )(T ) → eX (F )(T ) by assumption ′ ′ of the morpihsm F → F . Then eX (F ) ≃ eX (F ), since T is an enough set of Z-points in S. Thus F ≃ F ′ by Lemma 1.2.  Definition 1.6. Let R be a square ′ i W ′ / X′ (1.7)

  i W / X in S. We say that a presheaf F ∈ PSh(S) is R-excisive if ′ F (X) ≃ F (W ) ×F (X) F (X ). (1.8) Given a morphism V → X in S; we say that F is R-excisive over V if F is excisive with respect to the square R ×X V . Given a cd-structure N on a category S. Denote by the same symbol N the completely decom- posable topology N on S defined by the cd-structure, and call the squares of N by N-squares. It is well known and follows immediately from the definition that a presheaf F ∈ PSh(S) is an N-sheaf if and only if it is N-excisive for each N-square. Lemma 1.9 (Main lemma). Let N be a Grothendieck topology on a category S and Z be a subtopol- ogy of N. Assume that N is completely decomposable, and Z admits enough set of points given by a family of pro-objects in S. Let F be a Z-sheaf on S such that for each Z-point U the presheaf F is N-excisive over U. Then F is N-sheaf. Proof. Let R be an N-square (1.7) in S over the scheme X. Consider the presheaf F ′ on the category of pro-objects in SX defined by the assignment ′ {V → X} 7→ F (W ×X V ) ×F (X×X V ) F (X ×X V ). Since finite limits commute with filtering colimits F ′ is continuous, and since finite limits commute ′ with limits F in Z sheaf on SX . It follows by the assumption on F and Lm. 1.5, that F is R-excisive over any T ∈ TX . So we ′ have equivalences F (T ) ≃ F (T ) for each T ∈ TX . Hence by Lm. 1.5 the canonical morphism F → F ′ is Z-local equivalence. Thus F ≃ F ′ since both presheaves are Z-sheaves. Since F ′ is R-excisive by the definition, F is R-excisive. Thus F is R-excisive for each N-square R in S. 

2. preadditive categories of correspondences. Definition 2.1. An ∞-category A is called preadditive if it admits a zero object and the map X Y → X × Y a is an equivalence for any objects X, Y ∈ A. In this case it makes sense to talk about the direct sum of objects, which we denote by X ⊕ Y . preadditive categories are canonically enriched over E∞-monoids (see [28, Proposition 2.3]). An ∞-category A is called additive if all mapping spaces are group-like with respect to the E∞-monoid structure.

Example 2.2. (1) The 2-category of spans of morphisms of G-sets SpanG is preadditive, [7, Def. 5.7]. (2) Various ∞-categories of correspondences in motivic homotopy theory are preadditive. In fr particular, CorrS is preadditive, see [21], as well as the discrete category CorS from [41]. padd add We denote the ∞-category of preadditive or additive ∞-categories by Cat∞ or Cat∞ . Denote tr tr,gp gp gp PShΣ (S) = PShΣ(CorrS), PShΣ (S) = PShΣ(CorrS ) ≃ PShΣ (CorrS), see [28, Lemma 1.6] for the second equivalence, and see Section 3 for notation. ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 7

Definition 2.3. An ∞-category of correspondences CorrS over a scheme S ∈ Sch, is an ∞-category CorrS ∈ Cat∞ with a chosen object ptS, and an action of the monoidal category SmS,

SmS × CorrS → CorrS

(X,T ) 7→ (X ×S T ), such that the induced functor SmS → CorrS (2.4)

X 7→ X ×S ptS is essentially surjective. G The endofunctor −× m on CorrS induces the endofunctors ΩGm and ΣGm on PSh(CorrS) G G ΩGm F (X) ≃ fib(F (X × m) → F (X ×{1}), ΣGm F = cofib(F ×{1} → F × m), ∗ where F ×{1} ≃ F , the functor −× Gm on PSh(CorrS) is given by (−× Gm) , and the right- side morphism is induced by the morphism . and the latter functors agree with the Gm-loop and Gm-suspension on PSh(SmS) in the sense of natural equivalences ∗ ∗ ΣGm r ≃ r ΣGm , ΩGm r∗ ≃ r∗ΩGm . (2.5)

Definition 2.6. A family of ∞-categories of correspondences Corr(−) over noetherian separated base schemes S, is a functor ∗ Corr(−) : Sch → Cat∞; S 7→ CorrS; f 7→ f (2.7) from the category of noetherian separated base schemes to the ∞-category of preadditive ∞- categories equipped with a structure of a ∞-category of correspondences over S, and such that for any morphism f : S′ → S, and X ∈ SmS there is a natural equivalence ∗ ′ ∗ f (−× (X ×S S )) ≃ f (−× X). padd add If the functor (2.7) lands in Cat∞ or Cat∞ the family of ∞-categories of correspondences will be called additive or preadditive. For a preadditive family of ∞-categories of correspondences gp CorrS we denote by Corr(−) the corresponding additive one.

Definition 2.8. A family of ∞-categories of correspondences Corr(−) over schemes S ∈ Sch satisfies the property (Embed), if for any open immersion U → S in Sch, there is a fully faithful ∗ functor j# : CorrU → CorrS left adjoint to j .

Definition 2.9. A family of ∞-categories of correspondences CorrS over schemes S ∈ Sch satisfies the property (FinE), if it satisfies (Embed) and for any S ∈ Sch, and a generic point η ∈ S, for any X ∈ SmS there is the equivalence for the representable presheaves htr(X × η) ≃ lim htr (X × U ), η S −→ Uα S α α where Uα runes over the filtered set of Zariski neighbourhoods of η.

Given a preadditive family of ∞-categories of correspondences CorrS, a presheaf F ∈ PSh(CorrS) is called τ-sheaf, for a topology τ over S (resp. A1-invariant) if and only if it goes to the object of such type along the forgetful functor PSh(CorrS) → PSh(SmS). Denote by Shτ (CorrS) the sub- category of PSh(CorrS ) spanned by the τ-sheaves. Define Hτ (CorrS) as the subcategory spanned 1 by A -invariant τ-sheaves in PSh(CorrS). Let S1 1 ∧−1 S1 G∧−1 SHτ (CorrS)= Hτ (CorrS)[(S ) ], SHτ (CorrS)= SHτ (CorrS)[ m ]. tr tr S1,tr S1,tr S1 tr Denote Hτ (S)= Hτ (SmS)= Hτ (CorrS), SHτ (S)= SHτ (SmS )= SHτ (CorrS), SHτ (S)= tr tr SHτ (SmS)= SHτ (CorrS). Similarly we define the ∞-categories H (SmAff S)= H(Corr(SmAffS )), S1,tr S1 tr ( SH (SmAffS) = SH (Corr(SmAffS)), SH (SmAff S) = SH Corr(SmAff S)) with respect to the ∞-categories SmAffS and their images Corr(SmAff S) in CorrS instead of SmS and CorrS. ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 8

3. Etale´ excision and motivic localisation. 3.1. Etale´ excision theorem for framed presheaves.

Definition 3.1. Let F be an object of PSh(S) or PShtr(S), for a scheme S. We say that F is 1 1 A -invariant if for any X ∈ SmS the map F (X) → F (A × X) is an equivalence.

Let Corrk be an additive ∞-category of correspondences over a field k in the sense of Defini- tion 2.6, and suppose that the canonical functors Smk → Corrk passes through the graded classical category of framed correspondences Fr+(k) in the sense of [23].

Proposition 3.2 (Etale´ excision). Given a smooth scheme X over a field k, a Nisnevich square A1 tr R of the form (1.7), and a point x ∈ X, then any -invariant presheaf F ∈ PShΣ (k) is R-excisive over the local scheme Xx. i.e. the map ′ ′ ′ − F (Xx) → F (W ×X Xx) ×F (W × ′ X − ) F (Xp 1 x ) X p 1(x) ( ) is an equivalence.

Proof. It suffices to show that the map

∗ ′∗ i i πi Fib(F (Xx) → F (Ui−1(x))) → πi Fib(F (Vp−1(x)) → F ((U ×X V )(p′i)−1(x))) is an isomorphism for any i. Since taking homotopy groups commutes with filtered colimits hidden in the definition of values of a presheaf on essentially smooth schemes, the left-hand side is an ∗ ∗ ′∗ extension of Ker(πiF (i )) by Coker(πi+1F (i )) and the right-hand is an extension of Ker(πiF (i )) ′∗ by Coker(πi+1F (i )). The kernels are trivial by [24, Theorem 2.15(3)] and [17, Corollary 3.6(2)] and the maps

πiF (Ui−1(x))/πiF (Xx) → πiF ((U ×X V )(p′i)−1(x))/πiF (Vp−1 (x)) are isomorphisms by [24, Theorem 2.15(5)] in combination with the result of [18] and [17, Corol- lary 3.6(4)], so the result follows. 

3.2. Zariski and Nisnevich motivic localisations. Let k be a field.

Definition 3.3. An ∞-category of correspondences Corrk over a field k satisfies the property ´ A1 tr (EtEx) if any -invariant presheaf F ∈ PShΣ (S) is R-excisive over the essetially smooth local scheme Xx, for any Nisnevich square R of the form (1.7) over k. See Definition 1.6.

Example 3.4. Proposition 3.2 equivalently claims that the ∞-category Corrgp,fr(k) equipped with fr gp,fr the composite functor Smk → Corr (k) → Corr (k) satisfies (EtEx).´

Lemma 3.5. Given an ∞-category Corrk and a functor Smk → Corrk that satisfy (EtEx),´ then A1 tr an -invariant presheaf F ∈ PShΣ (k) is a Nisnevich sheaf if and only if F is a Zariski sheaf. In A1 fr particular, any group-like -invariant Zariski sheaf F ∈ PShΣ(S) is a Nisnevich sheaf. Proof. The first claim follows by the definition from Lemma 1.9. 

Theorem 3.6. Let k be a field, and Corrk be an ∞-category of correspondences over k that satisfies ´ tr tr fr,gp (EtEx). There is an equivalence of ∞-categories Hnis(k) ≃ Hzar(k). In particular, Hnis (k) ≃ tr,gp fr fr Hzar (k), SHnis(k) ≃ SHzar(k). tr tr Proof. The first claim follows from Lemma 3.5, since the ∞-categories Hnis(k) and Hzar(k) are the tr A1 full subcategories of in PShΣ (k) whose objects are group-like -invariant Nisnevich and Zariski sheaves, respectively. The case of framed presheaves follows because of Example 3.4 and the equivalence PShfr,gp(k) ≃ PSh(Corrfr,gp(k)) provided by [28, Lemma 1.6]. Note that the case of 1 Htr(k) implies the equivalences for SHS ,tr(k), SHtr(k).  ZARISKI-LOCAL FRAMED A1-HOMOTOPYTHEORY 9

4. Relative case over a base scheme For a positive dimensional base scheme S, the property (EtEx)´ in the sense of Definition 3.3 fails for the ∞-category Corrgp,fr(S). To explain the counterexample let us consider a non empty open subscheme V in a positive dimensional scheme S, and a closed subscheme Z in V , Z 6= V , and the Nisnevich square V ′ − Z / V ′ (4.1)

e   V − Z / V.

The argument in [16, §13] shows that (EtEx)´ fails for the base change of the square (4.1) over the local scheme Vz, z ∈ Z, basically because there are no framed correspondences from Vz and ′ ′ V ×V Vz to V − Z, nor to V − Z. Moreover, Theorem 3.6 is not expected to be true over S, because the morphism fib(V ′ − Z → V − Z) → fib(V ′ → V ) fr fr is an equivalence in SHnis(S) but it is not expected to be an equivalence in SHzar(S). However, there is a positive result in this direction generalizing Theorem 3.6 that employs a new Grothendieck topology on the category of schemes. This topology is stronger than the Zariski topology and has the property that the square (4.1) defines a covering.

4.0.1. Localisation property on affines.

Definition 4.2. We call by a family of topologies on the categories SmS or SmAffS over noetherian separated schemes a choice of topology for each S ∈ Sch such that the base change functors are continuous.

Let Corr(−) be a family of categories of correspondences, Definition 2.6. We consider ∞- tr categories Shτ (SmS) for all schemes S. For a morphism f : S1 → S2 in SmS, there is the direct image functor

tr tr f∗ : Shτ (SmS1 ) → Shτ (SmS2 ); f∗F (U) ≃ F (U ×S2 S1),U ∈ SmS2 , (4.3) and the (left adjoint) inverse image functor

∗ tr tr ∗ tr tr f : Shτ (SmS2 ) → Shτ (SmS1 ); f hS1 (X) ≃ hS1 (X ×S2 S1), that is induced via the left Kan extension of the functor

tr tr SmS2 → Shτ (SmS1 ); X 7→ hS1 (X ×S2 S1) tr along SmS2 → Shτ (SmS2 ). Assume in addition that for any closed immersion i: Z1 → Z2 over ! S, there is a functor i right adjoint to i∗. By [32, Theorem 5.5.2.9] it exists if the ∞-categories tr Shτ (SmSi ) are presentable, in particular, it is so if τ is regular bounded completely decomposable topology as in Example 4.11. ∗ ! tr Then the above functors f , f∗, and i induce functors on ∞-categories SHτ (−), and for any closed immersion of schemes Z → B in SchS, there are two adjunctions

S1,tr ⇄ S1,tr ! i∗ : SHτ (SmZ ) SHτ (SmB): i 1 1 (4.4) ∗ S ,tr ⇄ S ,tr j : SHτ (SmB) SHτ (SmB−Z ): j∗.

! ∗ S1,tr Similarly, we define the functors i∗,i , j , j∗ for the case of categories SmAffS, and SHτ (SmAff S), for any family of topologies τ on the categories SmAff S over noetherian separated schemes. Definition 4.5. Let Corr, τ and S be as above. We say that τ satisfies property (LocAff) with respect to Corr if the following holds: ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 10

(LocAff) For an affine scheme S, a closed immersion i: Z → S, and the open immersion S1,tr j : S − Z → S, for any F ∈ SHτ (SmAff S) the square

! / i∗i F / F (4.6)

   / ∗ ∗ / j∗j F

∗ ! ∗ is pullback, where i , i , j∗, j are as in (4.4). Let us formulate the following consequence of the property (LocAff), that is similar to the statement used in [30] for the deduction of the equivalence of SHfr(S) and SH(S) over base schemes from the base field. The result is not used formally in what follows, but is formulated to illustrate the property (LocAff); we use Lemma 4.30 based on a similar principle.

Lemma 4.7. If a family of topologies τ on SmS satisfies (LocAff), and (FinE), defined in Defini- tion 4.25 in what follows, then for an affine scheme S of finite Krull dimension the functor

1 1 SHS ,tr(SmAff ) → SHS ,tr(SmAff ) (4.8) τ S Y τ z z∈S is conservative, where for each z ∈ S the functor

S1,tr S1,tr SHτ (SmAff S) → SHτ (SmAff z); F 7→ Fz (4.9) ∗ ! is given by the assignment Fz = j i (F ), j : z → Z, i: Z → S the canonical morphisms for Z = z being the closure of z in S.

S1,tr S1,tr Proof. Since SHτ (SmAff S) ≃ SHτ (SmAff Sred ) by (LocAff) applied to the closed immersion Sred → S, if dim S is zero, (4.8) is an equivalence. Assume that the claim is proved for all schemes tr of dimension less then dim S. Let F ∈ SHτ (S) be a morphism such that Fz ≃ 0 for all z ∈ S. Let i: B → S be a closed immersion of positive codimension. For a point z ∈ B let Z = z be the closure in B, t: z → Z, and c: z → Z be canonical morphisms, then ! ∗ ! ! ∗ ! i (F )z ≃ t c i (F ) ≃ t (i ◦ c) (F )= Fz ≃ 0. ! ! Hence i (F ) ≃ 0 by the inductive assumption, since dim Z < dim S. Thus i∗i (F ) ≃ 0, and (LocAff) ∗ implies F ≃ j∗j F for j : S − Z → S. Taking the cofiltered limit along all Z as above and using Lemma 4.29 we get ∗ F ≃ lim j∗j F ≃ F ≃ 0 −→ M η η∈S(0) where η runes over generic points of S. 

Here is the list of examples of topologies that satisfy (Loc) or (LocAff) with respect to Corrfr(S). Example 4.10. By the Localisation theorem from [30] by Hoyois the property (Loc) defined in Section 2 holds for the Nisnevich topology, and (LocAff) follows similarly. Example 4.11. The property (LocAff) holds for the so called trivial fibre topology [16, Def. 3.1], tf-topology, that is completely decomposable bounded topology generated by the Nisnevich squares (4.1) such that e is affine and Y = V ×S Z for a closed subscheme Z ⊂ S, see ??. The claim can be proved by the arguments of the localisation theorem from [16, Section 14.2] 1 and follows by the latter result in view of the A -equivalence of framed correspondences Fr+(−, −) and Corrfr(−, −) on affine schemes [21, Corollary 2.3.25, Corollary 2.2.20], in combination with [16, Lemma A.7]. Example 4.12. As shown in what follows property (LocAff) holds for the Zariski fibre topology generated by the Zariski topology and trivial fibre topology; property (Loc) from Section 2 holds too because any smooth scheme has a smooth affine covering. ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 11

4.0.2. Weak lifting coverings property. In this subsection, we formulate a criterion on ν that guar- anties the functor i∗ : Htriv(SmAff Z ) → Htriv(SmAff S) for a closed immersion of affine schemes i: Z → S detects ν-sheaves. cci For any scheme S, consider the subcategory SmS in SmAff S spanned by the schemes with trivial tangent bundle. Lemma 4.13. The restriction functor cci Htriv(SmS ) ← Htriv(SmAff S) (4.14) is an equivalence. cci Proof. The left Kan extension along the embedding SmS → SmAff S induces the functor cci Htriv(SmS ) → Htriv(SmAff S) that is left inverse to (4.14). For any X ∈ SmAffS , consider the tangent bundle TX on X, and since X is affine, there is a bundle N on X such that N ⊕ TX is trivial. Then the tangent bundle of the scheme Tot(N) defined by N, i.e. the spectrum of the tensor algebra of the dual coherent cci sheaf, is trivial. So Tot(N) ∈ SmS .At the same time the canonical morphism h(Tot(N)) → h(X) 1 is an A -equivalence in PSh(SmAff S). So for any F ∈ Htriv(SmAff S), F (X) ≃ F (Tot(N)). Hence the functor (4.14) is conservative. The claim follows because a conservative functor that has a left inverse is an equivalence. 

Definition 4.15. Let ν be a family of topologies on the categories SmAffS over noetherian sepa- rated schemes. We say that ν has the property (rCovLift) if the following holds: cci (rCovLift) The topology ν induces the topology in the subcategory SmS for each S. Given a cci closed embedding Z → S of affine schemes, for any ν-covering XZ → XZ in SmZ , cci there is a ν-covering XS → XS in SmS , such that (1) XZ ≃ X ×eS Z, and (2) there exists a morphism XZe← XS ×S Z. e e Lemma 4.16. Let ν be a family of topologies that has the property (rCovLift). Then for any closed immersion of noetherian separated schemes Z → S the functor cci cci i∗ : PSh(SmZ ) → PSh(SmS ) detects ν-sheaves. cci cci Proof. For any ν-covering XZ → XZ in SmZ ; and a ν-covering XS → XS in SmS , e e XZ ≃ X ×S Z, XZ = XS ×S Z. Then e e Map cci (R(XS/XS),i∗(F )) ≃ Map cci (R(XZ /XZ ), F ), PSh(SmS ) PSh(SmZ ) Map cci (h(XeS),i∗(F )) ≃ Map cci (h(XeZ ), F ), PSh(SmS ) PSh(SmZ ) where R(XS/XS) ∈ PSh(Z) and R(XZ /XZ) ∈ PSh(S) are the covering sieves, and h(XS), h(XZ ) are the representablee presheaves. e cci Given a scheme XZ , consider the filtered categories I of ν-coverings (XZ )α → XZ in SmZ , and cci J of ν-coverings (XS)β → XS in SmS such that XS ×S Z ≃ XZ . Thene the base change induces functor e I → J; h(XS)β → XSi 7→ h(XZ )β → XZ i . e e By (rCovLift) the latter functor of filtered categories is final. Hence

Lν i∗(F )(XS ) = lim Map cci (R((XS )β/XS),i∗(F )) −→β∈J PSh(SmS ) e ≃ lim Map cci (R((XZ )β/XZ ), F ) −→β∈J PSh(SmZ ) e ≃ lim Map cci ((R(XZ )α/XZ), F ) −→α∈I PSh(SmZ ) = Lν F (XZ ), e ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 12

So Lν i∗F (XS ) ≃ i∗F (XS) ⇒ Lν F (XZ ) ≃ F (XZ ). Thus if i∗(F ) is a ν-sheaf, then F is a ν-sheaf.  Corollary 4.17. Let ν be a family of topologies that has the property (rCovLift). Then the functor

i∗ : Htriv(SmAff Z ) → Htriv(SmAff S) detects ν-sheaves. Proof. The claim follows by Lemma 4.13 and Lemma 4.16.  4.0.3. Weak lifting of Zariski, and Nisnevich, and trivial fibre coverings. In this subsection, we prove the property (rLiftCov) for Zariski and Nisnevich topologies. cci Lemma 4.18. For any closed immersion of affine scheme Z → S, and XZ ∈ SmZ there is a cci scheme XS ∈ SmS , XS ×S Z ≃ XZ , and moreover, if XZ is ´etale over Z then XS can be choosen being ´etale over S. AN Proof. Since XZ is affine over an affine scheme Z there is a closed immersion XZ → Z . Moreover, AN since the tangent bundle of XZ is trivial for any closed immersion XZ → Z the normal bundle is AN stably trivial, hence there is such a closed immersion XZ → Z that the normal bundle is trivial. N Choose a trivialisation of the co-normal bundle of XZ in AZ , and consider a set of regular Z Z AN functions fdim XZ +1,...fN on Z that are trivial of XZ and define the mentioned trivialisation of Z Z the normal bundle. Then XZ is a clopen subscheme in the vanishing locus Z(fdim XZ +1,...fN ). S S Z Z AN Choose lifting fdim XZ +1,...fN of fdim XZ +1,...fN on S and define ′ S S XS = Z(fdim XZ +1,...fN ). ′ Since XZ is smooth over Z, the morphism XZ → S is smooth over XZ , and hence it is smooth ′ over some Zariski neighbourhood of XZ , let us say XS − C1, C1 ∩ XZ = ∅. Since the tangent ′ bundle of XZ over Z is trivial, the tangent bundle of XS − C1 over S is trivial on some Zariski ′ neighbourhood of XZ . Moreover, if XZ → Z is ´etale, then XS → S is ´etale over XZ , and hence it ′ is ´etale over some Zariski neighbourhood of XZ in XS. ′ Let X −C be the intersection of the mentioned above Zariski neighbourhoods of XZ , C∩XZ = ∅. ′ Choose a function g on XS that equals unit on XZ , and vanishes on the scheme C and on the ′ subscheme Z(fdim XZ +1,...,N ) − XZ in XS. Then the scheme ′ XS := XS − Z(g) ′ is a main open subset of affine scheme, so XS is affine, Since XS is contained in XS − C, it follows cci that XS ∈ Sm ; and moreover, if XZ is ´etale over Z then XS is ´etale over S.  Lemma 4.19. The property (rCovLift), see Definition 4.15, holds for the Zariski, the Nisnevich topology and the trivial fibre topology on SmAff S over a noetherian separated scheme S.

Proof. Let Z → S be a closed immersion of affine schemes in Sch, and XZ → XZ be a ν-covering cci in SmZ , for ν being one of the above topologies. e cci cci The Nisnevich case. By Lemma 4.18 for any XZ ∈ SmZ , there is a scheme XS ∈ SmS , XS ×S Z ≃ XZ . Moreover, applying the same fact to the embedding XZ → XS we get that for cci cci cci any XZ ∈ SmXZ = SmXZ ∩ SmZ there is XS ∈ SmXS such that X ×S Z ≃ X. Even more, if the canonicale morphisms XZ → XZ is ´etale, thene Lemma 4.18 providese an ´etale morphisme XS → XS. Hence if XZ → XZ is ae Nisnevich covering, then XS ∐(XS −XZ ) gives a Nisnevich coveringe of XS. cci Choose ane affine covering U → S − Z, then XS ∐eU ×S Xe S ∈eSmS , and it is a Nisnevich covering of XS. Since if XZ → XZ is a trivial fibre covering,e thee constructed above Nisnevich covering is a trivial fibre covering,e the claim follows for the trivial fibre topology as well. cci ′ cci The Zariski case. By Lemma 4.18 for any XZ ∈ SmZ , there is a scheme XS ∈ SmS , XS ×S Z ≃ XZ . Given a Zariski covering (X − C ) → X , (4.20) [ Z i Z i ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 13 since XZ is affine there is a refinement of (4.20) given by the coproduct of main open subsets of the scheme XZ (X − Z(gZ )) → X ,gZ ∈ O (X ). a Z j Z j XZ Z j∈J B Choose a lifting gj ∈ OXS (XS) for each j, and let g be a functions that is unit on XZ and vanishes B on the vanishing clus of the set gj , j ∈ J. Then ′ cci XS := XS − g(XZ ) ∈ SmS , and the morphism X = (X − Z(gS)) → X′ − g(X ): X S a S j S Z S e j is the required covering.  4.1. Embedding, and continuity.

Definition 4.21. Let τ be a family of topologies on SmAff S for S ∈ Sch. We say that τ satisfies the property (Embed) if the following holds:

(Embed) For any S as above, and open immersion U ∈ S the canonical embeddin SmAffU → SmAffS preserves and detects ν-coverings. Example 4.22. The property (Embed) holds for the Zariski, Nisnevich, and the trivial fibre topologies.

Lemma 4.23. Let τ be a family of topologies on SmAff S for S ∈ Sch that has the property (Embed). Then for any open immersion of schemes j : U → S in Sch the functor ∗ j : Htriv(SmAff S) → Htriv(SmAff U ) (4.24) preserves τ-sheaves. Proof. 

Definition 4.25. Let τ be a family of topologies on SmAffS over noetherian separated S. We sat that τ has property (FinE) if the following holds:

(FinE) τ satisfies (Embed); for any S as above, X ∈ SmS, a generic point η ∈ S, and a ] ν-covering (Xη) → Xη, where Xη = X ×S η, there is an Zariski neighbourhood U of η and a ν-covering XU → X ×S U such that XU ×U η → Xη is a refinement of ] (Xη) → Xη. e e

Example 4.26. The (FinE) holds for any family of topologies τ on the categories SmAffS for S ∈ Sch satisfying (Embed) and such that τ is trivial on the categories SmAff k for all fields k, in particular, the trivial fibre topology from Example 4.11 has the property. Example 4.27. The Zariski and Nisnevich topologies over a noetherian separated scheme S have the property (FinE). Note that if (FinE) holds for topologies τ and ν then it holds for the topology τ ∪ ν.

Lemma 4.28. Let τ be a family of topologies on the categories SmAff S for S ∈ Sch that satisfies the property (FinE). Then the functor j∗ : PSh(S) → PSh(η) preserves τ-sheaves for each S ∈ Sch.

Proof. Any scheme in Smη is a generic fibre Xη = X ×S η of a scheme X ∈ SmS. Given a τ- ] covering (Xη) → Xη, there is a τ-covering XV → X ×S V , for an Zariksi neighbourhood V of η, ] that generic fibre (X)η = XV ×V η → X is ae refinement of (Xη) → Xη. If F ∈ PSh(S) is a ν-sheaf, then for any Zariskie neighbourhoode U of η in V

F (XV ×V U) ≃ F (Cˇ(XV ×V U)). Hence for a filtered system of U such that lim U =eη, we have α ←−α α j∗F (X ) ≃ lim F (X × U ) ≃ lim F (Cˇ(X × U)) ≃ j∗F (Cˇ((X) )). η −→ V V α −→ V V η α α e e ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 14

∗ ∗ Thus Lν j F ≃ F , and consequently j F is a τ-sheaf.  Lemma 4.29. Let U be a filtered system of open subschemes of a scheme S, such that lim U = α ←−α α S(0) is the union of generic points of S. The following natural equivalence holds for the functors ∗ ∗ (j )∗ : PSh(U ) → j∗ lim (j )∗(j ) ≃ j∗j The same applies to the case of ∞-categories Sh (−), α α −→α α α τ if a family of topologies τ on SmAffS satisfies the property (FinE). Moreover, the claim holds for tr Shτ (−) for a family of ∞-categories of correspondences CorrS over S ∈ Sch satisfying (FinE).

Proof. Consider the continuous functor on EssSmS defined by F and denote it by the same symbol. Then ∗ (0) ∗ lim(j )∗(j ) (F )(X) ≃ lim F (X × U ) ≃ F (X × S ) ≃ j∗j (F )(X) −→ α α −→ S α S α α that implies the first claim in the lemma. The second claim follows in view of Lemma 4.28. The ∗ third claim holds because for CorrS satisfying (FinE) there is a canonical equivalences r∗j ≃ ∗ j r∗. 

Lemma 4.30. Let CorrS be a preadditive family of ∞-categories of correspondences over S ∈ Sch satisfying (FinE), and τ be families of topologies on SmAffS for S ∈ Sch such that τ satisfies (LocAff) and (FinE), Definition 4.5, Definition 4.25, and ν is stronger then τ. Then for any S ∈ Sch of finite Krull dimension the functor

1 1 SHS ,tr(SmAff ) → SHS ,tr(SmAff ) τ S Y τ z z∈S tr,gp detects ν sheaves, and similarly for Hτ (SmAff S). Proof. The claim is tautological for the empty base scheme case (and even in the case of base fields). S1,tr Assume that claim for all base schemes of dimension less then S. Let F ∈ SHτ (SmAffS) that S1,tr image in SHτ (SmAff z) is a ν-sheaf for each z ∈ S. Consider the filtered system of closed subschemes Zα of positive codimension in S. By (LocAff) ! ∗ (iα)∗(iα) F ≃ fib(F → (jα)∗(jα) F ), where jα : S − Zα → S, and iα : Zα → S is the canonical embeddings. Then ! ∗ lim i∗i F ≃ fib(F → lim(j )∗(j ) F ), (4.31) −→ −→ α α α α

The projective limit of the schemes S − Zα equals the union of generic points of S. Denote it by S(0) and by j the canonical embedding to S, then by Lemma 4.29 ∗ ∗ lim(j )∗(j ) F ≃ j∗j F. −→ α α α ! ∗ By the assumption it follows that (iα)∗(iα) F , and j∗j F are ν-sheaves. By (4.31) it follows that ∗ ! F ≃ fib(j∗j F → lim i∗i F [1]). −→ α

tr,gp S1,tr Thus the claim is proved. The case of Hτ (SmAff S) follows from the case of SHτ (SmAff S). 

4.2. Inheriting of (LocAff). In this subsection, we formulate an enough criterion on a topology ν such that (LocAff) for a topology τ implies (LocAff) for τ ∪ ν, Lemma 4.37.

Lemma 4.32. Let τ and ν be families of topologies on SmAffS for S ∈ Sch such that the base change functors are continuous, and ν satisfy the property (Embed). Let Corr be a family of ∞- categories of correspondences that satisfies (Embed). Then for any open immersion j : U → S in Sch, the functors ∗ S1,tr ⇄ S1,tr j : SHτ (SmAff S) SHτ (SmAff U ): j∗ preserve ν-sheaves. ZARISKI-LOCAL FRAMED A1-HOMOTOPY THEORY 15

Proof. For any family of topology ν such that the base change functors are continuous for any PSh morphism of schemes f : S1 → S2 in Sch, the functor f∗ : PSh(S1) → PSh(S2) preserves ν- PSh tr sheaves. Then since f∗ r∗ ≃ r∗f∗, where r∗ : PShτ (−) → PSh(−), and f∗ stands for (4.3), it follows that the latter functor f∗ preserves ν-sheaves, and consequently the functor

S1,tr S1,tr f∗ : SHτ (SmAff S1 ) → SHτ (SmAff S2 )

∗ does too. So the claim for j∗ is proved, and to prove the claim for j it is enough to note that ∗ S1,tr S1,tr the functor j : SHτ (B) → SHτ (B − Z) is the canonical restriction, and hence it preserves ν-sheaves by (Embed). 

Lemma 4.33. Let τ be a family of topologies that satisfies the property (LocAff). Let ν be a family i of topology on SmAffS for S ∈ Sch such that for any closed inclusion of schemes Z → S in Sch the functor

i∗ : Htriv(SmAff Z ) → Htriv(SmAff S) (4.34)

j detects ν-sheaves, and for any open immersion of schemes U → S in Sch the functor

∗ j : Htriv(SmAff S) → Htriv(SmAff U ) (4.35) preserves ν-sheaves. Let Corr be a family of ∞-categories of correspondences that satisfies (Embed). Then (1) for any closed immersion Z → S as in Definition 4.5, both functors

S1,tr ⇄ S1,tr ! i∗ : SHτ (SmAffZ ) SHτ (SmAff S): i

preserve ν-sheaves, (2) and if S is affine, then for any point z ∈ S, the functor (4.9) preserves ν-sheaves.

∗ Proof. By Lemma 4.32 the functors j , j∗ in diagram (4.6) preserve ν-sheaves, and by the property ! (LocAff) it follows that the functor i∗i preserves too. If the functor (4.35) detects ν-sheaves, then the functor S1,tr S1,tr i∗ : SHτ (SmAff Z ) → SHτ (SmAff S) does, and it follows that i! preserves that ones. So (1) is proved. The claim of (2) follows because ∗ ! the functor F 7→ Fz by the definition is given by the composition j i where i: Z → S, Z is the closure of z, and j : z → Z is the canonical open immersion. 

Corollary 4.36. Let τ, and ν, and Corr be as in Lemma 4.33. Then τ ∪ ν has the property (LocAff).

! ∗ Proof. The claim follows from Lemma 4.33(1) immediate because the functors i , i∗, j , j∗ in (4.6) preserve the subcategories of (τ ∪ ν)-sheaves. 

Lemma 4.37. Let CorrS be a family ∞-categories of correspondences, and τ, and ν be families of topologis on SmAffS for S ∈ Sch such that • the family Corr that satisfies (Embed); • τ has the properties (Embed), (LocAff); • ν has the properties (Embed), (rCovLift); Definitions 4.5, 4.15, 4.21 and 4.25. Then the topology τ ∪ ν has the property (LocAff). In particular, the τ ∪ zar and τ ∪ nis have properties (LocAff).

Proof. The first claim holds follows by Corollaries 4.17 and 4.36 and lemmas 4.19 and 4.23.  REFERENCES 16

4.3. Result. Theorem 4.39 proved in this subsection proves Theorem C, and in view of Exam- ple 4.11 and Example 4.12 it implies Corollary 1. We use easy fact to reduce the case of SmS to SmAffS.

Lemma 4.38. Let ν be the Zariski or the Nisnevich topology on SchS over a noetherian separated scheme S. Then (1) the canonical restriction tr tr Shν (S) → Shν (SmAffS ) is an equivalence (2) for a Zariski covering S → S the inverse image functor e tr tr Shν (S) → Shν (S) is conservative. e Proof. The claim follows because for any local scheme U over S the canonical morpshim U → S passes through the scheme S.  e Theorem 4.39. Let CorrS be a family of preadditive ∞-categories of correspondences that satisfies (Embed), and (EtEx),´ Definitions 2.6, 2.8 and 3.3. Let τ be a topology on the category SchS over noetherian separated scheme S of finite Krull dimension that restricts to the category SchAffS and satisfies properties (LocAff) and (FinE), Definitions 4.5 and 4.25. Then the canonical functor tr,gp tr,gp Hzar∪τ (S) → Hnis∪τ (S), is an equivalence. Proof. By Theorem 3.6 the claim holds over fields. Consider the case of affine S. By the first point tr,gp of Lemma 4.38 the claim follows from the case of ∞-categories Hν∪τ (SmAff S), where ν = zar, nis. tr,gp By Lemma 4.37 the topologies τ∪ν on SchAffS satisfy property (LocAff). Let F ∈ Hzar∪τ (SmAff S), tr,gp we are going to show that F ∈ Hnis∪τ (SmAffS), i.e. it is a Nisnevich sheaf. By Theorem 3.6 F goes to the Nisnevich sheaf under the functor Htr,gp(SmAff ) → Htr,gp(SmAff ), ν∪τ S Y τ z z∈S By Lemma 4.30 applied to the topologies τ ∪ zar and τ ∪ nis in view of Example 4.27 the claim follows. The general case of the base scheme S follows by the second point of Lemma 4.38 applied to a Zariski covering Sβ → S with affine schemes Sβ.  `β Remark 4.40. Zariski fibre topology from Example 4.12 is the strongest subtopology of the Nis- nevich topology on SchS that equals the Zariski topology over residue fields; and it is expected that Zariski fibre topology is the weakest subtopology that is stronger then the Zariski topology and satisfies property (Loc). References

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Andrei Druzhinin, St. Petersburg Department of Steklov Mathematical Institute of Russian Acad- emy of Sciences, Fontanka, 27, 191023 Saint Petersburg, Russia

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka, 27, 191023 Saint Petersburg, Russia