In this article we show that the theory of framed motivic spaces introduced in [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19] satisfies localization: if 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ is a closed immersion of schemes, 
$j\colon {\rm U}\hookrightarrow {\rm S}$ is the complementary open immersion, and 
${\mathcal {F}} \in {\mathbf {H}}^{{\operatorname {fr}}}({\rm S})$ is a framed motivic space over 
${\rm S}$, then there is a cofiber sequence
\[ j_{\sharp}j^{*}{\mathcal{F}} \to {\mathcal{F}} \to i_{*}i^{*}{\mathcal{F}} \]
(see Theorem 10). Consequently, the theory of framed motivic spectra satisfies Ayoub's axioms [Reference AyoubAyo08], which implies that it admits a full-fledged formalism of six operations. Using this formalism, we show that the equivalence 
${\mathbf {SH}}^{{\operatorname {fr}}}({\rm S})\simeq {\mathbf {SH}}({\rm S})$, proved in [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19] for 
${\rm S}$ the spectrum of a perfect field, holds for any scheme 
${\rm S}$ (see Theorem 18).
The 
$\infty$-category 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$ of framed motivic spaces consists of 
$\mathbf {A}^1$-invariant Nisnevich sheaves on the 
$\infty$-category 
$\mathbf {Corr}^{\operatorname {fr}}(\operatorname {Sm}_{{\rm S}})$ of smooth 
${\rm S}$-schemes and framed correspondences. A framed correspondence between 
${\rm S}$-schemes 
${\rm X}$ and 
${\rm Y}$ is a span
over 
${\rm S}$, where 
$f$ is a finite syntomic morphism equipped with a trivialization of its cotangent complex in the 
${\rm K}$-theory of 
${\rm Z}$. Our result stands in contrast to the case of finite correspondences in the sense of Voevodsky, where the analog of the Morel–Voevodsky localization theorem remains unknown. The essential ingredient in our proof is the fact that the Hilbert scheme of framed points [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Definition 5.1.7] is smooth.
1. Review of the Morel–Voevodsky localization theorem
We start by reviewing the localization theorem of Morel and Voevodsky [Reference Morel and VoevodskyMV99, § 3, Theorem 2.21]. We refer to [Reference HoyoisHoy14, Appendix C] for the definition of the Morel–Voevodsky 
$\infty$-category 
${\mathbf {H}}({\rm S})$ for 
${\rm S}$ an arbitrary scheme. We shall denote by 
${\rm L}_\mathrm {nis}$, 
${\rm L}_{\mathbf {A}^1}$, and 
${\rm L}_{\rm {mot}}$ the localization functors enforcing Nisnevich descent, 
$\mathbf {A}^1$-invariance, and both, respectively.
Theorem 1 (Morel–Voevodsky) Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion of schemes, 
$j\colon {\rm U}\hookrightarrow {\rm S}$ the complementary open immersion, and 
${\mathcal {F}} \in {\mathbf {H}}({\rm S})$ a motivic space over 
${\rm S}$. Then the square
is cocartesian in 
${\mathbf {H}}({\rm S})$.
This theorem was proved in this generality in [Reference HoyoisHoy14, Proposition C.10], but we give here a more direct proof that was alluded to in that paper. In the sequel, we will actually use a slightly different form of this theorem; see Corollary 5 below.
Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion with open complement 
$j\colon {\rm U}\hookrightarrow {\rm S}$. For an 
${\rm S}$-scheme 
${\rm X}$ and an 
${\rm S}$-morphism 
$t\colon {\rm Z}\to {\rm X}$, we define the presheaf
\[ {\rm h} _{{\rm S}}({\rm X},t)\colon \operatorname{Sch}_{{\rm S}}^\mathrm{op}\to\mathrm{Set} \]by the cartesian square
where 
${\rm h} _{{\rm S}}\colon \operatorname {Sch}_{{\rm S}}\to \mathrm {PSh}{}(\operatorname {Sch}_{{\rm S}})$ is the Yoneda embedding. Explicitly,
\[ {\rm h} _{{\rm S}}({\rm X},t)({\rm Y}) = \begin{cases} \mathrm{Maps}_{{\rm S}}({\rm Y},{\rm X}) \times_{\mathrm{Maps}_{\rm Z}({\rm Y}_{\rm Z},{\rm X}_{\rm Z})} \{{\rm Y}_{\rm Z}\to {\rm Z}\xrightarrow t {\rm X}_{\rm Z}\}& \text{if ${\rm Y}_{\rm Z}\neq\varnothing$,} \\ * & \text{if ${\rm Y}_{\rm Z}=\varnothing$.} \end{cases} \]
If 
${\rm S}$ is a Henselian local scheme, we have the following well-known facts:
(a) if
${\rm X}$ is étale over ${\rm S}$, then ${\rm h} _{{\rm S}}({\rm X},t)({\rm S})$ is contractible;(b) if
${\rm X}$ is smooth over ${\rm S}$, then ${\rm h} _{{\rm S}}({\rm X},t)({\rm S})$ is connective (i.e. not empty).
Both assertions hold by definition of 
${\rm h} _{{\rm S}}({\rm X},t)$ if 
${\rm Z}=\varnothing$. Otherwise, 
$({\rm S},{\rm Z})$ is an affine Henselian pair where 
${\rm Z}$ has a unique closed point, so we can assume 
${\rm X}$ affine. Assertion (a) is then a special case of [Reference GrothendieckGro67, Proposition 18.5.4], and assertion (b) is a special case of [Reference GrusonGru72, Théorème I.8]. For general 
${\rm S}$, it follows immediately that:
(a′) if
${\rm X}$ is étale over ${\rm S}$, then ${\rm L}_\mathrm {nis}{\rm h} _{{\rm S}}({\rm X},t)$ is contractible;(b′) if
${\rm X}$ is smooth over ${\rm S}$, then ${\rm L}_\mathrm {nis}{\rm h} _{{\rm S}}({\rm X},t)$ is connective.
Assertion (b
$'$) is an abstract version of Hensel's lemma in several variables. The crux of the Morel–Voevodsky localization theorem is a refinement of (b
$'$) asserting that the motivic localization 
${\rm L}_{\rm {mot}} {\rm h} _{{\rm S}}({\rm X},t)$ is contractible.
Lemma 2 Let 
$f\colon {\rm X}\to {\rm Y}$ be a morphism of locally finitely presented 
${\rm S}$-schemes that is étale in a neighborhood of 
$t({\rm Z})$. Then the induced map 
${\rm h} _{{\rm S}}({\rm X},t)\to {\rm h} _{{\rm S}}({\rm Y},f\circ t)$ is a Nisnevich local isomorphism.
Proof. Since the presheaves 
${\rm h} _{{\rm S}}({\rm X},t)$ and 
${\rm h} _{{\rm S}}({\rm Y},f\circ t)$ transform cofiltered limits of quasicompact, quasiseparated (qcqs) schemes into colimits [Reference GrothendieckGro66, Théorème 8.8.2(i)], it suffices to show that the given map is an isomorphism on Henselian local schemes. Since 
${\rm h} _{{\rm S}}({\rm X},t)({\rm T})={\rm h} _{\rm T}({\rm X}_{\rm T},t_{\rm T})({\rm T})$, we are reduced to proving that the map 
${\rm h} _{{\rm S}}({\rm X},t)({\rm S})\to {\rm h} _{{\rm S}}({\rm Y},f\circ t)({\rm S})$ is an isomorphism when 
${\rm S}$ is Henselian local; we will show that its fibers are contractible. Let 
${\rm X}'\subset {\rm X}$ be an open neighborhood of 
$t({\rm Z})$ where 
$f$ is étale. Given a section 
$s\colon {\rm S} \to {\rm Y}$ extending 
$f\circ t$, we have a cartesian square
By assertion (a) above, 
${\rm h} _{{\rm S}}({\rm X}'\times _{\rm Y}{\rm S},(t,i))$ is contractible, as desired.
Theorem 3 (
$\mathbf {A}^1$-Hensel lemma)
Let 
${\rm S}$ be a scheme, 
${\rm Z} \subset {\rm S}$ a closed subscheme, 
${\rm X}$ an 
${\rm S}$-scheme, and 
$t\colon {\rm Z}\to {\rm X}$ an 
${\rm S}$-morphism. If 
${\rm X}$ is smooth over 
${\rm S}$, then 
${\rm L}_{\rm {mot}} {\rm h} _{{\rm S}}({\rm X},t)$ is contractible.
Proof. By Lemma 2, we can replace 
${\rm X}$ by any open neighborhood of 
$t({\rm Z})$ in 
${\rm X}$. Since the question is Nisnevich local on 
${\rm S}$, we can assume that 
${\rm S}$ and 
${\rm X}$ are both affine. Since 
${\rm L}_\mathrm {nis}{\rm h} _{{\rm S}}({\rm X},t)$ is connective, we can further assume that there exists a section 
$s\colon {\rm S}\to {\rm X}$ extending 
$t$. Then there exists an 
${\rm S}$-morphism 
$f\colon {\rm X}\to {\mathbf {V}} ({\mathcal {N}} _s)$, étale in a neighborhood of 
$s({\rm S})$, such that 
$f\circ s$ is the zero section of the normal bundle 
${\mathbf {V}} ({\mathcal {N}} _s)\to {\rm S}$. Using Lemma 2 again, we are reduced to the case where 
${\rm X}\to {\rm S}$ is a vector bundle and 
$t\colon {\rm Z}\to {\rm X}$ is the restriction of its zero section. In this case, an obvious 
$\mathbf {A}^1$-homotopy shows that 
${\rm L}_{\mathbf {A}^1}{\rm h} _{{\rm S}}({\rm X},t)$ is contractible.
Remark 4 The proof of Theorem 3 actually shows that 
${\rm L}_\mathrm {nis}{\rm L}_{\mathbf {A}^1} {\rm L}_\mathrm {nis} {\rm h} _{{\rm S}}({\rm X},t)\simeq *$.
Corollary 5 Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion with open complement 
$j\colon {\rm U}\hookrightarrow {\rm S}$. For every 
${\mathcal {F}} \in \mathrm {PSh}{}(\operatorname {Sm}_{{\rm S}})$, the square
is motivically cocartesian, that is, its motivic localization is cocartesian in 
${\mathbf {H}}({\rm S})$.
Proof. Since this square preserves colimits in 
${\mathcal {F}}$, we can assume that 
${\mathcal {F}} ={\rm h} _{{\rm S}}({\rm X})$ for some smooth 
${\rm S}$-scheme 
${\rm X}$. We must then show that the canonical map
\[ {\rm h} _{{\rm S}}({\rm X}) \sqcup_{{\rm h} _{{\rm S}}({\rm X}_{\rm U})}{\rm h} _{{\rm S}}({\rm U}) \to i_*{\rm h} _{\rm Z}({\rm X}_{\rm Z}) \]
is a motivic equivalence in 
$\mathrm {PSh}{}(\operatorname {Sm}_{{\rm S}})$. In fact, we will show that it is a motivic equivalence in 
$\mathrm {PSh}{}(\operatorname {Sch}_{{\rm S}})$. Writing the target as a colimit of representables, it suffices to show that for every morphism 
$f\colon {\rm T}\to {\rm S}$ and every map 
${\rm h} _{{\rm S}}({\rm T})\to i_*{\rm h} _{\rm Z}({\rm X}_{\rm Z})$, corresponding to a 
${\rm T}$-morphism 
$t\colon {\rm Z}_{\rm T} \to {\rm X}_{\rm T}$, the projection
\[ \big({\rm h} _{{\rm S}}({\rm X}) \sqcup_{{\rm h} _{{\rm S}}({\rm X}_{\rm U})}{\rm h} _{{\rm S}}({\rm U})\big) \times_{i_*{\rm h} _{\rm Z}({\rm X}_{\rm Z})} {\rm h} _{{\rm S}}({\rm T}) \to {\rm h} _{{\rm S}}({\rm T}) \]
is a motivic equivalence. This map is the image by the functor 
$f_\sharp \colon \mathrm {PSh}{}(\operatorname {Sch}_{\rm T})\to \mathrm {PSh}{}(\operatorname {Sch}_{{\rm S}})$ of the map
\[ {\rm h} _{\rm T}({\rm X}_{\rm T},t) \to {\rm h} _{\rm T}({\rm T})\simeq *. \]
Indeed, this follows from the projection formula 
$f_\sharp (f^*({\rm B})\times _{f^*({\rm A})} {\rm C}) \simeq {\rm B}\times _{\rm A} f_\sharp ({\rm C})$, which holds for any morphisms 
${\rm B}\to {\rm A}$ in 
$\mathrm {PSh}{}(\operatorname {Sch}_{{\rm S}})$ and 
${\rm C}\to f^*({\rm A})$ in 
$\mathrm {PSh}{}(\operatorname {Sch}_{\rm T})$, and the base change equivalence 
$f^*i_* \simeq i_{{\rm T}*}f_{\rm Z}^*$. Since 
$f_\sharp$ preserves motivic equivalences and 
${\rm X}_{\rm T}$ is smooth over 
${\rm T}$, Theorem 3 concludes the proof.
Proof of Theorem 1. The functors 
$j_\sharp$, 
$j^*$, and 
$i^*$ between 
$\infty$-categories of presheaves preserve motivic equivalences, as does the functor 
$i_*\colon \mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{\rm Z}) \to \mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{{\rm S}})$ by [Reference Bachmann and HoyoisBH20, Proposition 2.11]. Thus, for 
${\mathcal {F}} \in {\mathbf {H}}({\rm S})$, the given square is the motivic localization of the square of Corollary 5.
2. The localization theorem for framed motivic spaces
We now turn to the proof of localization for framed motivic spaces. We use the notation from [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19].
Lemma 7 The forgetful functor 
$\gamma _*\colon \mathrm {PSh}{}_\Sigma (\mathbf {Corr}^{\operatorname {fr}}(\operatorname {Sm}_{{\rm S}})) \to \mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{{\rm S}})$ detects Nisnevich and motivic equivalences.
Proof. This follows from [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Proposition 3.2.14].
Proposition 8 Let 
$f\colon {\rm T}\to {\rm S}$ be an integral morphism. Then the functor
\[ f_*\colon \mathrm{PSh}{}_\Sigma(\mathbf{Corr}^{\operatorname{fr}}(\operatorname{Sm}_{\rm T})) \to \mathrm{PSh}{}_\Sigma(\mathbf{Corr}^{\operatorname{fr}}(\operatorname{Sm}_{{\rm S}})) \]preserves Nisnevich and motivic equivalences.
Proof. By Lemma 7, this follows from the fact that the functor 
$f_*\colon \mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{\rm T}) \to \mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{{\rm S}})$ preserves Nisnevich and motivic equivalences [Reference Bachmann and HoyoisBH20, Proposition 2.11].
Corollary 9 Let 
$f\colon {\rm T}\to {\rm S}$ be an integral morphism. Then the functor
\[ f_*\colon {\mathbf{H}}^{\operatorname{fr}}({\rm T}) \to {\mathbf{H}}^{\operatorname{fr}}({\rm S}) \]preserves colimits.
Proof. It follows from Proposition 8 that 
$f_*$ preserves sifted colimits. It also preserves limits, hence finite sums since 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$ is semiadditive [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Proposition 3.2.10(iii)].
If 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ is a closed immersion, it follows from Corollary 9 that we have an adjunction
\[ i_*: {\mathbf{H}}^{\operatorname{fr}}({\rm Z}) \rightleftarrows {\mathbf{H}}^{\operatorname{fr}}({\rm S}): i^!. \]Theorem 10 (Framed localization) Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion with open complement 
$j\colon {\rm U}\hookrightarrow {\rm S}$. Then the null sequence
\[ j_\sharp j^* \to \mathrm{id} \to i_*i^* \]
of endofunctors of 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$ is a cofiber sequence. Dually, the null sequence
\[ i_*i^! \to \mathrm{id} \to j_*j^* \]
of endofunctors of 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$ is a fiber sequence.
Proof. It suffices to prove the first statement. Since all functors involved preserve colimits by Corollary 9, it suffices to check that the sequence is a cofiber sequence when evaluated on 
$\gamma ^*({\rm X}_+)$ where 
${\rm X}$ is smooth over 
${\rm S}$ and affine [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Proposition 3.2.10(i)]. By Proposition 8 and Lemma 7, it suffices to show that the map
\[ {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}) / {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}_{\rm U}) \to i_*{\rm h} ^{\operatorname{fr}}_{\rm Z}({\rm X}_{\rm Z}) \]
in 
$\mathrm {PSh}{}(\operatorname {Sm}_{{\rm S}})$ is a motivic equivalence, where 
${\rm h} ^{\operatorname {fr}}_{{\rm S}}({\rm X}) / {\rm h} ^{\operatorname {fr}}_{{\rm S}}({\rm X}_{\rm U})$ denotes the quotient in commutative monoids. Note that if 
${\rm Y}\in \operatorname {Sch}_{{\rm S}}$ is connected then
\[ {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}_{\rm U})({\rm Y}) = \begin{cases} * & \text{if Y$_{\rm Z}\neq\varnothing$,} \\ {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X})({\rm Y}) & \text{if Y$_{\rm Z}=\varnothing$.} \end{cases} \]It follows that the canonical map
\[ {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X})\sqcup_{j_\sharp{\rm h} ^{\operatorname{fr}}_{\rm U}({\rm X}_{\rm U})} {\rm h} _{{\rm S}}({\rm U}) \to {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}) / {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}_{\rm U}) \]
is an equivalence on connected essentially smooth 
${\rm S}$-schemes, hence it is a Zariski-local equivalence in 
$\mathrm {PSh}{}(\operatorname {Sm}_{{\rm S}})$.Footnote 1 We are thus reduced to showing that the map
\[ {\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X})\sqcup_{j_\sharp {\rm h} ^{\operatorname{fr}}_{\rm U}({\rm X}_{\rm U})} {\rm h} _{{\rm S}}({\rm U}) \to i_*{\rm h} ^{\operatorname{fr}}_{\rm Z}({\rm X}_{\rm Z}) \]
is a motivic equivalence. By [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Corollary 2.3.27] and the non-framed version of Proposition 8, we can replace 
${\rm h} ^{\operatorname {fr}}$ by 
${\rm h} ^\mathrm {nfr}$: it suffices to show that the map
\[ {\rm h} ^{\mathrm{nfr}}_{{\rm S}}({\rm X})\sqcup_{j_\sharp{\rm h} ^{\mathrm{nfr}}_{\rm U}({\rm X}_{\rm U})} {\rm h} _{{\rm S}}({\rm U}) \to i_*{\rm h} ^{\mathrm{nfr}}_{\rm Z}({\rm X}_{\rm Z}) \]
is a motivic equivalence in 
$\mathrm {PSh}{}(\operatorname {Sm}_{{\rm S}})$. By [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Theorem 5.1.5], the presheaf 
${\rm h} ^{\mathrm {nfr}}_{{\rm S}}({\rm X})$ on all 
${\rm S}$-schemes is ind-representable by smooth 
${\rm S}$-schemes and compatible with any base change 
${\rm S}'\to {\rm S}$. Considering 
${\rm h} ^{\mathrm {nfr}}_{{\rm S}}({\rm X})$ as a presheaf on smooth 
${\rm S}$-schemes, this implies that 
$i^*({\rm h} ^{\mathrm {nfr}}_{{\rm S}}({\rm X}))\simeq {\rm h} ^{\mathrm {nfr}}_{\rm Z}({\rm X}_{\rm Z})$ and 
$j^*({\rm h} ^{\mathrm {nfr}}_{{\rm S}}({\rm X}))\simeq {\rm h} ^{\mathrm {nfr}}_{\rm U}({\rm X}_{\rm U})$. Thus, the result follows from Corollary 5.
Corollary 11 Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion with open complement 
$j\colon {\rm U}\hookrightarrow {\rm S}$. Then
\[ {\mathbf{H}}^{\operatorname{fr}}({\rm U}) \xrightarrow{j_\sharp} {\mathbf{H}}^{\operatorname{fr}}({\rm S}) \xrightarrow{i^*} {\mathbf{H}}^{\operatorname{fr}}({\rm Z}) \]
is a cofiber sequence of presentable 
$\infty$-categories, that is, the functor 
$i_*\colon {\mathbf {H}}^{\operatorname {fr}}({\rm Z})\to {\mathbf {H}}^{\operatorname {fr}}({\rm S})$ is fully faithful with image 
$(j^*)^{-1}(0)$.
Proof. Theorem 10 implies that 
$j^*({\rm A})\simeq 0$ if and only if 
${\rm A}\simeq i_*i^*({\rm A})$. It also implies that the unit map 
$i_*\to i_*i^*i_*$ is an equivalence, hence also the counit map 
$i_*i^*i_*\to i_*$ by the triangle identities. It remains to show that 
$i_*$ is conservative. This follows immediately from the fact that every smooth 
${\rm Z}$-scheme admits an open covering by pullbacks of smooth 
${\rm S}$-schemes [Reference GrothendieckGro67, Proposition 18.1.1].
Remark 12 Similarly, the localization theorem holds for motivic spaces with finite étale transfers or with finite syntomic transfers, because the corresponding Hilbert schemes of points in 
$\mathbf {A}^n$ are smooth.
The localization theorem implies as usual the closed base change property and the closed projection formula, which states that 
$i_*\colon {\mathbf {H}}^{\operatorname {fr}}({\rm Z}) \to {\mathbf {H}}^{\operatorname {fr}}({\rm S})$ is an 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$-module functor, as well as 
${\rm S}^1$-stable and 
${\mathbf {T}}$-stable versions.
In the 
${\mathbf {T}}$-stable case, using the work of Ayoub [Reference AyoubAyo08] and Cisinski and Déglise [Reference Cisinski and DégliseCD19], we obtain for every separated morphism of finite type 
$f\colon {\rm X}\to {\rm Y}$ an exceptional adjunction
\[ f_!: {\mathbf{SH}}^{\operatorname{fr}}({\rm X}) \rightleftarrows {\mathbf{SH}}^{\operatorname{fr}}({\rm Y}): f^! \]satisfying the usual properties. In particular, framed motivic spectra satisfy proper base change and the proper projection formula.
Note that the cofiber sequence of Corollary 11 is not part of a recollement in the sense of [Reference LurieLur17, Definition A.8.1], because 
$i^*$ is not left exact and the pair 
$(i^*,j^*)$ is not conservative. These properties are, however, automatic in a stable setting.
Corollary 13 Let 
$i\colon {\rm Z}\hookrightarrow {\rm S}$ be a closed immersion with open complement 
$j\colon {\rm U}\hookrightarrow {\rm S}$. Then the following pairs of fully faithful functors are recollements:
(i)
${\mathbf {SH}}^{{\rm S}^1,{\operatorname {fr}}}({\rm Z})\xrightarrow {i_*} {\mathbf {SH}}^{{\rm S}^1,{\operatorname {fr}}}({\rm S}) \xleftarrow {j_*} {\mathbf {SH}}^{{\rm S}^1,{\operatorname {fr}}}({\rm U})$;(ii)
${\mathbf {SH}}^{{\operatorname {fr}}}({\rm Z})\xrightarrow {i_*} {\mathbf {SH}}^{{\operatorname {fr}}}({\rm S}) \xleftarrow {j_*} {\mathbf {SH}}^{{\operatorname {fr}}}({\rm U})$.
Corollary 14 Let 
${\rm S}$ be a scheme locally of finite Krull dimension. Then the following pullback functors are conservative:
(i)
${\mathbf {SH}}^{{\rm S}^1,{\operatorname {fr}}}({\rm S}) \to \prod _{s\in {\rm S}} {\mathbf {SH}}^{{\rm S}^1,{\operatorname {fr}}}(s)$;(ii)
${\mathbf {SH}}^{\operatorname {fr}}({\rm S}) \to \prod _{s\in {\rm S}} {\mathbf {SH}}^{\operatorname {fr}}(s)$.
Proof. We can assume 
${\rm S}$ qcqs and we prove the claim by induction on the dimension of 
${\rm S}$. For 
$s\in {\rm S}$, let 
$\iota _s\colon \operatorname {Spec} {\mathcal {O}} _{{\rm S},s}\to {\rm S}$ be the canonical map. Since 
$\iota _s$ is pro-smooth, the pullback functor
\[ \iota_s^*\colon \mathrm{PSh}{}_\Sigma(\mathbf{Corr}^{\operatorname{fr}}(\operatorname{Sm}_{{\rm S}}))\to \mathrm{PSh}{}_\Sigma(\mathbf{Corr}^{\operatorname{fr}}(\operatorname{Sm}_{{\mathcal{O}} _{{\rm S},s}})) \]
preserves 
$\mathbf {A}^1$-invariant Nisnevich sheaves and commutes with the internal Hom from compact objects (in particular, with 
$\Omega$ and 
$\Omega _{\mathbf {T}}$). Hence, for a framed motivic 
${\rm S}^1$-spectrum or 
${\mathbf {T}}$-spectrum 
${\rm E}=({\rm E}_n)_{n\geqslant 0}$ over 
${\rm S}$ and a qcqs smooth 
${\rm S}$-scheme 
${\rm X}$, the Zariski stalk of 
${\rm E}_n^{\rm X}$ at 
$s$ may be computed as 
$\iota _s^*({\rm E})_n({\rm X}\times _{{\rm S}}\operatorname {Spec}{\mathcal {O}} _{{\rm S},s})$. By the hypercompleteness of the Zariski 
$\infty$-topos of 
${\rm S}$ [Reference Clausen and MathewCM19, § 3], equivalences between Zariski sheaves on 
${\rm S}$ are detected on stalks. Since the family of functors 
${\rm E}\mapsto {\rm E}_n^{\rm X}({\rm S})$ is conservative, so is the family 
$\iota _s^*$ for 
$s\in {\rm S}$. We can therefore assume 
${\rm S}$ local. Then the result follows from Corollary 13 and the induction hypothesis.
Remark 15 Corollary 14 is also true if 
${\rm S}$ is locally Noetherian of arbitrary dimension; see the proof of [Reference AyoubAyo14, Proposition 3.24].
3. The reconstruction theorem over a general base scheme
Next, we extend the reconstruction theorem [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Theorem 3.5.12] to more general base schemes.
Lemma 16 Let 
$f\colon {\rm T}\to {\rm S}$ be a morphism of schemes. Then the canonical transformation
\[ f^*\gamma_* \to \gamma_*f^*\colon {\mathbf{H}}^{\operatorname{fr}}({\rm S}) \to {\mathbf{H}}({\rm T}) \]
is an equivalence, and similarly for 
${\mathbf {SH}}^{{\rm S}^1}$ and 
${\mathbf {SH}}$.
Proof. The stable statements follow from the unstable one, using the fact that the functors 
$\gamma _*$ and 
$f^*$ can be computed levelwise on prespectra. Since 
$f^*$ and 
$\gamma _*$ preserve sifted colimits and commute with 
${\rm L}_{{\rm mot}}$ [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Propositions 3.2.14 and 3.2.15], it suffices to check that the canonical map
\[ f^*{\rm h} ^{\operatorname{fr}}_{{\rm S}}({\rm X}) \to {\rm h} ^{\operatorname{fr}}_{\rm T}({\rm X}\times_{{\rm S}}{\rm T}) \]
is a motivic equivalence for every 
${\rm X}\in \operatorname {Sm}_{{\rm S}}$ affine, where we regard 
${\rm h} ^{\operatorname {fr}}_{{\rm S}}({\rm X})$ as a presheaf on 
$\operatorname {Sm}_{{\rm S}}$. By [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Corollary 2.3.27], we can replace 
${\rm h} ^{\operatorname {fr}}$ by 
${\rm h} ^\mathrm {nfr}$. But the map
\[ f^*{\rm h} ^\mathrm{nfr}_{{\rm S}}({\rm X}) \to {\rm h} ^\mathrm{nfr}_{\rm T}({\rm X}\times_{{\rm S}}{\rm T}) \]
is an isomorphism because 
${\rm h} ^\mathrm {nfr}_{{\rm S}}({\rm X})$ is a smooth ind-
${\rm S}$-scheme that is stable under base change [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Theorem 5.1.5].
Lemma 17 Let 
$p\colon {\rm T}\to {\rm S}$ be a proper morphism of schemes. Then the canonical transformation
\[ \gamma^* p_* \to p_*\gamma^*\colon {\mathbf{SH}}({\rm T}) \to {\mathbf{SH}}^{\operatorname{fr}}({\rm S}) \]is an equivalence.
Proof. If 
$p$ is a closed immersion, this follows from Theorem 10 and its non-framed version. If 
$p$ is smooth and proper, it follows from the ambidexterity equivalences 
$p_*\simeq p_\sharp \Sigma ^{-\Omega _p}$. Together with Zariski descent, this implies the result for 
$p$ locally projective. The general case (which we will not use) follows by a standard use of Chow's lemma; see [Reference Cisinski and DégliseCD19, Proposition 2.3.11(2)] and [Reference HoyoisHoy14, Proposition C.13] for details.
Theorem 18 (Reconstruction theorem) Let 
${\rm S}$ be a scheme. Then the functor
\[ \gamma^*\colon {\mathbf{SH}}({\rm S})\to{\mathbf{SH}}^{\operatorname{fr}}({\rm S}) \]
is an equivalence of symmetric monoidal 
$\infty$-categories
Proof. Since the right adjoint 
$\gamma _*$ is conservative [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Proposition 3.5.2], it suffices to show that 
$\gamma ^*$ is fully faithful, that is, that the unit transformation 
$\mathrm {id}\to \gamma _*\gamma ^*$ is an equivalence. By Zariski descent, we may assume 
${\rm S}$ qcqs. In this case, the 
$\infty$-category 
${\mathbf {SH}}({\rm S})$ is generated under colimits by the objects 
$\Sigma _{\mathbf {T}} ^n p_*\mathbf {1}_{\rm X}$ for 
$n\in \mathbf {Z}$ and 
$p\colon {\rm X}\to {\rm S}$ a projective morphism [Reference AyoubAyo08, Lemme 2.2.23]. By Lemma 17, we are thus reduced to proving that 
$\mathbf {1}_{{\rm S}}\to \gamma _*\gamma ^*\mathbf {1}_{{\rm S}}$ is an equivalence. By Lemma 16, we can now assume that 
${\rm S}=\operatorname {Spec}\mathbf {Z}$. By the non-framed version of Corollary 14 and again Lemma 16, the result follows from the cases 
${\rm S}=\operatorname {Spec}{\mathbf {Q}}$ and 
${\rm S}=\operatorname {Spec}{\mathbf {F}}_p$ for 
$p$ prime, which are known by [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Theorem 3.5.12].
Remark 19 The argument used in the proof of Theorem 18 can be axiomatized as follows. Let 
${\rm S}$ be a qcqs scheme of finite Krull dimension, let
\[ {\mathbf{A}} ,{\mathbf{B}} \colon (\operatorname{Sch}_{{\rm S}}^\mathrm{qcqs})^{\mathrm{op}} \to \infty {\rm -}\mathrm{Cat}^\mathrm{st} \]
be functors satisfying Ayoub's axioms [Reference AyoubAyo08, § 1.4.1], and let 
$\varphi \colon {\mathbf {A}} \to {\mathbf {B}}$ be a natural transformation that commutes with 
$f_\sharp$ for 
$f$ smooth. Suppose that:
(i) each
${\mathbf {A}} ({\rm X})$ is cocomplete and generated under colimits by objects of the form $f_\sharp f^*p^* ({\rm A})$ where $f\colon {\rm Y} \to {\rm X}$ is smooth, $p\colon {\rm X}\to {\rm S}$ is the structure map, and ${\rm A}\in {\mathbf {A}} ({\rm S})$;(ii)
$\varphi$ has a right adjoint that preserves colimits and commutes with $f^*$ for any $f$;(iii)
$\varphi _s\colon {\mathbf {A}} (s)\to {\mathbf {B}} (s)$ is fully faithful for every $s\in {\rm S}$.
Then 
$\varphi _{\rm X}\colon {\mathbf {A}} ({\rm X})\to {\mathbf {B}} ({\rm X})$ is fully faithful for every 
${\rm X}\in \operatorname {Sch}_{{\rm S}}^\mathrm {qcqs}$.
Since 
${\mathbf {SH}}^{\operatorname {fr}}({\rm S})\simeq {\mathbf {SH}}({\rm S})\otimes _{{\mathbf {H}}({\rm S})}{\mathbf {H}}^{\operatorname {fr}}({\rm S})$, the reconstruction theorem implies that the right-lax symmetric monoidal functor 
$\Omega ^\infty _{\mathbf {T}} \colon {\mathbf {SH}}({\rm S})\to {\mathbf {H}}({\rm S})$ factors uniquely as
Indeed, the 
$\infty$-groupoid of such factorizations is equivalent to that of colimit-preserving symmetric monoidal retractions of the functor 
$\gamma ^*\colon {\mathbf {SH}}({\rm S})\to {\mathbf {SH}}^{\operatorname {fr}}({\rm S})$. In particular, the underlying cohomology theory 
$\operatorname {Sm}_{{\rm S}}^\mathrm {op} \to {{\rm Spc}}$ of a motivic spectrum extends canonically to the 
$\infty$-category 
$\mathbf {Corr}^{\operatorname {fr}}(\operatorname {Sm}_{{\rm S}})^\mathrm {op}$. As proved in [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+20, Theorem 3.3.10], this enhanced functoriality of cohomology theories can be described using the Gysin morphisms constructed using Verdier's deformation to the normal cone (see [Reference Déglise, Jin and KhanDJK18]).
4. Application to motivic cohomology
In this final section, we obtain a simple presentation of the motivic cohomology spectrum in terms of framed correspondences. Let us denote by 
${\rm H}\mathbf {Z}_{{\rm S}}\in {\mathbf {SH}}({\rm S})$ Spitzweck's motivic cohomology spectrum over a base scheme 
${\rm S}$ [Reference SpitzweckSpi18]. By construction, it is stable under arbitrary base change, and when 
${\rm S}$ is a Dedekind domain it represents Bloch–Levine motivic cohomology. More precisely, for such 
${\rm S}$, the presheaf 
${\rm X}\mapsto \mathrm {Maps}_{{\mathbf {SH}}({\rm S})}(\Sigma ^\infty _{{\mathbf {T}} }{\rm X}_+,\Sigma ^n_{\mathbf {T}} {\rm H}\mathbf {Z}_{{\rm S}})$ on smooth 
${\rm S}$-schemes is the Zariski sheafification of Bloch's cycle complex 
${\rm X}\mapsto z^n({\rm X},*)$ (which is known to already be a Zariski sheaf when 
${\rm S}$ is semilocal [Reference LevineLev01, Theorem 1.7]). When 
${\rm S}$ is the spectrum of a field, 
${\rm H}\mathbf {Z}_{{\rm S}}$ is equivalent to Voevodsky's motivic cohomology spectrum.
For any commutative monoid 
${\rm A}$, the constant sheaf 
${\rm A}_{{\rm S}}$ on 
$\operatorname {Sm}_{{\rm S}}$ admits a canonical extension to 
$\mathbf {Corr}^\mathrm {f{}lf}(\operatorname {Sm}_{{\rm S}})$, where ‘
$\mathrm {f{}lf}$’ denotes the class of finite locally free morphisms: to a span
with 
$f$ finite locally free and a locally constant function 
$a\colon {\rm Y}\to {\rm A}$, we associate the locally constant function
\[ X\to {\rm A},\quad x\mapsto \sum_{z\in f^{-1}(x)} \deg_z(f)\cdot a(g(z)) \]
(see [Reference Bachmann and HoyoisBH20, Lemma 13.13]). In particular, 
${\rm A}_{{\rm S}}$ can be regarded as an object of 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$ via the forgetful functor 
$\mathbf {Corr}^{\operatorname {fr}}(\operatorname {Sm}_{{\rm S}}) \to \mathbf {Corr}^\mathrm {f{}lf}(\operatorname {Sm}_{{\rm S}})$.
If 
$f\colon {\rm T}\to {\rm S}$ is a morphism, there is an obvious map 
${\rm A}_{{\rm S}} \to f_* {\rm A}_{\rm T}$ in 
${\mathbf {H}}^{\operatorname {fr}}({\rm S})$, whence by adjunction a map 
$f^* {\rm A}_{{\rm S}} \to {\rm A}_{\rm T}$ in 
${\mathbf {H}}^{\operatorname {fr}}({\rm T})$.
Lemma 20 Let 
${\rm A}$ be a commutative monoid and 
$f\colon {\rm T}\to {\rm S}$ a morphism of schemes. Then the canonical map 
$f^*{\rm A}_{{\rm S}} \to {\rm A}_{\rm T}$ in 
${\mathbf {H}}^{\operatorname {fr}}({\rm T})$ is an equivalence.
Proof. We consider the following commutative triangle in 
$\mathrm {PSh}{}_\Sigma (\operatorname {Sm}_{\rm T})$:
The vertical map is a motivic equivalence by Lemma 16, and the diagonal map is trivially a Zariski equivalence. Hence, the lower horizontal map is a motivic equivalence. Since 
$\gamma _*$ detects motivic equivalences (Lemma 7), we are done.
Theorem 21 Let 
${\rm S}$ be a scheme. Then there is an equivalence of motivic 
${{\mathcal {E}} _\infty }$-ring spectra 
${\rm H}\mathbf {Z}_{{\rm S}} \simeq \gamma _* \Sigma ^\infty _{{\mathbf {T}} ,{\operatorname {fr}}}\mathbf {Z}_{{\rm S}}$.
Proof. By Lemmas 16 and 20, it suffices to prove this when 
${\rm S}$ is a Dedekind domain. In this case, there is an isomorphism of presheaves of commutative rings
\[ \Omega^\infty_{\mathbf{T}} {\rm H}\mathbf{Z}_{{\rm S}} \simeq \mathbf{Z}_{{\rm S}}. \]
We claim that this isomorphism is compatible with the framed transfers on either side, the ones on the left coming from Theorem 18. Since we are dealing with discrete constant sheaves, it suffices to compare the transfers for a framed correspondence of the form 
$\eta \leftarrow {\rm T}\rightarrow \eta$ where 
$\eta$ is a generic point of a smooth 
${\rm S}$-scheme. Thus we may assume that 
${\rm S}$ is a field, in which case we can compute the framed transfers on 
$\Omega ^\infty _{\mathbf {T}} {\rm H}\mathbf {Z}_{{\rm S}}$ using [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Proposition 5.3.6], verifying the claim.
By adjunction, we obtain a morphism of 
${{\mathcal {E}} _\infty }$-algebras
\[ \varphi_{{\rm S}}\colon \Sigma^\infty_{{\mathbf{T}} ,{\operatorname{fr}}} \mathbf{Z}_{{\rm S}} \to \gamma^*{\rm H}\mathbf{Z}_{{\rm S}} \]
in 
${\mathbf {SH}}^{\operatorname {fr}}({\rm S})$. We show that 
$\varphi _{{\rm S}}$ is an equivalence. By construction, 
$\varphi _{{\rm S}}$ is functorial in 
${\rm S}$. By Corollary 14(2), we may therefore assume that 
${\rm S}$ is the spectrum of a perfect field. In this case, the recognition principle [Reference Elmanto, Hoyois, Khan, Sosnilo and YakersonEHK+19, Theorem 3.5.14(i)] implies that 
$\varphi _{{\rm S}}$ exhibits 
$\gamma _* \Sigma ^\infty _{{\mathbf {T}} ,{\operatorname {fr}}}\mathbf {Z}_{{\rm S}}$ as the very effective cover of 
${\rm H}\mathbf {Z}_{{\rm S}}$. Since 
${\rm H}\mathbf {Z}_{{\rm S}}$ is already very effective [Reference Bachmann and HoyoisBH20, Lemma 13.7], we conclude that 
$\varphi _{{\rm S}}$ is an equivalence.
If 
${\rm S}$ is a Dedekind domain, the motivic spectrum 
${\rm H}\mathbf {Z}_{{\rm S}}\in {\mathbf {SH}}({\rm S})$ lies in the heart of the effective homotopy 
$t$-structure [Reference Bachmann and HoyoisBH20, Lemma 13.7]. It follows that it admits a unique strictly commutative monoid structure in 
${\mathbf {SH}}({\rm S})$, in the sense of [Reference HoyoisHoy18, § 7]. Hence, for any scheme 
${\rm S}$, 
${\rm H}\mathbf {Z}_{{\rm S}}\in {\mathbf {SH}}({\rm S})$ is a module over the Eilenberg–Mac Lane spectrum 
$\mathbf {Z}\in \rm {Spt}$. In particular, for any 
${\rm A}\in {\mathbf {Mod}}_{\mathbf {Z}}(\rm {Spt})$, we can form the tensor product 
${\rm HA}_{{\rm S}}={\rm H}\mathbf {Z}_{{\rm S}}\otimes _{\mathbf {Z}} {\rm A}$. This construction defines a symmetric monoidal functor
\[ {\mathbf{Mod}}_{\mathbf{Z}}(\rm{Spt}) \to {\mathbf{Mod}}_{{\rm H}\mathbf{Z}_{{\rm S}}}({\mathbf{SH}}({\rm S})), \quad A\mapsto {\rm HA}_{{\rm S}}. \]
When 
${\rm S}$ is the spectrum of a field and 
${\rm A}$ is an abelian group, 
${\rm HA}_{{\rm S}}$ is equivalent to Voevodsky's motivic Eilenberg–Mac Lane spectrum with coefficients in 
${\rm A}$.
Corollary 22 Let 
${\rm S}$ be a scheme and 
${\rm A}$ an abelian group (respectively, a ring; a commutative ring). Then there is a canonical equivalence of 
${\rm H}\mathbf {Z}_{{\rm S}}$-modules (respectively, of 
${\mathcal {A}} _\infty$-
${\rm H}\mathbf {Z}_{{\rm S}}$-algebras; of 
${{\mathcal {E}} _\infty }$-
${\rm H}\mathbf {Z}_{{\rm S}}$-algebras) 
${\rm HA}_{{\rm S}} \simeq \gamma _* \Sigma ^\infty _{{\mathbf {T}} ,{\operatorname {fr}}}{\rm A}_{{\rm S}}$.
Proof. By Lemmas 16 and 20, we may assume that 
${\rm S}$ is a Dedekind domain. Since the equivalence of Theorem 21 takes place in the heart of the effective homotopy 
$t$-structure, it can be uniquely promoted to an equivalence of 
${{\mathcal {E}} _\infty }$-rings in strictly commutative monoids. Hence, for any 
${\rm A}\in {\mathbf {Mod}}_{\mathbf {Z}}(\rm {Spt}_{\geqslant 0})$, we obtain an equivalence 
${\rm HA}_{{\rm S}}\simeq \gamma _* \Sigma ^\infty _{{\mathbf {T}} ,{\operatorname {fr}}}(\mathbf {Z}_{{\rm S}}\otimes _{\mathbf {Z}}{\rm A})$. To conclude, note that 
$\mathbf {Z}_{{\rm S}}\otimes _{\mathbf {Z}}{\rm A}\simeq {\rm A}_{{\rm S}}$ when 
${\rm A}$ is discrete.
