text:
\section{Introduction: review, and overview. }
The motivic homotopy theory studies invariants of algebraic varieties or schemes with respect to the motivic homotopies, coverings, and gluings for various Grothendieck topologies, and motivic suspensions and loops.
The perfectness of the base field is one of the assumptions that sometimes appear in the study.
The article is focused on a question that was one of the basic reasons of the assumption in transfers-based techniques and constructions of motivic categories such as Voevodsky's motives from the origin of the study \cite{Voe-hty-inv}.
To achieve the result we develop constructions of the compactified geometrical data such as relative curves and framed correspondences
over one-dimensional compactified base schemes that have a wider area of applications as we consider.

\subsection{Perfectness for motivic categories.}
The categories $\mathbf{DM}(S)$ \cite{Voe-motives,Voe-cancel,MVW,Cisinski-Deglise-Triangmixedmotives}, and $\mathbf{SH}(S)$ \cite{Morel-Voevodsky,Jardine-spt,morel-trieste,mot-functors} are well defined
and satisfy six functors formalism
at least over noetherian separated base schemes of finite Krull dimension
provided by \cite{Cisinski-Deglise-Triangmixedmotives}
by F.~D\'{e}glise $\&$ D-C.~Cisinski, and \cite{AyoubI} by J.~Ayoub
respectively.
The perfectness assumption usually appears in such situations as:
\begin{itemize}
\item[(p1)]
universal computational results on homotopies and homologies of wide classes of smooth schemes,
\item[(p2)]
such structural results as descriptions of the motivic localisation functors,  various filtrations, t-structures, and wight structures, and their harts, and constructions of the duality.
\end{itemize}
The major cases that do not assume perfectness:
\begin{itemize}
\item[(g1)]
the questions on the cohomology theories defined by the inverse image from a perfect field, in particular, all theories defined over $\mathbb Z$.
\item[(g2)]
the case of $\mathbb Z[1/p]$-coefficients, where $\operatorname{char} k=p$;
\item[(g3)]
the case of \'etale $\mathbb A^1$-motivic categories;
note that $\mathbf{SH}_\mathrm{\acute et}(k)[1/p]\simeq \mathbf{SH}_\mathrm{\acute et}(k)$.
\end{itemize}
One can see that (g1-g3) cover many applications of the $\mathbb A^1$-motivic homotopy theory, so the assumption could be considered as almost non-restrictive.
Nevertheless, the perfectness or the inverting of the characteristic assumptions are deep enough to play some roles till nowadays for all the listed below triangulated motivic categories,
starting from constructions oriented on Chow-motives and Weil-cohomologies:
\begin{itemize}
\item[1)]
Hanamura motives $\mathcal{D}(k)$ \cite{Homcohmotives} defined with $\mathbb Q$-coefficients;
\item[2)]
Nori motives $\mathrm{MM}_\mathrm{Nori}(k,\Lambda)$ \cite{Norimotives,nori1mot} defined for a field $k$ embeddable in $\mathbb C$;
\item[3)]
Levine motives $\mathbf{DM}(k,\Lambda)$ \cite{Levine2005};
\item[4)]
Voevodsky motives $\mathbf{DM}(k)$ \cite{Voe-motives};
\end{itemize}
and including the following list of categories oriented on the study of other classes of cohomology theories:
\begin{itemize}
\item[5)]
the Morel-Voevodsky stable motivic homotopy categories, and categories of modules over ring motivic spectra;
\item[6)]
various transfers based constructions and reconstructions in between $\mathbf{DM}(S)$ and $\mathbf{SH}(S)$;
the most universal at the present moment are
framed motivic categories \cite{Framed,five-authors,BigFrmotives,FramedGamma,five-authors2}
that allow to reconstruct the stable Morel-Voevodsky category;
\item[7)]
$\square$-homotopy categories \cite{MotiveModulePairs}, \cite{logDMk}
oriented on the study of non-$\mathbb A^1$-invariant cohomology theories;
\item[8)]
Cousin acyclic motivic categories \cite{tnpsCompactificationsacCousin}
oriented on the study of cohomology theories that satisfy the Gersten conjecture.
\end{itemize}
There could be even asked a question:
\begin{itemize}[leftmargin=20pt]
\item[]
\emph{Do the motivic categories detect such arithmetic property of the base field as perfectness;
is the effect different or similar for the listed above constructions?}
\end{itemize}
The present work shows the independence from the perfectness assumption of some structural results of point (6).
The result helps to extend the generality of the result in \cite{DKO:SHISpecZ,LGmLnisLA1tf} oriented on the study of
$\mathbf{DM}(S)$, and $\mathbf{SH}(S)$ over base schemes.
Let us note that the class of schemes $S$ with perfect residue fields do not contain the positive dimensional finite type schemes over positive-characteristic fields, and the inverting of the characteristic over a base scheme like in \cite{DegliseBondarko,ElmantoKhannonperfect} would mean the characteristic at each point of the base.

The result helps for the study of point (8) above that is $\mathbf{SH}_\mathrm{Cousin}(S)$ in \cite{tnpsCompactificationsacCousin}.

To prove the strict $\mathbb A^1$-homotopy invariance over an arbitrary field we make all the constructions of $\mathbb A^1$-homotopies being equipped with the compactifications and use essentially the compactified morphisms over compactified one-dimensional base scheme, namely the base scheme pair $(\mathbb A^1_k,\mathbb P^1_k)$ given by $\mathbb A^1_k$ with the compactification $\mathbb P^1_k$.
This shows that our work applies
to $\square$-motivic studies, point (7) above, and
allows us announce a progress.
We consider that the constructions of the present work implicitly provide the isomorphism $H^n_\nu(X,F)\to H^n_\tau(\square\times X,F)$,
where $\tau$ is the Nisnevich topology for \cite{MotiveModulePairs} and the strict Nisnevich one for \cite{logDMk},
and (1) $\nu=\tau$, $\partial X=\emptyset$, (2) $\nu$ denotes the cohomologies with finite support over $X$, $\partial X$ is smooth.
We left all the $\square$-homotopy questions here for the another work.
We could say that the argument covers in a sense the case of framed correspondences, but this notion is not defined in this setting yet.
The text is written in the setting of classical schemes and the compactifications appear on the technical level of constructions with the use of some temporal language of compactified framed correspondences.

\subsection{Perfectness in the motivic homotopy theory techniques and approaches}

Let us discuss the perfectness assumption in the following two basic techniques for the study of motivic cohomologies and their generalised versions appeared in the literature approximately the same time in the second half of 90s
and used and developed in parallel during the last two decades:
\begin{itemize}
\item[(t1)]
transfers technique developed by Voevodsky \cite{Voe-hty-inv,Voe-motives}, according to this approach
\[\text{motivic cohomoligies} = \text{cohomologies of singular motivic complexes}\]
with the use of $\mathrm{Cor}(k)$ correspondences originally,
generalised
later for other types of transfers,
and by the framed correspondences technique \cite{Voe-notes,Framed}
for the study of
$\mathrm{D}_{\mathbb A^1}(k)$, and $\mathbf{SH}(k)$;
\item[(t2)]
the technique of cycle modules initiated by M.~Rost \cite{Rost1996},
\[\text{motivic cohomoligies} = \begin{matrix}\text{cohomoligies of Gerten type complexes}\\\text{such as the complex}\\\text{associated with the the cycle module},\end{matrix}\]
or the technique of the unramified sheaves and
the Rost-Schmid complex \cite{MorelA1topologyfield} developed later
by F.~Morel, and Milnor-Witt cycle modules
defined by N.~Feld \cite{MWcyclemodules}.
\end{itemize}
The geometric property for perfect base fields used in both cases is the essential smoothness of points of reduced finite type schemes.

Voevodsky's article \cite{Voe-motives} uses perfectness for the construction of motives of smooth schemes, namely, for the universal property of the motivic complexes
\[X\mapsto M(X)\simeq C^* \mathbb Z^\mathrm{tr}(X)=\mathrm{Cor}(-\times\Delta_k^\bullet,X).\]
Namely, the perfectness is used for the strict homotopy invariance \cite[Theorem 5.6]{Voe-hty-inv}. The latter theorem allows to combine the Suslin-Voevodsky singular complex $\mathrm{Sing}_{\mathbb A^1}(-)=C^*(-)$ and the Nisnevich localisation.
Note that the resolution of singularities required also in \cite{Voe-motives} for the cancellation theorem was eliminated by Voevodsky in \cite{Voe-cancel}.

To eliminate the perfectness assumption
A.~Suslin showed in \cite{Sus-nonperftalk,Sus17nonperfect} that the strict homotopy invariance holds over any field $k$ with $\mathbb Z[1/p]$-coefficients, $p=\operatorname{char} k$.
Suslin deduced the result from the perfect base field case by proving the quasi-equivalence of motivic complexes with $\mathbb Z[1/p]$-coefficients for a purely inseparable extension $k^\prime/k$.
The article \cite{Sus17nonperfect} documents the result presented in the talk in 2008 in Russian language \cite{Sus-nonperftalk}.
To explain the proof we say that Suslin made a trace argument for purely inseparable finite extensions $K/k$ and showd that the composition of the inverse and direct images is the multiplication by $p^l$; as he said in the talk \cite{Sus-nonperftalk} "what is needed is an accurate analyse of Frobenius endomorphism action". Answering the question "Is it true, the result with integral coefficients?", Suslin \cite[32:15 min]{Sus-nonperftalk}
said "Yes, but the proof is unknown", and "Nevertheless, $\mathbb Z[1/p]$-case is enough for the most of known applications".

\par
The perfectness assumption or the inverting of the characteristic came later to other transfers-based studies.
The most universal is the theory of framed motives \cite{Voe-notes,Framed,BigFrmotives,FramedGamma}
based on \cite{framed-cancel,hty-inv,ConeTheGNP,surj-etale-exc,DrKyllfinFrpi00},
and the alternative $\infty$-categorical version \cite{five-authors,ElmantoKhannonperfect} based on the most of ingredients of the first one.
The strict homotopy invariance theorem is provided by \cite[Thereom 1.1]{hty-inv} by Garkusha${\&}$Panin with complements for the cases of finite base fields, and characteristic 2 given by \cite{surj-etale-exc,DrKyllfinFrpi00,five-authors}.

M.~Rost's cycle modules theory \cite{Rost1996} appeared in parallel to Voevodsky's works \cite{Voe-HomolSchcovariantmotives,Voe-HomolSch,Voe-motives}
and gave another approach for the study of motivic cohomologies.
As shown by F.~Deglese \cite{CycleModtstructurehart} the category of cycle modules is the heart of the homotopy t-structure on $\mathbf{DM}(k)$ for perfect $k$.
As mentioned in \cite{TransfersCycleModules} by Deglise that \cite{Rost1996} is more general study then \cite{Voe-motives}
in the sense that the perfectness is not assumed, in particular, for the $\mathbb A^1$-invariance of the cohomology theory associated with a given cycle module, and for the construction of transfers provided by \cite{TransfersCycleModules}.
Nevertheless, the perfectness assumption is needed in \cite{Rost1996} for the criterion, when a cycle premodule is a cycle module;
if we would want to apply the theory to the motivic cohomologies of a smooth $k$-scheme $X$,
there would be a question on a construction \[ X \mapsto \text{the associated cycle module}.\]
\par
The perfectness is assumed for the generalisation of the original cycle modules by Rost named Milnor-Witt cycle modules
defined by Feld \cite{MWcyclemodules}, that reconstructs the heart of $\mathbf{SH}(k)$.
In between the works by Rost and Feld the elements of similar techniques appeared, for example, in Barge-Morel's work \cite{BM_PullBack}, Morel's work \cite{Morel-sphere-spt}, M.~Schmid's thesis \cite{WittHomol}, J.~Fasel's study of Chow-Witt groups \cite{ChWitt,ChWittRing}, and Morel's book \cite{MorelA1topologyfield}.
As mentioned in the introduction
the only reason for the perfectness assumption in \cite{MorelA1topologyfield}
is the proof of strictly $\mathbb A^1$-invariance of strongly $\mathbb A^1$-invariant sheaves, because of the study of Rost-Schmid complexes in its turn.
As mentioned by Morel in \cite{MorelA1topologyfield}
the first version of the text did
not require the perfectness assumption,
but M.~Levine found a problem in the section on Rost-Schmid complexes;
nevertheless, it is mentioned that the argument possibly can be done for any field, but is too complicated technically.

\subsection{Reason for the assumption}\label{subsect:s1s2s3}
For both approaches (t1) and (t2)
it plays
a role the interaction or interactions of such structures/\-properties/\-constructions
as
\begin{itemize}
\item[(s1)] pushforwards, or transfers in particular;
\item[(s2)] $\mathbb A^1$-homotopies, and strict $\mathbb A^1$-invariance;
\item[(s3)] exactness of the Cousin complex or other Gersten type complexes,
and, in particular, the injectivity and the purity properties, for a presheaf or a cohomology theory;
\end{itemize}
We mean that the approaches start with some partial initial data, structures, and properties that exist and hold for cohomology theories,
namely:
\begin{itemize}
\item[(t1)] the structure of presheaves on $Sm_k$ that have transfers and $\mathbb A^1$-invariance property,
and
\item[(t2)] the values of the cohomology theory on the fields, that are residue fields of smooth schemes, equipped with differentials, traces for finite extensions, multiplication by some coefficients such as the Milnor K-theory, that satisfy some relations.
\end{itemize}
Starting with these data both approaches consequently piece-by-piece/step-by-step/case-by-case construct other data and structures and prove the properties of constructed cohomology theories such as $\mathbb A^1$-invariance, Gysin isomorphisms, pushforwards, and the exactness of the Cousin and Gersten complexes.
At some point, it was needed to use the perfectness.

Starting with the transfers structure and the $\mathbb A^1$-invariance property, Voevodsky's study \cite{Voe-hty-inv} deduces firstly some Zariski and Nisnevich excision isomorphisms, and construct some Gysin maps.
The perfectness is used for the Gysin isomorphisms
that allow identifying the Gersten and the Cousin complexes.
Let us note that on the one hand, the first use is contained in the proof of the strict homotopy invariance \cite[Theorem 5.6]{Voe-hty-inv},
that is formally applied later for the Gersten conjecture \cite[Theorem 4.37]{Voe-hty-inv},
and at the same time,
the Gersten complex or the coniveau spectral sequence
are implicitly used in the argument for \cite[Theorem 5.6]{Voe-hty-inv},
and this is the first appearance of the perfectness assumption.
\par
To describe (t2) we discuss Morel's technique used in \cite{MorelA1topologyfield} to get unstable results.
Morel's strategy in \cite{MorelA1topologyfield} starts with the notion of unramified sheaves,
and the perfectness is needed again for the proof of the strict $\mathbb A^1$-invariance of the associated sheaf.
Unramified sheaf can be considered morally as the data given by the 'beginning terms' of 'an $\mathrm{SL}$-oriented cycle module'.
For a given unramified sheaf it is proven firstly the strongly $\mathbb A^1$-invariance of the associated sheaf, and the trace homomorphism along finite extensions of fields are constructed.
Then the data is summarised to construct a certain Gersten type complex, namely the Rost-Schmid complex, that definition, the pushforwards construction, and the proof of $\mathbb A^1$-invariance of cohomologies assume perfectness.

\subsection{Our proof and viewpoint}
On one side, the obstruction had a little effort in the often studied $\mathbb A^1$-motivic invariants. On the other side, it required enough powerful 'lens' to look and contract it in general.

The obstruction in the arguments of \cite{Voe-hty-inv} and \cite{hty-inv} was hidden at infinity of the relative affine line.
The strict $\mathbb A^1$-invariance isomorphisms $H^n_\mathrm{nis}(\mathbb A^1\times X, F_\mathrm{nis})\simeq H^n_\mathrm{nis}(X,F_\mathrm{nis})$, $X\in \mathrm{Sm}_k$, were proved firstly for $X=\operatorname{Spec}K$,
and then by certain induction on $\operatorname{dim}X$ and $n\in \mathbb Z$ based on a hidden comparing of the Gersten complexes or the coniveau spectral sequences for $F_\mathrm{nis}$ and $\mathrm{pr}_*F_\mathrm{nis}$ on the small \'etale site over $X$, where $\mathrm{pr}\colon \mathbb A^1\times X\to X$.
This reasoning does not distinguish smooth and non-smooth compactifications of $\mathbb A^1\times X$;
the obstruction 'runs' along $\mathbb A^1_X$ to the infinity that is the center of the projection $\mathbb A^1\times X\to X$, and is a non-smooth point of the minimal compactifiaction.

To prove that cohomologies on $\mathbb A^1_U$ for a local henselian essentially smooth $U$ with a closed point $z$ are trivial we prove the triviality of cohomologies 'out of infinity' and 'at the infinity' on $\mathbb A^1_U$.
Our notion of cohomologies 'out of infinity' is not isomorphic but is morally close to 'the cohomologies with finite support' over $U$.
The cohomologies 'out of infinity' on $\mathbb A^1_U$ are defined at the beginning of \Cref{sect:outofinffinitesupSHI} as the inductive limit of cohomologies with respect to the Nisnevich topology over $\mathbb A^1_U/(\mathbb A^1_U-Z)$, where $Z$ is a closed subscheme finite over $U$, basing on the notion of the Nisnevich topology over a pair $X/(X-Z)$ defined in \Cref{sect:(un)NisXZ}.
The general notion of homotopies and cohomologies at infinity is defined recently in \cite{atinfinityDOD}.
For the case of $\mathbb A^1_U$ we mean just cohomologies on the complement $V=(\mathbb P^1_ U)_{(\infty,z)}-\infty\times U$ to the infinite divisor of the neighbourhood of the infinite point $(\infty,z)$ in $\mathbb P^1_U$.

To prove the claim on cohomologies 'out of infinity' we elaborate the techniques and constructions used in the original argument and known generalisations, see \Cref{remark:ExLpArg}.
Namely, we prove Zariski excision isomorphisms on $\mathbb A^1_U$ for closed subschemes finite over $U$, and combine them with the \'etale excision proved in \cite{surj-etale-exc,hty-inv} for any base field.
This is done in \Cref{th:FinSuppLocHensbaseafflinetriviality}, \Cref{sect:outofinffinitesupSHI}.
For this part of the reasoning, it is essentially used that $U$ is local henselian.

To prove the triviality of cohomologies at infinity we consider the affine line $\mathbb A^1_k$ equipped with the compactification $\mathbb P^1_k$ as the base scheme pair, and use essentially that $U$ is essentially smooth local scheme over $k$.
This is done in \Cref{th:GenFifLocEssSmSchemeonedimbasetriviality}.
We prove the claim by constructing in \Cref{cor:contractinggenpointhomoveretaSmX} a certain 'contracting' framed homotopy in $V$ over $\mathbb P^1_k$ that satisfy properties like in the injectivity theorems \cite{hty-inv} for the case of local schemes and certain additional property.
The homotopy $c$ should respect the support $Z$ of the cohomology class $e$ on $V$
that it contracts. Namely, the infinity section of the homotopy
$c$ does not intersect the closure of $Z$ in the compactification of $V$ over $\mathbb P^1_k-\infty$.
Here is a key point of the argument that uses the 'lens' of the compactifications.
Due to this, the inverse image of $e$ with respect to $c$ is contained in the cohomologies out of the infinity on the corresponding relative lines over the domain of the homotopy, which allows applying \Cref{th:FinSuppLocHensbaseafflinetriviality} already proven in \Cref{sect:outofinffinitesupSHI}.
Since the points of the compactified homotopy $c$ that we have to control are $0$, $1$, and $\infty$ the argument is written firstly for $k$ that has at least 3 elements, and then the case of $\mathbb F_2$ is deduced by the use of framed correspondences form $\mathrm{Spec}\; k$ to the spectra of co-prime extensions like in \cite{DrKyllfinFrpi00,five-authors}.

\Cref{lm:XetafDimFrS,lm:DimDim1,lm:CurveFrCor}.
To construct the required compactified homotopy on $V$
we control its behaviour using the closed point $(\infty,z)$ of the compactification $\mathbb P^1_k\times U$ contained in the closure of $V$.
We construct in \Cref{lm:XetafDimFrS,lm:DimDim1,lm:CurveFrCor}
the framed correspondence that is equipped with the compactification defined on the pair $(V,(\mathbb P^1_k\times U)_{(\infty,z)}))$ where the second scheme is used 'a compacitifaction' of $V$.
We call this type of compactification of the relative framed correspondences over $(\mathbb A^1,\mathbb P^1)$ in \Cref{subsect:dcopmactifiedframedcorrespondece} as 'focusing' and distinguish it terminologically with the word compactification used when the infinity divisor is of positive relative codimension.

The formal way how \Cref{th:FinSuppLocHensbaseafflinetriviality} is used for the triviality of the cohomology classes on $(\mathbb P^1\times U)_{(\infty,z)}-\infty\times U$ in \Cref{th:strhominv} is the injectivity theorem in \Cref{sect:InjThbyFinSupHom} that formulation respects the above notification on supports of cohomology classes. For this purpose following the same principles as in the original proof we prove that cohomologies out of infinty have framed transfers for any framed abelian presheaf.

It is natural in view of the above discussion that our argument improves Voevodsky's result on the Cousin and Gersten complexes, and covers the case of main open subsets with essentially smooth complement. This is exactly the type of the scheme that is a neighbourhood of a smooth point of infinity divisor of smooth compactification. At the same time it explains why the argument applies to the corresponding case of the question in the $\square$-motivic study

\subsection{Overview}
The proof of the strict homotopy invariance occupies \Cref{sect:(un)NisXZ,sect:outofinffinitesupSHI,sect:homotpiesatinfinity,sect:StrHomInvTheorem}.
The argument for strict homotopy invariance, \Cref{th:strhominv} in \Cref{sect:StrHomInvTheorem}, summarises results on the triviality of cohomoligies out the infinity and at the infinity from \Cref{sect:outofinffinitesupSHI,sect:homotpiesatinfinity}.
\Cref{sect:CousinExactness} deduces the result on the Cousin complexes.

\Cref{sect:Fr} recollects the definitions of framed correspondences, and presheaves; \Cref{sect-app:FrHomotopies} recollects some used framed correspondences and homotopies from other articles.
\Cref{sect-app:RelDim} contains the formulation of the result on the relative dimensions over one-dimensional schemes used in \Cref{sect:homotpiesatinfinity}.
\Cref{sect-app:SerreTh} contains  consequences of Serre's theorem on ample bundles used in \Cref{sect:outofinffinitesupSHI,sect:homotpiesatinfinity}.

\Cref{sect:HSHfrCorr}
discusses the Garkusha-Panin's theory of framed motives and motivic categories for other types of correspondences such as $\mathrm{Corr}^\mathrm{fr}(k)$, $\widetilde{\mathrm{Cor}}(k)$, $\mathrm{Cor}(k)$, in particular.
The formulations of cited results are extended with stuff on the geometric motives categories $\mathbf{H}_\mathrm{geom}(\mathrm{Corr})$ in \Cref{subsect:SHGeomCor}.

Let us say that the argument does not use the strict homotopy invariance over a perfect field, but uses the injectivity theorem and the \'etale excision isomorphism from \cite{hty-inv,surj-etale-exc} that written proofs hold for an arbitrary field.

\section{Introduction: results. }
\subsection{The strict homotopy invariance theorem}
\label{subsect:StrHomInv}
\begin{theorem*}[\Cref{cor:strhominv}, \Cref{cor:HniscongHzar}, \Cref{cor:ZarStrictHomotopyInvariance}]
Let $F$ be an $\mathbb A^1$-invariant quasi-stable framed additive presheaf of abelian groups over a field $k$, and $F_\mathrm{nis}$ denote the associated Nisnevich sheaf, and similarly $F_\mathrm{zar}$ for Zariski topology.
Then for any $X\in \mathrm{Sm}_k$, $n\in \mathbb Z$, the canonical projection induces the isomorphism of Nisnevich cohomologies
\begin{equation}\label{eq:strA1hominvNis}H^n_\mathrm{nis}(\mathbb A^1_k\times X,F_\mathrm{nis})\cong H^n_\mathrm{nis}(X,F_\mathrm{nis}),\end{equation}
and similarly for Zariski cohomologies
\begin{equation}\label{eq:HzarA1timesXFzarcongHzarXFzar}H^n_\mathrm{zar}(\mathbb A^1_k\times X,F_\mathrm{zar})\cong H^n_\mathrm{zar}(X,F_\mathrm{zar}),\end{equation}
and even more, there is the canonical isomorphism
\begin{equation}\label{eq:HniscongHzar}H^n_\mathrm{zar}(X,F_\mathrm{zar})\cong H^n_\mathrm{nis}(X,F_\mathrm{nis}).\end{equation}
In particular, the claim holds for presheaves with classical Voevodsky's transfers with respect to the finite correspondences $\mathrm{Cor}(k)$.
\end{theorem*}

\begin{remark}[Known earlier cases and novelty]
The information on the question known before:
(1) the case of $n=0$ by \cite{Voe-hty-inv,hty-inv};
(2) the case of presheaves $F$ that are inverse images of presheaves over perfect fields;
(3) the case of presheaves of $\mathbb Z[1/p]$-modules over a field for characteristic $p$ for $\mathrm{Cor}$-correspondences \cite{Sus17nonperfect};
(4) it was known by \cite{Sus-nonperftalk,five-authors} and \cite{SHzar} basing on \cite{hty-inv,surj-etale-exc} the equivalence of the natural isomorphisms
\eqref{eq:strA1hominvNis} and \eqref{eq:HzarA1timesXFzarcongHzarXFzar}, and the implication from \eqref{eq:strA1hominvNis} and \eqref{eq:HzarA1timesXFzarcongHzarXFzar} to \eqref{eq:HniscongHzar}, though the isomorphism \eqref{eq:HniscongHzar} itself was not.

The general case for both topologies was an entire mystery;
the present proof of \eqref{eq:strA1hominvNis} does not reduce the result to the above cases and does not use the above reductions.
\end{remark}

\begin{theorem*}[\Cref{th:CousinsubsetsLocEssSm}]
Let
$U$ be an essentially smooth local scheme,
and $f\in \mathcal O(U)$ be such that the vanishing locus $Z(f)$ is a smooth proper subscheme of $U$, possibly empty,
$\eta\in V$ be the generic point.
Denote $V=U-Z(f)$.

Then for a quasi-stable additive framed presheaf of abelian groups $F$ over $k$, and $E^l_Z(X)=H^l_{Z}(X,F_\mathrm{nis})$, where $F_\mathrm{nis}$ is the Nisnevich sheafification,
the Cousin complex
\[0\to E^l(V)\to E^l(\eta)\to \bigoplus_{x\in V^{(1)}}E_x^{l+1}(V)\to \dots \bigoplus_{x\in V^{(r)}}E_x^{l+r}(V)\to \dots \bigoplus_{x\in V^{(d)}}E_x^{l+d}(V)\to 0\]
is exact, $d=\operatorname{dim}V$. \par More generally, the claim holds
for a stable additive framed presheaf of $S^1$-spectra $F$ over $k$, and $E^l_Z(X)=\pi_{-l}\mathrm{fib}(F_\mathrm{nis}(X)\to F_\mathrm{nis}(X-Z))$, $l\in \mathbb Z$, where $F_\mathrm{nis}$ is the Nisnevich local replacement of $F$ on $\mathrm{Sm}_k$,
and in particular, for $\mathbf{SH}(k)$-representable cohomology theories $E^*_{-}(-)$.
\end{theorem*}
\begin{remark}
If the base field is perfect the above complex is isomorphic to the Gersten complex, as shown in \cite[Theorem 4.37]{Voe-hty-inv}.
\end{remark}

\begin{remark}[On the originality of the second theorem] The exactness of the Gersten complex on a local essentially smooth scheme over a perfect field for the case of $\mathrm{Cor}$-transfers is proven in \cite[Theorem 4.37]{Voe-hty-inv}, and the case of framed transfers follows from \cite{hty-inv} by the same argument.
\end{remark}

\subsection{Applications to motives}\label{subsect:intro:motives}

The results allow us to extend the generality of the theory of framed motives, and other transfers based studies,
see \Cref{sect:HSHfrCorr}. The basic application of these approaches for the study of categories $\mathbf{DM}(k)$ and $\mathbf{SH}(k)$ relates to (1) the computing of motives of smooth schemes or pairs (2) the homotopy t-structure, (3) the slice filtration, and the reconstructions of $\mathbf{DM}^\mathrm{eff}(k)$, $\mathbf{SH}^\mathrm{veff}(k)$, $\mathbf{SH}^\mathrm{eff}(k)$, $\mathbf{SH}(k)$. Moreover, the result allows to eliminate the assumption on the residue fields from \cite{DKO:SHISpecZ,LGmLnisLA1tf}.
We discuss the base field case, we cite some of the results and simultaneously complete some of the formulations.

Here are the central results of the theory of framed motives \cite{Framed,BigFrmotives} by G.~Garkusha and I.~Panin,
and the result of their joint work with P.A.~{\O}stv{\ae}r \cite{FramedGamma}.
\begin{namedtheorem}{Theorem F}\label{namedth:smotFib(Y)}
For any $Y\in \mathrm{Sm}_k$, (1) there are natural $\mathbb P^1$-stable and $S^1$-stable motivic equivalences
\begin{equation}\label{eq:replacementP1smotOmegaGmspectrumS1}
\Sigma^\infty_{\mathbb P^{\wedge 1}}Y_+\simeq \mathrm{Fr}(-\times \Delta_k^\bullet,\Sigma^\infty_T Y_+)_f,\quad
\Omega^\infty_{\mathbb G_m^{\wedge 1}}\Sigma^\infty_{\mathbb P^{\wedge 1}} Y_+ \simeq
\mathrm{Fr}(-\times \Delta_k^\bullet,\Sigma^\infty_{S^1} Y_+)_f
\end{equation}
and the right sides in positive degrees are motivically fibrant with Morel-Voevodsky model structure $\Omega_{\mathbb P^1}$-spectrum or $\Omega_{S^1}$-spectrum of motivic spaces;
(2)
there are equivalences of categories
\begin{equation}\label{eq:LequivalencesShfrHfrgpNislocalcat}\mathbf{SH}(k)\simeq \mathcal{SH}^\mathrm{fr}(k),\quad \mathbf{SH}^\mathrm{veff}(k)\simeq \mathcal H^\mathrm{fr}_\mathrm{gp}(k)\end{equation}
where the right sides are the categories of the big framed motives \cite{BigFrmotives} by Garkusha ${\&}$ Panin, and the group-like motivic $\Gamma$-spaces \cite{FramedGamma} by Garkusha ${\&}$ Panin ${\&}$ {\O}stv{\ae}r.
(3)
In particular, the stalks on an essentially smooth local henselian scheme $Y$ of
the motivic homologies and homotopies of $Y$ in $\mathrm{D}_{\mathbb A^1}(k)$ and $\mathbf{SH}(k)$
equal the homologies and homotopies of the following complex and simplicial space
\begin{equation*}\begin{aligned}
H_{i+j,j}^{\mathrm{D}_{\mathbb A^1}(k)}(U,Y)&=&H_i({\mathbb Z\mathrm{F}}(U\times\Delta^\bullet_k\times\mathbb G_m^{\wedge j^+},Y\times \mathbb G_m^{\wedge j^-})),\\ \pi_{i+j,j}^{\mathbf{SH}(k)}(Y)(U)&\simeq& \pi_i\mathrm{Fr}(U\times\Delta_k^\bullet\times\mathbb G_m^{\wedge j^+},Y\times \mathbb G_m^{\wedge j^-})^\mathrm{gp},
\end{aligned}\end{equation*} where $j = j^{+}-j^{-} $
for a local henselian essentially smooth $U$.
\end{namedtheorem}

Next let us discuss the results in the abstract setting of a correspondences category $\mathrm{Corr}$ that satisfy (Fr), (NL), (NC)  in sense of \Cref{def:CorrinftyCategory,def:FrNLNC} suggested in \cite{SHzar}.
The main example is
the additivisation of the pre-additive $\infty$-category of tangentially framed correspondences $\mathrm{Corr}^\mathrm{fr}$ introduced by
E.~Elmanto, M.~Hoyois, A.~Khan, V.~Sosnilo, M.~Yakerson
 in \cite{five-authors}.
An example of classical categories is
the class of categories of a cohomological correspondences $\mathrm{Cor}^A$ defined in \cite{ccorrs},
in particular, Voevodsky's correspondences $\mathrm{Cor}$, and
Milnor-Witt correspondences $\widetilde{Cor}$ \cite{MW-cplx,MW-ring-spt,MW-cancel}.
See \Cref{ex:CorA,ex:Cor,ex:widetildeCor} for details.

\begin{definition}
Define $\mathbf{H}_\mathrm{geom}(\mathrm{Corr})$ as
the localisation of the category of simplicial objects
\[\Delta \mathrm{Corr}[\check C_{X}(\widetilde X)\to X, \mathbb A^1\times X\to X]\]
with respect to morphisms of the written form
for all $X\in \mathrm{Sm}_k$, and Zariski coverings $\widetilde X\to X$, where $\check C_{X}(\widetilde X)$ is the simplicial object defined by the \v{C}ech construction.
\end{definition}
\begin{theorem*}
There is the commutative diagram of fully faithful functors and equivalences
\begin{equation}\label{eq:transfersreconstruction}\xymatrix{
&\mathbf{SH}^\mathrm{veff}(\mathrm{Corr})\ar[r] & \mathbf{SH}^\mathrm{eff}(\mathrm{Corr})\ar[rd]&
\\
\mathbf{H}_\mathrm{geom}(\mathrm{Corr})\ar[ru]\ar[r] &\mathbf{H}(\mathrm{Corr})\ar[u]^{\simeq}\ar[r]& \mathbf{SH}^{S^1}(\mathrm{Corr})\ar[u]^{\simeq}\ar[r]& \mathbf{SH}(\mathrm{Corr})
}.\end{equation}
For any $Y\in \mathrm{Sm}_k$, \begin{equation}\label{eq:loopspace}\Omega^\infty_{\mathbb P^1}\Sigma^\infty_{\mathbb P^1} Y = L_\mathrm{nis} L_{\mathbb A^1} h^{\mathrm{tr},\mathrm{gp}}(Y).\end{equation}
There are equivalences for the heart and the effective heart of the homotopy t-structure
\[\mathbf{SH}^{\mathrm{eff},\heartsuit}(\mathrm{Corr})\simeq \mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr}),\quad \mathbf{SH}^\heartsuit(\mathrm{Corr})\simeq \mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr})[\mathbb G_m^{\wedge -1}].\]
\end{theorem*}
\begin{remark}
We refer to \cite{SHzar} to make the reduction to the Zariski topology in the definition of $\mathbf{H}_\mathrm{geom}(\mathrm{Corr})$.
\end{remark}

\begin{remark}
The vertical equivalences in the above diagram ware proven in \cite{ElmantoKhannonperfect} for an arbitrary base field with $\mathbb Z[1/p]$-coefficients.
\end{remark}

Specialising in the cases of framed correspondences, and Milnor-Witt correspondences we extend generality of results
\cite{framed-MW}, \cite[Proposition 29]{MotivicTambarafunctors}, \cite[Theorem 5.14]{TowardsconservativityofGm–stabilization}.

\begin{corollary*}
For a field $k$ the heart of the homotopy t-structure on $\mathbf{SH}^{\mathrm{eff}}(k)$ is equivalent
\[\mathbf{SH}^{\mathrm{eff},\heartsuit}(k)\simeq \mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)\]
to the category $\mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)$ of $\mathbb A^1$-invariant Nisnevich sheaves of abelian groups with framed transfers such that the framed correspondences $\sigma_X$, $X\in \mathrm{Sm}_k$, act identically.
\end{corollary*}
\begin{corollary*}
The heart of the homotopy t-structure on $\mathbf{SH}(k)$ is equivalent to the category of homotopy modules, that is $\mathrm{SHI}^\mathrm{Ab}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]$, where $\mathrm{SHI}^\mathrm{Ab}_\mathrm{nis}(k)$ denote the category of strict $\mathbb A^1$-invariant abelian sheaves.
\end{corollary*}
\begin{remark}
The latter corollary is proven already over an arbitrary field by the original arguments from \cite{morel-trieste} in combination with \cite{Morel-connectivity}. We write the alternative proof using framed correspondences.
\end{remark}

\begin{theorem*}
There are equivalences of the hearts of homotopy t-structures
\[\mathbf{SH}^{\mathrm{eff},\heartsuit}(k)\simeq \widetilde{\mathbf{DM}}^{\mathrm{eff},\heartsuit}(k),\quad\mathbf{SH}^\heartsuit(k)\simeq \widetilde{\mathbf{DM}}^\heartsuit(k).\]
\end{theorem*}

\subsection*{Acknowledgement}
The author thank A.~Ananyevskiy for the consultation on hearts of homotopy t-structures and the generalities of results,
and V.~Sosnilo for the consultation of $\infty$-categorical definitions and constructions.
The author thank G.~Garkusha for the remarks and comments.
According to the accepted agreement the author should say here that he is a Young Russian Mathematics award winner and the author would like to thank its sponsors and jury.

\subsection{Conventions: notions, and notation}

Let us list used notation of standard objects, and definitions of some extra or modified notions which we use in the text.

\subsubsection{Functions, sections, schemes}

For a presheaf or a sheaf $F$ on a site $\mathcal C$ we denote by
$F(X)$ or $\Gamma(X,F)$ the set of sections on $X\in \mathcal C$.
In particular, for an invertible sheaf or a line bundle $\mathcal L$ on a scheme $X$ we write $\Gamma(U,\mathcal L)$ for the group of sections on an open subscheme $U$ in $X$.
We denote by $\mathcal O_X$ the sheaf of regular functions on a small Zariski site of $X$, and use the notation
\[\mathcal O(X)=\mathcal O_X(X)=\Gamma(X,\mathcal O_X).\]
For a closed subscheme $Z\subset X$, $\mathcal I_{Z/X}$ denotes the ideal sheaf in $\mathcal O_X$ given by functions that vanish on $Z$, and $I_{Z/X}\subset \mathcal O(X)$ denotes the global sections.
The vanishing locus of a sheaf of ideals $I\subset \mathcal O(X)$ is denoted by $Z(s)$.
The vanishing locus of a section $s\in \Gamma(X,\mathcal L)$ is $Z(s)=Z(I)$, where $I$ is generated locally by $s$ in view of the trivialisation of $V$; the common vanishing locus of a set of sections $s_j$, $j\in J$, is denoted by $Z(s_j)_{J\in J}$.
For an injective morphism $r\colon Z\to X$ we write \[s\big|_{Z}=r^*(s)\in r^*(F)\] for the inverse image of $s\in \Gamma(X,F)$ of a presheaf $F$ on a small Zariski site of $X$.

We denote by
$\mathcal O_{X,x}$ the local ring of a scheme $X$ at a point $x\in X$, and by
$\mathcal O^h_{X,x}$ the henselisation.
We use the notation
\[
X_x = \mathrm{Spec}\;\mathcal O_{X,x},
X^h_x = \mathrm{Spec}\;\mathcal O^h_{X,x}
\]
for the corresponding affine schemes.
The same notation
$\mathcal O_{X,Z}$,
$\mathcal O^h_{X,Z}$
we use for the localisation and the henselisation of $\mathcal O_X$ at $\mathcal I_{Z/X}$, for a closed subscheme $Z$ in $X$,
and write
$X_Z = \mathrm{Spec}\;\mathcal O_{X,Z}$,
$X^h_Z = \mathrm{Spec}\;\mathcal O^h_{X,Z}$.

\subsubsection{Categories of schemes, relative dimension}
Denote by $\mathrm{Sch}_S$ the category of schemes over a base scheme $S$, and by $\mathrm{Sm}_S$ the category of smooth schemes.
We usually denote the base scheme by $S$, and also by $B$, or $V$ depending on the context:
$V$ is usually used as the base for the relative affine line $\mathbb A^1\times V$,
and if we consider $\mathbb A^1\times V$ as a scheme over $\mathbb A^1$ or $\mathbb P^1$, then we usually denote the base scheme by $B$.

We say that $X\in \mathrm{Sch}_V$ is equidimensional of dimension $d$, if for each $z\in V$, $\dim X\times_V z=d$.
We say that a closed subscheme $X^\prime$ of $X$ has positive relative codimension, if $\codim (X^\prime\times_V z)/(X\times_V z)>0$ for all $z\in V$.

We denote the site given by a Grothendieck topology $\tau$ on a category $C$ by $C^\tau$.
In particular, we write $\mathrm{Et}_S^\mathrm{nis}$ and $\mathrm{Sm}_S^\mathrm{nis}$ for the sites of the Nisnevich topology on the categories of \'etale and smooth schemes over a scheme $S$ respectively.

\subsubsection{Cohomologies with respect to a topology.}\label{subsect:Htaunotations}
For a Grothendieck topology $\tau$ on some category $\mathcal C$, and
a presheaf $F$ on $\mathcal C$, we denote by $F_\tau$ the sheafification.
Denote by
\begin{equation}H_\tau^*(X,F_\tau)=H_\tau^*(X,F)\end{equation} the cohomologies of $F_\tau$ on $X\in \mathcal C$;
we write sometimes $F$ without subscript above to short notation.

We use also the notion for what we call reduced cohomoligies $\underline{H}^*_\tau(X,F)$ defined as follows:
\[\begin{cases}
\underline{H}^i_\tau(X, F)=0, i<-1,\\
\underline{H}^{-1}_\tau(X, F)=\ker(F(X)\to F_\tau(X)),\\
\underline{H}^0_\tau(X, F)=\coker(F(X)\to F_\tau(X)),\\
\underline{H}^i_\tau(X, F)=H^i_\tau(X,F), i>0.
\end{cases}\]
So the following conditions are equivalent
\[\left[\underline{H}^*_\tau(X,F)=0\right]\Longleftrightarrow\left[\begin{cases}H^0_\tau(X, F_\tau)=F(X), \\H^i_\tau(X, F_\tau)=0, i>0.\end{cases}\right]\]

Given morphism $\widetilde X\to X$, there are the \v{C}ech complex and the extended \v{C}ech complex:
\[\begin{array}{lllll}
\check C_{\widetilde X}(X,F)&=&[\dots 0\to0\to  &F(\widetilde X)\to& \dots\to F(\widetilde X^{\times (l-1)})\to F(\widetilde X^{\times l})\to \dots ],\\ \underline{\check C}_{\widetilde X}(X,F)&=&
[\dots 0\to F(X)\to &F(\widetilde X)\to& \dots\to F(\widetilde X^{\times (l-1)})\to F(\widetilde X^{\times l})\to \dots ]
\end{array}\]
where $F(\widetilde X)$ is located in the zeroth deg., and
$\underline{\check C}_{\widetilde X}(X,F)
\simeq
\mathrm{Cone}(
F(X)\to
\check C_{\widetilde X}(X,F)
)
.$

\subsubsection{The Nisnevich topology}
$\mathrm{Nis}^X$ denotes the Nisnevich topology over the scheme $X$. We usually use superscript or subscript '$\mathrm{nis}$'.
We use the notation $\mathrm{Nis}^X_Z$ for the subtopology of $\mathrm{Nis}^X$ that is trivial over the open subscheme $X-Z$, for a closed subscheme $Z$,
and also use the notation $\underline\mathrm{Nis}^X$ and $\underline\mathrm{Nis}^X_Z$ for certain subtopologies of $\mathrm{Nis}^X$ and $\mathrm{Nis}^X_Z$, see \Cref{sect:(un)NisXZ}.

\section{Framed correspondences and presheaves}

We start with a recollection of definitions of framed correspondences and framed presheaves introduced in \cite{Voe-notes,Framed,hty-inv}.
The definition of the notion of correspondences of pairs is slightly different.
In addition, we recall the notion of first-order framed correspondences introduced in \cite{SmModelSpectrumTP},
that is a type of framed correspondences in between of framed correspondences from \cite{Framed}, and normally framed correspondences, that were introduced by Neshitov in privately shared notes, and by the authors of \cite{five-authors}.

Throughout the section $S$ denotes the base scheme.

\subsection{Framed correspondences of schemes}\label{sect:Fr}

\begin{definition}
An \emph{explicit framed correspondence}
from a scheme $X$ to a scheme $Y$ over $S$
is a set of data $(Z,V,\varphi,g)$ organised in the commutative diagram
\[
\xymatrixrowsep{0.2in}
\xymatrixcolsep{0.6in}
\xymatrix{
& V\ar[ld]_{\textit{\'etale}}\ar[r]^{(\varphi,g)} & \mathbb A^n_Y \\
\mathbb A^n_X\ar[rd]& Z\ar[d]|{\textit{finite}}\ar[l]_<<<<<<<{\textit{cl. imm.}}\ar[u]\ar[r]^{g\big|_Z} & Y\ar[u]^{0}\\
& X
}\]
where the right-upper square is Cartesian, and the right vertical arrow is the zero section,
i.e.
\[V\times_{\mathbb A^n\times Y} (\{0\}\times Y)=V\times_{\mathbb A^n} \{0\} \simeq Z,\]
and $\varphi\colon V\to \mathbb A^n$, and $g\colon V\to Y$ are morphisms that define the morphism $(\varphi,g)$.
In other words, the set $(Z,V,\varphi,g)$ is given by an \'etale neighbourhood $V$ of a closed subscheme $Z$ in $\mathbb A^n_X$ finite over $X$,
$\varphi=(\varphi_1,\dots \varphi_n)$ is a set of regular functions on $V$, and $g\colon V\to Y$ is a regular map such that
the common vanishing locus $Z(\varphi)$ in $X$ equals $Z$.
The integer $n$ is called the \emph{level} of the correspondence.
\end{definition}
\begin{definition}
The pointed set $\mathrm{Fr}_n(X, Y )$ is the set of classes of explicit framed correspondences from $X$ to $Y$ of level $n$
up to the equivalence relation given by the choice of the \'etale neighbourhood $V$ for a given $Z$, pointed at the class of framed correspondences with empty $Z$.
In detail,
the equivalence relation identifies the correspondences $(Z,V_1,\varphi_1,g_1)$ and $(Z,V_2,\varphi_2,g_2)$, if
the inverse images of $(\varphi_1,g_1)$ and $(\varphi_2,g_2)$ to some common shrinking $V$ of the neighbourhoods $V_1$ and $V_2$ of $Z$ in $\mathbb A^n_X$ are equal.
\end{definition}

\begin{definition}
Define the category $\mathrm{Fr}_+(X, Y )$ with objects being the same as in $\mathrm{Sm}_S$, and morphisms given by pointed sets
\[\mathrm{Fr}_+(X,Y) = \varinjlim_n\mathrm{Fr}_n(X, Y ).\]
The composition of
$c_1\in \mathrm{Fr}_{n_1}(X_1, X_2)$, and
$c_2\in \mathrm{Fr}_{n_2}(X_2, X_3)$
is given by the correspondence
\[(Z_1\times_Y Z_2, V_1\times_Y V_2, \mathrm{pr}_1^*(\varphi_1), \mathrm{pr}_2^*(\varphi_2), g_2 \mathrm{pr}_2),\]
where the morphism
$V_1\times_{X_2} V_2\to \mathbb A^{n_1}_{X_1}\times\mathbb A^{n_2}$ is induced by the morphisms
\[
V_1\to \mathbb A^{n_1}_{X_1},
V_2\to \mathbb A^{n_2}_{X_2}\to \mathbb A^{n_2},
\]
and $\mathrm{pr}_1\colon V_1\times_{X_2} V_2\to V_1$, $\mathrm{pr}_2\colon V_2\times_{X_2} V_2\to V_2$.
\end{definition}

\subsection{Category of open pairs}

\begin{definition}\label{def:Smpair}
Let $S$ be a base scheme.
Given pairs $(X_1,U_1)$, $(X_2,U_2)$ with open immersions of $S$-schemes $U_1\to X_1$ and $U_2\to X_2$,
consider the set of morphisms $\mathrm{Sm}_S(X_1,X_2)$ and the subset
\[\mathrm{Sm}^\mathrm{arrow}( (X_1,U_1) , (X_2,U_2) ) = \{f\in \mathrm{Sm}(X_1,X_2)| f(U_1)\subset U_2\} .\]
Define the pointed set
\[\mathrm{Sm}^\mathrm{pair}( (X_1,U_1), (X_2,U_2) )  = \Cofib ( \mathrm{Sm}( X_1 , U_2 ) \to \mathrm{Sm}^\mathrm{arrow}( (X_1,U_1) , (X_2,U_2) ) ),\]
where the morphism at the right side is induced by the map $\mathrm{Sm}( X_1 , U_2 )\to \mathrm{Sm}( X_1 , X_2 )$ given by the composition with the canonical morphism $U_2\to X_2$.
\end{definition}

\begin{definition}\label{def:Frpair_+}
Define the category of pairs $\mathrm{Fr}_+^\mathrm{pair}(S)$ with the same objects as in $\mathrm{Sm}^\mathrm{pair}_S$ and morphisms $\mathrm{Fr}_+^{\mathrm{pair}}((X,U), (Y,W) )$ between $(X,U)$ and $(Y,W)$ defined as follows.

Define the subset $\mathrm{Fr}_n^{\mathrm{arrow}}((X,U), (Y,W) )\subset \mathrm{Fr}_n(X, Y )$
that elements are correspondences
$(Z,V,\varphi,g)$ such that
\[ U\times_X Z \subset Z\times_{Y}W ,\]
or equivalently
\[ (X\setminus U)\times_X Z \supset Z\times_Y (Y\setminus W).\]
Define the set
$\mathrm{Fr}_n^{\mathrm{pair}}((X,U), (Y,W) )$
as the quotient set, i.e. cokernel, of the morphism
$\mathrm{Fr}_n(X, W )\to \mathrm{Fr}_n^{\mathrm{arrow}}((X,U), (Y,W) )\colon c\mapsto j_Y\circ c$, where $j_Y\colon W\to Y$ is the canonical open immersion.
Define the set of morphisms
\[\mathrm{Fr}_+^\mathrm{pair}((X,U), (Y,W) ) = \varinjlim_n\mathrm{Fr}_n^\mathrm{pair}((X,U), (Y,W) ).\]
\end{definition}

\subsection{Linearised correspondences, and framed presheaves}

\begin{definition}\label{def:ZFstar}
For a category $\mathcal C$ enriched over pointed sets, denote by $\mathbb Z\mathcal C$ the category enriched over abelian groups that have the same objects and hom-groups being the quotient of the free abelian groups associated with the hom-sets in $\mathcal C$ such that the pointed element equals zero.
The \emph{additivisation} of a category $\mathcal C$ enriched over sets is the universal quotient category of $\mathbb Z\mathcal C$ such that
$\mathcal C(X_1\amalg X_2,Y) = \mathcal C(X_1,Y)\oplus \mathcal C(X_2,Y)$.
Define the categories ${\mathbb Z\mathrm{F}}_*(S)$ and ${\mathbb Z\mathrm{F}}_*^\mathrm{pair}(S)$ as the additivisations of the categories $\mathrm{Fr}_+(S)$ and $\mathrm{Fr}_+^\mathrm{pair}(S)$.
\end{definition}

\begin{definition}
A \emph{framed presheaf} is
a functor $\mathrm{Fr}_+(S)\to \mathrm{Set}$.
A framed presheaf is \emph{quasi-stable}, if the morphisms $\sigma_X^*$ are auto-morphisms of $F(X)$ for all $X$.
The same definition applies to the case of the category of open pairs $\mathrm{Fr}_+^\mathrm{pair}(S)$.
\end{definition}

\begin{definition}
A \emph{linear framed presheaf} over a base scheme $S$ or a \emph{framed additive presheaf of abelian groups} is a functor $F\colon \mathrm{Fr}_+(S)\to \mathrm{Ab}$ such that $F(X_1\amalg X_2)\simeq F(X_1)\oplus F(X_2)$.
Equivalently we can say that a linear framed presheaf defines a functor of categories enriched over abelian groups
\[F\colon {\mathbb Z\mathrm{F}}_*(S)\to \mathrm{Ab}.\]
The same definition applies to the case of $\mathrm{Fr}_+^\mathrm{pair}(S)$.
\end{definition}

\begin{proposition}
Consider the left Kan extension and the forgetful functors
\[\gamma^*\colon \mathbb Z\mathrm{Sh}(S)\to \mathbb Z\mathrm{Sh}(\Fr_+(S)),\; \gamma_*\colon \mathbb Z\mathrm{Sh}(\Fr_+(S))\to \mathbb Z\mathrm{Sh}(S).\]
Then for any $E\in \mathbb Z\mathrm{Sh}(k)$, and $F\in \mathbb Z\mathrm{Sh}(\Fr_+(S))$,
\[\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(S)}(E,\gamma_* F)\simeq
\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(\Fr_+(S))}(\gamma^* E,F).\]
\end{proposition}
\begin{proof}
Since the category $\mathbb Z\mathrm{Sh}(S)$ is generated via colimits by the representable sheaves $\mathbb Z(U)$, $U\in \mathrm{Sm}_S$, it is enough to prove the claim for $E$ of such type. For $E=\mathbb Z(U)$, $\gamma^* E = {\mathbb Z\mathrm{F}}_*(U)_\mathrm{nis}$, and the claim is \cite[Proposition 16.2]{hty-inv}.
\end{proof}

\begin{definition}\label{def:CohSupTransfer}
For $S$-schemes $U$, and $X$, and a framed correspondence $c\in \mathrm{Fr}_n(U,X)$,
for any closed subscheme $Z$ in $X$, and the closed subscheme $Y=c^{-1}(Z)$ in $U$, define the canonical morphism
on cohomologies with support
\[c_*\colon\underline{H}^*_Z(X,F)\to \underline{H}^*_Y(U,F)\]
for any linear presheaf with framed transfers $F$ over $S$
as the composite morphism
\begin{multline*}
H^l_{Z}(X,F)=
\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(S)}(\mathbb Z(X)/\mathbb Z(X-Z),F)\simeq\\
\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(\Fr_+(S))}({\mathbb Z\mathrm{F}}(X)/{\mathbb Z\mathrm{F}}(X-Z),F)\to
\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(\Fr_+(S))}({\mathbb Z\mathrm{F}}(U)/{\mathbb Z\mathrm{F}}(U-Y),F)\simeq\\
\mathrm{Ext}^l_{\mathbb Z\mathrm{Sh}(S)}(\mathbb Z(U)/\mathbb Z(U-Y),F)=
H^l_{Y}(U,F).
\end{multline*}
\end{definition}

\subsection{First-order framed correspondences}

We recall a notion introduced in \cite{SmModelSpectrumTP}.

\begin{definition}
\emph{A first-order framed correspondence} of level $n$
from $X$ to $Y$ is a set
$(Z,\tau,g)$ given by a closed subscheme $Z$ in $\mathbb A^n_X$,
a trivialisation of the conormal sheaf of $Z$
\[\tau\colon \mathcal I_{Z/\mathbb A^n_X}/\mathcal I^2_{Z/\mathbb A^n_X}\simeq \mathcal O_Z^{\times n},\]
and a regular map $g\colon \mathcal Z(I^2_{Z/\mathbb A^n_X})\to Y$.
In other words, the pair $(\tau,g)$ defines the morphism of schemes
\[(\tau,g)\colon Z(I^2_{Z/\mathbb A^n_X})\to \mathbb A^n\times Y\] such that
$Z(I^2_{Z/\mathbb A^n_X})\times_{\mathbb A^n_X}(0\times Y)\cong Z$.
Denote the pointed set of such $(Z,\tau,g)$ by $\mathrm{Fr}_n(X,Y)$.
\end{definition}

\begin{definition}
The pointed set $\mathrm{Fr}^\mathrm{1th}_n((X,U),(Y,V))$ of \emph{first-order framed correspondences of level $n$
between pairs $(X,U)$ to $(Y,V)$} is the quotient of the set of the elements
$(Z,\tau,g)\in \mathrm{Fr}^\mathrm{1th}_n(X,Y)$ such that
\[
((X\setminus U)\times_X Z)_\mathrm{red}\supset (Z\times_Y (Y\setminus V))_\mathrm{red},
\]
with respect to the subset of elements such that
\[\emptyset = Z\times_Y (Y\setminus V)\]
\end{definition}

\begin{definition}
Define the \emph{categories} $\mathrm{Fr}^\mathrm{1th}_+(S)$ and $\mathrm{Fr}^\mathrm{1th}_+(\mathrm{Sm}_S^\mathrm{pair})$ as the categories that morphisms are given by first-order correspondences.
The composition of $(Z_1,\tau_1,g_1)\in \mathrm{Fr}^\mathrm{1th}_{n_1}(X_0,X_1)$ with $(Z_2,\tau_2,g_2)\in \mathrm{Fr}^\mathrm{1th}_{n_2}(X_1,X_2)$ is
$(Z_{12}, \tau_{12},g_{12} )$, where
\[
Z_{12}= Z_1\times_{X_1}Z_2 , (\tau_{12},g_{12})\colon Z(\mathcal I^2_{Z_{12}}) \to Z(\mathcal I^2_{Z_{2}}) \to X_2
\]
and the first morphism is induced by $(\tau_2,g_2)$, and the second one is induced by $(\tau_1,g_1)$.
A first-order framed presheaf is a presheaf on $\mathrm{Fr}^\mathrm{1th}_+(S)$ or $\mathrm{Fr}^\mathrm{1th}_+(\mathrm{Sm}_S^\mathrm{pair})$.
\end{definition}
There is the natural map\begin{equation}\label{eq:Fr->nrFr}
\mathrm{Fr}_n(X,Y)\to \mathrm{Fr}^\mathrm{1th}_n(X,Y)\colon
(Z,V,\varphi,g)\mapsto (Z, \tau,g\big|_Z),
\end{equation}
where $\tau$ is induced by $\varphi\big|_{Z(\mathcal I^2_{\mathbb A^n_X}(Z)}$.
\begin{definition}
A \emph{first-order framed presheaf} is a presheaf on $\mathrm{Fr}^\mathrm{1th}_+(S)$ or $\mathrm{Fr}^\mathrm{1th}_+(\mathrm{Sm}^\mathrm{pair}_S)$.
A first-order framed presheaf is \emph{quasi-stable} if
the morphism $\sigma_X\in \mathrm{Fr}^\mathrm{1th}_1(X,X)$ defined as the image of $\sigma_X\in\mathrm{Fr}_1(X,X)$ induces auto-equivalence on $F(X)$ for each $X\in \mathrm{Sm}_S$.
\end{definition}
\begin{lemma}
Any quasi-stable first-order framed presheaf has a canonical structure of quasi-stable framed presheaf.
\end{lemma}
\begin{proof}
The claim follows because \eqref{eq:Fr->nrFr} defines the functor $\mathrm{Fr}_+(S)\to \mathrm{Fr}^\mathrm{1th}_+(S)$ and because the latter functor preserves morphisms of the form $\sigma_X$, $X\in \mathrm{Sm}_S$.
\end{proof}

\begin{lemma}
Suppose the base scheme $S$ is affine. Any $\mathbb A^1$-invariant quasi-stable framed presheaf $F$ on the category of affine schemes over $S$
is canonically equipped with the structure of a first-order framed presheaf and is quasi-stable is this sense.
\end{lemma}
\begin{proof}
To prove the claim we show that for any $c,c^\prime\in \mathrm{Fr}_n(X,Y)$,
the induced morphisms $c^*,{c^\prime}^*\colon F(Y)\to F(X)$ are equal,
if the images of $c$ and $c^\prime$ in $\mathrm{Fr}^\mathrm{1th}_n(X,Y)$ are equal.
Then since the morphism $\mathrm{Fr}_n(X,Y)\to \mathrm{Fr}^\mathrm{1th}_n(X,Y)$ is surjective, the first claim follows.
Let $c,c^\prime$ be as above.
Note that $g\big|_{Z(\mathcal I^2(Z))}=g^\prime\big|_{Z(\mathcal I^2(Z))}\colon Z(\mathcal I^2(Z))\to Y$,
denote this function by $g_{Z(\mathcal I^2(Z))}$, and denote by the same symbols the inverse images on ${\mathbb A^1\times Z(\mathcal I^2(Z))}$, ${0\times Z(\mathcal I^2(Z))}$, ${1\times Z(\mathcal I^2(Z))}$;
consider the map
\begin{gather}
\check{g}\colon
E\to Y,\label{eq:amalggGlassY}\\
\begin{array}{llccccc}
E&=&(0\times (\mathbb A^n_X)^h_Z)&\amalg_{0\times Z(\mathcal I^2(Z))}&(\mathbb A^1\times Z(\mathcal I^2(Z)))&\amalg_{1\times Z(\mathcal I^2(Z))}&(1\times (\mathbb A^n_X)^h_Z)\\
\check{g}&=&g&\amalg_{g_{0\times Z(\mathcal I^2(Z))}}& g_{\mathbb A^1\times Z(\mathcal I^2(Z))}&\amalg_{g_{1\times Z(\mathcal I^2(Z))}}& g^\prime
.\end{array}\nonumber
\end{gather}
Choose a lifting of \eqref{eq:amalggGlassY}
\[\tilde g\colon (\mathbb A^1\times\mathbb A^n_X)^h_{\mathbb A^1\times Z}\to Y,\]
that exists by \cite[Lemma 3.6]{FrRigidSmAffpairs}
since the canonical morphism $E\to (\mathbb A^1\times\mathbb A^n_X)^h_{\mathbb A^1\times Z}$ is an affine henseleian pair.
Consider the framed homotopy
\[h = (\mathbb A^1\times Z,\lambda \varphi+(1-\lambda) \varphi^\prime, \tilde g)\in \mathrm{Fr}_n(\mathbb A^1\times X, Y).\]
Then
$h\circ i_0=c$, $h\circ i_1=c^\prime$, and since $F$ is $\mathbb A^1$-invariant,
it follows that $c^*={c^\prime}^*$.
So the claim is proven.
The second claim of the lemma follows, because
for any $X$, the correspondence $\sigma_X\in \mathrm{Fr}_1(X,X)$ goes to $\sigma_X\in \mathrm{Fr}^\mathrm{1th}_n(X,X)$ along morphism \eqref{eq:Fr->nrFr}.
\end{proof}

\section{Topology over a quotient-space $X/(X-Z)$ and cohomologies} \label{sect:(un)NisXZ}

\subsubsection{Topologies.}

Denote by $\mathrm{Nis}^X$ the Nisnevich topology on $\mathrm{Sm}_X$ over a scheme $X$.
Let $Z$ be a closed subscheme.
\begin{definition}\label{def:NisXZ}
Define \emph{the topology $\mathrm{Nis}^X_Z$} as the strongest subtopology of $\mathrm{Nis}_X$ that restriction on $\mathrm{Sm}_{X-Z}$ is trivial.
\end{definition}
In other words,
the topology $\mathrm{Nis}^X_Z$ is generated by Nisnevich coverings $\widetilde U\to U$ for all $U\in \mathrm{Sm}_X$ such that there is a lifting in the diagram
\[\xymatrix{
&\widetilde U\ar[d] \\
U\times_X(X-Z)\ar@{-->}[ru]\ar[r]&U
.}\]

\begin{definition}\label{def:underNisV}
Let $X$ be a scheme.
Define the topology $\underline{\mathrm{Nis}}^X$ as the weakest topology
with coverings being \'etale morphisms $\widetilde U\to U$
such that for each $v\in U$, there is a lifting in the triangle
\begin{equation}\label{eq:UXhpvlifting}\xymatrix{
&\widetilde U\ar[d] \\
U\times_X X^h_{p(v)}\ar@{-->}[ru]\ar[r]&U,
}\end{equation}
where $p\colon U\to X$ is the structure morphism.
\end{definition}
The above definition in other words defines the strongest subtopology $\underline{\mathrm{Nis}}^X$
of $\mathrm{Nis}^X$ that is trivial on the schemes of the form $U\times_X X^h_x$ for $U\in \mathrm{Sm}_X$, $x\in X$, and $U\times_X x\neq \emptyset$.

\begin{definition}
Define \emph{the topology $\underline{\mathrm{Nis}}^X_Z$} on $\mathrm{Sm}_X$ as the intersection $\underline{\mathrm{Nis}}^X \cap \mathrm{Nis}^X_Z$, see \Cref{def:underNisV} above.\end{definition}

Let $\mathrm{Pre}(X)$ and
$\mathbb Z\mathrm{Pre}(X)$ denote the category of presheaves and the category of additive presheaves of abelian groups over the scheme $X$.

\begin{lemma}\label{lm:NisZXNisZpoints}
(1) The topologies $\mathrm{Nis}^X_Z$ and $\underline{\mathrm{Nis}}^X_Z$ have enough sets of points given respectively by the schemes of the form
\begin{equation}\label{eq:NisZpoints}U^h_v, \text{ for } v\in U,\, \text{ and } U\times_X (X-Z),\end{equation}
and
\begin{equation}\label{eq:XNisZpoints}U\times_X X^h_{p(v)}, \text{ for } v\in U,\, \text{ and } U\times_X (X-Z),\end{equation}
where $p\colon U\to X$ is the canonical morphism, for all $U\in \mathrm{Sm}_X$.
(2) An \'etale morphism $\widetilde X\to X$ is a $\mathrm{Nis}^X_Z$-covering or $\underline{\mathrm{Nis}}^X_Z$-covering if
for any scheme $V$ of the form \eqref{eq:NisZpoints} or \eqref{eq:XNisZpoints}, the triangle
\[\xymatrix{
&\widetilde X\ar[d]\\
V\ar@{-->}[ru]\ar[r]& X
}\]
admits a lifting.
\end{lemma}
\begin{proof}
(1) The case of $\mathrm{Nis}^X_Z$ follows immediately from the definition.
To prove the case of $\underline{\mathrm{Nis}}^X_Z$ one we note firstly that by the definition all the schemes \eqref{eq:XNisZpoints} are $\underline{\mathrm{Nis}}^X_Z$-paints.
Then for any $\underline{\mathrm{Nis}}^X_Z$-local isomorphism $F\to G$, the morphisms of stalks $F(V)\to G(V)$ are isomorphisms for $V$ of the form \eqref{eq:XNisZpoints}.
Conversely,
let  $F\to G$ be a morphism in $\mathrm{Pre}(X)$ such that $F(U\times_X X^h_{p(v)})\simeq G(U\times_X X^h_{p(v)})$ for each $v\in U$.
Then for each $v\in U$, there is an \'etale neighbourhood $\tilde X_v$ of $p(v)$ in $X$ such that $F(\tilde X_v)\simeq G(\tilde X_v)$.
Define the $\underline{\mathrm{Nis}}^X_Z$-covering of $U$ $\widetilde X= \coprod_{v\in U}\tilde X_v\amalg (X-Z)\to U$.
Then $F(\widetilde X)\simeq G(\widetilde X)$.
So if both $F$ and $G$ are $\underline{\mathrm{Nis}}^X_Z$-sheaves then $F\cong G$.
(2) The claim follows from the previous one.
\end{proof}

\subsubsection{Framed transfers for cohomologies}

Consider the subcategories of sheaves in $\mathbb Z\mathrm{Pre}(X)$ with respect to topologies $\mathrm{Nis}^X_Z$ and $\underline{\mathrm{Nis}}^X_Z$.
Denote the sheafifications of $F\in\mathbb Z\mathrm{Pre}(X)$ by $F_{\mathrm{Nis}^X_Z}$, $F_{\underline{\mathrm{Nis}}^X_Z}$.
\begin{lemma}\label{lm:XNisZNisZ(UfinX)cong}
For any $U\in \mathrm{Sch}_X$ finite over $X$,
and any additive presheaf of abelian groups $F\in \mathbb Z\mathrm{Pre}(X)$, there is a canonical isomorphism
$F_{\mathrm{Nis}^X_Z}(U)\cong F_{\underline{\mathrm{Nis}}^X_Z}(U)$.
\end{lemma}
\begin{proof}
The embedding of topologies $\underline{\mathrm{Nis}}^X_Z\to \mathrm{Nis}^X_Z$ leads to the spectral sequence
\[H^q_{\underline{\mathrm{Nis}}^X_Z}(U,H^p_{\mathrm{Nis}^X_Z}(F)_{\underline{\mathrm{Nis}}^X_Z})\Rightarrow H_{\mathrm{Nis}^X_Z}^{p+q}(U,F_{\mathrm{Nis}^X_Z}),\]
where $H^p_{\mathrm{Nis}^X_Z}(F)_{\underline{\mathrm{Nis}}^X_Z}$ is the sheafifiticaton of $H^p_{\mathrm{Nis}^X_Z}(-,F{\mathrm{Nis}^X_Z})$ with respect to $\underline{\mathrm{Nis}}^X_Z$.
Since $U$ is finite, for any $x\in X$, $U\times_{X} X^h_x=\coprod_{y\in U\times_X x}U^h_y$. Hence
$\underline{H}^p_{\mathrm{Nis}^X_Z}(U\times_{X} X^h_x,F_{\mathrm{Nis}^X_Z})=0$.
Since $\mathrm{Nis}_Z$ is trivial on $\mathrm{Sm}_{X-Z}$, $\underline{H}^p_{\mathrm{Nis}^X_Z}(U\times_{X} (X-Z),F_{\mathrm{Nis}^X_Z})=0$.
Thus by \Cref{lm:NisZXNisZpoints} $H^*_{\mathrm{Nis}^X_Z}(F)_{\underline{\mathrm{Nis}}^X_Z}=F_{\underline{\mathrm{Nis}}^X_Z}$.
\end{proof}

\begin{lemma}\label{lm:XNisZFr}
For any quasi-stable linear (first-order) framed presheaf $F$, the presheaf $H^*_{\underline\mathrm{Nis}^X_Z}(-,F_{\underline{\mathrm{Nis}}^X_Z})$ has a canonical structure of a quasi-stable (first-order) framed presheaf.
Even more, for a pair of closed subschemes $Z_1\subset Z_2$, the canonical morphism
$H^*_{\underline\mathrm{Nis}^X_Z}(-,F_{\underline{\mathrm{Nis}}^X_{Z_1}})\to H^*_{\underline\mathrm{Nis}^X_Z}(-,F_{\underline{\mathrm{Nis}}^X_{Z_2}})$ agrees with the framed transfers, and the morphism
$H^*_{\underline\mathrm{Nis}^X_Z}(-,F_{\underline{\mathrm{Nis}}^X_{Z}})\to H^*_{\mathrm{Nis}}(-,F_{\mathrm{Nis}})$ agrees too.
The same applies to the reduced cohomologies $\underline H^*_\nu(-,F_\nu)$.
\end{lemma}
\begin{proof}
For any $c\in \mathrm{Fr}_n(U_1, U_2)$ or $c\in \mathrm{Fr}^\mathrm{1th}_n(U_1, U_2)$ over $X$, and a morphism $\widetilde X\to X$
for each $l$, there is the canonical framed correspondence $\widetilde c\in \mathrm{Fr}_n(U_1\times_X \widetilde X, U_2\times_X \widetilde X^{\times l})$ or $\widetilde c\in \mathrm{Fr}^\mathrm{1th}_n(U_1\times_X \widetilde X, U_2\times_X \widetilde X^{\times l})$ given by the base change.
Hence for any quasi-stable linear framed or first-order framed presheaf $F$,
there is the morphism of \v{C}ech complexes with respect to the coverings $U_1\times_X\widetilde X\to U_1$, and $U_2\times_X \widetilde X\to U_2$.
Hence the presheaf is a quasi-stable linear (first-order) framed presheaf.
Thus the first claim is proven. The second claim follows because of the naturality of the above construction.
The case of $\underline H^*_\nu(-,F_\nu)$ is similar.
\end{proof}

\begin{lemma}\label{lm:restrNisZtopology}
For a morphism of schemes $T\to X$, and a closed subscheme $Z$ in $X$, there is a canonical isomorphism
\[H_{\mathrm{Nis}^T_{T\times_X Z}}^*(T,F)\cong H_{\mathrm{Nis}^X_Z}^*(T,F).\]
\end{lemma}
\begin{proof}
The claim follows since the topology $\mathrm{Nis}^X_Z$ restricted on $\mathrm{Sch}_T$ equals the topology $\mathrm{Nis}^T_{T\times_X Z}$.
\end{proof}

\begin{definition}\label{def:pushforwardtopcahngeinvimagefinhens}
Let $c=(T,\tau,g)\in \mathrm{Fr}^\mathrm{1th}_n((X,U), (Y,V))$ for $(X,U), (Y,V)\in \mathrm{Sm}^\mathrm{pair}_S$ over a base scheme $S$.
Let $W=Y\setminus V$, $Z= X\setminus U$.
Define the homomorphism
\[c^*\colon H^*_{\mathrm{Nis}^Y_{W}}(Y,F)\to H^*_{\mathrm{Nis}^X_Z}(X,F)\]
as the composite homomorphism
\begin{equation}\label{diag:pushforwardtopcahngeinvimagefinhens}\xymatrix{
&&H_{\mathrm{Nis}^{Y}_{W}}(T)\ar[d]_{}& H_{\mathrm{Nis}^{Y}_{W}}(Y)\ar[l]^{g^*}\\
&H_{\underline{\mathrm{Nis}}^{X}_{Z}}(T)\ar[d]_{\text{Lm}\,\ref{lm:XNisZFr}}\ar[r]^{\simeq}_{\text{Lm}\,\ref{lm:XNisZNisZ(UfinX)cong}} & H_{\mathrm{Nis}^{X}_{Z}}(T)\\
H_{\mathrm{Nis}^{X}_{Z}}(X)&H_{\underline\mathrm{Nis}^{X}_{Z}}(X)\ar[l]^{\simeq}_{{\text{Lm}\,\ref{lm:XNisZNisZ(UfinX)cong}}}
,}\end{equation}
where $H_\nu(-)=H^*_\nu(-,F_\nu)$,
and the second morphism in the sequence is provided by \Cref{lm:restrNisZtopology} and the equality $c^{-1}(W)=Z$.
The same applies to $c^*\colon\underline{H}^*_{\mathrm{Nis}^X_Z}(X,F)\to \underline{H}^*_{\mathrm{Nis}^Y_W}(Y,F)$.
\end{definition}

\begin{proposition}\label{prop:NisZFr}
For any quasi-stable linear first-order framed presheaf $F$ over a base scheme $S$,
the presheaf
\[(X,X-Z)\mapsto H^*_{\mathrm{Nis}^X_Z}(X,F),\]
where $\mathrm{Nis}^X_Z$ is the topology on $\mathrm{Sch}_X$ from \Cref{def:NisXZ},
has the canonical structure of the quasi-stable first-order framed presheaf.
The same applies to $(X,X-Z)\mapsto \underline{H}^*_{\mathrm{Nis}^X_Z}(X,F)$.
\end{proposition}
\begin{proof}
The transfer homomorphisms are given by \Cref{def:pushforwardtopcahngeinvimagefinhens}.
To prove the claim we need to show the functoriality, i.e. that
\begin{equation}\label{eq:compspushformwardsHNisXZ}(c_2 \circ c_1)^*\cong c_1^*\circ c_2^*\end{equation} for
two composible framed correspondences $c_1,c_2$.
Consider two subcategories in $\mathrm{Fr}^\mathrm{1th}_+(S)$:
(p) denote by $\mathrm{Fr}^\mathrm{1th}_\mathrm{p}(S)$
the subcategory that morphisms are correspondences $c=(T,\tau,g)$ such that $g\cong\mathrm{id}_{Z(\mathcal I^2(T))}$, and
(z) denote by $\mathrm{Fr}_0(S)$ the subcategory spanned by framed correspondences of level zero, that are given by morphisms of schemes.
Any morphism $c=(T,\tau,g)\in \mathrm{Fr}^\mathrm{1th}_n(X,Y)$ in $\mathrm{Fr}^\mathrm{1th}_+(S)$ admits the decomposition
\[c=g \circ c_p, c_p=(T,\tau,\mathrm{id}_{Z(\mathcal I^2(T)})\in\mathrm{Fr}^\mathrm{1th}_n(X,Z(\mathcal I^2(T)).\]
Because of this to prove the claim we can assume that $c_1$ and $c_2$ in \eqref{eq:compspushformwardsHNisXZ} belong to the union of the sets of morphisms $\mathrm{Fr}^\mathrm{1th}_\mathrm{p}(S)$, and $\mathrm{Fr}^\mathrm{1th}_0(S)$.
We consider four cases of types of $c_1$ and $c_2$.
\par (0)
For $c_1,c_2\in \mathrm{Fr}^\mathrm{1th}_0(S)$,
the claim holds by the functoriality of the pushforwards for regular morphisms of schemes.
\par (1)
For $c_1,c_2\in \mathrm{Fr}^\mathrm{1th}_\mathrm{p}(S)$, the claim follows by \Cref{lm:XNisZFr}.
\par (2)
The case of $c_1=(T,\tau,\mathrm{id}_{Z(\mathcal I^2(T)})\in \mathrm{Fr}^\mathrm{1th}_n(X_1,Z(\mathcal I^2(T))$, $c_2=g\in \mathrm{Fr}_0(Z(\mathcal I^2(T),Y)=\mathrm{Sch}_S(Z(\mathcal I^2(T)),Y)$ holds by \Cref{def:pushforwardtopcahngeinvimagefinhens}.
\par (3)
Let
\[\begin{array}{ll}
c_1=f&\in \mathrm{Fr}_0((X,Z),(X^\prime,Z^\prime))=\mathrm{Sch}^\mathrm{pair}_S((X,Z),(X^\prime,Z^\prime)), \\
c_2=(Y,\tau,\mathrm{id}_Y)&\in \mathrm{Fr}^\mathrm{1th}_n((X^\prime,Z^\prime),(Y,W))
.\end{array}\]
Then the claim follows because of the commutative diagram
\[\xymatrix{
H^*_{\mathrm{Nis}^Y_W}(T,F)\ar[d] & H^*_{\mathrm{Nis}^Y_W}(Y,F)\ar[d]\ar[l] \\
H^*_{\mathrm{Nis}^X_Z}(T,F) & H^*_{\mathrm{Nis}^{X^\prime}_{Z^\prime}}(Y,F)\ar[l] \\
H^*_{\underline\mathrm{Nis}^X_Z}(T,F)\ar[u]^{\simeq}_{\text{Lm}\,\ref{lm:XNisZNisZ(UfinX)cong}}\ar[d]^{\text{Lm}\,\ref{lm:XNisZFr}} & H^*_{\underline\mathrm{Nis}^{X^\prime}_{Z^\prime}}(Y,F)\ar[u]^{\simeq}_{\text{Lm}\,\ref{lm:XNisZNisZ(UfinX)cong}}\ar[l]\ar[d]^{\text{Lm}\,\ref{lm:XNisZFr}} \\
H^*_{\mathrm{Nis}^X_Z}(X,F) & H^*_{\mathrm{Nis}^{X^\prime}_{Z^\prime}}(X^\prime,F)\ar[l]^{f^*}
,}\]
where $T=Y\times_{X^\prime} X$.

The case of $(X,X-Z)\mapsto \underline{H}^*_{\mathrm{Nis}^X_Z}(X,F)$ is similar.
\end{proof}

\begin{lemma}\label{prop:CommCohSupTrans}
Let $V$ be a scheme over $S$, and let $U$ be an $V$-scheme.
Let $X$ be a $S$-scheme, and $c\in \mathrm{Fr}_n(U,X)$ be a framed correspondence over $S$.
Then for any linear presheaf with framed transfers $F$ over $S$,
there is the commutative square
\[\xymatrix{
\underline{H}^*_{\mathrm{Nis}_{Y}^{U}}(U,F)\ar[d] & \underline{H}^*_{\mathrm{Nis}_{Z}^{X}}(X,F)\ar[l]^{c^*}\ar[d] \\
\underline{H}^*_\mathrm{Nis}(U,F)&\underline{H}^*_{\mathrm{Nis}}(X,F)\ar[l]^{c^*}
,}\]
where $Y=c^{-1}(Z)$.
\end{lemma}
\begin{proof}
The claim follows because morphisms of diagram \eqref{diag:pushforwardtopcahngeinvimagefinhens} commute with the morphisms $H_\nu^*(-,F_\nu)\to H_{\mathrm{Nis}}^*(-,F_\mathrm{Nis})$ for $\nu$ as in \Cref{def:pushforwardtopcahngeinvimagefinhens}.
\end{proof}

\section{Cohomologies out of infinity on $\mathbb A^1_V$.}\label{sect:outofinffinitesupSHI}

In this section, we show some partial invariance of $\mathbb A^1$-invariant quasi-stable linear framed presheaves.
Namely, we introduce the notion of cohomologies out of the infinity and show that the cohomologies out of infinity on the relative affine line $\mathbb A^1_V$ over a local henselian scheme $V$ are trivial for a quasi-stable linear framed presheaf, \Cref{th:FinSuppLocHensbaseafflinetriviality}.
We say 'cohomologies out of infinity' and we could 'the universal cohomologies that are trivial at infinity, these are cohomologies with respect to the union of subtopologies $\mathrm{Nis}^{\mathbb A^1\times V}_Z$, defined in \Cref{sect:(un)NisXZ} for all closed subschemes $Z$ finite over $V$.

\begin{definition}\label{def:PropS}
Let $X$ be a scheme over a scheme $S$. Denote by $\mathrm{Prop}_S^X$ the topology on $\mathrm{Sch}_{X}$
generated by topologies $\mathrm{Nis}_{Z}^{X}$ for all closed subschemes $Z$ in $X$ proper over $S$, see \Cref{def:NisXZ}.
\end{definition}

\begin{example}\label{def:FinS}
Let $S$ be a scheme. We denote by \[\mathrm{Fin}_S=\mathrm{Prop}^{\mathbb A^1\times S}_S\] the topology on $\mathrm{Sch}_{\mathbb A^1\times S}$
generated by $\mathrm{Nis}_{Z}^{\mathbb A^1\times S}$ for all closed subschemes $Z$ in $\mathbb A^1\times S$ finite over $S$.
\end{example}

\subsection{Excision isomorphisms}

\begin{proposition}(\'Etale excision)[\protect{\cite[Theorem 3.10]{surj-etale-exc}}]\label{prop:EtExcision}
Let $\widetilde X$ be an \'etale neighbourhood of a closed subscheme $Z$ in an essentially smooth local henselian scheme $X$ over $k$.
Then for any quasi-stable framed linear presheaf $F$, the square
\[\xymatrix{
F(\widetilde X-Z)& F(\widetilde X)\ar[l]\\
F(X-Z)\ar[u]& F(X)\ar[l]\ar[u]
}\]
is Cartesian and coCartesian.
\end{proposition}

\begin{proposition}(Zariski excision with finite support)\label{lm:pairZarExA1Vhalffin}
Let $V$ be a local scheme. Let $Z_0,Z_1\to \mathbb A^1_V$ be closed immersions, $Z_0\cap Z_1=\emptyset$.
Assume that $Z_0$ is finite over $V$.

Then for any quasi-stable linear framed presheaf \[G\colon {\mathbb Z\mathrm{F}}_*^\mathrm{pair}(V)\to \mathrm{Ab},\]
see \Cref{def:Smpair,def:Frpair_+} for the category of open pairs,
the morphism
\begin{equation}\label{eq:j*GpairsA1V}j^*\colon G(\mathbb A^1_V,\mathbb A^1_V-Z_0)\to G(\mathbb A^1_V-Z_1,\mathbb A^1_V-Z_1-Z_0),\end{equation}
induced by the canonical open immersion of pairs
\[j\colon (\mathbb A^1_V-Z_1,\mathbb A^1_V-Z_1-Z_0)\to (\mathbb A^1_V, \mathbb A^1_V-Z_0)\]
is an isomorphism.
\end{proposition}
\begin{proof}
Denote by $\mathcal O(1)$ the canonical ample bundle on $\mathbb P^1$ and let $t_\infty$
denote the section that vanishing locus is $\infty$.
Denote by the same symbols the inverse images on $\mathbb P^1_V$, and schemes of the form $-\times\mathbb P^1_V$.
Denote $\infty_V=\infty\times V=\mathbb P^1_V\setminus \mathbb A^1_V$,
and let $\overline{Z}_1$ be the closure of $Z_1$ in $\mathbb P^1_V$.
Since $Z_0$ is finite over $V$ it is a closed subscheme in $\mathbb P^1_V$.
So \[Z_0\cap \overline{Z}_1 = Z_0\cap Z_1 = \emptyset.\]
Denote by $\delta\in \Gamma(\mathbb A^1_V\times_V \mathbb P^1_V,\mathcal O(1))$ a section that vanishing locus is the graph of the canonical immersion $\mathbb A^1_V\to \mathbb P^1_V$.
Since $Z_0$ and $\overline{Z}_1$ are finite over a local scheme, they are semi-local,
and
for any $l\in \mathbb Z$,
there are
invertible sections
\[\gamma_{1,\infty}\in \Gamma(\overline{Z}_1,\mathcal O(l)), \gamma_{Z_0}\in \Gamma(Z_0,\mathcal O(l-1)).\]

We are going to show that \eqref{eq:j*GpairsA1V} has left and right inverse morphisms.

(1) Firstly, we construct the left inverse.
By \cite[III, Corollary 10.7]{Hartshorne-AlG}
for large enough $l\in \mathbb Z$,
there exist sections
\[s_1\in \Gamma(\mathbb A^1\times \mathbb P^1_V,\mathcal O(l)),\;
s_0^\prime\in \Gamma(\mathbb A^1\times \mathbb P^1_V,\mathcal O(l-1))\]
such that
\[\begin{array}{ll}
s_1\big|_{\mathbb A^1\times Z_0}=\gamma_{Z_0}\delta\big|_{\mathbb A^1\times Z_0}, &
s_0^\prime\big|_{\mathbb A^1\times Z_0}=\gamma_{Z_0},\\
s_1\big|_{\mathbb A^1\times (\infty_V\cup\overline{Z}_1)}=\gamma_{1,\infty}, &
s_0^\prime\big|_{\mathbb A^1\times \infty_V}=\gamma_{1,\infty}\delta^{-1}\big|_{\mathbb A^1\times (\infty_V)},\\
& s_0^\prime\big|_{Z(\delta)}=t_\infty^{l-1}.
\end{array}\]

Define the section $s=s_1\lambda+s_0(1-\lambda)$, and
framed correspondences of open pairs over $V$
\[\begin{array}{lll}
c &=& (Z(s),\mathbb A^1\times \mathbb A^1_V\times_V \mathbb A^1_V, s/t_\infty^l, g)\\&\in&
\mathrm{Fr}_1( \mathbb A^1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0) , (\mathbb A^1_V, \mathbb A^1_V-Z_0) ),\\
c^\prime_0&=& (Z(s^\prime),0\times \mathbb A^1_V\times_V (\mathbb A^1_V-Z_0), s_0^\prime/t_\infty^l, g_0^\prime)\\&\in&
\mathrm{Fr}_1(0\times (\mathbb A^1_V,\mathbb A^1_V-Z_0), (\mathbb A^1_V-Z_0, \mathbb A^1_V-Z_0),\\
\tilde c_1&=& (Z(s_1),1\times \mathbb A^1_V\times_V (\mathbb A^1_V-Z_1), s_1/t_\infty^l, \tilde g_1)\\&\in&
\mathrm{Fr}_1(1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0), (\mathbb A^1_V-Z_1,\mathbb A^1_V-Z_1-Z_0)),
\end{array}\]
where $g$, $g_0^\prime$ and $\tilde g_1$ are given by the projections to the second multiplicand.
Define
\[\mathrm{id}^\tau = (Z(\delta),0\times\mathbb A^1_V\times_V\mathbb A^1_V, s_0/t_\infty^l,g_0)\in \mathrm{Fr}_1(0\times(\mathbb A^1_V,\mathbb A^1_V-Z_0),(\mathbb A^1_V,\mathbb A^1_V-Z_0)),\]
$g_0$ is induced by the projection to the second multiplicand.
Then the following equalities hold
\[c\circ i_1 = j \circ \tilde c_1, c\circ i_0 = \mathrm{id}^\tau + e\circ c_0^\prime\]
in ${\mathbb Z\mathrm{F}}_*^\mathrm{pair}(V)$,
where
\[\begin{array}{lll}
i_0\colon 0\times (\mathbb A^1_V,\mathbb A^1_V-Z_0)&\to& \mathbb A^1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0),\\
i_1\colon 1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0)&\to& \mathbb A^1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0),
\end{array}\]
and
\[e\colon (\mathbb A^1_V-Z_0, \mathbb A^1_V-Z_0)\to (\mathbb A^1_V, \mathbb A^1_V-Z_0).\]
Since $G$ is $\mathbb A^1$-invariant
the framed correspondence $\mathrm{id}^\tau$ induces the same endomorphism on $G(\mathbb A^1_V,\mathbb A^1_V-Z_0)$
as $\sigma_{(\mathbb A^1_V,\mathbb A^1_V-Z_0)}$ because of the $\mathbb A^1$-homotopy
\[(Z(\delta),(s_0\lambda + \delta t_0^{l-1})/t_\infty^l,g_0\circ \mathrm{pr})\in \mathrm{Fr}_1(\mathbb A^1\times(\mathbb A^1_V,\mathbb A^1_V-Z_0),(\mathbb A^1_V,\mathbb A^1_V-Z_0)),\]
where $\mathrm{pr}\colon \mathbb A^1\times(\mathbb A^1_V,\mathbb A^1_V-Z_0)\to (\mathbb A^1_V,\mathbb A^1_V-Z_0)$.
Since $G$ is additive, we have $i_0^* c^* = (\mathrm{id}^\tau)^*+ (e c_0^\prime)^*$.
The morphism $c^\prime$ is trivial morphism in the category of pairs $\mathrm{Fr}_+^\mathrm{pair}(V)$, and induces consequently the trivial morphism $(c^\prime)^*$ on the values of the presheaf $G$.
Thus
\[c_1^* j^* = (\mathrm{id}^\tau)^*+ (e c_0^\prime)^*=(\mathrm{id}^\tau)^*=\sigma^*_{(\mathbb A^1_V,\mathbb A^1_V-Z_0)},\]
Since the latter morphism is an automorphism, because $G$ is quasi-stable,
the morphism \eqref{eq:j*GpairsA1V} has the left inverse.

(1) Next, we construct the right inverse to \eqref{eq:j*GpairsA1V}.
Define $V_1=\mathbb A^1_V - Z_1$.
By \cite[III, Corollary 10.7]{Hartshorne-AlG},
for large enough $l\in \mathbb Z$,
there exist sections
\[s_1\in \Gamma(\mathbb A^1\times \mathbb P^1_V,\mathcal O(l)),\;
s_0^\prime\in \Gamma(V_1\times \mathbb P^1_V,\mathcal O(l-1))\]
such that
\[\begin{array}{ll}
s_1\big|_{\mathbb A^1\times_V Z_0}=\gamma_{Z_0}\delta\big|_{\mathbb A^1\times Z_0}, &
s_0^\prime\big|_{V_1\times_V Z_0}=\gamma_{Z_0},\\
s_1\big|_{\mathbb A^1\times (\infty_V\cup\overline{Z}_1)}=\gamma_{1,\infty}, &
s_0^\prime\big|_{V_1\times_V (\infty_V\cup\overline{Z}_1)}=\gamma_{1,\infty}\delta^{-1}\big|_{V_1\times_V (\infty_V\cup\overline{Z}_1))},\\
& s_0^\prime\big|_{Z(\delta)}=t_\infty^{l-1},\\
\end{array}\]

Define the sections
\[s_0 = \delta s_0^\prime, s=s_1\lambda+s_0(1-\lambda),\] and
framed correspondences of open pairs over $V$
\[\begin{array}{lll}
c &=& (Z(s),\mathbb A^1\times V_1\times_V V_1, s/t_\infty^l, g)\\ &\in& \mathrm{Fr}_1( \mathbb A^1\times (V_1,V_1-Z_0) , (V_1, V_1-Z_0) ),\\
c^\prime_0&=& (Z(s^\prime),0\times V_1\times_V (V_1-Z_0), s_0^\prime/t_\infty^l, g_0^\prime)\\ &\in& \mathrm{Fr}_1(0\times (V_1,V_1-Z_0), (V_1-Z_0, V_1-Z_0),\\
\tilde c_1&=& (Z(s_1),1\times \mathbb A^1_V\times_V V_1, s_1/t_\infty^l, g)\\ &\in& \mathrm{Fr}_1(1\times (\mathbb A^1_V,\mathbb A^1_V-Z_0), (V_1,V_1-Z_0)),
\end{array}\]
where $g$, $g_0^\prime$ and $\tilde g_1$ are given by the projections on the second multiplicand.
Define
\[\mathrm{id}^\tau = (Z(\delta),0\times V_1\times_V V_1, s_0,g_0)\in \mathrm{Fr}_1(1\times(V_1,V_1-Z_0),(V_1,V_1-Z_0)),\]
$g_0$ is induced by the projection to the second multiplicand.
Then the following equalities hold in ${\mathbb Z\mathrm{F}}_*^\mathrm{pair}(V)$
\[c\circ i_1 = j \circ \tilde c_1,\; c\circ i_0 = \mathrm{id}^\tau + e c_0^\prime\]
where
\[\begin{array}{lll}
i_0\colon 0\times (V,V_1-Z_0)&\to& \mathbb A^1\times (V_1,V_1-Z_0),\\
i_1\colon 1\times (V,V_1-Z_0)&\to& \mathbb A^1\times (V_1,V_1-Z_0),\\
e\colon (V_1-Z_0, V_1-Z_0)&\to& (V_1,V_1-Z_0).
\end{array}\]
Since $G$ is $\mathbb A^1$-invariant,
the framed correspondence $\mathrm{id}^\tau$
induces the same endomorphism on $G(V_1,V_1-Z_0)$
as $\sigma_{(\mathbb A^1_V,\mathbb A^1_V-Z_0)}$ because of the $\mathbb A^1$-homotopy
\[(Z(\delta),(s_0\lambda + \delta t_0^{l-1})/t_\infty^l,g_0\circ \mathrm{pr})\in \mathrm{Fr}_1(\mathbb A^1\times(V_1,V_1-Z_0),(V_1,V_1-Z_0)),\]
where $\mathrm{pr}\colon \mathbb A^1\times(V_1,V_1-Z_0)\to (V_1,V_1-Z_0)$.
Since $G$ is additive, we have $i_0^* c^* = (\mathrm{id}^\tau)^*+ (e c_0^\prime)^*$.
The object $(V_1-Z_0,V_1-Z_0)$ is trivial in the category of pairs, so the morphism $c^\prime_0$ is trivial, and the induced morphism $(c^\prime_0)^*$ too.
Thus
\[
j^* c_1^* = (\mathrm{id}^\tau)^* + (e c_0^\prime)^* = (\mathrm{id}^\tau)^*=\sigma^*_{(V_1,V_1-Z_0)}
.\]
Since $G$ is quasi-stable, the latter morphism above is an automorphism, hence the morphism \eqref{eq:j*GpairsA1V} has a right inverse.
\end{proof}

\begin{corollary}\label{lm:ZarExA1Vhalffin}
Let $V$ be a local scheme.
Let $Z$ be a closed subscheme in $\mathbb A^1_V$ finite over $V$,
and $X^\prime\subset X$ be two open neighbourhoods of $Z$ in $\mathbb A^1_V$.

Then for any quasi-stable linear framed presheaf \[G\colon {\mathbb Z\mathrm{F}}_*^\mathrm{pair}(V)\to \mathrm{Ab},\]
the canonical morphism
\begin{equation*}G(X^\prime,X^\prime-Z)\to G(X,X-Z),\end{equation*}
is an isomorphism.
See \Cref{def:Smpair,def:Frpair_+,def:ZFstar} for the category of open pairs ${\mathbb Z\mathrm{F}}_*^\mathrm{pair}(V)$.
\end{corollary}
\begin{proof}
By \Cref{lm:pairZarExA1Vhalffin} the claim, for any considered presheaf $G$, holds for $X=\mathbb A^1_V$.
The case of an arbitrary open neighbourhood $X$ of $Z$ follows by the composition
\begin{equation*}G(X^\prime,X^\prime-Z)\simeq G(\mathbb A^1_V,\mathbb A^1_V-Z_0)\simeq G(X,X-Z).\end{equation*}
\end{proof}
\begin{corollary}\label{lm:ZarExA1VhalffinF}
Under the assumptions of \Cref{lm:ZarExA1Vhalffin}
for any quasi-stable framed linear presheaf $F$ on $\mathrm{Sm}_V$,
the square
\[\xymatrix{
F(X^\prime-Z)& F(X^\prime)\ar[l]\\
F(X-Z)\ar[u]& F(X)\ar[l]\ar[u]
}\]
is Cartesian and coCartesian.
\end{corollary}
\begin{proof}
Consider the framed presheaves on the category of open pairs over $V$ given by
\begin{equation}\label{eq:KerCokerF(X->U)}\begin{array}{lcl}
(X,U)&\mapsto&\Ker(F(X)\to F(U)),\\
(X,U)&\mapsto&\Coker(F(X)\to F(U))
\end{array}\end{equation}
The claim follows by \Cref{lm:ZarExA1Vhalffin}, since the canonical open immersion of pairs
\[j\colon (\mathbb A^1_V-Z_1,\mathbb A^1_V-Z_1-Z_0)\to (\mathbb A^1_V, \mathbb A^1_V-Z_0)\]
induces the isomorphism
\begin{equation*}j^*\colon G(\mathbb A^1_V,\mathbb A^1_V-Z_0)\to G(\mathbb A^1_V-Z_1,\mathbb A^1_V-Z_1-Z_0)\end{equation*}
for $G$ being anyone from \eqref{eq:KerCokerF(X->U)}.
\end{proof}

\begin{corollary}\label{lm:ZarExlocA1Vhalffin}
Let $V$ be a local scheme, and $X$ be an open subscheme of $\mathbb A^1_V$.
Let $Z$ be a closed subscheme in $X$ that is finite over $V$, and is local.
Let $z\in Z$ be the closed point.

Then for any quasi-stable framed linear presheaf $F$ over $V$,
the square
\[\xymatrix{
F(X_z-Z)& F(X_z)\ar[l]\\
F(X-Z)\ar[u]& F(X)\ar[l]\ar[u]
}\]
is Cartesian and coCartesian,
where $X_z$ denotes the local scheme of $X$ at $z$.
\end{corollary}
\begin{proof}
The claim follows from \Cref{lm:ZarExA1VhalffinF}.
\end{proof}

\begin{remark}\label{remark:ExLpArg}
The basic principles of the above arguments are like in \cite{Voe-hty-inv}. The arguments were later developed for other types of transfers and at the same time the scheme of some arguments and the techniques were modified.

The use of category of pairs appeared in works \cite{hiWt,phiWtshv,shvNhiWt} on Witt correspondences,
and the use of sections linear bundles and Serre's theorem was suggested to the author by I. Panin.
The latter works did not contain the argument for excision on $\mathbb A^1_U$ over a local henselian $U$.

\cite{Framed} contains some similar principles, namely the use of category of pairs and the use of Serre's theorem for the choice sections,
and contained proof of some excision isomorphism on the relative affine line,
though the statement and the argument of the excision isomorphism on the relative affine line are different from the above one.

The used above argument for the excision on the relative affine line was used in \cite{GWStrHomInv}, and in \cite{ccorrs} later,
for some particular case, namely for the closed subscheme in the relative affine line given by the zero section, but the argument is the same.
\end{remark}

\subsection{Coverings and cohomologies}
\begin{theorem}\label{th:FinSuppLocHensbaseafflinetriviality}
Let $V$ be a local henselian scheme, and $F$ be an $\mathbb A^1$-invariant quasi-stable framed additive presheaf of abelian groups over $V$.
Let $X$ be an open subscheme of $V\times\mathbb A^1$.
Then \[\underline{H}^*_{\mathrm{Nis}^X_Z}(X,F)=0\]
for any closed subscheme $Z$ in $X$ finite over $V$.
\end{theorem}
\begin{proof}
If $Z$ is finite over a local henselian scheme $V$, the connected components of $Z$ are local henselian schemes.
We prove the claim by induction on the number of the connected components of $Z$.
If $Z=\emptyset$, i.e. the number is zero, the claim is a tautology, since the topology $\mathrm{Nis}^X_Z$ is trivial.
We can assume that $Z= Z_0\amalg Z_1$, where $Z_0$ is local henselian, and $Z_1$ has less number of connected components than $Z$.

Consider the long exact sequence
\begin{multline}\label{eq:HnisZXX-Z0X-Z1X-Z}
\dots\to \underline{H}^n_{\mathrm{Nis}^X_Z}(X-Z,F)\to \underline{H}^n_{\mathrm{Nis}^X_Z}(X,F)\to \\
\underline{H}^n_{\mathrm{Nis}^X_Z}(X-Z_0,F)\oplus \underline{H}^n_{\mathrm{Nis}^X_Z}(X-Z_1,F)\to \underline{H}^n_{\mathrm{Nis}^X_Z}(X-Z,F)\to\dots
.\end{multline}
For $Y$ being $X-Z$, or $X-Z_0$,
\[\underline{H}^n_{\mathrm{Nis}^X_Z}(Y,F)=\underline{H}^n_{\mathrm{Nis}^X_{Z_1}}(Y,F)=0,\]
where the first isomorphism holds because $Y\cap Z= Z_1$, and the second isomorphism holds because by the inductive assumption the claim holds for the topology $\mathrm{Nis}^{X}_{Z_1}$.
Note that $\underline{H}^n_{\mathrm{Nis}^X_Z}(X-Z_1,F)=\underline{H}^n_{\mathrm{Nis}^X_{Z_0}}(X-Z_1,F)$.
By \Cref{lm:ZarExlocA1Vhalffin,prop:EtExcision} applied to the closed subscheme $Z_0$, and the local scheme $X_z$, where $z$ is the closed point of $Z_0$,
it follows that
\[\underline{H}^n_{\mathrm{Nis}^X_{Z_0}}(X-Z_1,F)=0.\]
So the claim follows because of the long exact sequence \eqref{eq:HnisZXX-Z0X-Z1X-Z}.
\end{proof}

\begin{corollary}\label{cor:NisFalongA1hominv}
Let $S$ be a scheme.
Consider the topology $\underline\mathrm{Nis}^S\cup\mathrm{Fin}_S$ on $\mathrm{Sm}_{S\times\mathbb A^1}$ generated by the topologies $\underline\mathrm{Nis}^S$, and $\mathrm{Fin}_S$, see \Cref{def:underNisV}, and \Cref{def:FinS} respectively.
Then the projection $S\times\mathbb A^1\to S$ induces the canonical isomorphism \[H^*_{\underline\mathrm{Nis}^S\cup\mathrm{Fin}_S}(S\times\mathbb A^1,F)\simeq H^*_\mathrm{Nis}(S,F)\]
for any $\mathbb A^1$-invariant quasi-stable framed additive presheaf of abelian groups $F$ on $\mathrm{Sm}_S$.
\end{corollary}
\begin{proof}
Consider
the presheaves on the small \'etale site over $S$
\[\begin{array}{lclcl}
U&\mapsto& h^q_\mathrm{fnis}(U)&=& H^q_{\mathrm{fin}_U}(U\times\mathbb A^1,F)\\
U&\mapsto&h^q_\mathrm{nis}(U)&=& H^q_{\mathrm{nis}}(U\times\mathbb A^1,F).\end{array}\]
By \Cref{th:FinSuppLocHensbaseafflinetriviality} the canonical morphism of presheaves $h^q_\mathrm{fnis}\to h^q_\mathrm{nis}$ is a Nisnevich local isomorphism.
Then the morphism of
the spectral sequences
\[
\begin{array}{lcl}
H^p_{\mathrm{nis}}(S, h^q_\mathrm{fnis}) & \Rightarrow& H^{p+q}_{\underline\mathrm{Nis}^S\cup\mathrm{Fin}_S}(S\times\mathbb A^1,F) \\
H^p_{\mathrm{nis}}(S, h^q_\mathrm{nis}) & \Rightarrow& H^{p+q}_{\mathrm{nis}}(S\times\mathbb A^1,F)
\end{array}
\]
is an isomorphism.
\end{proof}

\section{Finite support injectivity}\label{sect:InjThbyFinSupHom}

In this section,
we apply the result of the previous section
to formulate some enough criterion used in the next sections
The criterion guaranties that the restriction homomorphism
\[H^*_\mathrm{nis}(X,F_\mathrm{nis})\to H^*_\mathrm{nis}(\eta,F_\mathrm{nis})\]
of the Nisnevich cohomologies of an $\mathbb A^1$-invariant quasi-stable framed linear presheaf $F$
from an irreducible scheme $X$ to the generic point $\eta$ is injective, see
We use it in the forms of \Cref{cor:preimFinFrCintr->Inj}
and \Cref{cor:preimFinCovFrA1hom}.

\begin{proposition}\label{prop:FrA1HomFinCovPreimage}
Let $S$ be a scheme over a field $k$, $X\in \mathrm{Sm}_k$, and $Z$ be a closed subscheme in $X$.
Assume there is a framed correspondence $c\in \mathrm{Fr}_N(S\times\mathbb A^1, X)$ such that $c^{-1}(Z)$ is finite over $S$.
Denote $c_0=c\circ i_0$, $c_1=c\circ i_1$, where $i_0\colon S\times 0\to S\times\mathbb A^1$, $i_1\colon S\times 1\to S\times\mathbb A^1$.
Then the morphisms
\[c_0^*, c_1^*\colon \underline{H}^i_{\mathrm{Nis}_{Z}^{X}}(X,F)\to \underline{H}^i_\mathrm{nis}(S,F),\]
are equal for all $i\geq 0$.
\end{proposition}
\begin{proof}
The claim follows because of the commutative diagram
\[
\xymatrix{
& \underline{H}^*_{\mathrm{Nis}_{Z}^{X}}(X,F)\ar[dl]\ar[d]\ar[dr] & \\
\underline{H}^*_{\mathrm{Nis}_{c^{-1}_0(Z)}^{S\times 0}} (S\times 0, F)\ar[d] &
\underline{H}^*_{\mathrm{Nis}_{c^{-1}(Z)}^{S\times\mathbb A^1}} (S\times\mathbb A^1, F)\ar[l]\ar[r]\ar[d] &
\underline{H}^*_{\mathrm{Nis}_{c^{-1}_1(Z)}^{S\times 1}} (S\times 1, F)\ar[d]\\
\underline{H}^*_\mathrm{nis} (S\times 0, F) &
\underline{H}^*_{\underline{\mathrm{Nis}}^S\cup\mathrm{Fin}_S} (S\times\mathbb A^1, F)\ar[l]_{\cong}\ar[r]^{\cong} &
\underline{H}^*_\mathrm{nis} (S\times 1, F)
,}
\]
where the topologies in the second row are defined in \Cref{def:NisXZ}, $\underline{\mathrm{Nis}}_S$ is defined in \Cref{def:underNisV}, and $\mathrm{Fin}_S$ in \Cref{def:FinS}.
Here
the arrows between the first two rows are given by \Cref{def:CohSupTransfer},
the arrows from the second row to the third one are inverse images for the canonical morphisms of sites.
The commutativity of the diagram
is provided by \Cref{prop:NisZFr,prop:CommCohSupTrans},
and the isomorphisms in the last row are proven by \Cref{cor:NisFalongA1hominv}.
\end{proof}

\begin{corollary}\label{cor:preimFinCovFrA1hom}
Let $S$ be a scheme over a field $k$, and $X\in \mathrm{Sm}_k$, and $Z$ be a closed subscheme in $X$.
Assume that
\begin{itemize}[leftmargin = 10pt]
\item[-]
there is a framed correspondence $c\in {\mathbb Z\mathrm{F}}_N(S\times\mathbb A^1, X)$ given by the difference of explicit framed correspondences $c=c^+-c^-$
such that schemes $(c^+)^{-1}(Z)$ and $(c^-)^{-1}(Z)$ are finite over $S$.
\item[-]
for some $\widetilde c_1\in {\mathbb Z\mathrm{F}}_N(S,X-Z)$,
\[c_1=j\circ \widetilde c_1,\]
where
$c_1=c\circ i_1$, $i_1\colon S\to S\times\mathbb A^1$ is the unit section,
$j\colon X-Z\to X$ is the canonical immersion.
\end{itemize}
Then for any additive $\mathbb A^1$-invariant quasi-stable framed presheaf of abelian groups $F$ on $\mathrm{Sm}_k$
the morphisms
\[c_0^*\colon \underline{H}^i_{\mathrm{Nis}^X_Z}(X,F)\to \underline{H}^i_\mathrm{nis}(X,F)\]
induced by $c_0$ vanish for all $i\geq 0$.
\end{corollary}
\begin{proof}
The claim follows, since homomorphism
$j^*\colon \underline{H}^i_{\mathrm{Nis}^X_Z}(X,F)\to \underline{H}^i_{\mathrm{nis}}(X-Z,F)$ given by the inverse image vanishes,
and $c_0^*=\widetilde c_1^*j^*$ by \Cref{prop:FrA1HomFinCovPreimage}.
\end{proof}
\begin{corollary}\label{cor:preimFinFrCintr->Inj}
Let $X\in \mathrm{Sm}_k$.
If the assumption of \Cref{cor:preimFinCovFrA1hom} holds for any $Z$ in of positive codimension, then \[\underline{H}^i_\mathrm{nis}(X,F)\cong 0,\]
for any additive $\mathbb A^1$-invariant quasi-stable framed presheaf $F$ of abelian groups on $\mathrm{Sm}_k$.
\end{corollary}
\begin{proof}
Let $\Theta$ denote the scheme that is the union of generic points of $X$.
Consider the topology $\mathrm{Nis}^{X/\Theta}:=\bigcup_{Z}\mathrm{Nis}^X_Z$, where $Z$ runes over all  closed subschemes $Z$ in $X$ of positive codimension.
Then $\underline{H}^i_{\mathrm{Nis}^{X/\Theta}}(X,F_{\mathrm{Nis}^{X/\Theta}})\cong \varinjlim_Z \underline{H}^i_{\mathrm{Nis}^X_Z}(X,F_{\mathrm{Nis}^X_Z})$.
By \Cref{cor:preimFinCovFrA1hom} we have that the canonical morphisms
$\underline{H}^i_{\mathrm{Nis}^X_Z}(X,F_{\mathrm{Nis}^X_Z})\to \underline{H}^i_{\mathrm{Nis}^X_Z}(X,F_{\mathrm{Nis}^X_Z})$
are trivial for all closed subschemes $Z$ of positive codimension.
Hence  the canonical morphism \begin{equation}\label{eq:HNisXThetaHNisX}\underline{H}^i_{\mathrm{Nis}^{X/\Theta}}(X,F_{\mathrm{Nis}^X_Z})\to \underline{H}^{i}_{\mathrm{Nis}^X}(X,F_{\mathrm{Nis}^X})\end{equation} is trivial.

$\mathrm{Nis}^{X/\Theta}$ is the maximal subtopology of $\mathrm{Nis}^X$ that is trivial on $\Theta$ because any covering that admits a lifting on $\Theta$ admits a lifting on some Zariksi neighbourhood.
Denote by ${H}^i_{\mathrm{Nis}^{X/\Theta}}(-,F_{\mathrm{Nis}^{X/\Theta}})_{\mathrm{Nis}^X}$ the $\mathrm{Nis}^X$-sheafification of the presheaf of $\mathrm{Nis}^{X/\Theta}$-cohomologies, where $i\in \mathbb Z$.
Because of the spectral sequence for the morphism of sites given by $\mathrm{Nis}^{X/\Theta}$ and $\mathrm{Nis}$
\[{H}^i_{\mathrm{Nis}^{X/\Theta}}(X,{H}^j_{\mathrm{Nis}^X}(-,F_{\mathrm{Nis}^X})_{\mathrm{Nis}^{X/\Theta}}) \Rightarrow H^{i+j}_{\mathrm{Nis}^X}(X,F_{\mathrm{Nis}^X}),\]
and since $\underline{H}^i_{\mathrm{Nis}^X}(\Theta,F)=\underline{H}^i_{\mathrm{nis}}(\Theta,F)\cong 0$,
it follows that the canonical morphism \eqref{eq:HNisXThetaHNisX}
is an isomorphism.

Thus $\underline{H}^i_{\mathrm{Nis}^X}(X,F)\cong 0$
\end{proof}

\section{Compactified correspondences and cohomoligies at $\infty$ on $\mathbb A^1_V$.}\label{sect:homotpiesatinfinity}

In this section, we prove the triviality of the cohomologies at infinity on the relative affine line $\mathbb A^1_V$ over a local essentially smooth scheme $V$, \Cref{th:GenFifLocEssSmSchemeonedimbasetriviality}. To do this we construct a framed $\mathbb A^1$-homotopy 'with finite supports' in \Cref{cor:contractinggenpointhomoveretaSmX} that allows to apply \Cref{cor:preimFinFrCintr->Inj}.

\subsection{Compactified correspondences}\label{subsect:dcopmactifiedframedcorrespondece}

\begin{definition}\label{def:reldimFr}
A \emph{$d$-dimensional framed correspondence} $\Phi$ from an affine $U$-scheme $X$ to a $U$-scheme $Y$ over a scheme $U$ is a set of data $(S,\varphi,g)$, where
\begin{itemize}
\item[(1)]
$S$ is an $X$-scheme of relative dimension $d$ over $X$ equipped with a closed embedding $S\to \mathbb A^n_X$,
\item[(2)]
$\varphi\in \mathcal O((\mathbb A^n_X)^h_{S})^{\oplus n-d}$ is a set of regular functions such that $S= Z(\varphi)$, and
\item[(3)]
$g\colon (\mathbb A^n_X)^h_{S}\to Y$ is a morphism of $S$-schemes.
\end{itemize}
\end{definition}

\begin{definition}\label{def:compactifiedfocusedFr}
Let $B$ a local irreducible one-dimensional scheme, $z\in B$ be the closed point, $\eta\in B$ be the generic point.

Let $V\in \mathrm{Sch}_B$, $T\in \mathrm{Sch}_\eta$, and $V_\eta=V\times_B \eta$.
Assume we are given with
\begin{itemize}
\item[(1)]
a projective equi-dimensional scheme $\overline{X}$ over $V$ of pure relative dimension $d$,
and an ample bundle $\mathcal O(1)$ on $\overline{X}$ with a section $t_\infty\in \Gamma(\overline{X}, \mathcal O(1))$
such that the closed subscheme $X_\infty=Z(t_\infty)$ in $\overline{X}$ has positive relative codimension over $V$;
\item[(2)]
a $d$-dimensional framed correspondence
$\Phi=(X_\eta, \varphi, g)$ from $V_\eta$ to $T$ over $\eta$,
\item[(3)]
an isomorphism
$X_\eta\cong(\overline{X}-X_\infty)\times_B \eta$;
\end{itemize}
The set of data
$(\overline{X},X_\infty,\Phi)$ is called
\emph{$V$-focused compactified $d$-dimensional $\eta$-framed correspondence} from $V_\eta$ to $T$.
\end{definition}

\begin{definition}\label{def:smoothcontain}
Let $B$ a local irreducible one-dimensional scheme, $z\in B$ be the closed point, $\eta\in B$ be the generic point.
Let $V\in \mathrm{Sch}_B$, $T\in \mathrm{Sch}_\eta$, and $V_\eta=V\times_B \eta$.

Let $r\colon V_\eta\to T$ be a morphism of schemes.
Let $(\overline{X},X_\infty,\Phi)$ be
$V$-focused compactified $d$-dimensional $\eta$-framed correspondence from $V_\eta$ to $T$.

We say that $(\overline{X},X_\infty,\Phi)$ \emph{$V$-smoothly contains} $r$ if
there is a closed subscheme $\Gamma$ in $X=\overline{X}-X_\infty$
such that \begin{itemize}
\item[(1)] the canonical projection $X\to V$ induces the isomorphism $\gamma\colon \Gamma\cong V$, and
the morphism $X\to V$ is smooth over $\Gamma$,
\item[(2)] $g \circ \gamma^{-1}_\eta=r$, where $\gamma_\eta\colon \Gamma\times_B \eta\cong V_\eta$.
\end{itemize}
\end{definition}

\subsection{Universal endo-correspondence and contracting correspondence of dimension one.}

\begin{proposition}\label{prop:eqXsubcurve}
Let $B$ be a local scheme.
Let $Z\in B$ be a closed subscheme, $U\subset B$ be the open complement.
Let $\overline{X}$ be a projective scheme over $B$, and $\mathcal O(1)$ denote an ample bundle on $\overline{X}$.

Let $Y\to \overline{X}$ and $D\to \overline{X}$ be closed immersions,
$F=\overline{F_U}$ be the the closure in $\overline{X}$ of the scheme $F_U=(Y\cap D)\times_B U$, and $\Delta\colon B\to X$ be a section
such that
\begin{itemize}
\item[(1)] $\overline{X}$ is equi-dimensional scheme over $B$ of pure relative dimension $n$;
\item[(2)]
$Y$ and $D$ have positive codimension in the closed fibre;
\item[(3)] $F$ is of codimension at least 2 in the closed fibre;
\item[(4)]
the $B$-scheme $\overline{X}$ is smooth over the subscheme $\Delta(B)$, and $\Delta(B)\cap D=\emptyset$.
\end{itemize}

Then for large enough $d_i$, there are sections
$s_i\in \Gamma(\overline{X},\mathcal O(d_i))$, $i=2,\dots n$, such that
\begin{itemize}
\item[(1)]
the vanishing locus $\overline{C}=Z(s_2,\dots, s_n)$ is of pure relative dimension one over $B$;
\item[(2)]
$\Delta(B)\subset \overline{C}$, and
the morphism $\overline{C}\to B$ is smooth over $\Delta(B)$;
\item[(3)]
$\overline{C}\cap D$ and $\overline{C}\cap Y$ are finite over $B$;
\item[(4)]
$(\overline{C}\cap Y\cap D)\times_B U=\emptyset$.
\end{itemize}
\end{proposition}
\begin{proof}
Step 1)
Consider the closed subscheme $\Delta(B)$ in $\overline{X}$,
and the first order thickening $\Delta(B)_2=Z(\mathcal I^2(\Delta(B))$.
Since the morphism $\overline{X}\to B$ is smooth over $\Delta(B)$, and $\Delta(B)$ is local, there is a trivialisation $N_{\Delta(B)/(\overline{X})}\simeq \mathbf 1^n_{\Delta(B)}$.
Since $\Delta(B)$ is local, there is a trivialisation $\mathcal O(1)\big|_{\Delta(B)}\simeq \mathcal O(\Delta(B))$.
Hence for any $d$, there are sections
\[\gamma_1,\dots \gamma_n\in \Gamma(\Delta(B)_2,\mathcal O(d))\] such that
\[Z(\gamma_1,\dots,\gamma_n)=\Delta(B).\]

Step 2)
Consider the schemes
\begin{equation}\label{eq:ovXzYzDzYDz}
\overline{X}\times_B z, \quad \overline{X}\times_B z, \quad Y\times_B z, \quad D\times_B z, \quad F\times_B z.
\end{equation}
Let $N$ be an irreducible component of anyone of schemes \eqref{eq:ovXzYzDzYDz}.
Let $S\subset \overline{X}\times_B z$ be a finite set of closed points
such that for any such $N$, the intersection $N\cap S$ is not empty.
Choose a section
\begin{equation}\label{eq:s_nDelta(B)S}
s_n\in \Gamma(\overline{X},\mathcal O(d)),\quad s_n\big|_{\Delta(B)_2}=\gamma_n,\quad s_n\big|_S\in \mathcal O(d_n)^\times.
\end{equation}
Define \[\overline{X}_{n-1}=Z(s_n)\subset\overline{X}=:\overline{X}_n.\]
Then it follows by the second condition on $s_n$ in \eqref{eq:s_nDelta(B)S} that
$\dim_z (\overline{X}_{n-1}\times_B z)=n-1$, and hence \begin{equation}\label{eq:dimBX(n-1)}\dim_B \overline{X}_{n-1}=n-1.\end{equation}
Moreover, if follows by the same condition that
\begin{equation}\label{eq:codimYzDzYDz(n-1)}
\begin{array}{lllll}
\codim_{\overline{X}_{n-1}\times_B z }(Y\cap \overline{X}_{n-1})&\times_B& z&\geq& 1, \\
\codim_{\overline{X}_{n-1}\times_B z} (D\cap \overline{X}_{n-1})&\times_B& z&\geq& 1, \\
\codim_{\overline{X}_{n-1}\times_Bz } (F\cap \overline{X}_{n-1})&\times_B& z&\geq& 2,
\end{array}
\end{equation}
and consequently
\begin{equation}\label{eq:codimYD(n-1)}
\codim_{\overline{X}_{n-1}\times_Bz } (F_{n-1}\cap \overline{X}_{n-1})\times_B z\geq 2,
\end{equation}
where $F_{n-1}= \overline{F_{U,n-1}}$, $F_{U,n-1}=(Y\cap D\cap \overline{X}_{n-1})\times_B U$.
Also we conclude that $\overline{X}_{n-1}$ is smooth on $\Delta(B)$ over $B$ by the first condition on $s_n$ in \eqref{eq:s_nDelta(B)S}.

By induction we construct sections $s_i\in\Gamma(\overline{X},\mathcal O(d_i))$ for $i=n,\dots 2$ and get the schemes
$X_{i}$, $Y_{i}$, $D_{i}$, $F_{i}$ satisfying conditions
\begin{gather}
\label{eq:dimBX(i)}
\dim_B \overline{X}_{i}=i, \\
\label{eq:codimYz(i)}
\codim_{\overline{X}\times_Bz}(Y_i\times_B z)\geq 1,\\
\label{eq:codimDz(i)}
\codim_{\overline{X}\times_Bz}(D_i\times_b z)\geq 1, \\
\label{eq:codimYD(i)}
\codim_{\overline{X}\times_Bz}(F_i\times_B z)\geq 2,
\end{gather}
where $Y_i=Y\cap\overline{X}_i$, $D_i=D\cap\overline{X}_i$, $F_i=\overline{F}_{i,U}$, $F_{i,U}=(Y\cap D\cap \overline{X}_i)\times_B U$,
and such that $\Delta\subset\overline{X}_i$, $\overline{X}_{i}\to B$ is smooth over $\Delta$.

Then the scheme $\overline{C}=\overline{X}_1=Z(s_2,\dots s_n)$
satisfies the claim of the proposition.
Indeed, assertion (1) in the proposition holds by \eqref{eq:dimBX(i)}.
Assertion (2) holds by the reason explained above.
Assertion (3) follows by \eqref{eq:codimYz(i)}, and \eqref{eq:codimDz(i)}.
Hence $F_{1}$ is a finite scheme over $B$, and
since by \eqref{eq:codimYD(i)}
$F_1\times_B z=\emptyset$, it follows by the Nakayama lemma that $F_1=\emptyset$.
Whence it follows that
\[(Y\cap D\cap\overline{C})\times_B U=F_1\times_B U=\emptyset.\]
So assertion (4) holds.
\end{proof}

\begin{lemma}\label{lm:XetafDimFrS}
Let $B$ be a one-dimensional irreducible local scheme, $z\in B$ be the closed point, $\eta\in B$ be the generic point.
Let $X\in \mathrm{Sm}_B$, $x\in X_z=X\times_B z$.
Let $d=\dim_{B}^{x} X$ be the relative dimension of $X$ at $x$ over $B$.
Let $Z$ be a closed subscheme of positive codimension in $X_\eta=X\times_B \eta$.

Then there is a $d$-dimensional
${X_{(x)}}$-focused compactified
$\eta$-framed correspondence
$(\overline{S},S_\infty,\Phi)$
from $({X_{(x)}})_\eta={X_{(x)}}\times_B \eta$ to $X_\eta=X\times_n \eta$,
where $\Phi=(S_\eta, \varphi, \mathrm{val})$, $S_\eta = S\times_B \eta$, $S = S-S_\infty$,
such that
\begin{itemize}
\item[(1)]
$(\overline{S},S_\infty,\Phi)$ ${X_{(x)}}$-smoothly contains the canonical map $({X_{(x)}})_\eta\to X_\eta$;
\item[(2)]
$\mathrm{val}^{-1}(Z)$ is of positive relative codimension in $\overline{X}_\eta$,
\item[(3)]
and \begin{equation}\label{eq:codimzovYXinfclaim}\codim_{\overline{S}\times_{X_{(x)}} x}(\overline{YX_\infty}\times_{X_{(x)}} x)\geq 2,\end{equation}
where $\overline{YX_\infty}$ is the closure of $(\mathrm{val}^{-1}(Z)\cap S_\infty)\times_B \eta$ in $\overline{S}$.
\end{itemize}
\end{lemma}
\begin{proof}

Step 0)
We can
assume without loss of generality that
$X\in \mathrm{SmAff}_B$, $X$ is irreducible, $\dim_B X=d$, $T_{X_\eta/\eta}\simeq \mathbf 1_{X_\eta}^d$, where $X_\eta =X\times_B\eta$, because we can
shrink the given scheme $X$ to a Zariski neighbourhood of $x$ with this properties.

Step 1)
Let
$X\to \mathbb A^{N^\prime}_B$ be a closed immersion, $N^\prime\in \mathbb Z$.
Let $\overline{X}$ denote the closure of $X$ in $\mathbb P^{N^\prime}_B$.
Then $\overline{X}$ is irreducible projective scheme over $B$.
Since $\dim B=1$,
the scheme $\overline{X}$ is equi-dimensional over $B$
by \ref{prop:OnedimBaseEquidimSch}.
Denote \[\overline{X}_\eta=\overline{X}\times_B \eta, \quad \overline{X}_z=\overline{X}\times_B z.\]
Consider the closed complement
$X_{\infty,\eta}=\overline{X}_\eta \setminus X_\eta$, and
denote by $X_{\overline{\infty}}$ the closure of $X_{\infty,\eta}$ in $\overline{X}$.
Note that
\[X_{\overline{\infty}}\times_B\eta = X_{\infty,\eta} \text{ and } X_{\overline{\infty}}\subset \overline{X}\setminus X.\]
Since $\dim B=1$,
it follows by \Cref{prop:OnedimBaseEquidimSch},
that $X_{\overline{\infty}}$ is equi-dimensional over $B$, and \[\dim (X_{\overline{\infty}} \times_B z)<\dim_B X=d.\]
So $X_{\overline{\infty}}$ is of positive relative codimension in $\overline{X}$ over $B$.

Step 2)
Since $\overline{X}$ is a projective scheme over $B$, there is an ample bundle $\mathcal O(1)$ on $\overline{X}$.
Let $F\subset \overline{X}_z$ be a finite set of
closed points that intersects non-emptily every irreducible component of
$\overline{X}_z-((X_{\overline{\infty}}\times_B z) \cup x)$.
Then for a large enough $d$, there is a section $t_\infty\in \Gamma(\overline{X},\mathcal O(d))$
\[t_\infty\big|_{X_{\overline{\infty}}}=0, t_\infty\big|_S\in \Gamma(F,\mathcal O(d)^\times).\]

Since $\dim_B ((X_{\overline{\infty}}\times_B z)\cup x)<\dim_B X$, it follows that
$\overline{X}_z$ equals the closure of $\overline{X}_z-((X_{\overline{\infty}}\times_B z) \cup x)$.
Hence $F$ intersects each irreducible component of $\overline{X}_z$.
Then \[Z(t_\infty)\supset X_{\overline{\infty}},\quad \dim_B Z(t_\infty) = n-1,\quad n=\dim_B X.\]
Let us
denote \[X^\prime_{\underline{\infty}}=Z(t_\infty),\quad X^\prime=\overline{X}-X^\prime_{\underline{\infty}},\]
and redenote $\mathcal O(1):=\mathcal O(d)$.

Step 3)
By step (2)
$X^\prime$ is an affine Zariski neighbourhood of $x$ in the projective $B$-scheme $\overline{X}$,
the complement
$X^\prime_{\underline{\infty}}$ is of positive relaitve codimension in $\overline{X}$ over $B$,
and $X^\prime_{\underline{\infty}}\supset X_{\overline{\infty}}$.
Consequently,
$X^\prime$ is smooth at $x$ over $B$,
and
there is the canonical open immersion
\[X^\prime_\eta\to X_\eta,\]
where $X^\prime_\eta =X^\prime\times_B \eta$.

Step 4)
Since the tangent bundle of $X_\eta$ is trivial,
we get
\[X^\prime_\eta\in \mathrm{SmAff}_\eta, T_{X^\prime_\eta}\simeq \mathbb 1^d_{X^\prime_\eta}.\]
Then
by \Cref{lm:framedscheme}
there is a
$d$-dimensional framed correspondence
\[(X^\prime_\eta)^\mathrm{fr}=(X^\prime_\eta,f_{d+1},\dots f_{N}, \tilde j)\]
where
$\tilde j\colon (\mathbb A^N_\eta)^h_{X^\prime_\eta}\to X_\eta$ is such that
$\tilde j\big|_{X^\prime_\eta}\colon X^\prime_\eta\to X_\eta$
is the canonical immersion.

Summarising with Step (3) we get $d$-dimensional
${X_{(x)}}$-focused compactified $\eta$-framed correspondence
\[
(\overline{X}, X^\prime_{\underline{\infty}}, (X^\prime_\eta)^{\mathrm{fr}} )\]
from $\eta$ to $X_\eta$.
The base change along the map ${X_{(x)}}\to B$ gives
the $d$-dimensional ${X_{(x)}}$-focused compactified $\eta$-framed correspondence
$(\overline{S},S_\infty, \Phi)$
from ${X_{(x)}}$ to $X$,
where $\overline{S}=\overline{X}\times_B {X_{(x)}}$, $S_\infty=X^\prime_{\underline{\infty}}\times_B {X_{(x)}}$,
and $\Phi$ is the base change of $(X^\prime_\eta)^{\mathrm{fr}}$.
This gives the initial claim of the lemma, we proceed with (1),(2), and (3).

Step 5)
The composite morphism
\[{X_{(x)}}\to X\to \overline{X}\]
induces the section $\Delta\colon {X_{(x)}}\to\overline{S}$.
Since the morphism $X\to B$ is smooth,
the morphism $\overline{S}\to {X_{(x)}}$ is smooth over $\Delta$.
Thus $(\overline{S},S_\infty, \Phi)$
${X_{(x)}}$-smoothly contains the canonical map $({X_{(x)}})_\eta\to X_\eta$.
This gives claim (1).

Step 6)
Since $Z$ is of positive codimension in $X_\eta$,
it follows that
$\mathrm{val}^{-1}(Z)=({X_{(x)}})_\eta\times j^{-1}(Z)=({X_{(x)}})_\eta\times (Z\cap X^\prime_\eta)$
is of positive codimension in $X^\prime_\eta$.
This gives claim (2).

Step 7)
Finally, to prove inequality \eqref{eq:codimzovYXinfclaim} of claim (3) we note that
$\mathrm{val}^{-1}(Z)\cap S_\infty=(\overline Z\cap X^\prime_{\underline{\infty}})\times_B (B-\eta)\times_B{X_{(x)}}$.
The claim follows,
since
\[\codim_{\overline{X}_\eta}(\overline{Z}\cap X^\prime_{\underline{\infty}})\times_B \eta\leq 2,\]
where $\overline Z$ is the closure of $Z$ in $\overline{X}$,
and consequently
\[\codim_{\overline{X}_z}(\overline{(\overline{Z}\cap X^\prime_{\underline{\infty}})\times_B \eta)}\times_Bz\leq 2.\]
\end{proof}

\begin{lemma}\label{lm:DimDim1}
Suppose
there is a $d$-dimensional ${X_{(x)}}$-focused
compactified framed correspondence
\[(\overline{S},S_\infty,\Phi),\;\; \Phi= (S_\eta, \varphi, \mathrm{val}_{S_\eta}),\;\;
S_\eta=(\overline{S}-S_\infty)\times_B\eta,\]
from the scheme $\mathcal V = ({X_{(x)}})_\eta={X_{(x)}}\times_B \eta$ to the scheme $X_\eta=X\times_B \eta$
such that \begin{itemize}
\item[(1)] $(\overline{S},S_\infty,\Phi)$ smoothly contains the canonical map $({X_{(x)}})_\eta\to X_\eta$,
\item[(2)] the closure $\overline{\mathcal Z}=\overline{\mathrm{val}^{-1}_S(Z)}$ of the scheme $\mathcal{Z}=\mathrm{val}^{-1}_S(Z)$ in $\overline{S}$ is of positive codimension over ${X_{(x)}}$,
the closure $\overline{YX_\infty}$ of $(\overline{\mathcal Z}\cap S_\infty)\times_B \eta$ in $\overline{S}$ has codimension at least 2 in the closed fibre over ${X_{(x)}}$,
\begin{equation}\label{eq:codimzovYXinf}\codim_{\overline{S}\times_{X_{(x)}} x}(\overline{YX_\infty}\times_{X_{(x)}} x)\geq 2.\end{equation}
\end{itemize}

Then there is a
one-dimensional ${X_{(x)}}$-focused compactified framed correspondence
\[(\overline{C},C_\infty,\Psi), \Psi=(C_\eta, \psi, \mathrm{val}_C), C_\eta = (\overline{C}-C_\infty)\times_B \eta,\]
from $\mathcal V = ({X_{(x)}})_\eta$ to $X_\eta$
such that
\begin{itemize}
\item[(1)] ${X_{(x)}}$-smoothly contains the canonical map $({X_{(x)}})_\eta\to X_\eta$,
\item[(2)] the closure $\overline{\mathrm{val}^{-1}_C(Z)}$ of $\mathrm{val}^{-1}_C(Z)$ in $\overline{C}$ is finite over ${X_{(x)}}$, and
\[(\overline{\mathrm{val}^{-1}_C(Z)}\cap C_\infty)\times_B \eta=\emptyset.\]
\end{itemize}
\end{lemma}
\begin{proof}
Denote by $\mathcal O(1)$ the ample line bundle on the projective scheme $\overline{S}$, and by $t_\infty$ the section on $\overline{S}$
such that $Z(t_\infty) = S_\infty$ provided by \Cref{def:compactifiedfocusedFr}.
We are going to apply \Cref{prop:eqXsubcurve}
to the scheme $\overline{S}$, and the subschemes \[D=S_\infty, \quad
Y=\overline{\mathrm{val}^{-1}_S(Z)}, \quad \Delta=\Delta({X_{(x)}}).\]
The assumptions (1) and (4) of \Cref{prop:eqXsubcurve} follow by the properties of the scheme $\overline{S}$, provided by \Cref{def:compactifiedfocusedFr} and the assumption (1) above.
The assumption (2) of \Cref{prop:eqXsubcurve} follows by the properties of $S_\infty$ provided by \Cref{def:compactifiedfocusedFr},
and by the assumption (2) above.
The assumption (3) of \Cref{prop:eqXsubcurve} holds by \eqref{eq:codimzovYXinf}

Then by \Cref{prop:eqXsubcurve}
there is a vector of sections $(s_2,\dots s_d)$, where $s_i\in \Gamma(\overline{S},\mathcal O(l))$ for a large enough $l$,
such that
\begin{itemize}\item[(1)]
the scheme \[\overline{C}=Z(s_2,\dots s_d)\subset \overline{S}\]
is of relative dimension $1$ over ${X_{(x)}}$,
\item[(2)]
there is the inclusion $\Delta({X_{(x)}})\subset \overline{C}$, and
the morphism $\overline{C}\to {X_{(x)}}$ is smooth over $\Delta({X_{(x)}})$,
\item[(3)]
$\overline{\mathrm{val}^{-1}_S(Z)}\cap \overline{C}$ is finite over ${X_{(x)}}$,
$C_\infty=S_\infty\cap \overline{C}$ is finite over ${X_{(x)}}$,
$(\overline{\mathrm{val}^{-1}_S(Z)} \cap C_\infty)\times_B \eta=\emptyset$.
\end{itemize}
By assumption we are given with a closed embedding $S_\eta\to \mathbb A^N_{\mathcal V}$,
a regular map $\mathrm{val}\colon (\mathbb A^N_{\mathcal V})^h_{S_\eta}\to X_\eta$,
and a vector of regular functions $\varphi\in \mathcal O((\mathbb A^N_{\mathcal V})^h_{S_\eta})^{\oplus N-d}$ such that $Z(\varphi)=S_\eta$.
By point (1) in the latter list above there are canonical embeddings $\overline{C}\to \overline{S}$, $C_\eta\to S_\eta$, where $C_\eta=C\times_B\eta$, $C=\overline{C}-C_\infty$. Define $\mathrm{val}_{C_\eta}\colon (\mathbb A^N_{\mathcal V})^h_{C_\eta}\to X_\eta$ as the composite morphism
\[(\mathbb A^N_\mathcal V)^h_{C_\eta}\to (\mathbb A^N_\mathcal V)^h_{S_\eta}\xrightarrow{\mathrm{val}_{S_\eta}} X_\eta.\]
Choose liftings $\varphi_j\in \mathcal O(\mathbb A^N_\mathcal V)$, $j=2,\dots ,d$, of the functions $s_j/t_\infty^l\in \mathcal O(S_\eta)$,
and denote by the same symbols the inverse images on $(\mathbb A^N_\mathcal V)^h_{C_\eta}$.
Thus $\Psi=(C_\eta, \varphi_2,\dots \varphi_d, \varphi, \mathrm{val}_{C_\eta})$ is a one-dimensional framed correspondence from $\mathcal V$ to $X_\eta$.
Moreover, $(\overline{C},C_\infty, \Psi)$ is an ${X_{(x)}}$-focused compactified framed correspondence because $\overline{C}$ is of pure relative dimension 1 and $C_\infty$ is finite over ${X_{(x)}}$ respectively by (1) and (3) in the list above.

By point (2) assertion (2) follows.
The assertion (3) follows by point (3) in the list above,
because $\overline{\mathrm{val}^{-1}_C(Z)}\subset \overline{\mathrm{val}^{-1}_S(Z)}\cap\overline{C}$ is finite over ${X_{(x)}}$,
and $\overline{\mathrm{val}^{-1}_C(Z)}\cap C_\infty\subset \overline{\mathrm{val}^{-1}_S(Z)}\cap C_\infty$ has the empty fiber over $\eta$.
\end{proof}

\begin{proposition}\label{prop:DimDim1}
Let $B$ be a one-dimensional irreducible local scheme, $z\in B$ be the closed point, $\eta\in B$ be the generic point.
Let $X\in \mathrm{Sm}_B$, $x\in X_z=X\times_B z$.
Let $d=\dim_{B}^{x} X$ be the relative dimension of $X$ at $x$ over $B$.
Let $Z$ be a closed subscheme of positive codimension in $X_\eta=X\times_B \eta$.

Then there is a
one-dimensional ${X_{(x)}}$-focused compactified framed correspondence
\[(\overline{C},C_\infty,\Psi), \quad \Psi=(C_\eta, \psi, \mathrm{val}_C), \quad C_\eta = (\overline{C}-C_\infty)\times_B \eta,\]
form $\mathcal V = ({X_{(x)}})_\eta$ to $X_\eta$
such that
\begin{itemize}
\item[(1)] ${X_{(x)}}$-smoothly contains the canonical map $({X_{(x)}})_\eta\to X_\eta$,
\item[(2)]  the closure $\overline{\mathrm{val}^{-1}_C(Z)}$ of $\mathrm{val}^{-1}_C(Z)$ in $\overline{C}$ is finite over ${X_{(x)}}$, and
\[(\overline{\mathrm{val}^{-1}_C(Z)}\cap C_\infty)\times_B \eta=\emptyset.\]
\end{itemize}
\end{proposition}
\begin{proof}

The claim follows by the combination of \Cref{lm:XetafDimFrS} and \Cref{lm:DimDim1}.
\end{proof}

\subsection{Finite correspondence homotopy, and the vanishing of cohomoligies at infinity.}

\begin{lemma}\label{lm:CurveFrCor}
Let $B$ be a one-dimensional local scheme, $z\in B$ be the closed point, $\eta\in B$ be the generic point, $X\in Sm_B$, $Z$ be a closed subscheme in $X_\eta=X\times_B \eta$.

Assume there is a one-dimensional compactified ${X_{(x)}}$-focused framed correspondence
$(\overline{C},C_\infty,\Psi)$
from $({X_{(x)}})_\eta$ to $X_\eta$
such that \begin{itemize}
\item[(1)]
$(\overline{C},C_\infty,\Psi)$ smoothly contains the canonical map $\mathrm{can}_\eta \colon ({X_{(x)}})_\eta\to X_\eta$,
\item[(2)]
the closure $\overline{\mathcal Z}$ of $\mathcal{Z}=\mathrm{val}^{-1}(Z)$ in $\overline{C}$ is finite over ${X_{(x)}}$,
and $(\overline{\mathcal Z}\cap C_\infty)\times_B\eta=\emptyset$,
where $\Psi=(C_\eta, \varphi, \mathrm{val})$, $C_\eta= (\overline{C}-C_\infty)\times_B \eta$.
\end{itemize}

Denote $\mathcal V=({X_{(x)}})_\eta={X_{(x)}}\times_B \eta$.
Then there are framed correspondences
$c\in \mathrm{Fr}_N(\mathcal V\times\mathbb A^1, X_\eta)$, $c_0^+,c_1\in \mathrm{Fr}_N(\mathcal V\times\mathbb A^1, X_\eta-Z)$, $N\in \mathbb Z$, such that \begin{itemize}
\item[(1)] $c \circ i_0=\sigma^N_{\mathcal V}\mathrm{can}_\eta+ c^+_0$, $\tilde c \circ i_1=j\circ c_1$, in ${\mathbb Z\mathrm{F}}_N(\mathcal V,X_\eta)$,
where $i_0,i_1\colon \mathcal V\to \mathcal V\times\mathbb A^1$ are the zero and unit sections, $j\colon X_\eta-Z\to X_\eta$ is the open immersion,
\item[(2)] $c^{-1}(Z)$ is finite over $\mathcal V$.
\end{itemize}
\end{lemma}
\begin{proof}
\item[1)]
Firstly, assume the residue field $\mathcal O(x)$ has at least three elements, $\#\mathcal O_z(z)>2$.
According to \Cref{def:reldimFr,def:compactifiedfocusedFr,def:smoothcontain}
we are given with
\begin{itemize}
\item[(d1)]
the projective equi-dimensional scheme $\overline{C}$ over ${X_{(x)}}$ of relative dimension one,
a closed subscheme $C_{\infty}$ finite over ${X_{(x)}}$,
a closed embedding $C_\eta\to \mathbb A^N_{\mathcal V}$ for some $N\in \mathbb Z$;
\item[(d2)]
a set of functions $e_i\in \mathcal O(\mathcal E)$, $i=2,\dots N$, and a regular map $\mathrm{val}\colon \mathcal E\to X_\eta$,
such that $C_\eta = Z(e_2,\dots,e_N)$,
where $\mathcal E$ is the henselisation $(\mathbb A^N_{\mathcal V})^h_{C_\eta}$;
\item[(d3)]
and a map $\Delta\colon {X_{(x)}}\to C$ such that $\mathrm{val}\circ\Delta$ equals to the canonical map $\mathrm{can}\colon {X_{(x)}}\to X$,
and the morphism $\overline{C}\to {X_{(x)}}$ is smooth over $\Delta({X_{(x)}})$.
\end{itemize}
Denote
\[\overline{C}_\eta=\overline{C}\times_B \eta=\overline{C}\times_{{X_{(x)}}} \mathcal V,\quad C_{\infty,\eta}= C_{\infty}\times_{{X_{(x)}}} \mathcal V,\]
and note that \[\mathcal{Z}=\mathrm{val}^{-1}(Z)=\overline{\mathcal Z}\times_B\eta.\]
Then
$\overline{C}_\eta$ is a projective scheme of relative dimension one over $\mathcal V$,
and $C_{\infty,\eta}$ and $\mathcal{Z}$ are closed subschemes finite over $\mathcal V$.
By (d3) the morphism $\overline{C}\to {X_{(x)}}$ is smooth over $\Delta({X_{(x)}})$.
It follows that there is a line bundle $\mathcal L(\Delta({X_{(x)}}))$ on $\overline{C}$ with a section $\delta$ that vanishing locus $Z(\delta)$ equals $\Delta({X_{(x)}})$.
We write $\mathcal L(\Delta)$ for $\mathcal L(\Delta({X_{(x)}}))$ and its restriction on $\overline{C}_\eta$.

Since $\mathcal{Z}$, and $\Delta({X_{(x)}})$ are finite over the local scheme ${X_{(x)}}$,
by \Cref{lm:semilocfinlocLinearbundle} the restrictions
$\mathcal O(1)\big|_{\mathcal{Z}}$ and $\mathcal L(\Delta)\big|_{\mathcal{Z}}$ are trivial.
Since $\#\mathcal O_z(z)>2$,
by \Cref{lm:b0b1binfty},
it follows that for any $d\in \mathbb Z_{\geq 0}$, there are invertible sections
\[\gamma_\infty, \gamma^+_0,\gamma_1\in \Gamma(\mathcal{Z}, \mathcal O(1)^\times), \quad
\gamma_\infty= \gamma_1-\gamma_0\delta\big|_\mathcal{Z}.\]
Since $C_{\eta,\infty}$ is finite over ${X_{(x)}}$ there is an invertible section
\[\beta_0^+\in \Gamma(C_{\eta,\infty},\mathcal O(1)^\times).\]

Since $\mathcal{Z}\cap C_{\eta,\infty}= \emptyset$ by the assumption (2) of the lemma, by Serre's theorem on ample bundles \cite[III, Corollary 10.7]{Hartshorne-AlG}
there are sections
$s_0^+\in \Gamma(\overline{C}_\eta, \mathcal L(\Delta(\mathcal V))(d)))$ and $s_1\in \Gamma(\overline{C}_\eta, \mathcal O(d))$
for large enough $d$ such that
\[\begin{array}{ll}
s_\infty\big|_{\mathcal{Z}}=\gamma_\infty, &
s_0^+\big|_{\mathcal{Z}}= \gamma^+_0, \\
s_\infty\big|_{C_{\eta,\infty}}=0, &
s_0^+\big|_{C_{\eta,\infty}}=(\beta_0^+)^d
.\end{array}\]
Define
\begin{gather*}
s_0=\delta s_0^+, \quad s_1=s_0+s_\infty,\\
s_\lambda = s_0(1-\lambda)+s_1\lambda.
\end{gather*}

Now define
\[\begin{array}{lll}
c=(Z(s_\lambda), e_1, e_2,\dots 2_N, g)\in \mathrm{Fr}_N(\mathcal V\times\mathbb A^1, X_\eta),\\
c_0^+ = (Z(s_0^+), e_1\big|_{\mathbb A^N_{\mathcal V\times 0}}, e_2,\dots 2_N, \mathrm{val}_{0})\in \mathrm{Fr}_N(\mathcal V,X_\eta-Z),\\
c_1 = (Z(s_1), s_1/t_\infty^d, e_2,\dots 2_N, \mathrm{val}_{1})\in \mathrm{Fr}_N(\mathcal V,X_\eta-Z),\\
\end{array}\]
where
$e_1\in \mathcal O(\mathbb A^N_{\mathcal V\times\mathbb A^1})$ is a lifting of
$s_\lambda/t_\infty^d\in \mathcal O(C_\eta\times\mathbb A^1)$,
the map $g\colon (\mathbb A^N_{\mathcal V\times\mathbb A^1})^h_{Z(s_\lambda)}\to X_\eta$ is a lifting of the map
\[Z(s_\lambda)\to C_\eta\times\mathbb A^1\to C_\eta\to X_\eta,\]
and the maps $g_0\colon (\mathbb A^N_\mathcal V)^h_{Z(s_0^+)}\to X_\eta-Z$ and $\mathrm{val}_1\colon (\mathbb A^N_{\mathcal V})^h_{Z(s_1)}\to X_\eta-Z$
are induced by $g$, because $s_0^+\big|_{\mathcal{Z}}=\delta^{-1}\gamma_0$ and $s_1\big|_{\mathcal{Z}}=\gamma_1$ are invertible.

Then
\[c\circ i_0= \mathrm{can}^\prime+ j\circ c_0^+, \quad c\circ i_1 = j\circ c_1,\]
where $\mathrm{can}^\prime = (\Delta(\mathcal V), e_1, e_2,\dots 2_N, g)$ equals
$\sigma^N_D\mathrm{can}$, and $D\in \mathrm{GL}_N(\mathcal V)$ is defined by the differentials of functions $e_i$.

Then
\[s_\lambda\big|_{C_{\eta,\infty}\times\mathbb A^1}= \delta\big|_{C_{\eta,\infty}}(\beta_0^+)^d\]
is an invertible section.
Hence $Z(s_\lambda)$ is finite over ${X_{(x)}}\times\mathbb A^1$ and is contained in $C_\eta$.
\[s_\lambda\big|_{\mathcal{Z}\times\infty}=s_\infty\big|_{\mathcal{Z}}=\gamma_\infty\]
is invertible, it follows that
$Z(s_\lambda\big|_{\mathcal{Z}\times\mathbb A^1})$ is contained in $\mathcal{Z}\times\mathbb A^1$ and finite over ${X_{(x)}}$.
Hence
\[c^{-1}(Z) = p(g^{-1}_{Z(s_\lambda)}(Z))=p(Z(s_\lambda\big|_{\mathcal{Z}\times\mathbb A^1}) )\] is finite over $\mathcal V$,
where $p\colon Z(s_\lambda)\to \mathcal V\times\mathbb A^1$ is a finite morphism induced by the projection $C_\eta\times\mathbb A^1\to \mathcal V\times\mathbb A^1$.

\item[2)]

Consider the case of an arbitrary residue field $k=\mathcal O_z(z)$ at $z$ in $B$.

Let $q_0,q_1\in \mathbb Z_{>0}$
\[(q_0,q_1)=1, \quad (q_0,\chark k)=1, \quad (q_2,\chark k)=1.\]
Define the polynomials
\[f_i= t^{q_i-1}+t^{q_i-2}+\dots +1=(t^{q_i}-1)/(t-1)\in R[t], R=\mathcal O_B(B).\]
Then $f_0$ and $f_1$ are separable over the closed point $z$.
Denote by $B_i=Z(f_i)$ the closed subschemes in $\mathbb A^1_B$, and let $p_i\colon B_i\to B$ denote the canonical projections.
Then $p_i$ are finite \'etale morphisms of schemes.
Define
\[e_i=(Z(f_i)\times_B\mathcal V,f_i,r_i)\in \mathrm{Fr}_1(\mathcal V, B_i\times_B\mathcal V)\]
and
$c = c^{B_0}\circ e^{\mathbb A^1}_0 + c^{B_1}\circ e^{\mathbb A^1}_1\in {\mathbb Z\mathrm{F}}_{2+N}(\mathcal V\times\mathbb A^1, X_\eta)$.
Then
\begin{multline*}
c\circ i_0 =
c^{B_0}_0\circ e_0\circ (b_0)_\varepsilon - c^{B_1}_0\circ e_1\circ (b_1)_\varepsilon = \\
\sigma^N\mathrm{can}^{B_0}\circ e_0\circ (b_0)_\varepsilon - \sigma^N\mathrm{can}^{B_1}\circ e_1\circ (b_1)_\varepsilon = \\
\sigma^N \mathrm{can}\circ (p_0\circ e_0\circ (b_0)_\varepsilon- p_1\circ e_1\circ (b_1)_\varepsilon),
\end{multline*}
where $p_i^{\mathcal V}\colon B_i\times_B\mathcal V\to \mathcal V$.
Finally, since by \Cref{lm:nvarepsilonclass} $[p_i\circ e_i]=[(q_i)_{\varepsilon}]\in \overline{\mathbb Z\mathrm{F}}_1(\mathcal V, \mathcal V)$,
it follows by \Cref{lm:classtrivsigma} that
\[[c\circ i_0] = [\sigma^{2+N}\mathrm{can}]\in\overline{\mathbb Z\mathrm{F}}_{2+N}(\mathcal V, X_\eta).\]
\end{proof}

\begin{proposition}\label{cor:contractinggenpointhomoveretaSmX}
Let $B$ be one-dimensional irreducible local scheme, with the closed point $z\in B$ and the generic point $\eta\in B$.
Let $X\in \mathrm{Sm}_B$, $x\in X\times_B z$.

Denote $\mathcal V={X_{(x)}}\times_B \eta$.
Then for any closed subscheme $Z$ in $X\times_B \eta$, there are framed correspondences
$c\in {\mathbb Z\mathrm{F}}_N(\mathcal V\times\mathbb A^1, X_\eta)$, $c^\prime\in {\mathbb Z\mathrm{F}}_N(\mathcal V\times\mathbb A^1, X_\eta-Z)$
such that \begin{itemize}
\item[(1)] $c \circ i_0=\sigma^N_{\mathcal V}\mathrm{can}_\eta$, $c\circ i_0 = j\circ c^\prime$,
where $i_0,i_1\colon \mathcal V\to \mathcal V\times\mathbb A^1$ are the zero and unit sections, $\mathrm{can}_\eta\colon \mathcal V\to X_\eta$ is the canonical morphism, $j\colon X_\eta-Z\to X_\eta$ is the open immersion,
\item[(2)] $c^{-1}(Z)$ is finite over $\mathcal V$.
\end{itemize}
\end{proposition}
\begin{proof}
The claim follows by the consequent use of \Cref{prop:DimDim1,lm:CurveFrCor}.
\end{proof}

\begin{theorem}\label{th:GenFifLocEssSmSchemeonedimbasetriviality}
Let $X$ be a smooth scheme over a one-dimensional irreducible base scheme $B$. Let $x\in X$ be a point over a closed point $z\in B$.
Let $F$ be an $\mathbb A^1$-invariant quasi-stable framed presheaf of abelian groups over the generic point $\eta\in B$.
Then \[\underline{H}^l(X_x\times_B \eta, F)\cong 0\]
\end{theorem}
\begin{proof}
The claim follows by the combination of \Cref{cor:preimFinFrCintr->Inj} with
\Cref{cor:contractinggenpointhomoveretaSmX}.
\end{proof}
\section{Strict homotopy invariance}\label{sect:StrHomInvTheorem}
In this section,
we combine \Cref{th:FinSuppLocHensbaseafflinetriviality} and \Cref{th:GenFifLocEssSmSchemeonedimbasetriviality}
on cohomologies out and at the infinity on $\mathbb A^1_V$ for an essentially smooth local henselian $k$-schemes $V$,
to prove the strict homotopy invariance theorem over the field $k$.
Let $F$ be a quasi-stable homotopy invariant framed additive presheaf of abelian groups over $k$.
\begin{theorem}\label{th:strhominv}
Let $V$ be a local henselian essentially smooth scheme over $k$.
Then \[\begin{cases} H^0_\mathrm{nis}(V\times\mathbb A^1, F)=F(V\times\mathbb A^1),  \\ H^i_\mathrm{nis}(V\times\mathbb A^1, F)=0, i>0.\end{cases}\]
\end{theorem}
\begin{proof}
The morphism of sites given by the embedding of the topology $\mathrm{Fin}_{S}$ on $\mathrm{Sm}_{\mathbb A^1\times V}$ into the Nisnevich topology leads to the spectral sequence
\begin{equation}\label{eq:H(Fin<-VNis)}
H^p_{\mathrm{Fin}_V} (\mathbb A^1\times V, H^q_{\mathrm{nis}}(F)_{\mathrm{Fin}_V} )\Rightarrow H^{r}_\mathrm{nis}(\mathbb A^1\times V, F_\mathrm{nis}),r=p+q,
\end{equation}
where ${H}^*_\mathrm{nis}(F)_{\mathrm{Fin}_V}$ denotes the sheafification with respct to $\mathrm{Fin}_V$ of the presheaf $H^*_\mathrm{nis}(-,F_\mathrm{nis})$.
By \Cref{lm:NisZXNisZpoints} the topology $\mathrm{Fin}_V$ has enough set of points given by schemes of the forms
\[U^h_v, \, U\times_{\mathbb P^1\times V}(\mathbb P^1\times V)_{(\infty,z)}\]
 for $v\in U$, and where $z$ is the closed point of $V$.
Since the schemes $U^h_v$ are Nisnevich points, $\underline H^*_\mathrm{nis}(U^h_v,F)\cong 0$.
By \Cref{th:GenFifLocEssSmSchemeonedimbasetriviality} \[\underline H^*_\mathrm{nis}(U\times_{\mathbb P^1\times V}(\mathbb P^1\times V)_{(\infty,z)},F)\simeq 0.\]
Combining the latter two equivalences we conclude
$\underline{H}^*_\mathrm{nis}(F)_{\mathrm{Fin}_V}\cong 0$, and consequently
\[{H}^*_\mathrm{nis}(F)_{\mathrm{Fin}_V}\cong F_{\mathrm{Fin}_V}.\]
Thus \eqref{eq:H(Fin<-VNis)} implies \[H^p_{\mathrm{Fin}_V} (\mathbb A^1\times V, F_{\mathrm{Fin}_V} )\cong H^{p}_\mathrm{nis}(\mathbb A^1\times V, F_\mathrm{nis});\]
and the claim of the theorem follows, since $\underline{H}^*_{\mathrm{Fin}_V}(\mathbb A^1\times V, F)\cong 0$ by \Cref{th:FinSuppLocHensbaseafflinetriviality}.
\end{proof}
Let us reformulate the result in the following standard form.
\begin{theorem}\label{cor:strhominv}
The presheaves $h_\mathrm{nis}^i(F)= H^i_\mathrm{nis}(-,F)$, $i\geq 0$, on the category $\mathrm{Sm}_k$ are $\mathbb A^1$-homotopy invariant.
\end{theorem}
\begin{proof}
Let $X\in \mathrm{Sm}_k$. The claim is to prove that
\begin{equation}\label{eq:H(X)H(XtimesA1} H^i_\mathrm{nis}(X\times\mathbb A^1,F)\simeq H^i_\mathrm{nis}(X), i\geq 0.\end{equation}
Consider the morphism of small \'etale sites
\[p\colon \mathrm{Et}_{X\times\mathbb A^1}^\mathrm{nis}\to \mathrm{Et}_X^\mathrm{nis},\]
and consider the restrictions of $F$ on the mentioned sites.
It follows by \Cref{th:strhominv} that
\[Rp_*^i(F)=0 \text{ for } i>0, Rp_*(F)=F,\]
where $Rp_*^i(F)$ denotes the higher direct image of $F$ along $p$.
Then the spectral sequence
\[
H_\mathrm{nis}^i(X\times\mathbb A^1,F) \Leftarrow H^p_\mathrm{nis}(X,Rp^{i-p}_*(F))
\]
implies \eqref{eq:H(X)H(XtimesA1}.
\end{proof}

\begin{corollary}\label{cor:HniscongHzar}
The canonical morphism induces the isomorphism
\[H^l_\mathrm{nis}(X,F_\mathrm{nis})\cong H^l_\mathrm{zar}(X,F_\mathrm{zar})\] for all $l\in \mathbb Z$ and $X\in \mathrm{Sm}_k$.
\end{corollary}
\begin{proof} We repeat the argument from \cite{Sus-nonperftalk}. By \Cref{cor:strhominv} the presheaves $H^*_\mathrm{nis}(-,F_\mathrm{nis})$ are $\mathbb A^1$-invariant, hence by \cite[Theorem 3.11]{hty-inv} the canonical morphism $H^*_\mathrm{nis}(U,F_\mathrm{nis})\to H^*_\mathrm{nis}(\eta,F_\mathrm{nis})$ is injective for any essentially smooth local scheme $U$ with the generic point $\eta$. Hence the presheaf $\underline{H}^*_\mathrm{nis}(-,F_\mathrm{nis})$ is Zariski locally trivial. Then because of the spectral sequence $H^{p}_\mathrm{zar}(X, H^{q}_\mathrm{nis}(-,F_\mathrm{nis})_\mathrm{zar})\Rightarrow H^{p+q}_\mathrm{nis}(X,F_\mathrm{nis})$ the claim follows.\end{proof}

\begin{corollary}\label{cor:ZarStrictHomotopyInvariance}
The canonical projection induces the isomorphism
$H^l_\mathrm{zar}(\mathbb A^1\times X,F_\mathrm{zar})\simeq H^l_\mathrm{zar}(X,F_\mathrm{zar})$
for all $X\in \mathrm{Sm}_k$.
\end{corollary}
\begin{proof}
The claim follows by \Cref{cor:HniscongHzar} and \Cref{cor:strhominv}.
\end{proof}

\section{Motives over non-perfect fields}\label{sect:HSHfrCorr}

\Cref{cor:strhominv} extends the theory of framed motives \cite{framed-notes,Framed,framed-cancel,hty-inv,ConeTheGNP,surj-etale-exc,DrKyllfinFrpi00,five-authors,BigFrmotives,FramedGamma,SmAffOpPairs,GNThomSpectra,SmModelSpectrumTP} to the case of a non-perfect base field,
because the perfectness assumption was required only in the argument for the strict homotopy invariance theorem \cite[Theorem 1.1, \S 17]{hty-inv}.
Thus all the results in the above works hold for non-perfect fields by \Cref{th:strhominv}.
Moreover, the strict homotopy invariance result applies to correspondences categories that admit a functor from the category of framed motives such as the original Voevodsky's correspondences $\mathrm{Cor}(k)$,
$K_0$ and $K$-correspondences \cite{WalkerKmotives,Suslin-Grayson,K-motives},
Witt and Grothendieck-Witt correspondences \cite{GWStrHomInv,DruDMGWeff,GW-cancel},
Milnor-Witt correspondences \cite{Calmes-Fasel,MWmotives,MW-cplx,MW-cancel,MW-ring-spt,hominvAnstdcor}, cohomological correspondences motives $\mathrm{Cor}^{A}(k)$ \cite{ccorrs},
and others.
In particular, this provides the localisation and embedding theorems for $\mathbf{DM}(k)$, and mentioned generalisations or modifications.

In this section, we start with brief retell on Garkusha-Panin theory of framed motives
to demonstrate the role of the strict homotopy invariance theorem, \Cref{th:strhominv}.
The detailed argument is written in \Cref{subsect:addinfyCor,subsect:SHGeomCor}
for a technically easier context of an abstract $\infty$-categories of correspondences that admit a functor from the graded category of framed correspondences $\mathrm{Fr}_*(k)$.
The axiomatic setting applies to the $\infty$-category $\mathrm{Corr}^\mathrm{fr}(k)$ and the version of the theory of framed motivic spaces and spectra by E.~Elmanto, M.~Hoyois, A.~Khan, V.~Sosnilo, M.~Yakerson in \cite{five-authors}
by using of the cancellation theorem, \Cref{citeth:Canc}.
Simultaneously this covers other examples of more special types of transfers listed above.
\Cref{subsect:hthSHCor} summarises results on the homotopy t-structure harts of the mentioned motivic categories.

\subsection{Garkusha-Panin's framed motives}\label{section:GPFr}
The initial generality of \cite{Framed} was an infinite perfect field $k$ of characteristic different from $2$. All the assumptions appeared because of the strict homotopy invaraiance theorem from \cite{hty-inv}. The proof of the surjectivity of the \'etale excision in \cite{surj-etale-exc} eliminated the assumption $\mathrm{char}\, k\neq 2$. The argument that deduced results for finite base fields from the case infinite base field case was initially presented in a preliminarly form in the appendix of the arXiv version of \cite{SmAffOpPairs} and later moved to \cite{DrKyllfinFrpi00}, where it was combined with results that extended the generality of Neshitov's proof from \cite{Nesh-FrKMW} to cover the case of odd finite characteristic perfect fields.
So the generality of the theory \cite{Framed} at the moment is an arbitrary perfect base field.

We retell some main ingredients used for the proof of results of \cite{Framed,BigFrmotives,FramedGamma}
summarised in Theorem F from \Cref{subsect:intro:motives}.
We refer the reader to the sources for the proofs, we present this retell to demonstrate the role of \Cref{th:strhominv} and convince the reader that it is enough.

Recall that $\mathrm{Fr}_+(k)$ for a base field $k$ denotes the graded category of framed correspondences \cite[Definition 2.3]{Framed},
and let
$\mathrm{Fr}(-,X)=\varinjlim_{l}\mathrm{Fr}_l(-,X)$
be the stable framed correspondences presheaves \cite[Definition 2..8]{Framed}.
Let $\Delta_k^\bullet$ denote the co-simplicial scheme given by affine simplices $\Delta^l_k$,
and let $(-)_f$ denote the fibrant replacement within the injective Nisnevich local model structure on the category of simplicial Nisnevich sheaves over $k$.
The main results of the first article \cite[Theorem 4.1, Theorem 11.7]{Framed}
that play key role for \cite{BigFrmotives,FramedGamma} provide
the constructions of the stable motivic equivalences
\begin{equation*}\label{eq:replacementssmot}
\Sigma^\infty_{\mathbb P^{\wedge 1}}Y_+\simeq \mathrm{Fr}(-\times \Delta_k^\bullet,\Sigma^\infty_T Y_+)_f,\quad
\Sigma^\infty_{\mathbb G_m^{\wedge 1}}\Sigma^\infty_{S^1} Y_+ \simeq \mathrm{Fr}(-\times \Delta_k^\bullet,\Sigma^\infty_{\mathbb G_m^{\wedge 1}}\Sigma^\infty_{S^1} Y_+)_f
\end{equation*}
such that the right sides are
stably motivically fibrant $\mathbb P^1$-spectrum and bi-spectrum respectively for any
Nisnevich locally connected simplicial smooth scheme $Y$ over the base field $k$,
and have this property in positive degrees with respect to $\mathbb P^1$ or $S^1$ for any $Y\in \mathrm{Sm}_k$.

The additivity theorem or in other words $S^1$-cancellation theorem provided by
\cite[Theorem 6.5]{Framed}
claims that the Segal $\Gamma$-space $K\mapsto \mathrm{Fr}(\Delta^\bullet_B\times -, K\otimes Y)$, where $K\mapsto \Gamma^\mathrm{op}$, is special. Consequently, the $S^1$-spectrum of presheaves $\mathrm{Fr}(\Delta^\bullet_B\times -, \Sigma^\infty_{S^1}Y_+)_f$ is a fibrant $\Omega_{S^1}$-spectrum in positive degrees within Nisnevich local model structure for any $Y\in \mathrm{Sm}_B$ and is a fibrant $\Omega_{S^1}$-spectrum in all degrees for Nisnevich locally connected simplicial scheme $Y$.
The original argument does not require assumptions on the base scheme $B$, see \cite[Appnedix B]{SmAffOpPairs} for more details.
Then the strict homotopy invariance theorem \Cref{cor:strhominv}
provides the following fibrancy property.
\begin{theorem}[$S^1$-spectra form of the strict homotopy invariance]
\label{citeth:Lmotrep}
The $S^1$-spectrum \[M_\mathrm{fr}(Y)_f = \mathrm{Fr}(\Delta^\bullet_k\times -, \Sigma^\infty_{S^1}Y)\] is fibrant within stable motivic model structure on $S^1$-spectra over a field $k$ for a Nisnevich locally connected simplicial scheme $Y$.
\end{theorem}
\begin{proof} The claim follows by the argument of \cite[Corollary 7.5]{Framed}.
\end{proof}
The latter $S^1$-spectrum is called \emph{framed motive} of $Y$.
Next, we formulate the cancellation theorem for framed motives.
\begin{theorem}[Cancellation]\label{citeth:Canc}
The canonical morphism of
$S^1$-spectra of pointed simplicial presheaves over a field $k$
\[
\mathrm{Fr}(-\times\Delta_k^\bullet,\Sigma^\infty_{S^1} Y))\to
\Omega_{\mathbb G_m}\mathrm{Fr}(-\times\Delta_k^\bullet,\Sigma_{\mathbb G_m}\Sigma^\infty_{S^1} Y)
\]
is a Nisnevich local equivalence in positive degrees.
\end{theorem}
\begin{proof}
The claim in other terms is the Nisnevich local equivalence
of $S^1$-spectra $M_\mathrm{fr}(Y_+)_f\to M_\mathrm{fr}(Y_+\wedge \mathbb G_m)_f(X_+\wedge \mathbb G_m)$, where the right side is defined in view of the left Kan extension of the functor $M_\mathrm{fr}(Y_+\wedge \mathbb G_m)_f$ from $\mathrm{Sm}_k$ to simplicial schemes.
The claim follows word by word
by the arguments of \cite{framed-cancel} with the use of
the injectivity theorem \cite[Theorem 3.11]{hty-inv} that holds for an arbitrary field,
and \Cref{cor:strhominv}.
\end{proof}
The above set of results implies the fibrancy of the right side (s,t)-bispectrum in \eqref{eq:replacementssmot}.
The claim for $\mathbb P^1$-spectrum
follows with the use of also one result called cone theorem \cite{ConeTheGNP}.
The latter theorem claims the canonical stable motivic equivalence
\begin{equation}\label{eq:Cone(A1/(A1-0))eq}
\mathrm{Fr}(-,\Sigma^\infty_{S^1}X\times \mathbb A^1)/
\mathrm{Fr}(-,\Sigma^\infty_{S^1}X\times \mathbb G_m)
\simeq
\mathrm{Fr}(-,\Sigma^\infty_{S^1}X_+\wedge T),
\end{equation}
where $T=(\mathbb A^1/\mathbb G_m)$.
The claim follows by the arguments of \cite{ConeTheGNP} with the use of \Cref{citeth:Lmotrep} and \Cref{citeth:Canc}.
Alternatively, we can combine the case of perfect fields with the published article \cite{SmAffOpPairs} that states the result for the space $X/U$ for smooth affine $X$ and open subscheme $U$ over an infinite field.

The fibrancy of the right sides in \eqref{eq:replacementssmot} is proven. The stable motivic equivalences \eqref{eq:replacementssmot} follow due to Voevodsky's lemma \cite[\S 3]{Framed} the equivalence \eqref{eq:Cone(A1/(A1-0))eq} and the machinery of symmetric spectra.

\begin{proof}[References for the proof of Theorem F from \Cref{subsect:intro:motives}.]
The first equivalence in the point (1) of the theorem follows by the argument of \cite[Theorem 4.1]{Framed} presented in \cite[\S 9]{Framed}
with the use of
\Cref{citeth:Lmotrep} and \Cref{citeth:Canc} and the equivalence \eqref{eq:Cone(A1/(A1-0))eq} in combination with Voevodsky's lemma \cite[\S 3]{Framed}.
The second equivalence follows by the arguments of \cite[Theorem 11.4]{Framed}, and other results of \cite[\S 11]{Framed}.

Point (2) follows by the arguments of \cite{BigFrmotives} and \cite{FramedGamma} with the use of point (1).
Point (3) is a consequence of (1).
\end{proof}
\begin{remark}The results of the works \cite{GNThomSpectra} and \cite{SmModelSpectrumTP}, in particular, extends as well.\end{remark}

\subsection{Additive $\infty$-categories of correspondences}\label{subsect:addinfyCor}
We define in the subsection some axiomatic formalism, that is used to formulate the results. The defined formalism is more special in comparison with the one introduced in \cite{SHzar}.
\begin{definition}\label{def:CorrinftyCategory}
A \emph{correspondences category} $\mathrm{Corr}$ is
an additive $\infty$-category equipped with
a monoidal functor
\begin{equation}\label{eq:-timesX}\mathrm{Sm}_k\to \mathrm{Func}(\mathrm{Corr},\mathrm{Corr}); X\mapsto -\times X,\end{equation}
and an object
$\mathrm{pt}_k\in \mathrm{Corr}$
such that the functor
\begin{equation}\label{eq:SmkCorrk}e\colon \mathrm{Sm}_k\to \mathrm{Corr}; X\mapsto \mathrm{pt}_k\times X.\end{equation}
is essentially surjective on objects.
Denote by $h_\mathrm{tr}(X)\in \mathrm{Pre}(\mathrm{Corr})$ the representbale presheaf $U\mapsto \mathrm{Corr}(U,X)$.
We sometimes denote $e(X)$ by $X$, and $e_*h_\mathrm{tr}(X)\in \mathrm{Pre}(\mathrm{Sm}_k)$ by $h_\mathrm{tr}(X)$.
Also we use notation $\mathrm{Pre}^\mathrm{tr}(\mathrm{Sm}_S)=\mathrm{Pre}(\mathrm{Corr})$.
\end{definition}

\begin{notation}
Denote by $\mathrm{Pre}^R_\Sigma(\mathrm{Corr})$ the category of additive functors $\mathrm{Corr}\to R$ for a presentable $\infty$-category $R$.
Denote by $\mathrm{HI}^R_\Sigma(\mathrm{Corr})$ the subcategory spanned by $\mathbb A^1$-invariant objects,
and by $\mathrm{Shv}^R_\tau(\mathrm{Corr})$ the subcategory of $\tau$-sheaves for a topology $\tau$ on $\mathrm{Sm}_k$.
For the case of $R=\mathbf{SH}$, that is the $\infty$-category of $S^1$-spectra,
we write $\mathrm{Pre}^{S^1}_\Sigma(\mathrm{Corr})$.
For $R$ being the $\infty$-category of simplicial sets, we omit the superscript and write $\mathrm{Pre}_\Sigma(\mathrm{Corr})$.
\end{notation}
\begin{definition}
For any object $X\in \mathrm{Sm}_S$, the endofunctor $-\times X$ on $\mathrm{Corr}$ induces the inverse and direct image endofunctors on $\mathrm{Pre}_\Sigma^R(\mathrm{Corr})$ that we denote $F\mapsto F\times X$, and $F\mapsto F^{X} = F(-\times X)$.
Moreover,
given a diagram of schemes $\mathcal X\colon \Gamma\to \mathrm{Sm}_S$,
we define the endofunctors $-\times\mathcal X$ and $(-)^{\mathcal X}$ on $\mathrm{Pre}_\Sigma^R(\mathrm{Corr})$
as the colimits of the diagrams
that are composite functors for the sequences
\[\Gamma \to \mathrm{Sm}_S\to \mathrm{Func}(\mathrm{Corr},\mathrm{Corr}) \rightrightarrows \mathrm{Func}(\mathrm{Pre}_\Sigma^R(\mathrm{Corr}),\mathrm{Pre}_\Sigma^R(\mathrm{Corr})),\]
where the latter arrows take a functor to the inverse image and direct image respectively.
\end{definition}
In particular, there are functors $F^{\mathbb A^1} = F(-\times\mathbb A^1)$, $F^{\Delta^\bullet_k}= F(-\times \Delta^\bullet_k)$;
the endofunctor $-\times \mathbb G_m^{\wedge 1}$ we denote also by $-\wedge \mathbb G_m^{\wedge 1}$, and by $\Sigma_{\mathbb G_m^{\wedge 1}}$.
We use notation $\Omega_{\mathbb G_m^{\wedge 1}}F= F(-\wedge \mathbb G_m^{\wedge 1})= F^{\mathbb G_m^{\wedge 1}}$.
Define the $\infty$-category
$\mathrm{Pre}^R(\mathrm{Corr})[\mathbb G_m^{\wedge -1}]=\mathrm{Pre}^R(\mathrm{Corr})[(-\times\mathbb G_m^{\wedge 1})^{- 1}]$, and similarly for $\infty$-categories
$\mathrm{Shv}^R_\mathrm{nis}(\mathrm{Corr})$.

\begin{definition}
Define $\mathbf{H}^R_\tau(\mathrm{Corr})=\mathrm{Shv}_\tau^R(\mathrm{Corr}) \cap \mathrm{HI}^R_\Sigma(\mathrm{Corr})$.
Define $\mathbf{SH}^{S^1}_\tau(\mathrm{Corr})=\mathbf{H}^{\mathbf{SH}}_\tau(\mathrm{Corr})$,
and $\mathbf{SH}_\tau(\mathrm{Corr})=\mathbf{SH}^{S^1}_\tau(\mathrm{Corr})[\mathbb G_m^{\wedge -1}]$,
where $\mathbb G_m^{\wedge -1}$ means the inverting of the suspension endofunctor $\Sigma_{\mathbb G_m^{\wedge 1}}$ on $\mathrm{Pre}_\Sigma(\mathrm{Corr})$.
We omit the subscript for the case $\tau=\mathrm{nis}$.
\end{definition}

\begin{definition}\label{def:FrNLNC}
Define the following properties of  correspondences categories $\mathrm{Corr}$ as above:
\begin{itemize}
\item[(Fr)]
Functor \eqref{eq:-timesX} lifts to a functor $\mathrm{Fr}_*(k)\to \mathrm{Corr}$ such that for any $X\in \mathrm{Sm}_k$,
$\sigma_X\sim \mathrm{id}_X\in \pi_0\mathrm{Map}_\mathrm{Corr}(X,X)$,
where $\mathrm{Fr}_*(k)$ denotes the $\infty$-category that is the nerve of the classical one.
\item[(NT)]
For any Nisnevich covering $\widetilde X\to X$, the canonical embedding morphism
\[h^{\widetilde X}_\mathrm{tr}(X)\to h_\mathrm{tr}(X)\] is a Nisnevich local equivalence,
where $h^{\widetilde X}_\mathrm{tr}(X)$ denotes the subpresheaf that is the image in the set-theoretic sense of the morphism $h_\mathrm{tr}({\widetilde X})\to h^{\widetilde X}_\mathrm{tr}(X)$.
\item[(NC)]
The natural transformation $\mathrm{Id}_{\mathbf{SH}^{S^1}(\mathrm{Corr})}\to \Omega_{\mathbb G_m^{\wedge 1}}\Sigma_{\mathbb G_m^{\wedge 1}}$ in $\mathbf{SH}^{S^1}(\mathrm{Corr})$ is an equivalence.
\end{itemize}
\end{definition}

\begin{example}\label{exCorrfr}
The additivisation of the pre-additive $\infty$-category of tangentially framed correspondences $\mathrm{Corr}^\mathrm{fr}$ introduced by
E.~Elmanto, M.~Hoyois, A.~Khan, V.~Sosnilo, M.~Yakerson
is a correspondences category.
As shown in \cite{five-authors} it satisfies (Fr) and (NL) for any base scheme.
(NC) holds over fields,
that follows by \Cref{citeth:Canc}.
\end{example}

\begin{example}\label{ex:CorA}
The category of cohomological correspondences $\mathrm{Cor}^A$, that is defined in \cite{ccorrs}
for an abstract axiomatic $\mathrm{SL}$-orientable cohomology theory on $\mathrm{Sm}_k$ like cohomology theories introduced by Panin and Smirnov in \cite{Panin-Smirnov}.
Recall that $c\in \mathrm{Cor}^A$ is given by a class in $A^{\operatorname{dim}X}_Z(U\times_k X)$ for some closed subscheme $Z$ finite over $X$.

The property (Fr) is proven in \cite{ccorrs}.
To check (NC) we use that standard argument.
We have to show that for any $c$ as above, and $U$ local henselian, and a Nisnevich covering $\widetilde X\to X$, there is a lifting $\tilde c$ along the morphism $\mathrm{Cor}^A(U,\widetilde X)\to \mathrm{Cor}^A(U,X)$. Since $Z$ is finite over local henselian $U$, the connected components of $Z$ are local henselian, so without loss of generality, we can assume that $Z$ is local henselian. Hence $Z$ admits a lifting along the Nisnevich covering $U\times_k\widetilde X\to U\times_k X$. So the claim follows, because
$A^{\operatorname{dim}X}_Z(U\times_k \widetilde)\cong A^{\operatorname{dim}X}_Z(U\times_k X)$.

To show (NC) we note that the cancellation theorem \cite[Theorem 9.0.17]{ccorrs}
implies the equivalence of endofunctors
$\mathrm{Id}\to \Omega_{\mathbb G_m^{\wedge 1}}^\mathrm{pre}\Sigma_{\mathbb G_m^{\wedge 1}}$ on $\mathrm{HI}_\Sigma^{S^1}(\mathrm{Cor}^A(\mathrm{SmAff}_k))$
and the claim holds because by \Cref{lm:HNCVopensubsetA1} it follows that
$\Omega_{\mathbb G_m^{\wedge 1}}F\simeq \Omega_{\mathbb G_m^{\wedge 1}}^\mathrm{pre}F$
for any $F\in \mathbf{SH}^{S^1}(\mathrm{Cor}^A(k))$.
\end{example}

\begin{example}\label{ex:Cor}
Voevodsky's correspondences $\mathrm{Cor}$ category is $\mathrm{Cor}^A$ from \Cref{ex:CorA}
for $A$ being Chow groups.
\end{example}

\begin{example}\label{ex:widetildeCor}
Milnor-Witt correspondences $\widetilde{Cor}$ category is $\mathrm{Cor}^A$ from \Cref{ex:CorA}
for $A$ being Chow-Witt groups.

Let us explain what do we mean under the theory of Chow-Witt groups over a field $k$.
The theory of Chow-Witt groups for perfect fields of characteristic different from 2 is developed in \cite{ChWitt,ChWittRing} by J.~Fasel.
The case of an arbitrary perfect field is provided by N.~Feld's work \cite{MWcyclemodules} as the theory associated with the Milnor-Witt cycle module $\mathrm{K}^\mathrm{MV}$ over $k$.
The theory of Chow-Witt groups on $\mathrm{Sm}_k$ over a field $k$ induces
by the continuity the theory on the category of essentially smooth scheme over $k$.
Consequently, this defines a theory over any base scheme that is an essentially smooth scheme over $k$.
In particular, this defines a theory over any field $k$,
since any field $k$ is an inductive limit of a finitely generated field over a finite field, which is perfect.
\end{example}

\subsection{Geometric motives and the stable motivic localisation}\label{subsect:SHGeomCor}

Let $\mathrm{Corr}$ be an additive $\infty$-category of correspondences $\mathrm{Corr}$, and $R$ be a presentable $\infty$-category.
\begin{definition}
Define $\mathbf{H}_\mathrm{geom}(\mathrm{Corr})$ as
the localisation of the $\infty$-category $\Delta \mathrm{Corr}=\mathrm{Func}(\Delta^\mathrm{op}, \mathrm{Corr})$
with respect to morphisms of the form
\[\cofib((X_0\cap X_1)\to (X_0\amalg X_1))\to X, \mathbb A^1\times X\to X,\]
$X\in \mathrm{Sm}_k$, $X_0\amalg X_1\to X$ is a Zariski covering, and $\check C_{X}(\widetilde X)$ is the \v{C}ech simplicial scheme.
\end{definition}
\begin{remark}\label{rem:DMgeom}
The $\infty$-category $\mathbf{H}^{\mathbf{SH}}_\mathrm{geom}(\mathrm{Cor}(k))\simeq\operatorname{Stab}\mathbf{H}_\mathrm{geom}(\mathrm{Cor}(k))$
is equivalent to the category $\mathbf{DM}_\mathrm{geom}(k)$ form \cite{Voe-motives}.
\end{remark}

\begin{lemma}\label{lm:LnistrLnis}
Suppose that $\mathrm{Corr}$ satisfies (NL),
then $L_\mathrm{nis} e_*\simeq e_* L^\mathrm{tr}_\mathrm{nis}$,
where $e_*\colon \mathrm{Pre}^R(\mathrm{Corr})\to \mathrm{Pre}^R(\mathrm{Sm}_k)$ denotes the canonical functor.
In particular, the category $\mathrm{Shv}_\mathrm{nis}^\mathrm{Ab}(\mathrm{Corr})$ is an abelian classical category.
\end{lemma}
\begin{proof}
The argument is contained in \cite{five-authors}.
By the construction and definition of $\mathrm{Shv}^R_\mathrm{nis}(\mathrm{Corr})$ the functor $e_*$
restricts to $\mathrm{Shv}^R_\mathrm{nis}(\mathrm{Corr})\to \mathrm{Shv}^R_\mathrm{nis}(\mathrm{Sm}_k)$.
By (NL) the functor $e_*$ is Nisnevich exact.
Hence for any $F\in \mathrm{Pre}(\mathrm{Corr})$,
$e_* L^\mathrm{tr}_\mathrm{nis}\in \mathrm{Shv}_\mathrm{nis}(\mathrm{Sm}_k)$, and the morphism $e_* F\to e_* L^\mathrm{tr}_\mathrm{nis}$ is a Nisnevich local equivalence.
The first claim follows.
The second claim follows because (1) the category is an additive classical category,
(2) the kernel functor $\ker^\mathrm{tr}_\mathrm{nis}$ in $\mathrm{Shv}_\mathrm{nis}^\mathrm{Ab}(\mathrm{Corr})$ is given by the restriction of the kernel $\ker^\mathrm{tr}$ in $\mathrm{Pre}^\mathrm{Ab}(\mathrm{Corr})$,
and (3) the cokerenl $\coker^\mathrm{tr}$ in $\mathrm{Shv}_\mathrm{nis}^\mathrm{Ab}(\mathrm{Corr})$ is given by $\coker^\mathrm{tr}_\mathrm{nis}(f)=L_\mathrm{nis}^\mathrm{tr}\coker^\mathrm{tr}$.
\end{proof}

\begin{definition}\label{def:NHC}
We say that a scheme $X\in \mathrm{Sm}_k$ satisfies (NHC), if for each $E\in \mathrm{HI}^\mathrm{Ab}_\Sigma(\mathrm{Corr})$, $H_\mathrm{nis}^l(X\times U,E_\mathrm{nis})=0$, if $l>0$, and $H^0_\mathrm{nis}(X\times U,E_\mathrm{nis})=E_\mathrm{nis}(X\times U)$
for any essentially smooth local henselian $U$,
where $E_\mathrm{nis}$ denotes the associated Nisnevich sheaf on $\mathrm{Sm}_k$.
\end{definition}

\begin{lemma}\label{lm:HNCVopensubsetA1}
(Fr) implies (NHC) for all open subsets of $\mathbb A^1_k$.
\end{lemma}
\begin{proof}
Let $F\in\mathrm{HI}^\mathrm{Ab}_\Sigma(\mathrm{Corr})$, and $F_\mathrm{nis}$ be the associated Nisnevich sheaf on $\mathrm{Sm}_k$.
(NHC) for $\mathbb A^1_k$ follows by \Cref{th:strhominv}.
To prove the general case note firstly that by \Cref{th:strhominv} and \cite[Theorem 3.9]{hty-inv} $H^l_\mathrm{nis}(V,F_\mathrm{nis})=0$, $l>0$, $H^0_\mathrm{nis}(V,F_\mathrm{nis})=F(V)$ for any open subset $V$ of $\mathbb A^1_k$. Then by \Cref{th:strhominv}  again and \cite[Theorem 3.11]{hty-inv} the claim follows, since $H^l_\mathrm{nis}(V\times U,F_\mathrm{nis})$ maps injectively to $H^l_\mathrm{nis}(V\times\eta,F_\mathrm{nis})$, where $\eta$ is the generic point of a local henselian essentially smooth $U$.
\end{proof}

\begin{lemma}\label{lm:F-timesXpipre-timesXNispish-timesX}
Let $F\in \mathrm{Shv}_\mathrm{nis}^\mathrm{Ab}(\mathrm{Corr})$.
Let $X\in \mathrm{Sm}_k$ satisfy (NHC).
The following conditions are equivalent:
\begin{itemize}
\item[(1)] the canonical morphism $F\to F(-\times X)$ is an equivalence in $\mathrm{Pre}(\mathrm{Corr})$,
\item[(2)] the canonical morphism of presheaves $\pi^\mathrm{pre}_i(F)\to \pi^\mathrm{pre}(F)(-\times X)$ is a Nisnevich local isomorphism for any $i\in \mathbb Z$,
\item[(3)] the canonical morphism of the Nisnevich sheaves $\pi^\mathrm{shv}_i(F)\to \pi^\mathrm{shv}_i(F)(-\times X)$ is an isomorphism for any $i\in \mathbb Z$.
\end{itemize}
\end{lemma}
\begin{proof}
Since (1) means that $\pi^\mathrm{pre}_i(F)(-)\to \pi^\mathrm{pre}(F)(-\times X)$ is an isomorphism for any $i\in \mathbb Z$, it implies (2).
Conversely, (2) means that the canonical morphism $F(-)\to F(-\times X)$ in $\mathrm{Pre}(\mathrm{Corr})$ is a Nisnevich local equivalence, and since both presheaves $F$ and $F(-\times X)$ are contained in $\mathrm{Shv}_\mathrm{nis}(\mathrm{Corr})$, it follows that $F(-)\simeq F(-\times X)$ in $\mathrm{Pre}(\mathrm{Corr})$. So (2) implies (1).
To prove the equivalence of (2) and (3)
for each $i\in \mathbb Z$ and essentially smooth local henselian $U$,
consider the commutative diagrams
\[\xymatrix{
\pi^\mathrm{pre}_i(F)(U)\ar[r]\ar[d]^{=}& \pi^\mathrm{pre}(F)(U\times X)\ar[d]^{\simeq}\\ \pi^\mathrm{shv}_i(F)(U)\ar[r]& \pi^\mathrm{shv}(F)(U\times X)
}\]
where the right vertical arrow is an isomorphism by \Cref{th:strhominv}.
Since (2) and (3) mean respectively that the first and the second horizontal arrows are isomorphisms, the claim follows.
\end{proof}
\begin{corollary}\label{cor:A1invpishv}
Given $\mathrm{Corr}$ such that $\mathbb A^1_k$ satisfies (NHC),
an object $F\in \mathrm{Shv}_\mathrm{nis}(\mathrm{Corr})$ is $\mathbb A^1$-invariant
if and only if the associated Nisnevich sheaves $\pi^\mathrm{shv}_{i}F$ are $\mathbb A^1$-invariant.
\end{corollary}
\begin{proof}
The claim follows by the equivalence of (1) and (3) in \Cref{lm:F-timesXpipre-timesXNispish-timesX}.
\end{proof}
Note that for each
$F\in \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathbf{SH}^{S^1}(\mathrm{Corr})$,
$\pi^\mathrm{shv}_{i}F\in \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}(\mathrm{Shv}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr}))$.
\begin{corollary}\label{cor:OmegaGmpishv}
Given $\mathrm{Corr}$ such that (NHC) holds for $\mathbb G_m$,
then $F\in \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathbf{SH}^{S^1}(\mathrm{Corr})$ is in $\mathbf{SH}(\mathrm{Corr})$
if and only if for each $i\in \mathbb Z$,
$\pi^\mathrm{shv}_{i}F\in \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}(\mathrm{Shv}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr}))$ is in
$\mathrm{Shv}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr})[\mathbb G_m^{\wedge 1}]$.
\end{corollary}
\begin{proof}
The claim follows by the equivalence of (1) and (3) in \Cref{lm:F-timesXpipre-timesXNispish-timesX} applied to $X=\mathbb G_m$ and $X=\mathrm{pt}_k\in \mathrm{Sm}_k$.
\end{proof}

\begin{corollary}\label{cor:PreSigmaSHI}
Given $\mathrm{Corr}$ such that $\mathbb A^1_k$ satisfies (NHC),
for any $\mathbb A^1$-invariant $F\in \mathrm{Pre}_{\mathrm{gp},\Sigma}(\mathrm{Corr})$, the sheaf $L^\mathrm{tr}_\mathrm{nis} F\in \mathrm{Shv}_\mathrm{nis}(\mathrm{Corr})$ is $\mathbb A^1$-invariant.
\end{corollary}
\begin{proof}
The claim follows by the equivalence of (1) and (2) in \Cref{lm:F-timesXpipre-timesXNispish-timesX}, and because
the morphisms $\pi^\mathrm{pre}_i(F)\to \pi^\mathrm{pre}_i(L^\mathrm{tr}_\mathrm{nis} F)$ are Nisnevich local equivalences by \Cref{lm:LnistrLnis}.
\end{proof}

\begin{theorem}\label{csth:smotcomputed}
Given a correspondences  category $\mathrm{Corr}$, then
\par (1)
if (Fr) holds,
there is the canonical equivalence
$L_\mathrm{mot}^\mathrm{tr} \simeq L_\mathrm{nis}^\mathrm{tr} L_{\mathbb A^1}^\mathrm{tr}$
of the endofunctors on $\mathrm{Pre}_{\mathrm{gp},\Sigma}(\mathrm{Corr})$.
\par (2)
if (Fr) and (NL) hold,
then for any $F\in \mathrm{Pre}_{\mathrm{gp},\Sigma}(\mathrm{Corr})$, there is the canonical equivalence
$L_\mathrm{mot} e_* F \simeq L_\mathrm{nis} L_{\mathbb A^1} \gamma_* F$, where $e_*\colon \mathrm{Pre}_{\Sigma}(\mathrm{Corr})\to \mathrm{Pre}(\mathrm{Sm}_k)$ denotes the canonical functor.
\par (3)
if (Fr), and (NL), and (NC) hold,
the canonical functors
\[\mathbf{H}_\mathrm{geom}(\mathrm{Corr})\to \mathbf{SH}^{S^1}(\mathrm{Corr})\to \mathbf{SH}(\mathrm{Corr})\] are fully faithful.
\end{theorem}
\begin{proof}
By (Fr) and \Cref{th:strhominv} $X=\mathbb A^1$ satisfies (NHC).
Then (1) follows immediate from \Cref{lm:HNCVopensubsetA1}, \Cref{cor:PreSigmaSHI}.
(2) follows by \Cref{lm:LnistrLnis}.
(NC) implies that the second arrow in (3) is fully faithful.
The functor
$\Delta \mathrm{Corr}\to \mathrm{Pre}_\Sigma(\mathrm{Corr})$ is fully faithful,
and consequently the composite
$\mathbf{SH}_\mathrm{geom}(\mathrm{Corr})\to \mathbf{H}_\mathrm{zar}(\mathrm{Corr})=\mathrm{HI}_\Sigma(\mathrm{Corr})\cap \mathrm{Shv}_\mathrm{zar}(\mathrm{Corr})$
too.
To conclude point (3) we use that $\mathbf{SH}(\mathrm{Corr})\simeq \mathbf{SH}_\mathrm{Zar}(\mathrm{Corr})$ by \cite{SHzar}.
\end{proof}

\subsection{Harts of homotopy t-structures}\label{subsect:hthSHCor}

Given a stable $\infty$-category $R$, and a correspondences category $\mathrm{Corr}$,
a t-structure on $R$ induces the ones on the stable $\infty$-categories $\mathrm{Pre}^R(\mathrm{Corr})$, $\mathrm{Pre}^R(\mathrm{Sm}_k)$ that we call homotopy t-structures.

Given a stable $\infty$-category $R$, and a correspondences category $\mathrm{Corr}$, a t-structure on $R$ induces the ones on the stable $\infty$-categories
$\mathrm{Pre}^R(\mathrm{Corr})$, $\mathrm{Pre}^R(\mathrm{Sm}_k)$ that we call homotopy t-structures and denote $\tau^{\mathrm{tr},\mathrm{pre}}$, $\tau^\mathrm{pre}$.
By \Cref{lm:Ltstructureonesideuniversal} there is the t-structure $\tau^\mathrm{shv}=L_\mathrm{nis}(\tau^\mathrm{pre})$ on $\mathrm{Shv}_\mathrm{nis}^R(\mathrm{Sm}_k)$, since $L_\mathrm{nis}$ preserves $\mathrm{Pre}^R(\mathrm{Sm}_k)_{< 0}$.
Moreover, by \Cref{lm:LnistrLnis} it follows that if (NisTr) holds, then $L_\mathrm{nis}^\mathrm{tr}$ preserves $\mathrm{Pre}^R(\mathrm{Corr})_{< 0}$.
Then by \Cref{lm:Ltstructureonesideuniversal} there is the t-structure $\tau^{\mathrm{tr},\mathrm{shv}}=L_\mathrm{nis}(\tau^{\mathrm{tr},\mathrm{pre}})$.
Summarising we get the following lemma.
\begin{lemma}\label{lm:tstructureShvtr}
Given $\mathrm{Corr}$ that satisfy (NisTr), and a stable $\infty$-category $R$ with filtered limits and equipped with a t-structure, then
there is a t-structure $\tau$ on $\mathrm{Shv}^R_\mathrm{nis}(\mathrm{Corr})$ such that (1) $F\in \mathrm{Shv}^R_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$ if and only if the stalks of $F$ at local henseleian essentially smooth schemes are in $R_{\geq 0}$ (2) the heart of $\tau$ is $\mathrm{Shv}^{R^\heartsuit}_\mathrm{nis}(\mathrm{Corr})$
\end{lemma}
\begin{proof}
By the above define $\tau=L_\mathrm{nis}(\tau^{\mathrm{tr},\mathrm{pre}})$ in sense of \Cref{lm:Ltstructureonesideuniversal}. Then (1) follows by \Cref{lm:LnistrLnis}, and \eqref{eq:tstructureS}. (2) follows by \Cref{lm:Ltstructureonesideuniversal}, since $\mathrm{Shv}^{R^\heartsuit}_\mathrm{nis}(\mathrm{Corr})=L^\mathrm{tr}_\mathrm{nis}(\mathrm{Pre}^{R^\heartsuit}(\mathrm{Corr}))$.
\end{proof}

\begin{proposition}\label{prop:tstructures}
(1)
The homotopy t-structure on $\mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})$
induces the t-structure on $\mathbf{SH}^{S^1}(\mathrm{Corr})$
such that
\begin{equation}\label{eq:tstructurerestrictionSHCorrkS1}\begin{array}{lcl}
\mathbf{SH}^{S^1}(\mathrm{Corr})_{\geq 0} &=& \mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0} \cap \mathrm{Pre}^{S^1}_{\mathbb A^1}(\mathrm{Corr}),\\
\mathbf{SH}^{S^1}(\mathrm{Corr})_{\leq 0} &=& \mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\leq 0} \cap \mathrm{Pre}^{S^1}_{\mathbb A^1}(\mathrm{Corr}),
\end{array}\end{equation}
and the truncation functor are induced by the restriction.
Moreover, for any $F\in \mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$, the image $L^\mathrm{tr}_\mathrm{mot}F\in \mathbf{SH}^{S^1}(\mathrm{Corr})$ is contained in $\mathbf{SH}^{S^1}(\mathrm{Corr})_{\geq 0}$.
(2)
The levelwise homotopy t-structure on $\mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathrm{Shv}^{S^1}(\mathrm{Corr})$
induces the homotopy t-structure on $\mathbf{SH}(\mathrm{Corr})$ such that
\begin{equation}\label{eq:tstructurerestrictionSHCorrk}\begin{array}{lcl}
\mathbf{SH}(\mathrm{Corr})_{\geq 0} &=& \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathrm{Shv}^{S^1}(\mathrm{Corr})_{\geq 0} \cap \mathrm{Pre}^{S^1}_{\mathbb A^1}(\mathrm{Corr})[\mathbb G_m^{\wedge 1}],\\
\mathbf{SH}(\mathrm{Corr})_{\leq 0} &=& \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathrm{Shv}^{S^1}(\mathrm{Corr})_{\leq 0} \cap \mathrm{Pre}^{S^1}_{\mathbb A^1}(\mathrm{Corr})[\mathbb G_m^{\wedge 1}].
\end{array}\end{equation}
and the truncation functor are induced by the restriction.
Suppose $\mathrm{Corr}$ satisfies (Fr), then for any $F\in \mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$, the image $L^\mathrm{tr}_\mathrm{smot}F\in \mathbf{SH}(\mathrm{Corr})$ is contained in $\mathbf{SH}^{S^1}(\mathrm{Corr})_{\geq 0}$.
\end{proposition}
\begin{proof}
By \Cref{cor:A1invpishv} and \Cref{cor:OmegaGmpishv} the homotopy t-structure truncation functors on $\mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})$ and on $\mathrm{Spt}_{\mathbb G_m^{\wedge 1}}\mathrm{Shv}^{S^1}(\mathrm{Corr})$ induces by the restriction a t-structures truncation functors on $\mathbf{SH}^{S^1}(\mathrm{Corr})$ and $\mathbf{SH}(\mathrm{Corr})$.
The induced t-structures are the homotopy t-structures, because the first equalities of \eqref{eq:tstructurerestrictionSHCorrkS1}, \eqref{eq:tstructurerestrictionSHCorrk} hold by the definitionn of the homotopy t-structures.

\par To prove the second claim in point (1) for the given $F\in \mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$ consider the truncation
$F_{\geq 0}$ with respect to the homotopy t-structure on $\mathrm{Pre}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$. Then $F\to F_{\geq 0}$ is a Nisnevich local equivalence.
By \Cref{th:strhominv} in view of \cite[Lemma 25]{LGmLnisLA1tf} the functor $L^\mathrm{tr}_{\mathbb A^1}$ preserves the Nisnevich local equivalences. Hence $L^\mathrm{tr}_{\mathbb A^1}F\to L^\mathrm{tr}_{\mathbb A^1}F_{\geq 0}$ is a Nisnevich local equivalence. Then since $L^\mathrm{tr}_{\mathbb A^1}F_{\geq 0}\in \mathrm{Pre}^{S^1}(\mathrm{Corr})_{\geq 0}$, $L^\mathrm{tr}_\mathrm{nis} L^\mathrm{tr}_{\mathbb A^1} F\simeq L^\mathrm{tr}_\mathrm{nis} L^\mathrm{tr}_{\mathbb A^1}F_{\geq 0}\in \mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})_{\geq 0}$, so the claim is done by \Cref{csth:smotcomputed}(1,2).
The claim in (2) follows similarly because $\Omega_{\mathbb G_m^{\wedge 1}}$ preserves the Nisnevich local equivalences on $\mathrm{Pre}^{S^1}_{\mathbb A^1}(\mathrm{Corr})$, since $\mathbb G_m$ satisfies (HNC) by \cite[]{LGmLnisLA1tf} because $\mathrm{Corr}$ satisfies (Fr).
\end{proof}

The homotopy t-structure on $\mathrm{Shv}^{S^1}_\mathrm{nis}(\mathrm{Corr})$ is the t-structure such that
$F\in \mathbf{SH}^{S^1}(\mathrm{Corr})_{\geq 0}$ whenever $F(X^h_x)\in \mathbf{SH}_{\geq 0}$ for all $X\in \mathrm{Sm}_k$ and $x\in X$.
The homotopy t-structure on $\mathbf{SH}^{S^1}(\mathrm{Corr})$
is the t-structure such that
$F\in \mathbf{SH}(\mathrm{Corr})_{\geq 0}$ whenever $\Omega^\infty_{\mathbb G_m}(\Sigma^t_{\mathbb G_m} F)(X^h_x)\in \mathbf{SH}_{\geq 0}$ for all $X\in \mathrm{Sm}_k$, $x\in X$, and $t\in \mathbb Z$.
\begin{theorem}\label{csth:t-structures}
Given correspondences category $\mathrm{Corr}$ that satisfies (Fr), (NisTr), and (NisCanc), the following holds:

(1)
$\mathbf{SH}^{S^1,\heartsuit}(\mathrm{Corr})\simeq \mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr})$,
where the latter category denotes the subcategory of the abelian category $\mathrm{Shv}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr})$ spanned by $\mathbb A^1$-invariant objects.

(2) $\mathbf{SH}^{\heartsuit}(\mathrm{Corr})\simeq \mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr})[\mathbb G_m^{\wedge -1}]$,
and the latter category is the abelian category with the objects given by sequences $(F_l\in \mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\mathrm{Corr}),l\in \mathbb Z)$, equipped with isomorphisms $F_{l}\cong \Omega_{\mathbb G_m^{\wedge 1}}F_{l+1}$.
\end{theorem}
\begin{proof}
The claim follows by the argument like in \cite{CycleModtstructurehart} for $\mathbf{DM}(k)$, see also  \cite{TowardsconservativityofGm–stabilization} for the framed correspondences case with the use of \cite{hty-inv,DrKyllfinFrpi00,surj-etale-exc}.
We presented details for the $\infty$-categorical revision of the argument in the arXiv:2108.01006 version of the text, we skip them here to economy the space and because the deduction of the theorem from the results of the previous section is standard and does not depend on the definition of the correspondences category and properties of the base field or scheme.
\end{proof}

Consider the additive category ${{\mathbb Z\mathrm{F}}}_*(k)$ that is the additivisation of $\mathrm{Fr}_+(k)$ in sense of \Cref{def:ZFstar}.
Denote by $\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)$
the classical additive category
that is the homotopy category of $\mathrm{Corr}^{\mathrm{tfr},\mathrm{gp}}(k)$.
So the representable presheaves are
\[h^{\mathbb Z\mathrm{tfr}}(X)(U)=\pi_0(h^{\mathrm{tfr},\mathrm{gp}}(X)(U)).\]
Define the quotient categories $\overline{{\mathbb Z\mathrm{F}}}_*(k)$, $\overline{{\mathbb Z\mathrm{F}}}(k)$ and $\pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)$ of the classical additive categories
${{\mathbb Z\mathrm{F}}}_*(k)$, ${{\mathbb Z\mathrm{F}}}(k)$ and $\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)$
with respect to the relations $s\circ\mathrm{pr}=\mathrm{id}_{\mathbb A^1\times X}$, where $\mathrm{pr}\colon \mathbb A^1_X\to X$, and $s\colon X\to \mathbb A^1_X$ is the zero section.
Then
\[\pi_0^{\mathbb A^1}h^{\mathbb Z\mathrm{tfr}}(X)(U)=\pi_0(h^{\mathrm{tfr},\mathrm{gp}}(X)(U\times\Delta_k^\bullet)).\]
There is the sequence of functors of classical categories
\[\overline{{\mathbb Z\mathrm{F}}}_*(k)\to \overline{{\mathbb Z\mathrm{F}}}(k)\to \pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)\]
because the homotopy classes of the mages of $\sigma_X$ and $\mathrm{id}_X$ in the mapping space the functor $\mathrm{Corr}^{\mathrm{tfr}}_k(X,X)$
with respect to the functor $\mathrm{Fr}_+(k)\to \mathrm{Corr}^{\mathrm{tfr}}(k)$
 are equal.
\begin{lemma}\label{lm:ZFtfr}
$\overline{{\mathbb Z\mathrm{F}}}(-,X)\simeq \pi_0^{\mathbb A^1}h^{\mathbb Z\mathrm{tfr}}(X)(-)$.
\end{lemma}
\begin{proof}
By \cite[Corollaries 2.3.20, 2.3.25]{five-authors}
the morphism
$\mathrm{Fr}(-\times\Delta^\bullet,X)\to h^\mathrm{tfr}(X)(-\times\Delta^\bullet)$
is an equivalence on affine schemes.
Hence
${\mathbb Z\mathrm{F}}(-\times\Delta^\bullet,X)\simeq h^{\mathbb Z\mathrm{tfr}}(X)(-\times\Delta^\bullet)$
on affine schemes, and
\[\pi_0({\mathbb Z\mathrm{F}}(-\times\Delta^\bullet,X))\simeq \pi_0(h^{\mathbb Z\mathrm{tfr}}(X)(-\times\Delta^\bullet)).\]
\end{proof}
\begin{lemma}\label{lm:ZF}
$\pi_0^{\mathbb A^1}h^{{\mathbb Z\mathrm{F}}}(X)(-)\simeq \overline{{\mathbb Z\mathrm{F}}}_k(-,X)$.
\end{lemma}
\begin{proof}
Consider the representable presheaves in $\overline{\mathbb Z\mathrm{F}}_*(k)$.
The presheaf $\pi_0^{\mathbb A^1}h^{{\mathbb Z\mathrm{F}}}(X)(-)$ is the quotient presheaf of
$\overline{\mathbb Z\mathrm{F}}_{*,k}(-,X)$ with respect to the relations
\begin{equation}\label{eq:hZFA1}c=c\circ \sigma_U\text{ for each }c\in \overline{\mathbb Z\mathrm{F}}_{*,k}(U,X).\end{equation}
The presheaf $\overline{\mathbb Z\mathrm{F}}_k(-,X)$ is the quotient presheaf of
$\overline{\mathbb Z\mathrm{F}}_{*,k}(-,X)$ with respect to the relations
\begin{equation}\label{eq:hZFA1ovZF}c=\sigma_X\circ c\text{ for each }c\in \overline{\mathbb Z F}_{*,k}(U,X).\end{equation}
Relations \eqref{eq:hZFA1} and \eqref{eq:hZFA1ovZF} are equivalent up to the naive $\mathbb A^1$-homotopy equivalence because
$\pi_0(\mathrm{SL}_n(\Delta^\bullet_k))$ is trivial for any field $k$.
\end{proof}
\begin{lemma}\label{lm:HIZFHIZtr}
There are equivalences of categories
\[\begin{array}{lll}
\overline{{\mathbb Z\mathrm{F}}}(\mathrm{SmAff}_k)&\simeq& \pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(\mathrm{SmAff}_k),\\
\mathrm{HI}^{\mathbb Z\mathrm{F}}(\mathrm{SmAff}_k)&\simeq& \mathrm{HI}^{\mathbb Z\mathrm{tfr}}(\mathrm{SmAff}_k),\\
\mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)&\simeq& \mathrm{HI}^{\mathbb Z\mathrm{tfr}}_\mathrm{nis}(k).
\end{array}\]
\end{lemma}
\begin{proof}
The functor
$\mathrm{Fr}_+(k)\to \mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)$
induces the functor
$\overline{{\mathbb Z\mathrm{F}}}(k)\to \pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k),$
that is an equivalence because the induced moresphim of prepresentable presheaves
\[\pi_0^{\mathbb A^1}h^{{\mathbb Z\mathrm{F}}}(X)(-)\cong \overline{{\mathbb Z\mathrm{F}}}(-,X)\cong \pi_0^{\mathbb A^1}h^{\mathbb Z\mathrm{tfr}}(X)(-)\]
is an isomorphism in $\mathrm{Pre}^\mathrm{Ab}(\mathrm{Sm}_k)$
by \Cref{lm:ZF} and \Cref{lm:ZF}.
Consequently, the categories of presheaves and Nisnevich sheaves on
$\overline{{\mathbb Z\mathrm{F}}}(\mathrm{SmAff}_k)$ and  $\pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(\mathrm{SmAff}_k)$ are equivalent.
Finally, note that the categories of Nisnevich sheaves on the latter categories are equivalent to the ones on
$\overline{{\mathbb Z\mathrm{F}}}(k)$ and  $\pi_0^{\mathbb A^1}\mathrm{Corr}^{\mathbb Z\mathrm{tfr}}(k)$.
\end{proof}
\begin{corollary}\label{cor:SHeffhatHIZFnis}
There is the equivalence
$\mathbf{SH}^{\mathrm{eff},\heartsuit}(k)\simeq \mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)$.
\end{corollary}
\begin{proof}
It follows by \Cref{csth:t-structures} applied to $\mathrm{Corr}^\mathrm{tfr}$ that $\mathbf{SH}^{\mathrm{eff},\heartsuit}(k)\simeq \mathrm{HI}^{\mathbb Z\mathrm{tfr}}_\mathrm{nis}(k)$.
and the claim follows by \Cref{lm:HIZFHIZtr}.
\end{proof}
\begin{corollary}\label{cor:SHeffharttildeDMeffhart}
There is the equivalence
$\mathbf{SH}^{\mathrm{eff},\heartsuit}(k)\simeq \widetilde{\mathbf{DM}}^{\mathrm{eff},\heartsuit}(k)$.
\end{corollary}
\begin{proof}
Consider quotient category $\overline{\widetilde{\mathrm{Cor}}}(k)$ of the category $\widetilde{\mathrm{Cor}}(k)$
with respect to the relations $s\circ\mathrm{pr}=\mathrm{id}_{\mathbb A^1\times X}$, where $\mathrm{pr}\colon \mathbb A^1_X\to X$, and $s\colon X\to \mathbb A^1_X$ is the zero section.
The functor
$\overline{{\mathbb Z\mathrm{F}}}(k)\to \overline{\widetilde{\mathrm{Cor}}}(k)$
is an equivalence because it induces the isomorphism of representable presheaves by \cite[Theorem 8.11]{framed-MW}. Hence
\[\mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)\simeq \mathrm{HI}^\mathrm{MW}_\mathrm{nis}(k),\]
where $\mathrm{HI}^\mathrm{MW}_\mathrm{nis}(k)=\mathrm{HI}^\mathrm{Ab}_\mathrm{nis}(\overline{\widetilde{\mathrm{Cor}}}(k))$.
By \Cref{csth:t-structures} applied to $\widetilde{Cor}(k)$ there is equivalence $\widetilde{\mathbf{DM}}^{\mathrm{eff},\heartsuit}(k)\simeq\mathrm{HI}^\mathrm{MW}_\mathrm{nis}(k)$.
The claim follows by \Cref{cor:SHeffhatHIZFnis}.
\end{proof}

\begin{corollary}
The heart of the homotopy t-structure on $\mathbf{SH}(k)$ is equivalent to the category of homotopy modules $\mathrm{SHI}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]$.
\end{corollary}
\begin{proof}
By \Cref{csth:t-structures} and \Cref{lm:HIZFHIZtr}
$\mathbf{SH}^\heartsuit(k)\simeq \mathrm{HI}^{\mathbb Z\mathrm{tfr}}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]\simeq \mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}].$

Consider the functors
\[\mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]\xrightarrow{u} \mathrm{SHI}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]\xrightarrow{\mathrm{EM}} \mathbf{SH}(k)\xrightarrow{\tau} \mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}],\]
where
$u$ is forgetful,
$\mathrm{EM}$ takes $F=(F_l,l\in \mathbb Z)$ to the $\mathbb G_m$-spectrum $HF$ of Eilenberg-MacLane $S^1$-spectra sheaves $\mathrm{H}F_l\in \mathbf{SH}^{S^1}(k)$,
$\tau(F) = (\pi_{l,l}(F),l\in \mathbb Z)$.
For any $F\in \mathrm{SHI}_\mathrm{nis}(k)$, the spectrum $L_\mathrm{nis} HF$ is $\mathbb A^1$-invariant, and Nisnevich local.
Hence $\pi_0\mathrm{H}F=F$.
Consequently, for $F=(F_l,l\in \mathbb Z)\in \mathrm{SHI}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]$,
\[\pi_{l,l}\mathrm{H}F=\varinjlim_{j} \pi_{j,j}(\Omega_{\mathbb G_m^{\wedge 1}}F_{l+j})=\varinjlim_{j} \Omega_{\mathbb G_m^{\wedge 1}}F_{l+j}=F_l.\]
Thus \[u \tau \mathrm{EM}\simeq \mathrm{id}_{\mathrm{SHI}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]}.\]
Since $u$ is conservative, the functors $u$ and $\tau \mathrm{EM}$ define equivalence
\[\mathrm{HI}^{\mathbb Z\mathrm{F}}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}]\simeq \mathrm{SHI}_\mathrm{nis}(k)[\mathbb G_m^{\wedge -1}].\]
\end{proof}

\section{Cousin complex}\label{sect:CousinExactness}

In this section, we apply the results of \Cref{sect:homotpiesatinfinity,sect:StrHomInvTheorem}
deduce the result on Cousin complexes on the open complements of smooth divisors in essentially smooth schemes.
We repeat the standard argument used in \cite{Voe-hty-inv} for the case of local schemes.

Let $\mathrm{Corr}$ be a correspondences category that satisfies (Fr) and (NL), see \Cref{def:CorrinftyCategory,def:FrNLNC}.
Let $F$ be an $\mathbb A^1$-invariant additive functor $\mathrm{Corr}\to \mathbf{SH}$.
For $X\in \mathrm{Sm}_k$, and a closed subscheme $Z$ in $X$,
denote $E^l_Z(X)=\pi_{-l}\operatorname{fib}(L_\mathrm{nis} F(X)\to L_\mathrm{nis} F(X-Z))$,
where $L_\mathrm{nis}$ denote the associated Nisnevich local sheaf.
Denote by $\mathrm{Corr}^\mathrm{pair}$ the subcategory in the category of arrows of $\mathrm{Corr}$ spanned by the morphisms induced by open immersions $X-Z\to X$ in $\mathrm{Sm}_k$, then there is the functor \begin{equation}\label{eq:E(X/X-Z)}\mathrm{Corr}^\mathrm{pair}\to \mathbf{SH}; ((X-Z)\to X)\to E^l_Z(X).\end{equation}
By \Cref{cor:PreSigmaSHI} and \Cref{lm:LnistrLnis} since $\mathrm{Corr}$ satisfies (Fr) and (NL),
$L_\mathrm{nis} F$ defines $\mathbb A^1$-invariant functor $\mathrm{Corr}\to \mathbf{SH}$.
Hence the presheaves \eqref{eq:E(X/X-Z)} are $\mathbb A^1$-invariant.

\begin{lemma}\label{lm:extensionofsupperts}
Let $U$ be essentially smooth local scheme over $k$,
and $V=U-Z(f)$ for $f\in \mathcal O(U)$ such that $Z(f)$ is smooth.
Then for any closed subscheme $W$ of codimension $r$ in $V$,
the morphism
\[E^l_W(V)\to \varinjlim_{\operatorname{codim}_V W^\prime=r-1} E^l_{W^\prime}(V)\]
is trivial for all $l$,
where $W^\prime$ runs over the filtered set of closed subschemes of codimension $r-1$.
\end{lemma}
\begin{proof}
For any $X\in \mathrm{Sm}_k$, $x\in X$, and a closed subscheme $Y$ such that $U=X_x$, $W=Y_x$,
by \cite[Theorem 2.11]{hty-inv}, if $Z(f)=\emptyset$, and \Cref{cor:contractinggenpointhomoveretaSmX}, if $Z(f)\neq\emptyset$,
there is \[c^\prime\in {\mathbb Z\mathrm{F}}_N(V,X-Y), c=(Z,\mathcal V,\varphi,g)\in {\mathbb Z\mathrm{F}}(V\times\mathbb A^1, X)\] such that $c\circ i_0=\sigma^N_V\mathrm{can}$, $c\circ i_1=j\circ c^\prime$,
where $i_0,i_1\colon V\to V\times\mathbb A^1$ is the zero and unit sections, $\mathrm{can}\colon V\to X$ is the canonical morphism, $j\colon X-Y\to X$ is the open immersion. Denote by $\widetilde W$ the image in $V$ of the preimage of $Y$ along the morphisms
\[V\leftarrow V\times \mathbb A^1 \leftarrow Z\xrightarrow{g} X,\]
then $\operatorname{codim}{\widetilde Y}\geq r-1$.
Since for each $l$, $E^l_{-}(-)$ defines an $\mathbb A^1$-invariant functor $\mathrm{Corr}^\mathrm{pair}\to \mathrm{Ab}$,
the morphisms $(j\circ c^\prime)^*, \mathrm{can}^*\colon E^l_Y(X)\to E^l_{\widetilde W}(V)$ are equal.
Since $(j\circ c^\prime)^*=0$, it follows that $\mathrm{can}^*=0$.
Hence the morphism \[j^*\colon E^l_Y(X)\to \varinjlim_{\operatorname{codim}W^\prime=r-1}E^l_{W^\prime}(V)\] is trivial.
Since the latter claim holds for all $ X$ and $Y$ as above,
the morphism \[j^*\colon E^l_W(V)\to \varinjlim_{\operatorname{codim}_V W^\prime=r-1} E^l_{W^\prime}(V)\] is trivial.

The case of empty $Z(f)$ follows by the same argument with the use of the injectivity on local essentially smooth schemes \cite[Theorem 3.11]{hty-inv}.
\end{proof}
\begin{corollary}\label{cor:shortexactseuence}
Under the assumptions of \Cref{lm:extensionofsupperts},
the short sequence
\[0\to \varinjlim_{\operatorname{codim}_V W^\prime=r-1} E^l_{W^\prime}(V) \to \varinjlim_{\operatorname{codim}_V W^\prime=r-1} E^l_{W^\prime}(V-W)\to E^{l+1}_W(V)\to 0\]
is exact. Recall that $W$ is any closed subscheme of $V$, $\operatorname{codim}{W}=r$.
\end{corollary}
\begin{proof}
\end{proof}
\begin{theorem}\label{th:CousinsubsetsLocEssSm}
Let
$F\in \mathbf{SH}^{S^1}(\mathrm{Corr})$, and $E^l_{Z}(X)=\pi_{-l}\operatorname{fib}(F(X)\to F(X-Z))$.
The Cousin complex
\[0\to E^l(V)\to E^l(\eta)\to \bigoplus_{x\in V^{(1)}}E_x^{l+1}(V)\to \dots \bigoplus_{x\in V^{(r)}}E_x^{l+r}(V)\to \dots \bigoplus_{x\in V^{(d)}}E_x^{l+d}(V)\to 0\]
is exact for the scheme $V=U-Z(f)$ for an essentially smooth local scheme $U$ of dimension $d$, and $f\in \mathcal O(U)$ such that $Z(f)$ is smooth, where $\eta$ denotes the generic point of $V$,
in particular, the claim holds for $\mathbf{SH}^{S^1}(\mathrm{Corr})$-representable cohomology theories.
\end{theorem}
\begin{proof}
Note that $E_x^l(V)=E_x^l(V_x)$ by the definition.
It follows from \Cref{cor:shortexactseuence} that the short sequences
\[0\to
\varinjlim_{\operatorname{codim}{W}=r-1}E_W^{l+r-1}(V)\to \bigoplus_{x\in V^{(r-1)}}E_x^{l+r-1}(V_x)\to \bigoplus_{\operatorname{codim}{W}=r}E_{W^\prime}^{l+r}(V)\to 0\]
are exact.
\end{proof}

redpajama_set_name: RedPajamaArXiv