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This article is cited in 1 scientific paper (total in 1 paper) On an analogue of the fundamental Voevodsky theorem D. N. Tyurinab a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences b Leonard Euler International Mathematical Institute at Saint Petersburg (SPB LEIMI), St. Petersburg
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    § 1. IntroductionThe Voevodsky motivic category, which was originally introduced in [1], is build from elementary bricks, whose role is played by homotopy invariant Nisnevich sheafs with transfers, which were introduced and studied in [2]. This required from Voevodsky a detailed study of homotopy invariant presheaves with transfers (see [1]). Voevodsky predicted that, in the case of stable motivic homotopy theory $\operatorname{SH}(k)$, a similar role is played by homotopy invariant quasi-stable Nisnevich sheafs of abelian groups with framed transfers. The latter were introduced in the unpublished fundamental Voevodsky notes [3]. A sketch of these notes later appeared in the paper [4] by Garkusha and Panin. For this, they proposed in [5] an interpretation of Nisnevich sheafs of abelian groups with framed transfers as Nisnevich $\mathbb{Z}F_{\ast}$-sheafs of abelian groups. One of the merits of the category of $\mathbb{Z}F_{\ast}$-correspondences of [5] is that it is additive (unlike the Voevodsky category of $\operatorname{Fr}_{\ast}$-correspondences). For homotopy invariant quasi-stable $\mathbb{Z}F_{\ast}$-presheaves of abelian groups, Garkusha and Panin [5] proved an analogue of Voevodsky theorem (Theorem 3.1.12 in [1]) In particular, for any homotopy invariant quasi-stable $\mathbb{Z}F_{\ast}$-presheave of abelian groups $F$ on $\mathrm{Sm}/k$, the associated Nisnevich sheaf as well as all the cohomology presheafs $H^n_{\mathrm{Nis}}(-,F_{\mathrm{Nis}})$ are equipped with a canonical structure of $\mathbb{Z}F_{\ast}$-presheafs, and, in addition, are homotopy invariant and quasi-stable. As a corollary, the contravariant functor
$$
\begin{equation*}
H^{\bullet}\colon \mathrm{SmOp}/k\to \operatorname{Gr}\mathcal{A}b,\qquad (X,X-Z)\mapsto \bigoplus_{n\in\mathbb{N}}(H^n_{\mathrm{Nis}})_{Z}(X,F_{\mathrm{Nis}})
\end{equation*}
\notag
$$
can be considered as a Panin–Smirnov cohomology theory (see [6]). Hence, according to Theorem 9.1 in [7], if $X\in \mathrm{Sm}/k$ is irreducible, $x\in X$ is a point of codimension $d$, $U$ is a local affine scheme $\operatorname{Spec}(O_{X,x})$, and $\eta$ is a generic point of $U$, then the Cousin complex
$$
\begin{equation*}
0\to H^0(U)\to H^0(\eta)\to\bigoplus_{y\in U^{(1)}}H^1_y(X)\to\dots\to H^d_x(U)\to 0,
\end{equation*}
\notag
$$
is exact. In this paper, we show that if $F$ is an $\mathbb{A}^1$-invariant quasi-stable $\mathbb{Z}F_{\ast}$-presheaf on $\mathrm{Sm}/k$, then, for any $y\in U$ of codimension $m$ and any $n\geqslant 0$, the following isomorphisms take place:
$$
\begin{equation*}
(H^n_{\mathrm{Nis}})_y(U,F_{\mathrm{Nis}})= \begin{cases} F_{-m}(\operatorname{Spec}(k(y))), &n=m, \\ 0, & n\neq m. \end{cases}
\end{equation*}
\notag
$$
In particular, the corresponding Gersten complex of the Nisnevich sheaf $F_{\mathrm{Nis}}$
$$
\begin{equation*}
0\to F_{\mathrm{Nis}}(U)\to F(\eta)\to\bigoplus_{y\in U^{(1)}}F_{-1}(k(y))\to\dots\to F_{-d}(k(x))\to 0\,
\end{equation*}
\notag
$$
is exact (cf. Theorem 4.37 in [3]). The author thanks the excellent environment of the International Mathematical Center at POMI (St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences). The author also expresses his deepest gratitude to I. A. Panin for his invaluable help in posing the problem and shaping the proofs. § 2. Main definitionsIn this section we will recall some basic definitions from [3] and some important definitions from [5]. Let $k$ be a field of zero characteristic. By $\mathrm{Sm}/k$ we denote the category of $k$-smooth schemes of finite type, and by $\mathrm{Sm}'/k$ we denote the category of essentially $k$-smooth schemes (in our setting, by an essentially $k$-smooth scheme we mean a Noetherian $k$-smooth scheme $X$ which can be represented as an inverse limit $\lim_{i\in I}X_i$ such that each morphsim $X_i\to X_j$ is an etale affine morphsim of smooth $k$-schemes). In particular, examples of $k$-smooth schemes include $k$-schemes of the form $\operatorname{Spec}(A_{M})$, where $A$ is a smooth $k$-algebra of finite type and $M\subset A$ is a multiplicative system. The natural embedding of categories $\mathrm{Sm}/k\hookrightarrow \mathrm{Sm}'/k$ will be denoted by $\operatorname{inc}$. Throughout, all morphisms are supposed to be $k$-morphisms. Let $X$, $Y$ be $k$-smooth schemes, and $n\geqslant 0$. An explicit framed correspondence of level $n$ from $Y$ to $X$ is the following data:
Two explicit framed correspondences $\Phi=(Z,V,\varphi,g)$ and $\Phi'=(Z',V',\varphi',g')$ of the same level are said to be equivalent if $Z$ and $Z'$ coincide as sets and there is an etale neighborhood $W$ of $Z$ in $U\times_{\mathbb{A}^{n}_{Y}}U'$ on which $g\circ \operatorname{pr}$ agrees with $g'\circ \operatorname{pr}'$ and $\varphi\circ \operatorname{pr}$ agrees with $\varphi'\circ \operatorname{pr}'$. The equivalence class of explicit framed correspondences of level $n$ is called a framed correspondence of level $n$. In particular, each morphism $Y\to X$ can be considered as a framed correspondence of level $0$ presented by its graph. The set of all framed correspondences of level $n$ from $Y$ to $X$ is denoted by $\operatorname{Fr}_{n}(Y,X)$. For any $n\geqslant 0$, we consider $\operatorname{Fr}_{n}(Y,X)$ as a pointed set with basepoint $0_{n}$ corresponding to the empty set $V=\varnothing$. Now let $Y$, $X$ and $S$ be smooth $k$-schemes Then, for each pair of natural numbers $(n,m)$, there is an associative composition map (see [3])
$$
\begin{equation}
\operatorname{Fr}_n(Y,X)\times \operatorname{Fr}_m(X,S)\to \operatorname{Fr}_{n+m}(Y,S),
\end{equation}
\tag{2.1}
$$
which, in the case of regular morphisms $Y\to X$ and $X\to S$, coincides with the usual composition map (for details, see [3] and [5]). For each $Y$, $X\in \mathrm{Sm}/k$, by $\mathbb{Z}[\operatorname{Fr}_{n}(Y,X)]$ denote the free abelian group generated by all framed correspondences of level $n$ from $Y$ to $X$. Let $A$ be its subgroup generated by all elements of type
$$
\begin{equation*}
(Z\sqcup Z',V,\varphi,g)-(Z,V\setminus Z',\varphi|_{V\setminus Z'},g|_{V\setminus Z'})-(Z',V\setminus Z,\varphi|_{V\setminus Z},g|_{V\setminus Z}).
\end{equation*}
\notag
$$
Set $\mathbb{Z}F_n(Y,X):=\mathbb{Z}[\operatorname{Fr}_n(Y,X)]/A$. By $\mathbb{Z}F_{\ast}$ we denote the additive category whose objects are of $\mathrm{Sm}/k$ and the morphisms have the form
$$
\begin{equation*}
\operatorname{Hom}_{\mathbb{Z}F_{\ast}}(Y,X)=\bigoplus_{n\geqslant0}\mathbb{Z}F_n(Y,X).
\end{equation*}
\notag
$$
The composition is induced by maps (2.1). In particular, there is a natural embedding of categories $\operatorname{in}\colon \mathrm{Sm}/k\hookrightarrow\mathbb{Z}F_{\ast}$. A $\mathbb{Z}F_{\ast}$-presheaf of abelian groups is an additive contravariant functor from $\mathbb{Z}F_{\ast}$ to the category of abelian groups $Ab$. A $\mathbb{Z}F_{\ast}$-presheaf $F$ is called a $\mathbb{Z}F_{\ast}$-sheaf (in the corresponding topology) if the preshead $\operatorname{in}^{\ast}(F)$ is a sheaf on $\mathrm{Sm}/k$ (in the corresponding topology). A $\mathbb{Z}F_{\ast}$-presheaf is called $\mathbb{A}^{1}$-invariant if, for any $X\in \mathrm{Sm}/k$, the projection $X\times\mathbb{A}^{1}\to X$ induces an isomorphsim $F(X)\to F(X\times\mathbb{A}^{1})$. A $\mathbb{Z}F_{\ast}$-sheaf is called $\mathbb{A}^{1}$-invariant if it is $\mathbb{A}^{1}$-invariant as a $\mathbb{Z}F_{\ast}$-presheaf. A $\mathbb{Z}F_{\ast}$-presheaf is called quasi-stable if, for any $X\in \mathrm{Sm}/k$, the framed correspondence $\sigma_{X}:=(X\times 0,X\times\mathbb{A}^{1},t,\operatorname{pr}_{X})\in \operatorname{Fr}_{1}(X,X)$ induces an isomorphism $\sigma^{\ast}_{X}\colon F(X)\to F(X)$. Note, that for any $f\in F_{0}(Y,X)$, we have the equality $\sigma_{X}\circ f=f\circ\sigma_{Y}$. A $\mathbb{Z}F_{\ast}$-sheaf is called quasi-stable if it is quasi-stable as a $\mathbb{Z}F_{\ast}$-presheaf. Definition 2.1. We say that a $\mathbb{Z}F_{\ast}$-presheaf $F$ satisfies property $(\ast)$ (is an $(\ast)$-presheaf) if it is both $\mathbb{A}^{1}$-invariant and quasi-stable. Note that the $\mathbb{Z}F_{\ast}$-presheafs satisfying property $(\ast)$ form an abelian Grothendieck category. In what follows, if $G$ is a presheaf of abelian groups on $\mathrm{Sm}/k$, then, for simplicity, the direct image $\operatorname{inc}_{*}(G)$ relative to the embedding $\operatorname{inc}\colon \mathrm{Sm}/k\hookrightarrow \mathrm{Sm}'/k$ will be denoted by the same symbol. Note, that in this case, the presheafs $\operatorname{inc}^{*}(\operatorname{inc}_{*}(G))$ and $G$ coincide on $\mathrm{Sm}/k$. § 3. Auxiliary resultsIn this section, we recall some useful results obtained by Garkusha and Panin [5] for $(\ast)$-presheafs. The key result for this paper is the following theorem. Theorem 3.1 (see [3], Lemma 4.5, and [5], Theorem 1.1, Corollary 2.17). Let $F$ be an $(\ast)$-presheaf. Then the corresponding Nisnevich sheaf $F_{\mathrm{Nis}}$ allows a unique structure of a $\mathbb{Z}F_{\ast}$-presheaf correlating with the natural morphism $F\to F_{\mathrm{Nis}}$. In addition, the sheaf $F_{\mathrm{Nis}}$ and all the cohomology presheafs $X\to H^{n}_{\mathrm{Nis}}(X,F_{\mathrm{Nis}})$ are also $(\ast)$-presheafs. We also need the following auxiliary results. Theorem 3.2 (see [5], Theorem 3.15, (1)). Let $V$ be a non-empty open subset of an affine line $\mathbb{A}^{1}$ and $W\hookrightarrow V$ be a non-empty open embedding. Then, for any $(\ast)$-presheaf $F$, the natural homomorphism $F(V)\to F(W)$ is injective. Theorem 3.3 (see [5], Theorem 3.15, (2)). Under the assumptions of Theorem 3.2, for any proper closed subset $S\subset W$, the corresponding homomorphism is an isomorphism. This property of a presheaf $F$ is called the Zariski affine excision or simply the affine excision. Remark 3.4. Note that originally Theorems 3.2 and 3.3 were proposed for open subsets of an affine line $\mathbb{A}^{1}$ over the field $k$. Nevertheless, by invoking some additional arguments, these results can be extended to the case of an affine line $\mathbb{A}_{K}^{1}$ over an essentially $k$-smooth extension $K/k\in \mathrm{Sm}'/k$ of the field $k$. In what follows, when referring to these theorems, we will automatically consider their extended variants. Theorem 3.5 (see [5], Theorem 3.15, (3)). Let $X\in \mathrm{Sm}/k$ be irreducible, $x\in X$, and $U:=\operatorname{Spec}(\mathcal{O}_{X,x})$. Then, for any $(\ast)$-presheaf $F$, the homomorphism $F(U)\to F(\operatorname{Spec}(k(X))$ induced by the canonical morphism $\operatorname{Spec}(k(X))\to U$ is injective. Theorem 3.6 (see [5; Theorem 3.15, (3$'$)]). Under the assumptions of Theorem 3.5, let $U^{h}_{x}$ be the henselization of $U$ at the point $x$. Then, for any $(\ast)$-presheaf $F$, the homomorphism $F(U^{h}_{x})\to F(\operatorname{Spec}(k(U^{h}_{x})))$ induced by the canonical morphism $\operatorname{Spec}(k(U^{h}_{x}))\to U^{h}_{x}$ is injective. Corollary 3.7. Under the assumptions of Theorem 3.5, for any non-empty open subset $V\subset U$ and for any $(\ast)$-presheaf $F$, the natural homomorphism $F(U)\to F(V)$ is injective. Proof. We set $V=V'\cap U$, where $V'$ is the corresponding open subset of $X$. In particular, we have $k(V')=k(X)$. Consider the commutative diagram By Theorem 3.5, the map $F(U)\to F(\operatorname{Spec}(k(X))$ is injective. The proof now follows from the fact that the homomorphism $F(\operatorname{Spec}(k(X)))\to F(\operatorname{Spec}(k(V')))$ is an isomorphism. This proves the corollary. Note that, under the assumptions of Theorems 3.5 and 3.6, the groups $F(U)$ and $F(U^{h}_{x})$ are the stalks of $F$ at the point $x$ of the scheme $X$ in the corresponding (Zariski and Nisnevich, respectively) topologies (see Remark 2.25 in [5]). Now let $\pi\colon X'\to X$ be an etale morphism of smooth $k$-schemes. Let $V'\subset X'$ and $V\subset X$ be open subsets such that $V'=\pi^{-1}(V)$ and the restriction $\pi\colon S'=(X'-V')\to S=(X-V)$ is an isomorphism. The corresponding commutative diagram is called an elementary distinguished square or a Nisnevich square (see also § 3.1 in [8]). Recall that any sheaf in the Nisnevich topology on $\mathrm{Sm}/k$ sends this diagram to a Cartesian square.Suppose now that both $X'$ and $X$ are irreducible and $S$ is also $k$-smooth. We fix some point $x'\in S'$ together with its image $\pi(x')=x$ in $S$. By $U'$ and $U$ we denote, respectively, the spectra of the local rings $\mathcal{O}_{X',x'}$ and $\mathcal{O}_{X,x}$. Theorem 3.8 (see [5; Theorem 3.15, (5)]). Under the above assumptions, the homomorphism induced by $\pi$ is an isomorphism for any $(\ast)$-presheaf $F$. This property of the presheaf $F$ is called the Nisnevich etale excision or simply the etale excision. Corollary 3.9. Let $F$ be an $(\ast)$-presheaf, $V$ an open subset of $\mathbb{A}_{K}^{1}$, where $K\in \mathrm{Sm}'/k$. Then $F_{\mathrm{Nis}}(V)=F(V)$. Proof. We first note that, by Theorem 3.2, the restriction of $F$ to $\mathbb{A}_{K}^{1}$ is a Zariski sheaf, that is, for any open subset $V\subset\mathbb{A}^{1}$, we have the equality $F(V)=F_{\mathrm{Zar}}(V)$. Consider the commutative diagram induced by the embedding of a generic point $\operatorname{Spec}(K(V))\hookrightarrow V$. Each element of $F_{\mathrm{Zar}}(V)$ can be considered as a collection of elements of $F(\operatorname{Spec}(\mathcal{O}_{V,y}))$ over all $y\in V$, and so it follows from Theorem 3.5 that the homomorphism $F_{\mathrm{Zar}}(V)\rightarrow F(\operatorname{Spec}(K(V)))$ is injective. Hence so is the homomorphsim $F_{\mathrm{Zar}}(V)\to F_{\mathrm{Nis}}(V)$.
So, we need to show the surjectivity of map $F(V)=F_{\mathrm{Zar}}(V)\to F_{\mathrm{Nis}}(V)$. Let $a$ be some element of $F_{\mathrm{Nis}}(V)$. Since the stalks $F(\operatorname{Spec}(K(V)))$ and $F_{\mathrm{Nis}}(\operatorname{Spec}(K(V)))$ coincide, there exists an open set $W\subset V$ and an element $a'\in F(W)$ such that $a|_{W}=a'$ in $F_{\mathrm{Nis}}(W)$. Consider the commutative diagram By Theorem 3.1, both presheafs $F$ and $F_{\mathrm{Nis}}$ satisfy $(\ast)$. Hence by Theorem 3.2 the top and bottom rows of this diagram are short exact sequences. Since we have already shown that the maps $F(V)\to F_{\mathrm{Nis}}(V)$, $F(W)\to F_{\mathrm{Nis}}(W)$ are injective, it is enough to show that the map $F(W)/F(V)\to F_{\mathrm{Nis}}(W)/F_{\mathrm{Nis}}(V)$ is an isomorphism.For simplicity, we can suppose that $W$ differs from $V$ by a single closed point $y\in V$. Applying in succession the affine excision (see Theorem 3.3) to all closed points of $W$, we get the isomorphisms
$$
\begin{equation*}
\begin{aligned} \, F(W)/F(V) &\cong F(\operatorname{Spec}(K(V)))/F(\operatorname{Spec}(\mathcal{O}_{V,y})), \\ F_{\mathrm{Nis}}(W)/F_{\mathrm{Nis}}(V) &\cong F_{\mathrm{Nis}}(\operatorname{Spec}(K(V)))/F_{\mathrm{Nis}}(\operatorname{Spec}(\mathcal{O}_{V,y})). \end{aligned}
\end{equation*}
\notag
$$
We have $F(\operatorname{Spec}(K(V)))=F_{\mathrm{Nis}}(\operatorname{Spec}(K(V)))$, and so it suffices to show that $F(\operatorname{Spec}(\mathcal{O}_{V,y}))$ coincides with $F_{\mathrm{Nis}}(\operatorname{Spec}(\mathcal{O}_{V,y}))$. Consider the diagram
On one hand, this diagram can be considered as an inductive limit of Nisnevich squares. Since $F_{\mathrm{Nis}}$ is a Nisnevich sheaf, we get the Cartesian square On the other hand, applying the etale excision (see Theorem 3.8) to the $(\ast)$-presheaf $F$, we get another Cartesian square Thus, $F(\mathcal{O}_{V,y})$ coincides with $F_{\mathrm{Nis}}(\mathcal{O}_{V,y})$. This concludes the proof.Now let $F$ be a Nisnevich sheaf. For brevity, by $H^{n}(X,F_{\mathrm{Nis}})$ we denote the $n$th Nisnevich cohomology group $H^{n}_{\mathrm{Nis}}(X,F_{\mathrm{Nis}})$. Proposition 3.10 (see Proposition 17.2 in [5]). Let $Y$ be in $\mathrm{Sm}'/k$, which we represent as a filtered limit $\lim_{i\in I} Y_i$ of $k$-smooth schemes $Y_{i}$ over a small filtered category $I$ such that all the transition morphisms $\varphi_{ij}\colon Y_i\to Y_j$ are affine and etale. Then, for any Nisnevich sheaf $F$ and any $n\geqslant 0$, the canonical map is an isomorphism. More generally, let $\{\varphi_i\colon Y_i\to Y\}_{i\in I}$ be the set of canonical morphisms and let $Z\subset Y$ be a closed subset. Then there exists $i_{0}\in I$ and a closed set $Z_{i_{0}}\subset Y_{i_{0}}$ such that $Z_{i_{0}}=\varphi_{i_{0}}^{-1}(Z)$. Moreover, if, for any $j\geqslant i_{0}$, we denote $\varphi_{ji_{0}}^{-1}(Z_{i_{0}})$ by $Z_{j}$, then, for any $n\geqslant 0$, the canonical map is also an isomorphism.Proposition 3.11. Let $F$ be a Nisnevich sheaf satisfying $(\ast)$ and $V$ be an open subset of an affine line $\mathbb{A}^{1}_{K}$, where $K\in \mathrm{Sm}'(k)$. Then the cohomology group $H^{1}(V,F)$ is trivial. Proof. Let $a$ be an arbitrary element of $H^{1}(V,F)$. Since the Nisnevich topology is trivial at the generic point of $V$, it follows from Proposition 3.10 that there exists an open subset $W\subset V$ such that $a|_{W}$ is trivial. As in the proof of Corollary 3.9, one can assume that $V$ differs from $W$ by a single closed point $y\in V$. Consider the commutative diagram This diagram can be considered as an inductive limit of Nisnevich squares. Since $F$ is a Nisnevich sheaf, we have a part of an exact sequence
Here, $H^{1}(V^{h}_{z},F)=0$, and by Theorems 3.3 and 3.8, the map is surjective. Hence so is the map $F(W)\oplus F(V^h_{z})\to F(V^h_{z}-\{z\})$. So, the map $ H^1(V,F)\to H^1(W,F)$ is injective and hence $a$ is trivial. This concludes the proof.§ 4. Cohomologies of the sheaf $(F_{-1})_{\mathrm{Nis}}$Let $F$ be an $(\ast)$-presheaf on $\mathrm{Sm}/k$. Since it is $\mathbb{A}^{1}$-ivariant, for any $X\in \mathrm{Sm}/k$ the embedding $\tau\colon \mathbb{G}_m\hookrightarrow\mathbb{A}^1$ induces the homomorphism $\tau^{*}\colon F(X)\cong F(X\times\mathbb{A}^{1})\to F(X\times\mathbb{G}_{m})$. The presheaf $X\to F(X\times\mathbb{G}_{m})$ is denoted by $F^{\mathbb{G}_{m}}$ and the presheaf $X\to F^{\mathbb{G}_{m}}(X)/F(X)$ is denoted by $F_{-1}$. Obviously, these new presheafs also satisfy $(\ast)$, and if $F$ is a Nisnevich sheaf, then so is $F^{\mathbb{G}_{m}}$. We define the presheaf $F_{-n}$ as the result of $n$ times application of this operation to $F$. Let $p$ be the embedding of rational point $\{1\}$ into $\mathbb{G}_{m}$. Since $F$ is an $\mathbb{A}^{1}$-invariant presheaf, the commutative diagram induces, for any $X\in \mathrm{Sm}/k$, the commutative diagram where the left vertical arrow is an isomorphism. Thus, the homomorphsim $\tau^*\colon F\to F^{\mathbb{G}_m}$ has a section, and therefore, induces a split exact sequence of $(\ast)$-presheafs
$$
\begin{equation}
0\to F\leftrightarrows F^{\mathbb{G}_m}\to F_{-1}\to0.
\end{equation}
\tag{4.2}
$$
In particular, this implies that if $F$ is a Nisnevich sheaf, then so is $F_{-1}$. Theorem 4.1. For any $(\ast)$-presheaf $F$ and any $n\geqslant 0$, there is a canonical isomorphism of $(\ast)$-presheafs Proof. We first consider the case $n=0$.
Note that the canonical homorphism of $(\ast)$-presheafs $F\to F_{\mathrm{Nis}}$ induces the homomorphism of $(\ast)$-presheafs $F_{-1}\to (F_{\mathrm{Nis}})_{-1}$. Since $(F_{\mathrm{Nis}})_{-1}$ is still a Nisnevich sheaf by Theorem 3.1, this homomorphism can be extended to the homomorphism of Nisnevich $(\ast)$-sheafs In order to prove that this is an isomorphism, it is enough to consider the corresponding exact sequence of $(\ast)$-presheafs
$$
\begin{equation*}
0\to \ker\to (F_{-1})_{\mathrm{Nis}}\to (F_{\mathrm{Nis}})_{-1}\to \operatorname{coker}\to 0
\end{equation*}
\notag
$$
and show that (Nisnevich) stalks of $(\ast)$-presheafs $\ker$ and $\operatorname{coker}$ are trivial at any point $x$ of any irreducible $X\in \mathrm{Sm}/k$. In view of Theorem 3.6, it is enough to check that the homomorphism $(F_{-1})_{\mathrm{Nis}}\to (F_{\mathrm{Nis}})_{-1}$ induces a group isomorphism $(F_{-1})_{\mathrm{Nis}}(\operatorname{Spec}(K))\to (F_{\mathrm{Nis}})_{-1}(\operatorname{Spec}(K))$ for an arbitrary field extension $K/k\in \mathrm{Sm}'(k)$.
By definition, we have the equalities
$$
\begin{equation*}
\begin{aligned} \, (F_{-1})_{\mathrm{Nis}}(\operatorname{Spec}(K)) &=F_{-1}(\operatorname{Spec}(K)), \\ (F_{\mathrm{Nis}})_{-1}(\operatorname{Spec}(K)) &= F_{\mathrm{Nis}}(\mathbb{G}_{m,K})/F_{\mathrm{Nis}}(\operatorname{Spec}(K)). \end{aligned}
\end{equation*}
\notag
$$
Applying Corollary 3.9, we get the equality $F_{\mathrm{Nis}}(\mathbb{G}_{m,K})=F(\mathbb{G}_{m,K})$. In addition, we also have the equality $F(\operatorname{Spec}(K))=F_{\mathrm{Nis}}(\operatorname{Spec}(K))$. Therefore, the right-hand sides of the equalities coincide. Now consider the case $n>0$. Applying (4.2) to $F_{\mathrm{Nis}}$ and using the already proved isomorphism $(F_{-1})_{\mathrm{Nis}}\cong(F_{\mathrm{Nis}})_{-1}$ for $n=0$, we obtain a split exact sequence of Nisnevich $(\ast)$-sheafs
$$
\begin{equation*}
0\to F_{\mathrm{Nis}}\to (F_{\mathrm{Nis}})^{\mathbb{G}_m}\to (F_{\mathrm{Nis}})_{-1}\cong(F_{-1})_{\mathrm{Nis}}\to0.
\end{equation*}
\notag
$$
Since (4.2) is splitting, the long exact sequence of cohomologies also splits into short exact sequences
$$
\begin{equation}
0\to H^n(-,F_{\mathrm{Nis}})\to H^n(-,(F_{\mathrm{Nis}})^{\mathbb{G}_m})\to H^n(-,(F_{-1})_{\mathrm{Nis}})\to0\,
\end{equation}
\tag{4.4}
$$
for any $n>0$. By Theorem 3.1, all the corresponding presheafs of cohomologies also satisfy $(\ast)$.
Now let $X\in \mathrm{Sm}/k$ be irreducible over $k$. Let $\pi$ be the projection $X\times\mathbb{G}_{m}\to X$. For the sheaf $F_{\mathrm{Nis}}$ on $X\times\mathbb{G}_m$, the corresponding Leray spectral sequence is as follows:
$$
\begin{equation*}
H^n(X,R^m(\pi)_*(F_{\mathrm{Nis}}))\Rightarrow H^{n+m}(X\times\mathbb{G}_m, F_{\mathrm{Nis}}).
\end{equation*}
\notag
$$
If $m=0$, then by definition $R^0(\pi)_*(F_{\mathrm{Nis}})=(\pi)_*(F_{\mathrm{Nis}})=(F_{\mathrm{Nis}})^{\mathbb{G}_m}$. Let us show that, for any $m>0$, the corresponding sheaf $R^{m}(\pi)_{*}(F_{\mathrm{Nis}})$ is trivial on $X$. Indeed, by Proposition 3.10, its Nisnevich stalk at an arbitrary point $x\in X$ is equal to $H^{m}(\mathbb{G}_{m}\times U^{h}_{x}, F_{\mathrm{Nis}})$, where by $U$ we denote $\operatorname{Spec}(\mathcal{O}_{X,x})$. Applying Theorem 3.6 and Proposition 3.10 to the $(\ast)$-presheaf $Y\to F(Y\times\mathbb{G}_{m})$, we get the embedding
$$
\begin{equation*}
H^m(\mathbb{G}_m\times U^h_x, F_{\mathrm{Nis}})\hookrightarrow H^m(\mathbb{G}_{m,k(U^h_x)}, F_{\mathrm{Nis}}).
\end{equation*}
\notag
$$
At the same time, by Proposition 3.11, the cohomologies $H^{m}(\mathbb{G}_{m,k(U^{h}_{x})}, F_{\mathrm{Nis}})$ are trivial for any $m>0$. Therefore, the same is true for the cohomology groups $H^m(\mathbb{G}_m\times U^h_x, F_{\mathrm{Nis}})$. Thus, for any $n>0$, the Leray spectral sequence induces the isomorphism
$$
\begin{equation*}
H^n(X,(F_{\mathrm{Nis}})^{\mathbb{G}_m})\cong H^n(X\times\mathbb{G}_m, F_{\mathrm{Nis}}).
\end{equation*}
\notag
$$
Substituting this isomorphism into the exact sequence (4.4), we get the assertion of the theorem for arbitrary $n$:
$$
\begin{equation*}
H^n(X,(F_{-1})_{\mathrm{Nis}})\cong H^n(X\times\mathbb{G}_m, F_{\mathrm{Nis}})/H^n(X,F_{\mathrm{Nis}})=H^n(X,F_{\mathrm{Nis}})_{-1}.
\end{equation*}
\notag
$$
This proves Theorem 4.1. § 5. The case of affine spaceLet $K/k$ be an essentially $k$-smooth field extension. Then if a presheaf $F$ is $\mathbb{A}^{1}$-invariant over $k$, then it is also $\mathbb{A}^{1}_{K}$-invariant over $K$, that is, for any $X\in \operatorname{Sm}'/k$, the projection $\operatorname{pr}:X\times_{K}\mathbb{A}^{1}_{K}\to X_{K}$ induces an isomorphism $F(X_{K})\to F(X\times_{K}\mathbb{A}^{1}_{K})$. Throughout this section, all schemes are supposed to be $K$-linear, so we simply write $\mathbb{A}^{1}$ instead $\mathbb{A}^{1}_{K}$, implying $\mathbb{A}^{1}_{K}$-invariance of the corresponding presheafs and $K$-linearity of morphisms. Lemma 5.1. Let $d>1$ and $G$ be a Nisnevich $(\ast)$-sheaf. Then the embedding $\{(1,\dots,1)\}\hookrightarrow\mathbb{A}^d-\{0\}$ induces an isomorphism $G(\operatorname{Spec}(K))\cong G(\mathbb{A}^{d}-\{0\})$. Proof. The proof is by induction on $d$. First, consider the case $d=2$. We split $\mathbb{A}^{2}-\{0\}$ into two open subsets $(\mathbb{A}^1\times\mathbb{G}_m)\cup(\mathbb{G}_m\times\mathbb{A}^1)$ whose intersection is $\mathbb{G}_{m}\times\mathbb{G}_{m}$. By Theorem 3.1, all the cohomology groups of $G$ are $\mathbb{A}^{1}$-invariant, therefore, the groups $H^1(\mathbb{A}^1\times\mathbb{G}_m,G)$, $H^1(\mathbb{G}_m\times\mathbb{A}^1,G)$ vanish by Proposition 3.11. Thus, the initial part of the corresponding Mayer–Vietoris sequence looks like
$$
\begin{equation*}
\begin{aligned} \, 0 &\to G(\mathbb{A}^2-\{0\})\to G(\mathbb{A}^1\times\mathbb{G}_m)\oplus G(\mathbb{G}_m\times\mathbb{A}^1) \\ &\xrightarrow{\varrho} G(\mathbb{G}_m\times\mathbb{G}_m)\to H^1(\mathbb{A}^2-\{0\},G)\to0, \end{aligned}
\end{equation*}
\notag
$$
and the following isomorphisms take place:
$$
\begin{equation*}
\begin{aligned} \, G(\mathbb{A}^2-\{0\}) &\cong\ker\bigl(\varrho\colon G(\mathbb{A}^1\times\mathbb{G}_m)\oplus G(\mathbb{G}_m\times\mathbb{A}^1)\to G(\mathbb{G}_m\times\mathbb{G}_m)\bigr), \\ H^1(\mathbb{A}^2-\{0\},G) &\cong\operatorname{coker}\bigr(\varrho\colon G(\mathbb{A}^1\times\mathbb{G}_m)\oplus G(\mathbb{G}_m\times\mathbb{A}^1)\to G(\mathbb{G}_m\times\mathbb{G}_m)\bigr). \end{aligned}
\end{equation*}
\notag
$$
Since sequence (4.2) splits and the $(\ast)$-presheaf $G$ is $\mathbb{A}^{1}$-invariant, we get
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, G(\mathbb{A}^1\times\mathbb{G}_m)\oplus G(\mathbb{G}_m\times\mathbb{A}^1) &\cong G(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K))\oplus G(\operatorname{Spec}(K)) \\ &\qquad\oplus G_{-1}(\operatorname{Spec}(K)), \end{aligned} \\ G(\mathbb{G}_m\times\mathbb{G}_m) \cong G(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K))\oplus G_{-2}(\operatorname{Spec}(K)). \end{gathered}
\end{equation*}
\notag
$$
In addition, the restriction of $\varrho$ to $G_{-1}(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K))$ gives an isomorphism
$$
\begin{equation*}
G_{-1}(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K))\to G_{-1}(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K)),
\end{equation*}
\notag
$$
and the restriction of $\varrho$ to $G(\operatorname{Spec}(K))\oplus G(\operatorname{Spec}(K))$ gives the map
$$
\begin{equation*}
G(\operatorname{Spec}(K))\oplus G(\operatorname{Spec}(K))\to G(\operatorname{Spec}(K)), \qquad(a,b)\mapsto\varrho'(a)-\varrho'(b),
\end{equation*}
\notag
$$
where $\varrho'\colon G(\operatorname{Spec}(K))\to G(\operatorname{Spec}(K))$ is the isomorphism induced by the embedding $\{(1,1)\}\hookrightarrow\mathbb{A}^{2}$.Thus, the case $d=2$ is proved.
Now let $d>2$. We split $\mathbb{A}^{d}-\{0\}$ into two open subsets
$$
\begin{equation}
V:=((\mathbb{A}^{d-1}-\{0\})\times\mathbb{A}^1)\cup W:=(\mathbb{A}^{d-1}\times\mathbb{G}_m),
\end{equation}
\tag{5.1}
$$
whose intersection $V\cap W$ is $(\mathbb{A}^{d-1}-\{0\})\times\mathbb{G}_{m}$. The corresponding Mayer–Vietoris sequence is as follows:
$$
\begin{equation*}
\begin{aligned} \, 0 &\to G(\mathbb{A}^d-\{0\})\to G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{A}^1)\oplus G(\mathbb{A}^{d-1}\times\mathbb{G}_m) \\ &\to G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{G}_m)\to\cdots. \end{aligned}
\end{equation*}
\notag
$$
By the induction hypothesis, we have the equalities
$$
\begin{equation*}
\begin{aligned} \, &G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{A}^1)\oplus G(\mathbb{A}^{d-1}\times\mathbb{G}_m)\cong G(\mathbb{A}^{d-1}-\{0\})\oplus G(\mathbb{G}_m) \\ &\qquad\cong G(\operatorname{Spec}(K))\oplus G(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K)), \\ &G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{G}_m)\cong G(\mathbb{A}^{d-1}-\{0\})\oplus G_{-1}(\mathbb{A}^{d-1}-\{0\}) \\ &\qquad \cong G(\operatorname{Spec}(K))\oplus G_{-1}(\operatorname{Spec}(K)). \end{aligned}
\end{equation*}
\notag
$$
Proceeding as in the case $d=2$, we get the isomorphism
$$
\begin{equation*}
\begin{aligned} \, &G(\mathbb{A}^d-\{0\}) \\ &\qquad\cong\ker\bigl(G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{A}^1)\oplus G(\mathbb{A}^{d-1}\times\mathbb{G}_m)\to G((\mathbb{A}^{d-1}-\{0\})\times\mathbb{G}_m)\bigr) \\ &\qquad\cong G(\operatorname{Spec}(K)). \end{aligned}
\end{equation*}
\notag
$$
This concludes the proof. Now we are ready to state the main theorem of this section for an affine space. Theorem 5.2. Let $K/k$ be an essentially $k$-smooth field extension of the field $k$. Then, for any $(\ast)$-presheaf $F$ and for any $n,d>0$, the following isomorphisms take place: Proof. Note that by Theorem 4.1 there are canonical isomorphisms $ (F_{-n})_{\mathrm{Nis}}\cong (F_{\mathrm{Nis}})_{-n}$ of $(\ast)$-presheafs. In particular, one can always substitute $(F_{\mathrm{Nis}})_{-n}$ for $F_{-n}$ in (5.2).
We first consider the cases $n=0$ and $d=0$. If $n=0$, then be considering the localization sequence
$$
\begin{equation}
\cdots\to H^n_{\{0\}}(\mathbb{A}^d,F_{\mathrm{Nis}})\to H^n(\mathbb{A}^d,F_{\mathrm{Nis}})\to H^n(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}})\to\cdots
\end{equation}
\tag{5.3}
$$
if follows that the group $H^0_{\{0\}}(\mathbb{A}^d,F_{\mathrm{Nis}})$ is isomorphic to the kernel of the homomorphism
$$
\begin{equation*}
\bigl\{F_{\mathrm{Nis}}(\mathbb{A}^d)=H^0(\mathbb{A}^d,F_{\mathrm{Nis}})\bigr\}\to \bigl\{F_{\mathrm{Nis}}(\mathbb{A}^d-\{0\})=H^0(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}})\bigr\}.
\end{equation*}
\notag
$$
Therefore, if $d=0$, then $\mathbb{A}^{d}-\{0\}=\emptyset$ and $H^{0}_{\{0\}}(\mathbb{A}^{0},F_{\mathrm{Nis}})\cong F_{\mathrm{Nis}}(\operatorname{Spec}(K))$. If $d>0$, then setting $x:=(1,0,\dots,0)\in\mathbb{A}^{d}$, we get the commutative diagrams Here, the right-hand map of the second commutative diagram is an isomorphism, because $F_{\mathrm{Nis}}$ remains a $(\ast)$-presheaf. It follows, it particular, that the homomorphism $F_{\mathrm{Nis}}(\mathbb{A}^d)\to F_{\mathrm{Nis}}(\mathbb{A}^d-\{0\})$ is injective, and hence $H^0_{\{0\}}(\mathbb{A}^d,F_{\mathrm{Nis}})=0$.Now let $d=0$. Then, for any $n$, the group $H^{n}(\mathbb{A}^{d}-\{0\},F_{\mathrm{Nis}})$ becomes trivial. Now it follows from the localization sequence that $H^{n}_{\{0\}}(\mathbb{A}^{0},F_{\mathrm{Nis}})\cong H^{n}(\mathbb{A}^{0},F_{\mathrm{Nis}})$. Thus, $H^{n}_{\{0\}}(\mathbb{A}^{0},F_{\mathrm{Nis}})$ is trivial for any $n>0$. Let us consider the case $n,d>0$. The presheaf $H^{n}(-,F_{\mathrm{Nis}})$ is $\mathbb{A}^{1}$-invariant by Theorem 3.1, and so, for any $n>0$, we have the equality $H^{n}(\mathbb{A}^{d},F_{\mathrm{Nis}})=0$. Applying the localization sequence (5.3), we get the formula
$$
\begin{equation}
H^n_{\{0\}}(\mathbb{A}^d,F_{\mathrm{Nis}})= \begin{cases} H^{n-1}(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}}), &n>1, \\ F_{\mathrm{Nis}}(\mathbb{A}^d-\{0\})/F_{\mathrm{Nis}}(\mathbb{A}^d), & n=1. \end{cases}
\end{equation}
\tag{5.4}
$$
In particular, the conclusion of Theorem 5.2 for the case $n=1$ follows directly from Lemma 5.1. Now suppose that $n>1$. The proof is again by induction on $d$. The case $d=0$ is already proved. We use (5.1) and write the corresponding Mayer–Vietoris sequence (here, we again used the $\mathbb{A}^1$-invariance of the corresponding $(\ast)$-presheafs of cohomologies).By Theorem 4.1 and from splitting of sequence (4.2), for any $i\geqslant0$ we have the isomorphisms
$$
\begin{equation*}
\begin{aligned} \, &H^i((\mathbb{A}^{d-1}{-}\,\{0\})\times\mathbb{G}_m,F_{\mathrm{Nis}}) \cong H^i((\mathbb{A}^{d-1}-\{0\}),F_{\mathrm{Nis}})\oplus H^i((\mathbb{A}^{d-1}-\{0\}),F_{\mathrm{Nis}})_{-1}\! \\ &\qquad\cong H^i((\mathbb{A}^{d-1}-\{0\}),F_{\mathrm{Nis}})\oplus H^i((\mathbb{A}^{d-1}-\{0\}),(F_{-1})_{\mathrm{Nis}}). \end{aligned}
\end{equation*}
\notag
$$
Substituting $i=n-2,n-1$ and using the fact that by Proposition 3.11 the group $H^{n-1}(\mathbb{G}_{m},F_{\mathrm{Nis}})$ is trivial for any $n>1$, we get can reduce our Mayer–Vietoris sequence to the form
$$
\begin{equation}
H^{n-2}(\mathbb{G}_m,F_{\mathrm{Nis}})\to H^{n-2}((\mathbb{A}^{d-1}-\{0\}),(F_{-1})_{\mathrm{Nis}})\to H^{n-1}(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}})\to 0.
\end{equation}
\tag{5.5}
$$
So, for $n>2$, sequence (5.5) assumes the form
$$
\begin{equation*}
0\to H^{n-2}((\mathbb{A}^{d-1}-\{0\}),(F_{-1})_{\mathrm{Nis}})\to H^{n-1}(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}})\to 0
\end{equation*}
\notag
$$
and the required conclusion of Theorem 5.2 is immediate from (5.4) and the induction hypothesis.
If $n=2$, then sequence (5.5) takes form
$$
\begin{equation*}
\begin{aligned} \, &F_{\mathrm{Nis}}(\mathbb{G}_m)\cong F_{\mathrm{Nis}}(\operatorname{Spec}(K))\oplus (F_{-1})_{\mathrm{Nis}}(\operatorname{Spec}(K))\to (F_{-1})_{\mathrm{Nis}}(\mathbb{A}^{d-1}-\{0\}) \\ &\qquad\to H^{n-1}(\mathbb{A}^d-\{0\},F_{\mathrm{Nis}})\to 0. \end{aligned}
\end{equation*}
\notag
$$
So, if $d\neq2$, then by Lemma 5.1, the map $F_{\mathrm{Nis}}(\mathbb{G}_m)\to (F_{-1})_{\mathrm{Nis}}(\mathbb{A}^{d-1}-\{0\})$ is injective, and all the cohomology groups $H^{n-1}(\mathbb{A}^{d}-\{0\},F_{\mathrm{Nis}})$ are trivial. If $d=2$, then the required isomorphism $H^{n-1}(\mathbb{A}^{d}-\{0\},F_{\mathrm{Nis}})\cong (F_{-2})(\operatorname{Spec}(K))$ follows from the splitting $(F_{-1})_{\mathrm{Nis}}(\mathbb{G}_{m})\cong (F_{-1})_{\mathrm{Nis}}(\operatorname{Spec}(K))\oplus (F_{-2})_{\mathrm{Nis}}(\operatorname{Spec}(K))$. This concludes the proof of the theorem. § 6. Proof of the main theoremBefore stating the main result, we introduce another auxiliary construction. Let $X\in \mathrm{Sm}/k$ be $k$-smooth, $S\subset X$ be a closed irreducible smooth subscheme of $X$. We also set $V:=X-S$. In this case, we say that the pair $(X,S)$ is also smooth. Denote by $i$ and $j$ the embeddings $S\subset X$ and $V\subset X$, respectively. For any $(\ast)$-presheaf $F$, there is an exact sequence of $(\ast)$-presheafs on $X$ Note that, for any open subset $W\subset X$, by definition there is a canonical isomorphism $\operatorname{coker}(W)\cong F(W-S)/F(W)$. Applying Zariski sheafication, we get an exact sequence of Zariski sheafs on $X_{\mathrm{Zar}}$:
$$
\begin{equation*}
F_{\mathrm{Zar}}\to (j_*(j^*(F)))_{\mathrm{Zar}}\to\operatorname{coker}_{\mathrm{Zar}}\to0.
\end{equation*}
\notag
$$
By $F_{(X,S)}$, we denote the Zariski sheaf $i^{*}(\operatorname{coker}_{\mathrm{Zar}})$ on $S_{\mathrm{Zar}}$. Lemma 6.1. If the pairs $(X',S')$, $(X,S)$ in the Nisnevich square (4.1) are smooth, then the morphism $\pi\colon S'\to S$ induces an isomorphism of the Zariski sheafs $F_{(X,S)}$ and $F_{(X',S')}$ Proof. It suffices to check that, for any $x'\in S'\to x\in S$, the stalk of $F_{(X,S)}$ at $x$ is isomorphic to the stalk of $F_{(X',S')}$ at $x'$. Since stalks are preserved by inverse image, we get
$$
\begin{equation*}
\begin{aligned} \, (F_{(X,S)})_x &\cong F(\operatorname{Spec}(\mathcal{O}_{X,x})-S)/ F(\operatorname{Spec}(\mathcal{O}_{X,x})), \\ (F_{(X',S')})_{x'} &\cong F(\operatorname{Spec}(\mathcal{O}_{X',x'})-S')/ F(\operatorname{Spec}(\mathcal{O}_{X',x'})). \end{aligned}
\end{equation*}
\notag
$$
The required result now follows directly from Theorem 3.8. Corollary 6.2. Let $(X,S)$ be a smooth pair and $S$ have codimension $d$ in $X$. Let $x$ be the generic point of $S$. Then there exists an open neighborhood $V\subset X$ of $x$ such that, for any $(\ast)$-presheaf $F$, there is the following isomorphism of Zariski sheafs on $(V\cap S)_{\mathrm{Zar}}$: Proof. Since the pair $(X,S)$ is smooth, there exists an open neighborhood $x\in V$ in $X$ and an etale morphism $\eta\colon V\to\mathbb{A}^{m}$ such that $\eta^{-1}(\mathbb{A}^{m-d}\times\{0\})=S\cap V$. For simplicity, we can assume that $V$ coincides with the whole $X$. Let $\eta_{0}$ be the restriction $\eta|_S\colon S\to\mathbb{A}^{m-d}\times\{0\}$. We construct $\eta$ and $\eta_0\times Id_{\mathbb{A}^d}$ to the Cartesian square: Note, that the preimage $\lambda^{-1}(S)$ is isomorphic to the direct product $S\underset{\mathbb{A}^{m-d}}{\times}S$. Thus, the restriction $\lambda|_{\lambda^{-1}(S)}\colon \lambda^{-1}(S)\to S$ has a diagonal section $\Delta\colon S\to\lambda^{-1}(S)$. Being a base change of an etale map, $\lambda$ it is also etale, and therefore, $\Delta(S)$ is a connected component of $\lambda^{-1}(S)$. Thus, $\lambda^{-1}(S)-\Delta(S)$ is closed in $\lambda^{-1}(S)$, and therefore, in $W_{0}$. Let $W$ be the difference $W_{0}-(\lambda^{-1}(S)-\Delta(S))$. We have two Nisnevich squares: Now to complete the proof it suffices to twice apply Lemma 6.1. Lemma 6.3. Let $(X,S)$ be a smooth pair and let $i$ be the corresponding embedding $i\colon S\hookrightarrow X$. Then, for any $(\ast)$-presheaf $F$ and for any $n\geqslant0$, the Zariski sheafs $i^{*}(H^{n+1}_{S}(-,F_{\mathrm{Nis}})_{\mathrm{Zar}})$ and $H^{n}(-,F_{\mathrm{Nis}})_{(X,S)}$ are isomorphic on $S_{\mathrm{Zar}}$. Proof. Let $V$ be an open subset of $X$. The exact localization sequence
$$
\begin{equation*}
\cdots\to H^n(V,F_{\mathrm{Nis}})\to H^n(V-S,F_{\mathrm{Nis}})\to H^{n+1}_S(V,F_{\mathrm{Nis}})\to\cdots
\end{equation*}
\notag
$$
induces the homomorphism of presheafs
$$
\begin{equation*}
H^n(-\setminus S,F_{\mathrm{Nis}})/H^n(-,F_{\mathrm{Nis}})\to H^{n+1}_S(-,F_{\mathrm{Nis}})
\end{equation*}
\notag
$$
and, as a corollary, a homomorphism of the Zarisky sheafs $H^{n}(-,F_{\mathrm{Zar}})_{(X,S)}\to i^{*}(H^{n+1}_{S}(-,F_{\mathrm{Nis}})_{\mathrm{Zar}})$. To show that this map is an isomorphism, it is enough to check how it acts on stalks. By Proposition 3.10, for any point $x$, the corresponding stalks homomorphism is induced by the localization sequence
$$
\begin{equation*}
\cdots\to H^n(U,F_{\mathrm{Nis}})\to H^n(U-S,F_{\mathrm{Nis}})\to H^{n+1}_S(U,F_{\mathrm{Nis}})\to\cdots,
\end{equation*}
\notag
$$
where, as above, $U$ is $\operatorname{Spec}(\mathcal{O}_{X,x})$. From Corollary 3.7 and Theorem 3.1 it follows that the map $H^{n}(U,F_{\mathrm{Nis}})\longrightarrow H^{n}(U-S,F_{\mathrm{Nis}})$ is injective. Thus, the localization sequence splits into short exact sequences Consequently, the stalks $H^n(U-S,F_{\mathrm{Nis}})/H^n(U,F_{\mathrm{Nis}})$ and $H^{n+1}_S(U,F_{\mathrm{Nis}})$ are isomorphic for any $n\geqslant0$. This concludes the proof. Now let $X\in \mathrm{Sm}/k$ and $x\in X$ be some point of codimension $d$. For an arbitrary $(\ast)$-presheaf $F$, let $H^{n}_{x}(X,F_{\mathrm{Nis}})$ denote the direct limit $\lim_{x\in V} H^n_{\overline{\{x\}}\cap V}(V,F_{\mathrm{Nis}})$ over all open neighborhoods $x\in V$. In particular, one can always replace $X$ with an open neighborhood of $x$. We are now ready to prove the main result of this paper: Proof. We first consider the cases $n=0$ and $d=0$. If $n=0$, then it follows from the exact localization sequence of the pair $(U,x)$ that $H^{0}_{x}(X,F_{\mathrm{Nis}})$ is isomorphic to the kernel of $F_{\mathrm{Nis}}(U)\to F_{\mathrm{Nis}}(U-x)$. So, if $d=0$, then $U=x$, and, as a corollary, $H^0_x(X,F_{\mathrm{Nis}})\cong F_{\mathrm{Nis}}(U)\cong F(\operatorname{Spec}(k(x)))$. If $d>0$, then by Corollary 3.7, the kernel of the homomorphism $F_{\mathrm{Nis}}(U)\to F_{\mathrm{Nis}}(U-x)$ is trivial. Now, if $d=0$ and $n>0$, it follows from the localization sequence and since $H^{n}((U-x),F_{\mathrm{Nis}})$ is trivial that $H^{n}_{x}(X,F_{\mathrm{Nis}})$ is isomorphic to $H^{n}(U,F_{\mathrm{Nis}})$. But it also follows from Theorem 3.5 that $H^{n}(U,F_{\mathrm{Nis}})$ can be embedded into $H^n(k(U),F_{\mathrm{Nis}})=(H^n(k(U),F_{\mathrm{Nis}}))_{\mathrm{Nis}}$. Thus, if $n>0$, then the group $H^{n}_{x}(X,F_{\mathrm{Nis}})$ is trivial.
Now suppose that $n,d>0$. Let $S$ be the closure of $x$ in $X$. Since the field $k$ is perfect, we can choose an open neighborhood $x\in V$ such that the pair $(V,S\cap V)$ is smooth. Since $X$ can always be reduced to an open neighborhood of $x$, we can assume that the pair $(X,S)$ is also smooth. Applying Corollary 6.2 to the $(\ast)$-presheaf $H^{n-1}(-,F_{\mathrm{Nis}})$, we get an isomorphism
$$
\begin{equation*}
H^{n-1}(-,F_{\mathrm{Nis}})_{(X,S)}\cong H^{n-1}(-,F_{\mathrm{Nis}})_{(\mathbb{A}^d\times S,\{0\}\times S)}
\end{equation*}
\notag
$$
of Zariski sheafs on $S_{\mathrm{Zar}}$.
Next, denoting by $i$ and $i'$, respectively, the embeddings $S\hookrightarrow X$ and $(\{0\}\times S)\hookrightarrow(\mathbb{A}^{d}\times S)$ and applying Lemma 6.3 twice, we get an isomorphism
$$
\begin{equation*}
i^{\ast}(H^n_S(-,F_{\mathrm{Nis}}))\cong (i')^{\ast}(H^n_{\{0\}\times S}(-,F_{\mathrm{Nis}}))
\end{equation*}
\notag
$$
of Zariski sheafs on $S_{\mathrm{Zar}}$. In particular, the stalks of these sheafs at $x\in S$ are isomorphic as well. By definition, the stalk of the left-hand sheaf coincides with the group $H^{n}_{x}(X,F_{\mathrm{Nis}})$, and the stalk of the right-hand sheaf coincides with the group $H^{n}_{\{0\}}(\mathbb{A}^{d}_{k(x)},F_{\mathrm{Nis}})$. Now the required result is immediate from Theorem 5.2.
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\Bibitem{Tyu25} |
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