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Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field A. E. Druzhinin St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 2, DOI: https://doi.org/10.4213/sm9762 
Bibliographic databases:
    § 1. IntroductionThroughout this article, except the introduction, we consider schemes over a field $k$. The Cousin complex $\mathrm{Cous}^{n,l}(V,E)$ on a scheme $V$ which is associated with a bigraded cohomology theory $E^{*,*}$ is the complex of abelian groups
$$
\begin{equation}
\begin{aligned} \, \notag \dotsb &\to 0\to E^{n,l}(V) \to \bigoplus_{z\in V^{(0)}} E^{n,l}(z)\to \bigoplus_{z\in V^{(1)}} E^{n+1,l}_z(V)\to \dotsb \\ &\to\bigoplus_{z\in V^{(c)}} E^{n+c,l}_z(V)\to\dots\to \bigoplus_{z\in V^{(d)}} E^{n+d,l}_z(V)\to 0\to \dotsb, \end{aligned}
\end{equation}
\tag{1.1}
$$
where $d=\dim V$ is the Krull dimension, $V^{(c)}$ denotes the set of points of codimension $c$, and the $E^{n+c,l}_z(V)=E^{n+c,l}_z(V_z)$ are the cohomologies with supports in $z$ of the local scheme $V_z$ at $z$, while the differentials of the complex are defined by the boundary homomorphisms, that is, differentials, of the cohomology theory. For a cohomology theory $E^{*}$ that has one grading index, the index $l$ in the complex (1.1) is dropped. The main question discussed in this article is as follows. What can be a sufficiently wide class of schemes over a field $k$ and cohomology theories $E^*$ such that the Cousin complex $\mathrm{Cous}^n(V,E)$ is acyclic for all $n$? When the scheme $V$ is local regular and $E^*$ is the Quillen algebraic $K$-theory, this question was formulated by Gersten [1] and is known as Gersten’s conjecture. Gersten’s conjecture was proved by Quillen in his fundamental article [2] for local schemes of the form $X_x$, where $x$ is a point of a smooth scheme $X \in \mathrm{Sm}_k$ over a field $k$. Next, it was proved by Panin [3] for all local regular equicharacteristic schemes $V$. For convenience we call the question on the acyclicity of the Cousin complex $\mathrm{Cous}^n(X_x,E)$ for all schemes of the form $X_x$ as above and all integers $n$ the geometric $E^*$-version of Gersten’s conjecture. After the pioneering articles [2] by Quillen and [4] by Bloch and Ogus and the revolutionary paper [5] by Voevodsky, the geometric $E^*$-version of Gersten’s conjecture (where the ground field $k$ was arbitrary) was proved in Panin’s article [6], § 9, for an arbitrary cohomology theory $E^*$ in the sense of Panin and Smirnov (see Definition 2.1 in [7]). Note that [6] followed Morel [8], [9]. In the latter articles the geometric $E^*$-version of Gersten’s conjecture was proved for all $\mathbf{SH}^{S^1}(k)$-representable cohomology theories (under the implicit assumption that the field $k$ is infinite). In [10] by Garkusha and Panin, extended by [11] by this author and Panin, and in the article [12] by this author and Kylling, the results of [5] were transferred to radditive $\mathbb{A}^1$-invariant presheaves of abelian groups with framed transfers. Note that the presheaves with framed transfers were introduced by Voevodsky in his fundamental article [5]. The radditive $\mathbb{A}^1$-invariant presheaves of abelian groups with framed transfers were introduced and systematically studied in [10]. Using the approach of [6], [8] and [9], in this article we obtain a generalization of the statements mentioned above to a more general class of schemes over a field. To formulate our main result (Theorem 4), we denote the category of smooth schemes of finite type over a field $k$ by $\mathrm{Sm}_k$. Let $X\in \mathrm{Sm}_k$ and $x\in X$, let $D$ be a strict normal crossing divisor in $X_x$ and $Z_\alpha$, where $\alpha\in A$, be a finite family of proper closed subschemes in $\mathbb{A}^1_k$. Let
$$
\begin{equation*}
V=(X_x-D)\times\prod_\alpha(\mathbb{A}^1_k-Z_\alpha).
\end{equation*}
\notag
$$
Then for any cohomology theory $E$ in the sense of Definition 1 the Cousin complex $\mathrm{Cous}^n(V,E)$ defined by formula (1.1) is acyclic. In particular (Corollary 7), this result holds for cohomology theories representable in the Morel $S^1$-stable motivic homotopy category $\mathbf{SH}^{S^1}(k)$ defined in [13], and for bigraded cohomology theories representable in the Voevodsky stable motivic homotopy category $\mathbf{SH}(k)$ defined in [14]. The scheme $V$ of the form described in Theorem 4 is cohomologically trivial in the sense of Definition 2.9 in [15]. Previously, for cohomology theories associated with an arbitrary $\mathbb A^1$-invariant radditive presheaf of abelian groups with framed transfers over an arbitrary field $k$, this author [16] considered the case of Theorem 4 for the scheme $X_x - D$ and a smooth divisor $D$. The new result extends the class of schemes and the class of cohomology theories at the same time. The fact that the complex $\mathrm{Cous}^n(X_x - D,E)$ is acyclic for a strict normal crossing divisor $D$ in a smooth scheme over a field leads naturally to an expectation that the complex is acyclic for certain cohomology theories for a regular scheme $X$ and a strict normal crossing divisor $D$ in $X$. We mean the generalization of Gersten’s conjecture for the Thomason-Trobaugh $\mathrm{K}$-theory (see [17]) and Schlichting’s hermitian $\mathrm{K}$-theory (see [18]), and the analogous statement for stable motivic homotopy groups and Voevodsky’s algebraic cobordisms $\mathrm{MGL}$ [14] which is formulated below. Conjecture 1. Let Then for any local regular scheme $U$ and strict normal-crossings divisor $D$ in $U$, and any integers $l$ and $n$ the complex $\mathrm{Cous}^{n,l}(V,E)$ is acyclic for the scheme ${V = U - D}$. In the formulations of parts (2)–(4) of this conjecture we have used Voevodsky’s stable motivic homotopy category $\mathbf{SH}(U)$ (see [14]) of the local regular scheme $U$ and bigraded presheaves of $\mathbb A^1$-homotopy groups on the category of smooth finite-type schemes over $U$. Remark 1. We believe that Theorem 4 and Conjecture 1 hold not only for strict normal-crossing divisors, but also for normal-crossing ones, and even for a wider class of divisors. This and more general cases are not discussed in our article. A related statement on the acyclicity of the Cousin complex on local regular schemes is the Grothendieck-Serre conjecture (see [20] and [21]) on torsors over reductive group schemes. It claims that the map of pointed sets is trivial, where $\eta$ is a generic point of a local regular scheme $U$. The adjoint Nisnevich conjecture (see [22]) claims that a similar property holds for the scheme $V = U - D$, where $D$ is a regular divisor. The Grothendieck-Serre conjecture has been proved in the equicharacteristic case: by Panin and Fedorov [23] for schemes over an infinite field and by Panin [24] for scheme over an arbitrary field. See details in Panin’s talk [25] at the International Congress of Mathematician in Rio de Janeiro. Nisnevich’s conjecture was proved by Fedorov in [26] in the equicharacteristic case.We state a natural question in the context of Theorem 4, which is related to Conjecture 1. The author does not know whether or not the answer is affirmative even for the group schemes $\mathrm{GL}_n$ ($n\geqslant 2$), $\mathrm{SO}_n$ ($n\geqslant 3$), and $\mathrm{PGL}_n$ ($n\geqslant 2$). This is why it is more appropriate to call it a ‘question’ than a ‘conjecture’. Question 1. Let $U$ be a local regular scheme and $D$ be a divisor with strict normal crossings in $U$. Let $G$ be a reductive group scheme over $U$. Is the kernel of the map of pointed sets $H^1_{\mathrm{et}}(V,G) \to H^1_{\mathrm{et}}(\eta,G)$ trivial, where $\eta$ is a generic point or $U$ and $V = U - D$? Both the conjecture and the question are presented for illustrative purposes and to demonstrate a wider context of the discussion. This article contains the proof of Theorem 4, which provides a formal analogy to the conjecture and the question, while it relates to a much simpler case. Denote by $\mathrm{Sm}_S$, $\mathrm{SmAff}_S$ and $\mathrm{EssSm}_S$ the categories of smooth, smooth affine and essentially smooth schemes over the scheme $S$. Following [27], by essentially smooth schemes we mean schemes that are filtered limits of smooth schemes with affine étale transition morphisms. Throughout most of the text, namely in §§ 3–5 and 7, $S = \operatorname{Spec} k$ for a field $k$. In §§ 2 and 6, $S$ is formally an arbitrary scheme and an affine noetherian scheme, while in applications $S\in \mathrm{SmAff}_k$ or $S\in \mathrm{EssSm}_k$ for some field $k$. AcknowledgementBeing a winner-recipient of the “Young Russian Mathematics” competition the author is thankful to the jury and sponsors of the competition. § 2. Cohomology theoriesThe notion of a cohomology theory defined on algebraic varieties or schemes which was introduced by Panin and Smirnov [28], [7], [29] accumulates a number of standard properties that are common to many known examples and hold, in particular, for theories representable in the $S^1$-stable motivic homotopy category of Morel-Voevodsky $\mathbf{SH}^{S^1}(S)$ over a noetherian scheme $S$ of finite Krull dimension. The definition below can formally be applied to an arbitrary base scheme $S$; in §§ 6 and 7 it is applied to schemes $S$ that are in their turn smooth affine over a field $k$, while in the other sections $S = \operatorname{Spec} k$. First recall the definition of the category of open pairs. The category of open pairs $\mathrm{SmOP}_S$ over a scheme $S$, as defined in [19], § 2, is the category with objects $(X,U)$, where $X$ is a smooth scheme over $S$ and $U$ is an open subscheme. Morphisms $(X_1,U_1) \to (X_2,U_2)$ are $S$-morphisms $X_1\to X_2$ that induce the morphism ${U_1\to U_2}$. For a closed subset $Z$ in $X$, and an arbitrary presheaf $E$ on $\mathrm{SmOP}_S$ put $E_Z(X)=E((X,X-Z))$. Definition 1 (see [7], Definition 2.1, and [19], § 2). A cohomology theory $E$ on $\mathrm{Sm}_S$ is a contravariant functor from $\mathrm{SmOP}_S$ to the category of abelian groups $\mathrm{Ab}$ which is equipped with natural homomorphisms of differentials for all $X\in \mathrm{Sm}_S$ and all closed subsets $Z$ of $X$ and which satisfies the following localization, excision, and homotopy invariance axioms. $\bullet$ (Localisation.) For any scheme $X\in \mathrm{Sm}_S$ and any closed subset $Z$, the sequence of abelian groups
$$
\begin{equation}
\dots\to E_Z(X)\to E(X)\to E(U)\xrightarrow{\partial} E_Z(X)\to\dotsb
\end{equation}
\tag{2.1}
$$
is exact. $\bullet$ (Excision.) For the morphism of pairs $(X',X'-Z')\to (X,X-Z)$ induced by an étale morphism $X'\to X$ that induces an isomorphism of reduced closed subschemes $Z'\stackrel{\simeq}{\to} Z$, the induced morphism is an isomorphism of abelian groups.$\bullet$ (Homotopy invariance.) For any $X\!\in\! \mathrm{Sm}_S$ the canonical projection ${\mathbb{A}^1\!\times\! X\xrightarrow{p} X}$ induces an isomorphism of abelian groups We denote the Zariski and Nisnevich sheaves associated with $E(-)$ on the category $\mathrm{Sm}_S$ by $E_\mathrm{Zar}$ and $E_{\mathrm{Nis}}$. Remark 2. Let $E$ be a cohomology theory on $\mathrm{Sm}_S$. Then for any $X\in \mathrm{Sm}_S$ and closed subsets $Z\subset W\subset X$ there is a natural homomorphism of differentials that is equal to $\partial_{(X,X-Z)}$ for $W=X$ and to the composition in the general case. Lemma 1 (see [28], §§ 2.2.3, 2.2.6 and 2.2.1). Let $S$ be a scheme, and let $E$ be a cohomology theory on $\mathrm{Sm}_S$. Then for any $X\in \mathrm{Sm}_S$ and any closed subsets $Z\subset W\subset X$ there is a long exact sequence of abelian groups and there is an isomorphism induced by the projection $\mathbb{A}^1\times X\to X$. § 3. Cousin complex and strict homotopy invarianceLet $k$ be a field. Definition 2. Let $X\in \mathrm{Sm}_k$, let $Z$ be a closed subscheme $X$ and $E$ be a cohomology theory on $\mathrm{Sm}_k$. Denote by $\mathrm{C}_Z(X,E)$ the complex of abelian groups For any presheaf $F$ on $\mathrm{Sm}_k$, we denote by $F_\mathrm{Zar}$ and $F_\mathrm{Nis}$ the associated sheaves with respect to the Zariski and the Nisnevich topologies, respectively. Definition 3. Note that $\mathrm{C}_Z(-,E)$ and $\mathrm{Cous}_Z(-,E)$ define complexes of sheaves on the small Zariski and Nisnevich sites over the scheme $X$. Then $\mathrm{C}_Z(-,E)_\mathrm{Zar}$, $\mathrm{Cous}_Z(-,E)_\mathrm{Zar}$, $\mathrm{C}_Z(-,E)_\mathrm{Nis}$ and $\mathrm{Cous}_Z(-,E)_\mathrm{Nis}$ denote the associated complexes of Zariski and Nisnevich sheaves. Theorem 1 (see [6], Theorems 9.1 and 9.2). Let $k$ be a field and let $X\in \mathrm{Sm}_k$ and $x\in X$. Let $E$ be a cohomology theory on $\mathrm{Sm}_k$. Then the complex $\mathrm{Cous}(X_x,E)$ is acyclic. Moreover, the morphism is a flasque resolvent of the sheaf $E_{\mathrm{Zar}}$ on the small Zariski site over $X$. Theorem 2 (extended version of Theorem 10.2 in [6]). Let $k$ be a field, and let $X\in \mathrm{Sm}_k$. Then the morphism is a flasque resolvent of the sheaf $E_{\mathrm{Nis}}$ on the small Nisnevich site over $X$ for any $E$ as above. There exist isomorphisms $E_{\mathrm{Nis}}\cong E_\mathrm{Zar}$ and $H^l_\mathrm{Nis}(-,E_{\mathrm{Nis}})\cong H^l_\mathrm{Zar}(-,E_\mathrm{Zar})$, where $l> 0$. Recall the strict homotopy invariance theorem from [6]. Theorem 3 (see [6], Theorem 1.1). Let $k$ be a field, let $X\in \mathrm{Sm}_k$, and let $E$ be a cohomology theory on $\mathrm{Sm}_k$. Then for each $X\in \mathrm{Sm}_k$ the homomorphism induced by the projection morphism $p\colon X\times\mathbb{A}^1\to X$ is an isomorphism. We collect the results mentioned above in the following equivalent form, which is used in the next sections. Corollary 1. Let $k$ be a field. Let $E$ be a cohomology theory on $\mathrm{Sm}_k$. Then there are isomorphisms Proof. Theorem 2 implies that the cohomologies of the complex $\mathrm{C}(X,E)$ are isomorphic to $H_\mathrm{Nis}^*(X,E_{\mathrm{Nis}})$. The second claim follows from the strict homotopy invariance theorem ([6], Theorem 1.1); see Theorem 3. Here is also one corollary. Corollary 2. For any $X$ in $\mathrm{Sm}_k$ and any cohomology theory $E$ on $\mathrm{Sm}_k$ the canonical restriction homomorphism $E_{\mathrm{Nis}}^n(X)\to E^n(X^{(0)})$ is injective, where $X^{(0)}$ is the coproduct of the generic points of the scheme $X$. Proof. Let $\bigoplus_{x\in X^{(0)}}E^n(x)\cong E^n(X^{(0)})$; then the statement follows from the isomorphism
$$
\begin{equation*}
E_{\mathrm{Nis}}^n(X)\simeq \ker\biggl(\bigoplus_{x\in X^{(0)}}E^n(x)\to \bigoplus_{x\in X^{(1)}} E^n_x(X) \biggr),
\end{equation*}
\notag
$$
which holds because of Theorem 2. § 4. Cohomologies on $ {\mathbb{G}}_m^{\times l}$Let $k$ be a field. While Lemma 2 is formulated over an arbitrary scheme $S$, all other statements relate to the category of smooth schemes over $k$. Recall a fact based on the principles of [9] and [6]. Lemma 2. Let $X$ be a smooth scheme over a scheme $S$, and $Z$ be a closed subscheme in $\mathbb{A}^1\times X$ which is finite over $X$. Then the canonical composition morphism is trivial for any cohomology theory $E$ on $\mathrm{Sm}_S$, where $W = p(Z)\subset X$ and $p\colon {\mathbb{A}^1\times X}\to X$ is the canonical projection. Proof. Consider the sequence of morphisms
$$
\begin{equation}
E_{Z}(\mathbb P^1\times X)\to E_{Z}(\mathbb{A}^1\times X)\to E_{\mathbb{A}^1\times W}(\mathbb{A}^1\times X) \stackrel{p^*}{\cong} E_W(X),
\end{equation}
\tag{4.1}
$$
where the left-hand morphism is an isomorphism by the excision axiom in Definition 1, and the right-hand isomorphism is provided by the $\mathbb{A}^1$-invariance of the cohomology theory $E$ in the sense of (2.2). Moreover, the $\mathbb{A}^1$-invariance of the cohomology theory $E$ just mentioned implies that the composition (4.1) equals the composition
$$
\begin{equation}
E_{Z}(\mathbb P^1\times X)\to E_{Z\cap (\infty\times X)}(\infty\times X)\to E_{W}(\infty\times X)\cong E_{W}(X),
\end{equation}
\tag{4.2}
$$
since the noncommutative triangle in the category of open pairs of schemes goes to the commutative triangle of abelian groups along the functor given by $E$. Since the intersection of the closed subschemes $Z$ and $(\infty\times X)$ in $\mathbb{P}^1\times X$ is empty, the left-hand morphism in the sequence (4.2) is zero. Consequently, the composition (4.2) equals zero. The lemma is proved. Lemma 3. Let $V$ be an open subset in $\mathbb{A}^1_k$ over a field $k$ and $Z$ be a proper closed subset in $V$. Let $X$ be a local essentially smooth scheme over $k$ and $x\in X$ be a closed point. Then the canonical morphism $E_{Z\times x}(V\times X)\to E_{V\times x}(V\times X)$ is trivial for any cohomology theory $E$ на $\mathrm{Sm}_k$. Proof. The following commutative diagram holds: The right-hand vertical arrow is trivial by Lemma 2. Consequently, the left-hand vertical arrow is trivial. The lemma is proved. Here is a generalisation of Corollary 9.5 in [6]. Lemma 4. For any open immersion $V\hookrightarrow \mathbb{A}^1_k$ and any cohomology theory $E$ over a field $k$, the complex $\mathrm{Cous}(V,E)$ is acyclic. Moreover, for any scheme $X\in \mathrm{Sm}_k$ and any point $x\in X$, $\mathrm{Cous}_{V\times x}(V\times X_x,E)$ is acyclic. This follows from Corollary 9.5 in [6], since by Lemma 3, for any closed subscheme $Z$ of the scheme $V\times x$ the sequence
$$
\begin{equation*}
0\to E_Z(V\times X_x)\to E(V\times X_x)\to E(V\times X_x-Z)\to 0
\end{equation*}
\notag
$$
is exact. Lemma 5. For any set of open immersions $V_b\hookrightarrow \mathbb{A}^1_k$ over the field $k$, where ${b=1,\dots, l}$, and any cohomology theory $E$, there is a quasi-isomorphism of complexes $\mathrm{C}(V\times X,E)\simeq \mathrm{C}(X,E^V)$, where $V=V_1\times \dots \times V_l$ and $E^V(-)\cong E(-\times V)$. Proof. The claim follows similarly to Theorem 10.1 in [6]. We repeat the argument briefly. Let $l=1$; then by Lemma 4, for any $x\in X$ the canonical morphisms $E_{V\times x}(V\times X)\to \mathrm{C}_{V\times x}(V\times X,E)$ are quasi-isomorphisms; consequently, the morphism is a quasi-isomorphism, and is a quasi-isomorphism too, since $E^V(X) = E(V\times X)$; the general case follows by induction on $l$. The lemma is proved. Corollary 3. Let $V_b\subset \mathbb{A}^1_k$, where $b=1,\dots, l$, be open subsets, and let $V=V_1\times \dots \times V_l$. Let $E$ be a cohomology theory on $\mathrm{Sm}_k$ in the sense of Definition 1. Then $\mathrm{Cous}(V,E)\simeq 0$. This follows from the quasi-isomorphisms $\mathrm{Cous}(V,E)\simeq \mathrm{Cous}(\mathrm{pt}_k,E^V) \simeq 0$, the first of which is a consequence of Lemma 5. Lemma 6. Let $V_b\subset \mathbb{A}^1_k$, where $b=1,\dots, l$, be open subsets over a field $k$, and let $V=V_1\times \dots \times V_l$. Then $H^*(V\times X,E_{\mathrm{Nis}})\simeq H^*(X,(E^V)_{\mathrm{Nis}})$ for any cohomology theory $E$ on $\mathrm{Sm}_k$, where $E^V(-)\cong E(-\times V)$. The claim follows from Lemma 5 and Corollary 1. § 5. Cohomologies on $X_x-D$Let $k$ be a field. Definition 4. We say that for some $n\in \mathbb Z$ claim $\mathbf{Cou}^{n}$ holds over the field $k$ if for any cohomology theory $E$ on $\mathrm{Sm}_k$, any $X\in \mathrm{Sm}_k$, $x\in X$, and any reduced strict normal-crossing divisor $D$ in $X_x$ such that $D=D_0\cup\dots \cup D_{l}$, where $D_0, \dots, D_l$ are the irreducible components and $l<n$, there are isomorphisms Lemma 7. Assume that $\mathbf{Cou}^{n}$ holds for some $n\in \mathbb Z$. Then for any cohomology theory $E$, any scheme $X\in \mathrm{Sm}_k$, any point $x\in X$, and a reduced strict normal-crossing divisor with $n$ irreducible components the morphism $E(X_x-D) \to E(V)$ is injective for any dense open subset $V$ of $X_x-D$. Proof. Since $E(X_x-D)\cong E_{\mathrm{Nis}}(X_x-D)$ by assumption, the injectivity of the homomorphism holds by Corollary 2. Hence for any dense open subscheme $V$ in $X_x-D$ the homomorphism $E(X_x-D) \to E(V)$ is injective. The lemma is proved. Definition 5. For any abelian presheaf $F$ on $\mathrm{Sm}_k$ and any $X\in\mathrm{Sm}_k$ set We use similar notation for presheaves on $\mathrm{SmOP}_k$. Lemma 8. Assume that $\mathbf{Cou}^{n}$ holds for some $n\in \mathbb Z$. Then for any cohomology theory $E$, any $X\in \mathrm{Sm}_k$ and $x\in X$, and a reduced strict normal-crossing divisor $D$ which has $n$ irreducible components, there are isomorphisms where Proof. Since $E^{\mathbb{G}_m^{\wedge 1}}$ is a cohomology theory by Theorem 3 and because of the assumption $\mathbf{Cou}^{n}$ as applied to the divisor ${\widetilde D}$ and the cohomology theory $E^{\mathbb{G}_m^{\wedge 1}}$ there are isomorphisms
$$
\begin{equation*}
H^v({\widetilde D},E^{\mathbb{G}_m^{\wedge 1}})\cong 0\quad\text{and}\quad H^0({\widetilde D},E^{\mathbb{G}_m^{\wedge 1}})\cong E({\widetilde D}\wedge\mathbb{G}_m)
\end{equation*}
\notag
$$
for $v>0$. Consequently, by Lemma 6, for any $v>0$. The claim follows from the sequence isomorphisms for all $v\in \mathbb Z$. The lemma is proved. Proof. The proof uses induction on $n$. The case $n=0$ was established by Theorem 1. Assume that the claim holds for all integers less then a fixed $n>0$. Set
By Theorem 3 the presheaf on $\mathrm{SmOP}_k$ given by
$$
\begin{equation}
(X,X-Z)\mapsto \bigoplus_{l}H^l_{Z}(X,E_{\mathrm{Nis}})
\end{equation}
\tag{5.2}
$$
is a cohomology theory on $\mathrm{Sm}_k$. Hence for any $v\in \mathbb Z$, by the inductive assumption as applied to the cohomology theory (5.2) and the scheme $U^{-1}$, the homomorphism
$$
\begin{equation*}
H^v(U,E_{\mathrm{Nis}}) \leftarrow H^v(U^{-1},E_{\mathrm{Nis}})
\end{equation*}
\notag
$$
is injective by Lemma 7. Because of the long exact sequence
$$
\begin{equation*}
\dots\leftarrow H^{v+1}_{\widetilde D}(U^{-1},E_{\mathrm{Nis}})\leftarrow H^v(U,E_{\mathrm{Nis}}) \leftarrow H^v(U^{-1},E_{\mathrm{Nis}})\leftarrow\dotsb
\end{equation*}
\notag
$$
we obtain the short exact sequences
$$
\begin{equation}
0\leftarrow H^{v+1}_{\widetilde D}(U^{-1},E_{\mathrm{Nis}})\leftarrow H^v(U,E_{\mathrm{Nis}}) \leftarrow H^v(U^{-1},E_{\mathrm{Nis}})\leftarrow 0.
\end{equation}
\tag{5.3}
$$
Now applying the inductive assumption to the scheme $U^{-1}$ we obtain the isomorphisms for all $v>0$. Next, by the inductive assumption and Lemma 8, for any $v>0$ there is an isomorphism
$$
\begin{equation*}
H^{v+1}_{\widetilde D}(U^{-1},E_{\mathrm{Nis}})\cong 0.
\end{equation*}
\notag
$$
Thus, the second isomorphism in (5.1) is proved.
By the inductive assumption as applied to the scheme $U^{-1}$ and the cohomology theory $E$ and by Lemma 7 the morphisms $E(U^{-1})\to E(U)$ are injective. Then the long exact sequence (2.1) leads to the short exact sequences
$$
\begin{equation}
0\leftarrow E_{D}(U^{-1})\leftarrow E(U) \leftarrow E(U^{-1})\leftarrow 0 .
\end{equation}
\tag{5.4}
$$
Combining with the short exact sequence (5.3) we obtain a morphism of short exact sequences Here the right-hand vertical arrow is an isomorphism by the inductive assumption as applied to $U^{-1}$ and $E$. The left-hand vertical arrow is an isomorphism by Lemma 8. Consequently, the middle vertical arrow is an isomorphism, and the first isomorphism in (5.1) follows. The proposition is proved. Theorem 4. Let $k$ be a field. Let $X\in \mathrm{Sm}_k$, $x\in X$, and let $D$ be a strict normal-crossing divisor in $X$. Let $V_\alpha$, $\alpha\in J$, be a finite set of open subschemes in $\mathbb{A}^1_k$. Put $U=(X_x-D)\times \prod_{\alpha\in J}V_\alpha$. Then for any cohomology theory $E$ in the sense of Definition 1 or [29] the complex $\mathrm{Cous}(U,E)$ defined by (1.1) is acyclic. In particular, the claim holds for cohomology theories representable in the Morel $S^1$-stable motivic homotopy category $\mathbf{SH}^{S^1}(k)$. Proof. The case $J=\varnothing$, $U=X_x-D$ follows by Corollary 1 and Proposition 1. The case $J\neq\varnothing$ follows by Lemma 5. The theorem is proved.
§ 6. Inverse image functor with compact supportsLet $S$ be an affine noetherian scheme. Consider the category of smooth $S$-schemes that has a trivial tangent bundle, denote it by $\mathrm{Sm}^\mathrm{cci}_S$, and denote by $\mathrm{SmOP}^\mathrm{cci}_S$ the full subcategory in $\mathrm{SmOP}_S$ spanned by the pairs $(X,U)$ such that $X\in \mathrm{Sm}^\mathrm{cci}_S$. Definition 6. A cohomology theory on $\mathrm{Sm}^\mathrm{cci}_S$ is a functor $E\colon \mathrm{SmOP}^\mathrm{cci}_S\to \mathrm{Ab}$ that has properties similar to the ones in Definition 1. Denote the category of such functors by $\mathfrak{C}(\mathrm{SmOP}^\mathrm{cci}_S)$. Remark 3. The restriction of any cohomology theory on $\mathrm{Sm}_S$ gives a cohomology theory on $\mathrm{Sm}^\mathrm{cci}_S$. We cite the following result, which is used below. Lemma 9 (see [30], Lemma 6.3). For any scheme $X\in \mathrm{Sm}^\mathrm{cci}_Z$ there is a scheme $\widetilde X\in \mathrm{Sm}^\mathrm{cci}_S$ such that $\widetilde X\times_S Z\cong X$. We also use the following result proved in various sources. Lemma 10 (see [31], Theorem I.8, and [32]). Let $C\hookrightarrow U$ be an affine henselian pair and $s\colon V\to U$ be a smooth affine morphism of schemes, and let $c\colon C\to V$ be a morphism of schemes such that the square given by the solid arrows of the diagram is commutative. Then there is a diagonal morphism of schemes shown by the dashed arrow above such that the diagram is commutative. Moreover, if the morphism $s$ is étale, then $l$ is unique. Note that the second claim of this lemma is a reformulation of the definition of henselization. Let $Z$ be closed subscheme of the scheme $S$. For any scheme $\widetilde X\in\mathrm{Sm}^\mathrm{cci}_S$ set $\widetilde X_Z=\widetilde X\times_S Z$, and set $(\widetilde X)^h_Z=(\widetilde X)^h_{X_Z}$, which means the henselization of $\widetilde X$ in $X_Z$. Corollary 4. For any morphism of schemes $X\to Y\in \mathrm{Sm}^\mathrm{cci}_Z$ and any schemes $\widetilde X,\widetilde Y\in \mathrm{Sm}^\mathrm{cci}_S$ such that $\widetilde X\times_S Z\cong X$ and $\widetilde Y\times_S Z\cong Y$, there is a morphism $\widetilde f\colon (\widetilde X)^h_Z\to \widetilde Y$ that goes to $f$ under the base change along $r$. Moreover, if the morphism $f$ is étale, then $\widetilde f$ is unique. The first claim follows from Lemma 10 as applied to the henselian pair ${X\,{\to}\, (\widetilde X)^h_Z}$, the morphisms $V = \widetilde Y\times_S (\widetilde X)^h_Z\to (\widetilde X)^h_Z$ and the morphism $X\to V$ defined by the morphisms $X\to Y\to \widetilde Y$ and $X\to (\widetilde X)^h_Z$. Lemma 11. Let $r\colon Z\to S$ be a closed immersion of affine schemes. Then there is a functor such that for all schemes $X\in \mathrm{Sm}^\mathrm{cci}_Z$, closed subschemes $Z\subset X$ and $\widetilde X\in \mathrm{Sm}^\mathrm{cci}_S$ such that $\widetilde X\times_S Z\cong X$ there is a canonical morphism $r^{!} E_{C}(X)\cong E_{C}(\widetilde{X})$. Proof. (1) By Lemma 9, for any $X\in \mathrm{Sm}^\mathrm{cci}_{Z}$ there is $\widetilde X\,{\in}\,\mathrm{Sm}^\mathrm{cci}_{Z}$ such that ${\widetilde X\!\mathbin{\times_S}\! Z\!\cong\! X}$. Moreover, for any such $\widetilde X_1$ and $ \widetilde X_2$ there is a canonical isomorphism $(\widetilde X_1)^h_{\widetilde X_1\times_S Z}\cong (\widetilde X_2)^h_{\widetilde X_2\times_S Z}$. So we obtain a well-defined map of the sets of objects of the categories such that
(2) Let $f\colon X\to Y$ be a morphism in the category $\mathrm{SmOP}^\mathrm{cci}_Z$, let $C\subset X$ and $B\subset Y$ be closed subschemes such that $C\subset f^{-1}(B)$, and let $\widetilde X,\widetilde Y\in \mathrm{SmOP}^\mathrm{cci}_S$, $\widetilde X\times_S Z\cong X$, $\widetilde Y\times_S Z\cong Y$. By Corollary 4 there is a morphism $\widetilde f\colon (\widetilde X)^h_Z\to \widetilde Y$ that goes to $f$ under the base change along $r$. Moreover, if $\widetilde f_1,\widetilde f_2\colon (\widetilde X)^h_Z\to \widetilde Y$ are morphisms that go to $f$ under the base change, then by Corollary 4 there is a morphism $\widetilde f\colon (\widetilde X\times\mathbb{A}^1)^h_Z \to \widetilde Y$ such that
$$
\begin{equation*}
\widetilde f\big|_{(\widetilde X\times 0)^h_Z}=f_1,\qquad \widetilde f\big|_{(\widetilde X\times 1)^h_Z}=f_2.
\end{equation*}
\notag
$$
Hence by the excision and homotopy invariance axioms in Definition 1 the homomorphisms
$$
\begin{equation*}
(\widetilde f_1)^*,(\widetilde f_2)^*\colon E((\widetilde X)^h_Z, (\widetilde X)^h_Z-C)\to E(Y-B)
\end{equation*}
\notag
$$
are equal. Thus we obtain a well-defined map of the sets of morphisms that takes $f$ to a homomorphism $\widetilde f^*\colon E(\widetilde Y,\widetilde Y-B)\to E(\widetilde X,\widetilde X-C)$. So we obtain a well-defined functor
$$
\begin{equation*}
r^!\colon \mathrm{SmOP}^{\mathrm{cci}}_Z\to \mathrm{Ab}^\mathrm{op},\qquad (X,X-C)\mapsto E(\widetilde X,\widetilde X-C).
\end{equation*}
\notag
$$
(3) The excision and homotopy invariance axioms for $r^*(E)$ hold because of similar properties of $E$; to prove the localisation axiom we use Lemma 1. The lemma is proved. § 7. Cohomologically trivial schemesLet $k$ be a field. Definition 7. The scheme $V$ over a field $k$ is cohomologically trivial if the complex $\mathrm{Cous}(V, E)$ is acyclic for any cohomology theory $E$ on $\mathrm{Sm}_k$. A scheme $V$ is called universally cohomologically trivial if for any field extension $K/k$, where $K=k(X)$ for $X\in\mathrm{Sm}_k$, the scheme $V\times_k\operatorname{Spec} K$ is cohomologically trivial. Example 1. (1) By Corollary 3 the scheme of the form $\mathbb{A}^1_k-Z$, where $Z\subset \mathbb{A}^1_k$ is a closed subset, is universally cohomologically trivial; (2) By Theorem 4 the scheme of the form $X_x-D$, where $X\in \mathrm{Sm}_k$, $x\in X$, and $D$ is a strict normal-crossing divisor in $X_x$, is cohomologically trivial. Lemma 12. For any cohomologically trivial scheme $V$ in the sense of Definition 7 and any cohomological theory $E$ on $\mathrm{Sm}_k$, for all $l>0$ the groups $H^l(V,E_{\mathrm{Nis}})$ are trivial, where $E_{\mathrm{Nis}}$ is the sheafification of the presheaf $E$ on $\mathrm{Sm}_k$. The claim follows from the definition and Corollary 1. Lemma 13. Any scheme that is cohomologically trivial in the sense of Definition 7 is cohomologically trivial in the sense of Definition 2.9 in [15]. This follows from Lemma 12, since any strict homotopy invariant sheaf $F$ defines the cohomology theory $E(-)=\bigoplus_{l} H^l(-,F)$. Lemma 14. Let $E$ be a cohomology theory on $\mathrm{Sm}_k$ and $V$ be a universally cohomologically trivial scheme over the field $k$. Then for any essentially smooth scheme $X$ over $k$ and any point $x\in X$ the complex $\mathrm{Cous}_{x\times V}(X\times V, E)$ is acyclic. Proof. Since the scheme $X$ in the formulation can be replaced by the local scheme $X_x$ of $X$ at $x$, without loss of generality the point $x$ can be a closed point of an affine essentially smooth scheme $X$. Note further that the cohomology theory $E$ on $\mathrm{Sm}_k$ defines a cohomology theory over $X$ in view of the functor $\mathrm{Sm}_X\to \mathrm{Sm}_k$, and consider the morphism of big étale sites defined by the functor $x\colon \mathrm{Sm}_X\to \mathrm{Sm}_x\colon W\to W\times_X x$. Moreover, for any $U\in \mathrm{Sm}_k$ the complex $\mathrm{Cous}(U,E)$ is uniquely defined by the restriction of $E$ to $\mathrm{SmOP}^\mathrm{cci}_k$, while $x$ induces a functor $x\colon \mathrm{SmOP}^\mathrm{cci}_X\to \mathrm{SmOP}^\mathrm{cci}_x$. This functor $x$ induces by Lemma 11 a functor $x^!$ from the category of cohomology theories on $\mathrm{SmOP}^\mathrm{cci}_X$ to the category of cohomology theories on $\mathrm{SmOP}^\mathrm{cci}_x$, while there is an equivalence Then and the second quasi-isomorphism is a consequence of the definition, since $x^!(E)$ is a cohomology theory over $k(x)$ and $V\times x$ is a cohomologically trivial scheme. The lemma is proved. Theorem 5. Let $E$ be a cohomology theory on $\mathrm{Sm}_k$ and $V$ be a universally cohomologically trivial scheme over the field $k$, Then for any scheme $X$ over $k$ and any point $x\in X$ the canonical morphisms where $E^V(-)\cong E(-\times V)$, are quasi-isomorphisms. Proof. Since $E(X\times V)\cong E^V(X)$, the above two quasi-isomorphisms are equivalent. By Lemma 14 there are equivalences
$$
\begin{equation*}
\mathrm{C}_{x\times V}(X\times V, E) \simeq E_{x\times V}(X\times V)
\end{equation*}
\notag
$$
for all points $x\in X$. The claim of theorem follows just like Lemma 5 follows from Lemma 4. Corollary 5. Let $V$ be a cohomologically trivial scheme, and let $V_\alpha$, where $\alpha\in A$ and $A$ is a finite set, be a family of universally cohomologically trivial schemes over the field $k$. Then the scheme $\prod_{\alpha\in A} V_\alpha$ is cohomologically trivial and $V\times\prod_{\alpha\in A} V_\alpha$ is cohomologically trivial. This follows by induction from Theorem 5. Corollary 6. Let This is a particular case of the previous corollary in accordance with Example 1. Corollary 7. A scheme $V$ of the form described in Theorem 4 is cohomologically trivial in the sense of Definition 2.9 in [15]. The claim follows from Corollary 6 and Lemma 13. Question 2. 1) Are the schemes $X_x$ and $X_x-D$, where $X\in \mathrm{Sm}_k,$ $x\in X$, and $D$ is a strict normal-crossing divisor or a normal-crossing divisor are universally cohomologically trivial? 2) Are the classes of cohomologically trivial and universally cohomologically trivial schemes equal?
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