2.1 Ind-objects of cofinite type

Let $p$ be a prime number. Let ${\mathcal{A}}$ be an artinian (${\mathcal{U}}_{0}$-)small abelian category such that any object of ${\mathcal{A}}$ is killed by some power of $p$. Note that no nonzero object of ${\mathcal{A}}$ is divisible, which will be used frequently later on. Denote its ind-category by $\text{I}{\mathcal{A}}$ (or more precisely, the ${\mathcal{U}}_{0}$-indcategory, where index sets are ${\mathcal{U}}_{0}$-small). The category $\text{I}{\mathcal{A}}$ is abelian by [Reference Kashiwara and SchapiraKS06, Theorem 8.6.5(i)]. Note that objects of ${\mathcal{A}}$ need not be artinian in $\text{I}{\mathcal{A}}$. For instance, the additive algebraic group scheme $\mathbf{G}_{a}$ over a field of characteristic $p$ contains the ind-finite-étale subgroup $\overline{\mathbb{F}_{p}}$. We say that an object $A\in \text{I}{\mathcal{A}}$ is of cofinite type if the $p^{n}$-torsion part $A[p^{n}]$ is in ${\mathcal{A}}$ for all $n\geqslant 0$. In this case, the equality $A=\varinjlim A[p^{n}]$ gives a presentation of $A$ as a filtered direct limit of objects of ${\mathcal{A}}$. Denote by $\text{I}^{f}{\mathcal{A}}$ the full subcategory of $\text{I}{\mathcal{A}}$ consisting of objects of cofinite type.

Proof. Let $A\in \text{I}^{f}{\mathcal{A}}$. To show $A/p^{n}A\in {\mathcal{A}}$, it is enough to show this for $n=1$. The object $A/pA$ is the union of the increasing sequence of subobjects $A[p^{m}]/p(A[p^{m+1}])$ each in ${\mathcal{A}}$. It is enough to show that this sequence stabilizes, or $A[p^{m-1}]+p(A[p^{m+1}])=A[p^{m}]$ for large $m$. This is equivalent that the decreasing sequence

$$\begin{eqnarray}\cdots \overset{p}{{\hookrightarrow}}A[p^{m+1}]/A[p^{m}]\overset{p}{{\hookrightarrow}}A[p^{m}]/A[p^{m-1}]\overset{p}{{\hookrightarrow}}\cdots\end{eqnarray}$$

stabilizes. It indeed does since each object in the sequence belongs to the artinian category ${\mathcal{A}}$.◻

Proposition 2.1.3. For any $A\in \text{I}^{f}{\mathcal{A}}$, the decreasing sequence

$$\begin{eqnarray}\cdots {\hookrightarrow}p^{2}A{\hookrightarrow}pA{\hookrightarrow}A\end{eqnarray}$$

stabilizes.

Proof. For any $n$, we have $p^{n}:A/(A[p^{n}]+pA)\overset{{\sim}}{\rightarrow }p^{n}A/p^{n+1}A$. The isomorphism

$$\begin{eqnarray}A/pA\cong \underset{n}{\varinjlim }\,(A[p^{n}]+pA)/pA\end{eqnarray}$$

in $\text{I}{\mathcal{A}}$ factors through $(A[p^{n}]+pA)/pA$ for some $n$ since $A/pA\in {\mathcal{A}}$. Hence $A/pA\cong (A[p^{n}]+pA)/pA$ for all large $n$. For such $n$, we have $A=A[p^{n}]+pA$ and $p^{n+1}A=p^{n}A$. Hence the sequence stabilizes.◻

For $A\in \text{I}^{f}{\mathcal{A}}$, we define

$$\begin{eqnarray}\displaystyle & \displaystyle A_{\text{div}}=\mathop{\bigcap }_{n}p^{n}A\quad (=p^{n}A\text{ for some }n), & \displaystyle \nonumber\\ \displaystyle & \displaystyle A_{/\text{div}}=A/A_{\text{div}}. & \displaystyle \nonumber\end{eqnarray}$$

Proposition 2.1.4. If $A\in \text{I}^{f}{\mathcal{A}}$, then $A_{\text{div}}\in \text{I}^{f}{\mathcal{A}}$ is divisible and$A_{/\text{div}}\in {\mathcal{A}}$. The sequence

$$\begin{eqnarray}0\rightarrow A_{\text{div}}\rightarrow A\rightarrow A_{/\text{div}}\rightarrow 0\end{eqnarray}$$

is exact. The object $A_{\text{div}}$ is the largest divisible subobject of $A$. The object $A_{/\text{div}}$ is the largest quotient of $A$ that belongs to ${\mathcal{A}}$.

Proof. By the previous proposition, an object of ${\mathcal{A}}$ is artinian in $\text{I}^{f}{\mathcal{A}}$ since an object of $\text{I}^{f}{\mathcal{A}}$ embeddable into an object of ${\mathcal{A}}$ does not have divisible part and hence itself is in ${\mathcal{A}}$. It is enough to show that a decreasing sequence $A_{0}\supset A_{1}\supset \cdots \,$ of divisible objects in $\text{I}^{f}{\mathcal{A}}$ stabilizes. Since $A_{n}[p]\in {\mathcal{A}}$ is artinian for any $n$, there is some $m$ such that $A_{m}[p]=A_{m+1}[p]=\cdots \,$. From the exact sequence

$$\begin{eqnarray}0\rightarrow A_{n}[p]\rightarrow A_{n}[p^{2}]\overset{p}{\rightarrow }A_{n}[p]\rightarrow 0,\end{eqnarray}$$

we know that $A_{m}[p^{2}]=A_{m+1}[p^{2}]=\cdots \,$ for the same $m$. Inductively, we have $A_{m}=A_{m+1}=\cdots \,$.◻

Denote the procategory of ${\mathcal{A}}$ by $\text{P}{\mathcal{A}}$ and the ind-category of $\text{P}{\mathcal{A}}$ by $\text{I}\text{P}{\mathcal{A}}$. For $A\in \text{I}^{f}{\mathcal{A}}$, we define

$$\begin{eqnarray}\displaystyle & \displaystyle TA=\underset{n}{\varprojlim }\,A[p^{n}]\in \text{P}{\mathcal{A}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle VA=\underset{m}{\varinjlim }\,TA\in \text{I}\text{P}{\mathcal{A}}, & \displaystyle \nonumber\end{eqnarray}$$

where the direct limit in the second definition is for multiplication by $p^{m}$. We call $TA$ the Tate module of $A$ and $VA$ the rational Tate module of $A$. For example, if $A$ is $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ in the category of torsion abelian groups, then $TA=\mathbb{Z}_{p}$ in the category of profinite abelian groups and $VA=\mathbb{Q}_{p}$ as an ind-object of profinite abelian groups. For general ${\mathcal{A}}$, if $A\in {\mathcal{A}}$, then the system $\{A[p^{n}]\}$ that defines $TA$ has essentially zero transition morphisms, hence $TA=0$. For each $m$, we consider the morphism $TA\rightarrow A$ in $\text{I}\text{P}{\mathcal{A}}$ given by $(a_{n})_{n}\mapsto a_{m}$. They form a morphism $VA\rightarrow A$ in $\text{I}\text{P}{\mathcal{A}}$.

Proof. Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be exact in $\text{I}^{f}{\mathcal{A}}$. For any $n$, we have an exact sequence

$$\begin{eqnarray}0\rightarrow A[p^{n}]\rightarrow B[p^{n}]\rightarrow C[p^{n}]\rightarrow A/p^{n}A\rightarrow B/p^{n}B\rightarrow C/p^{n}C\rightarrow 0\end{eqnarray}$$

in ${\mathcal{A}}$. The inverse limit in $n$ in $\text{P}{\mathcal{A}}$ gives an exact sequence

$$\begin{eqnarray}0\rightarrow TA\rightarrow TB\rightarrow TC\rightarrow A_{/\text{div}}\rightarrow B_{/\text{div}}\rightarrow C_{/\text{div}}\rightarrow 0\end{eqnarray}$$

in $\text{P}{\mathcal{A}}$. (Note that Mittag-Leffler conditions are not relevant here since filtered inverse limits in a procategory are exact by definition.) Since each $(\;\cdot \;)_{/\text{div}}\in {\mathcal{A}}$ is killed by multiplication by some power of $p$, the direct limit in multiplication by $m$ gives the desired exact sequence $0\rightarrow VA\rightarrow VB\rightarrow VC\rightarrow 0$ in $\text{I}\text{P}{\mathcal{A}}$.◻

Proposition 2.1.7. Let $A\in \text{I}^{f}{\mathcal{A}}$. Then we have an exact sequence

$$\begin{eqnarray}0\rightarrow TA\overset{p^{n}}{\rightarrow }TA\rightarrow A_{\text{div}}\cap A[p^{n}]\rightarrow 0\end{eqnarray}$$

in $\text{P}{\mathcal{A}}$ for any $n\geqslant 0$ and an exact sequence

$$\begin{eqnarray}0\rightarrow TA\rightarrow VA\rightarrow A_{\text{div}}\rightarrow 0\end{eqnarray}$$

in $\text{I}\text{P}{\mathcal{A}}$. We have $TA=T(A_{\text{div}})$ and $VA=V(A_{\text{div}})$.

Proof. The multiplication-by-$p^{n}$ map gives an exact sequence $0\rightarrow A[p^{n}]\rightarrow A\rightarrow p^{n}A\rightarrow 0$. We have $T(A[p^{n}])=0$ and $A[p^{n}]_{/\text{div}}=A[p^{n}]$ since $A[p^{n}]\in {\mathcal{A}}$ has no divisible part. Hence the long exact sequence in the proof of the previous proposition for this sequence is

$$\begin{eqnarray}0\rightarrow TA\rightarrow T(p^{n}A)\rightarrow A[p^{n}]\rightarrow A_{/\text{div}}\rightarrow (p^{n}A)_{/\text{div}}\rightarrow 0.\end{eqnarray}$$

in $\text{P}{\mathcal{A}}$. Since $A/p^{n}A\in {\mathcal{A}}$, we have $T(p^{n}A)=TA$. From this, we get the first exact sequence. The direct limit in $n$ gives the second exact sequence. We have $T(A_{/\text{div}})=0$ since $A_{/\text{div}}\in {\mathcal{A}}$. It follows that $TA=T(A_{\text{div}})$ and $VA=V(A_{\text{div}})$.◻

Next, let ${\mathcal{A}}$ be an artinian abelian category such that any object is killed by multiplication by some positive integer (not necessarily a prime power). For each prime $p$, denote by ${\mathcal{A}}_{p}$ the full subcategory of ${\mathcal{A}}$ of objects killed by a power of $p$. This is an artinian abelian category. Any object of ${\mathcal{A}}$ can be canonically decomposed as $A=\bigoplus _{p}A_{p}$, where each summand $A_{p}$ belongs to ${\mathcal{A}}_{p}$ and $A_{p}=0$ for almost all $p$. We call $A_{p}$ the $p$-primary part of $A$. Consider the ind-category $\text{I}{\mathcal{A}}$. Any object of $\text{I}{\mathcal{A}}$ can be canonically decomposed as $A=\bigoplus _{p}A_{p}$ (a filtered direct limit of finite partial sums), where each summand $A_{p}$ belongs to $\text{I}{\mathcal{A}}_{p}:=\text{I}({\mathcal{A}}_{p})$.

We say that an object $A\in \text{I}{\mathcal{A}}$ is of cofinite type if the $n$-torsion part $A[n]$ is in ${\mathcal{A}}$ for all $n\geqslant 1$. In this case, the equality $A=\varinjlim A[n]$ gives a presentation of $A$ as a filtered direct limit of objects of ${\mathcal{A}}$. Denote by $\text{I}^{f}{\mathcal{A}}$ the full subcategory of $\text{I}{\mathcal{A}}$ consisting of objects of cofinite type. An object $A\in \text{I}{\mathcal{A}}$ is of cofinite type if and only if $A_{p}$ is of cofinite type for all $p$. Note that $\text{I}^{f}{\mathcal{A}}$ is not necessarily artinian. For instance, the object $\bigoplus _{p}\mathbb{Z}/p\mathbb{Z}$ in the category of torsion abelian groups of cofinite type is not artinian. For $A\in \text{I}^{f}{\mathcal{A}}$, we define

$$\begin{eqnarray}A_{\text{div}}=\bigoplus _{p}(A_{p})_{\text{div}}\in \text{I}{\mathcal{A}}\qquad A_{/\text{div}}=A/A_{\text{div}}\in \text{I}{\mathcal{A}},\end{eqnarray}$$

where $(A_{p})_{\text{div}}\in \text{I}{\mathcal{A}}_{p}$ is previously defined. We also define

$$\begin{eqnarray}TA=\underset{n}{\varprojlim }\,A[n]\in \text{P}{\mathcal{A}},\qquad VA=\underset{m}{\varinjlim }\,TA\in \text{I}\text{P}{\mathcal{A}},\end{eqnarray}$$

where the inverse limit in the first definition is over multiplication by $n\geqslant 1$ and the direct limit in the second is over multiplication by $m\geqslant 1$. Also define $T_{p}A=T(A_{p})$, $V_{p}A=V(A_{p})$.

Contrary to the $p$-primary case, $A_{/\text{div}}$ might not be in ${\mathcal{A}}$ for general $A\in \text{I}^{f}{\mathcal{A}}$. An example is $\bigoplus _{p}\mathbb{Z}/p\mathbb{Z}$. This is related to the Mittag-Leffler condition. Here is a general fact.

Proof. We have

$$\begin{eqnarray}A_{/\text{div}}=\bigoplus _{p}A_{p}\bigg/\bigoplus _{p}(A_{p})_{\text{div}}=\bigoplus _{p}(A_{p})_{/\text{div}}.\end{eqnarray}$$

The result follows from this. ◻

Note that divisibly ML objects in $\text{I}^{f}{\mathcal{A}}$ do not form an abelian subcategory. For instance, the direct sum of the exact sequences $0\rightarrow \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow \mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow 0$ for primes $p$ gives an exact sequence $0\rightarrow \bigoplus _{p}\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 0$, where the last two terms are divisible but the first term is not divisibly ML.

2.2 Ind-algebraic groups of cofinite type

Let $k$ ($\in {\mathcal{U}}_{0}$) be a perfect field of characteristic $p>0$. We quickly recall some notation about perfections of algebraic groups from [Reference SuzukiSuz14, Section 2.1]. Let $\text{Alg}/k$ be the category of quasi-algebraic groups over $k$ (commutative, as assumed throughout the paper) in the sense of Serre [Reference SerreSer60]. Recall that a quasi-algebraic group is the perfection (inverse limit along Frobenii) of a (not necessarily connected) algebraic group [Reference SerreSer60, Section 1.2, Définition 2; Section 1.4, Proposition 10]. For example, the perfection of the additive group $\operatorname{Spec}k[x]$ is $\operatorname{Spec}k[x,x^{1/p},x^{1/p^{2}},\ldots \,]$, which we simply call the additive group and denote by $\mathbf{G}_{a}$ by abuse of notation. We say that a quasi-algebraic group is a unipotent group, a torus or a (semi-)abelian variety if it is the perfection of such a group. We call an object of the procategory $\text{P}\text{Alg}/k$ a proalgebraic group, an object of the ind-category $\text{I}\text{Alg}/k$ an ind-algebraic group and an object of the ind-procategory $\text{I}\text{P}\text{Alg}/k=\text{I}(\text{P}\text{Alg}/k)$ an ind-proalgebraic group. Let $\text{Alg}_{\text{uc}}/k$ be the full subcategory of $\text{Alg}/k$ consisting of groups whose identity component is unipotent. Let $\text{L}\text{Alg}/k$ be the category of perfections of smooth group schemes over $k$. A finitely generated étale group is an étale group with a finitely generated group of geometric points. A lattice is a finitely generated étale group with no torsion. Let $\text{FEt}/k\subset \text{FGEt}/k\subset \text{Et}/k$ be the categories of finite étale groups, finitely generated étale groups, étale groups, respectively, over $k$. For any prime $l$ (possibly equal to $p$), we denote the full subcategory of $\text{FEt}/k$ of groups of $l$-power order by $\text{FEt}_{l}/k$. For $A\in \text{L}\text{Alg}/k$, we denote its identity component by $A_{0}\in \text{Alg}/k$ and set $\unicode[STIX]{x1D70B}_{0}(A)=A/A_{0}\in \text{Et}/k$ ([Reference Demazure and GabrielDG70, II, Section 5, Proposition 1.8] plus perfection, noting that perfection does not change the underlying topological space). The endofunctor $A\mapsto A_{0}$ on $\text{Alg}/k$ extends to endofunctors on $\text{P}\text{Alg}/k$, $\text{I}\text{Alg}/k$ and $\text{I}\text{P}\text{Alg}/k$, still denoted by $A\mapsto A_{0}$. We say that $A\in \text{I}\text{P}\text{Alg}/k$ is connected if $A=A_{0}$. For $A\in \text{I}\text{P}\text{Alg}/k$, we define $\unicode[STIX]{x1D70B}_{0}(A)=A/A_{0}\in \text{I}\text{P}\text{FEt}/k$.

The category $\text{Alg}_{\text{uc}}/k$ is an artinian abelian category such that any object is killed by multiplication by some positive integer. Hence we can apply the results and notation in the previous subsection to ${\mathcal{A}}=\text{Alg}_{\text{uc}}/k$.

Proof. We may assume that $A$ is unipotent. Consider the sequence

$$\begin{eqnarray}\displaystyle 0 & \rightarrow & \displaystyle A_{0}[p^{n}]\rightarrow A[p^{n}]\rightarrow \unicode[STIX]{x1D70B}_{0}(A)[p^{n}]\nonumber\\ \displaystyle & \rightarrow & \displaystyle A_{0}/p^{n}A_{0}\rightarrow A/p^{n}A\rightarrow \unicode[STIX]{x1D70B}_{0}(A)/p^{n}\unicode[STIX]{x1D70B}_{0}(A)\rightarrow 0\nonumber\end{eqnarray}$$

for any $n\geqslant 0$. Since $A[p^{n}]$ is quasi-algebraic and $\unicode[STIX]{x1D70B}_{0}(A)[p^{n}]$ is étale, the image of $A[p^{n}]\rightarrow \unicode[STIX]{x1D70B}_{0}(A)[p^{n}]$ is finite and hence $A_{0}[p^{n}]$ is quasi-algebraic. Therefore $A_{0}\in \text{I}^{f}\text{Alg}_{\text{uc}}/k$ and so $A_{0}/p^{n}A_{0}$ is quasi-algebraic by Proposition 2.1.1. This implies that $\unicode[STIX]{x1D70B}_{0}(A)[p^{n}]$ is quasi-algebraic (i.e., finite étale) and $\unicode[STIX]{x1D70B}_{0}(A)\in \text{I}^{f}\text{Alg}_{\text{uc}}/k$.◻

We define $\text{L}^{f}\text{Alg}_{\text{uc}}/k$ to be the full subcategory of $\text{I}^{f}\text{Alg}_{\text{uc}}/k$ of objects whose identity component is quasi-algebraic (i.e., belongs to $\text{Alg}_{\text{uc}}/k$). Equivalently, an object $A\in \text{L}^{f}\text{Alg}_{\text{uc}}/k$ is an extension of a torsion étale group of cofinite type ($=\unicode[STIX]{x1D70B}_{0}(A)$) by a connected unipotent quasi-algebraic group ($=A_{0}$). Such objects can naturally be viewed as objects of $\text{L}\text{Alg}/k$; see [Reference SuzukiSuz13, Section 2.1, Footnote 3].

An example of a connected divisible group in $\text{I}^{f}\text{Alg}_{\text{uc}}/k$ is the direct limit $\mathop{\varinjlim }\nolimits_{n}W_{n}$ of (perfections of) groups of $p$-typical Witt vectors of finite length. We have $T\,\mathop{\varinjlim }\nolimits_{n}W_{n}=\mathop{\varprojlim }\nolimits_{n}W_{n}$, which is the group $W$ of $p$-typical Witt vectors of infinite length.

Proposition 2.2.5. Let $A\in \text{I}^{f}\text{Alg}_{\text{uc}}/k$ and define $A_{0\cap \text{div}}:=A_{0}\cap A_{\text{div}}$. Then $A_{0\cap \text{div}}/A_{0\text{div}}\in \text{FEt}_{p}/k$. We have exact sequences

$$\begin{eqnarray}\displaystyle & \displaystyle 0\rightarrow A_{0\cap \text{div}}\rightarrow A_{0}\rightarrow (A_{/\text{div}})_{0}\rightarrow 0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle 0\rightarrow A_{0\cap \text{div}}\rightarrow A_{\text{div}}\rightarrow (\unicode[STIX]{x1D70B}_{0}A)_{\text{div}}\rightarrow 0. & \displaystyle \nonumber\end{eqnarray}$$

The group $A_{0\cap \text{div}}$ in the proposition is in general nonzero even if $A\in \text{L}^{f}\text{Alg}_{\text{uc}}/k$. It is $\mathbb{Z}/p\mathbb{Z}$ if $A$ is the cokernel of the diagonal embedding of $\mathbb{Z}/p\mathbb{Z}$ into $\mathbf{G}_{a}\oplus (\mathbb{Q}_{p}/\mathbb{Z}_{p})$. In particular, $A_{0\cap \text{div}}$ can be neither connected nor divisible. On the other hand, $A_{0\text{div}}$ is always connected and divisible.

For example, if $A=\mathbf{G}_{a}\oplus \mathbb{Q}/\mathbb{Z}\oplus \mathbb{Z}/p\mathbb{Z}$, then $A_{0+\text{div}}=\mathbf{G}_{a}\oplus \mathbb{Q}/\mathbb{Z}$ and $\unicode[STIX]{x1D70B}_{0}A_{\text{div}}=\mathbb{Z}/p\mathbb{Z}$. To see a more nontrivial example, one may work out the subgroup $A$ of $\mathop{\varinjlim }\nolimits_{n}W_{n}=\{(\ldots ,x_{-2},x_{-1},x_{0})\mid x_{n}=0\text{ for almost all }n\}$ defined by $(F-1)x_{n}=(F-1)^{2}x_{-1}=0$ for all $n\leqslant -2$, where $F$ is the Frobenius. Then $A_{0}\cong \mathbf{G}_{a}$ is defined by $x_{n}=0$ for all $n\leqslant -1$, $A_{\text{div}}\cong \mathbb{Q}_{p}/\mathbb{Z}_{p}$ by $(F-1)x_{n}=0$ for all $n\leqslant 0$, $A_{0+\text{div}}$ by $(F-1)x_{n}=0$ for all $n\leqslant -1$, and $\unicode[STIX]{x1D70B}_{0}A_{/\text{div}}\cong \mathbb{Z}/p\mathbb{Z}$.

Here is a connection diagram between the named subgroups of $A\in \text{I}^{f}\text{Alg}_{\text{uc}}/k$:

All the arrows are inclusions. The label of an arrow means that the subquotient there is of that type. For example, $A_{0+\text{div}}/A_{\text{div}}$ is connected quasi-algebraic. The symbol $\bigoplus _{l}\text{FEt}_{l}$ means that the (sub)quotient there is a direct sum of finite étale $l$-primary groups for primes $l$ (with $l=p$ allowed), which is finite if and only if $A$ is divisibly ML. The upper and lower sides of the square have the same subquotient, and the left and right sides of the square have the same subquotient. The two step subquotient $A_{0}/A_{0\text{div}}$ is connected quasi-algebraic and, similarly, $A_{\text{div}}/A_{0\text{div}}$ is étale divisible. In particular, we have $\unicode[STIX]{x1D70B}_{0}(A_{0+\text{div}})=\unicode[STIX]{x1D70B}_{0}(A)_{\text{div}}$ and $(A_{0+\text{div}})_{/\text{div}}=(A_{/\text{div}})_{0}$. Later we will see in Proposition 3.2.3 that the cohomology object $G$ in any positive degree of the complex $R\boldsymbol{\unicode[STIX]{x1D6E4}}(X,{\mathcal{A}})$ mentioned in Introduction belongs to $\text{I}^{f}\text{Alg}_{\text{uc}}/k$. With some more effort, we will see in Theorem 3.4.1 that $G_{0\text{div}}=0$ (i.e., $G\in \text{L}^{f}\text{Alg}_{\text{uc}}/k$) and $G/G_{0+\text{div}}$ is finite (i.e., $G$ is divisibly ML).

2.3 The ind-rational pro-étale site and some derived limits

We quickly recall some definitions and notation about the ind-rational (pro-)étale site from [Reference SuzukiSuz14, Section 2.1]. We say that a $k$-algebra is rational if it is a finite direct product of perfections (direct limit along Frobenii) of finitely generated fields over $k$, and ind-rational if it is a filtered union of rational $k$-subalgebras. We denote the category of rational (resp. ind-rational) $k$-algebras with $k$-algebra homomorphisms by $k^{\text{rat}}$ (resp. $k^{\text{ind}\text{rat}}$). (More precisely, we only consider ${\mathcal{U}}_{0}$-small ind-rational $k$-algebras. Hence $k^{\text{ind}\text{rat}}$ is a ${\mathcal{U}}$-small ${\mathcal{U}}_{0}$-category.) We can endow $k^{\text{ind}\text{rat}}$ with the étale topology. That is, an étale covering of $k^{\prime }\in k^{\text{ind}\text{rat}}$ is a finite family $\{k_{i}^{\prime }\}$ of étale $k^{\prime }$-algebras such that the product $\prod k_{i}^{\prime }$ is faithfully flat over $k^{\prime }$. The resulting site is the ind-rational étale site $\operatorname{Spec}k_{\text{et}}^{\text{ind}\text{rat}}$. We can also endow $k^{\text{ind}\text{rat}}$ with the pro-étale topology of Bhatt and Scholze [Reference Bhatt and ScholzeBS15]. That is, a pro-étale covering of $k^{\prime }\in k^{\text{ind}\text{rat}}$ is a finite family $\{k_{i}^{\prime }\}$ of $k^{\prime }$-algebras such that each $k_{i}^{\prime }$ is a filtered direct limit of étale $k^{\prime }$-algebras and the product $\prod k_{i}^{\prime }$ is faithfully flat over $k^{\prime }$. The resulting site is the ind-rational pro-étale site $\operatorname{Spec}k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}$. The category of sheaves of sets (resp. abelian groups) on $\operatorname{Spec}k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}$ is denoted by $\operatorname{Set}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ (resp. $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$). (Here, the target categories $\operatorname{Set}$ and $\operatorname{Ab}$ for sheaves are the categories of ${\mathcal{U}}$-small sets and abelian groups.) The category of complexes in $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is denoted by $\operatorname{Ch}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, its homotopy category by $K(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ and its derived category by $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. See [Reference Kashiwara and SchapiraKS06, Chapter 18] for details about unbounded derived categories of sheaves on sites. The cohomology of $k^{\prime }\in k^{\text{ind}\text{rat}}$ with coefficients in $G\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is denoted by $R\unicode[STIX]{x1D6E4}(k_{\text{pro}\text{et}}^{\prime },G)$, with $n$th cohomology $H^{n}(k_{\text{pro}\text{et}}^{\prime },G)$ when $G\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. The sheaf-Hom, $n$th sheaf-Ext and derived sheaf-Hom functors are denoted by $\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}$, $\mathbf{Ext}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}^{n}$ and $R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}$, respectively. Similar notation applies to $\operatorname{Spec}k_{\text{et}}^{\text{ind}\text{rat}}$. For $G\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, we define its Serre dual [Reference SuzukiSuz14, Section 2.4] to be

$$\begin{eqnarray}G^{\text{SD}}=R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(G,\mathbb{Z}).\end{eqnarray}$$

See the list of examples of Serre duals in the paragraph after the proof of [Reference SuzukiSuz14, Proposition (2.4.1)]. A key example there is $\mathbf{G}_{a}^{\text{SD}}\cong \mathbf{G}_{a}[-2]$, which comes from $\mathbf{G}_{a}\overset{{\sim}}{\rightarrow }\mathbf{Ext}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}^{1}(\mathbf{G}_{a},\mathbb{Z}/p\mathbb{Z})$ defined by sending $c\in \mathbf{G}_{a}$ to the pullback of the Artin–Schreier extension class $0\rightarrow \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbf{G}_{a}\rightarrow \mathbf{G}_{a}\rightarrow 0$ by the multiplication-by-$c$ map $\mathbf{G}_{a}\rightarrow \mathbf{G}_{a}$ to the final term. In particular, $\mathbf{G}_{a}$ is Serre reflexive. We say that $G$ is Serre reflexive if the canonical morphism $G\rightarrow G^{\text{SD}\text{SD}}$ is an isomorphism. For $G\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, we denote its torsion part by $G_{\text{tor}}$ and set $G_{/\text{tor}}=G/G_{\text{tor}}$. We say that a sheaf $F\in \operatorname{Set}(k_{\text{et}}^{\text{ind}\text{rat}})$ is locally of finite presentation if it commutes with filtered direct limits as a functor $k^{\text{ind}\text{rat}}\rightarrow \operatorname{Set}$ [Reference SuzukiSuz14, Section 2.4]. In this case, $F$ is automatically a sheaf for the pro-étale topology.

The natural Yoneda functor $\text{Alg}/k\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ extends to an additive functor $\text{I}\text{P}\text{Alg}/k\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. (Here, as before, we consider the ${\mathcal{U}}_{0}$-procategory of $\text{Alg}/k$ and the ${\mathcal{U}}_{0}$-indcategory of the ${\mathcal{U}}_{0}$-category $\text{P}\text{Alg}/k$.) This functor is exact by [Reference SuzukiSuz14, Proposition (2.1.2)(e)], which is fully faithful and induces a fully faithful embedding $D^{b}(\text{I}\text{P}\text{Alg}/k){\hookrightarrow}D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ by [Reference SuzukiSuz14, Proposition (2.3.4)]. If $G\in \text{I}\text{P}\text{Alg}/k$ is an extension of an object of $\text{I}\text{Alg}_{\text{uc}}/k$ by an object of $\text{P}\text{Alg}/k$, then we denote

$$\begin{eqnarray}G^{\text{SD}^{\prime }}=\mathbf{Ext}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}^{1}(G,\mathbb{Q}/\mathbb{Z}).\end{eqnarray}$$

By [Reference SuzukiSuz14, Proposition (2.4.1)(b)], we have $G^{\text{SD}}=G^{\text{SD}^{\prime }}[-2]$. The same proposition shows that if $G$ is connected unipotent proalgebraic or ind-algebraic, then $G^{\text{SD}^{\prime }}$ is connected unipotent ind-algebraic or proalgebraic, respectively, and $G\overset{{\sim}}{\rightarrow }G^{\text{SD}^{\prime }\text{SD}^{\prime }}$. In particular, $G$ is Serre reflexive. By [Reference SuzukiSuz14, Proposition (2.4.1)(b)], if $G$ is semiabelian, then $G^{\text{SD}^{\prime }}$ is the Pontryagin dual of the Tate module $TG$. If $G$ is not necessarily connected, then we denote $G_{0}^{\text{SD}^{\prime }}=(G_{0})^{\text{SD}^{\prime }}$. If $G$ is an extension of a torsion étale group by a profinite-étale group, then its Pontryagin dual is denoted by $G^{\text{PD}}$, which can be given by $\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(G,\mathbb{Q}/\mathbb{Z})$, or by $G^{\text{SD}}[1]$ by [Reference SuzukiSuz14, Proposition (2.4.1)(b)]. For example, if $G=\mathbb{Q}_{p}$, then $G^{\text{PD}}\cong \mathbb{Q}_{p}$. If the torsion part of $G\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is such a group, then we denote $G_{\text{tor}}^{\text{PD}}=(G_{\text{tor}})^{\text{PD}}$. If $G$ is a lattice over $k$, then its dual lattice is denoted by $G^{\text{LD}}$, which can be given by $\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(G,\mathbb{Z})$. If the torsion-free quotient of $G\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is such a group, then we denote $G_{/\text{tor}}^{\text{LD}}=(G_{/\text{tor}})^{\text{LD}}$.

For an abelian (${\mathcal{U}}$-)category, we say that it has exact products if the product $\prod _{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}A_{\unicode[STIX]{x1D706}}$ of any family of objects $\{A_{\unicode[STIX]{x1D706}}\}$ (with $\unicode[STIX]{x1D6EC}\in {\mathcal{U}}$) exists and for any family of surjections $\{A_{\unicode[STIX]{x1D706}}{\twoheadrightarrow}{B_{\unicode[STIX]{x1D706}}\}}_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}}$ (with $\unicode[STIX]{x1D6EC}\in {\mathcal{U}}$), the morphism $\prod A_{\unicode[STIX]{x1D706}}\rightarrow \prod B_{\unicode[STIX]{x1D706}}$ is surjective.

Proposition 2.3.1. The category $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ has enough projectives and exact products. For every sequence

$$\begin{eqnarray}\cdots \overset{\unicode[STIX]{x1D711}_{3}}{\rightarrow }A_{3}\overset{\unicode[STIX]{x1D711}_{2}}{\rightarrow }A_{2}\overset{\unicode[STIX]{x1D711}_{1}}{\rightarrow }A_{1}\end{eqnarray}$$

in $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, its derived inverse limit is represented by the complex $\prod _{n}A_{n}\rightarrow \prod _{n}A_{n}$ in degrees $0$ and $1$, where the morphism sends $(a_{n})_{n}$ to $(a_{n}-\unicode[STIX]{x1D711}_{n}(a_{n+1}))_{n}$. If the sequence is Mittag-Leffler, then its derived inverse limit is zero in positive degrees.

Let $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ be the procategory of $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. (Here we consider the ${\mathcal{U}}$-procategory since $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is a ${\mathcal{U}}$-category.) A filtered inverse system $\{{A_{\unicode[STIX]{x1D706}}\}}_{\unicode[STIX]{x1D706}}$ in $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ as an object of $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is denoted by . The category $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is a Grothendieck category and hence has enough injectives. Therefore the procategory $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ also has enough injectives by [Reference Kashiwara and SchapiraKS06, Exercise 13.6(ii)]. As $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ has exact products (and exact filtered inverse limits), it follows from [Reference Kashiwara and SchapiraKS06, Theorem 14.4.4] that any additive functor from $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ to an abelian category admits a right derived functor on (unbounded) derived categories, which can be calculated by K-injective (or homotopically injective) complexes in $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. In particular, the inverse limit functor $\varprojlim :\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ admits a right derived functor $R\varprojlim :D(\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}))\rightarrow D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. For a more detailed treatment of $R\varprojlim$, see [Reference Kashiwara and SchapiraKS06, Proposition 13.3.15, Corollary 13.3.16, Example 13.3.17(i)].Footnote ¹

For a sheaf $A\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, the multiplication maps by integers $n\geqslant 1$ yield a pro-object  with index set $\mathbb{Z}_{{\geqslant}1}$ ordered by divisibility. Note that there is a cofinal map $\unicode[STIX]{x1D70E}$ from the set of positive integers with the usual ordering ${\leqslant}$ to the set of positive integers with the divisibility ordering. For example, we can take $\unicode[STIX]{x1D70E}(n)$ to be the $n$th power of the product of the first $n$ primes. Hence the pro-object  is isomorphic to the sequence $\cdots \rightarrow A_{\unicode[STIX]{x1D70E}(2)}\rightarrow A_{\unicode[STIX]{x1D70E}(1)}$. The assignment  defines an exact functor $\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})\rightarrow \text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ and hence a triangulated functor $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})\rightarrow D(\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}))$. By composing it with $R\varprojlim$, we have a triangulated endofunctor on $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, which we denote by $A\mapsto R\,\mathop{\varprojlim }\nolimits_{n}A$. The objects  and $R\,\mathop{\varprojlim }\nolimits_{n}A$ are uniquely divisible. We have a projection  to the $n=1$ term in $D(\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}))$. This induces a morphism $R\,\mathop{\varprojlim }\nolimits_{n}A\rightarrow A$ in $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. It is an isomorphism if and only if $A$ is uniquely divisible. We have $R\,\mathop{\varprojlim }\nolimits_{n}A=0$ if $A$ is killed by multiplication by some positive integer.

For $A\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, we define $A_{\text{div}}$ to be the image of the natural morphism $\mathop{\varprojlim }\nolimits_{n}A\rightarrow A$. It is the maximal divisible subsheaf of $A$. We denote $A_{/\text{div}}=A/A_{\text{div}}$. We define

$$\begin{eqnarray}TA=\underset{n}{\varprojlim }\,A[n],\qquad VA=(TA)\otimes \mathbb{Q},\qquad \hat{A}=\underset{n}{\varprojlim }\,(A\otimes \mathbb{Z}/n\mathbb{Z}),\end{eqnarray}$$

where $A[n]$ is the $n$-torsion part of $A$. If $A\in \text{I}^{f}\text{Alg}_{\text{uc}}/k$, then Section 2.1 gives an object $TA\in \text{P}\text{Alg}_{\text{uc}}/k$, while the above definition gives an object$TA\in \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. They are compatible under the Yoneda functor $\text{P}\text{Alg}_{\text{uc}}/k{\hookrightarrow}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. A similar relation holds for $VA$. We denote $TA_{\text{div}}=T(A_{\text{div}})$, $VA_{\text{div}}=V(A_{\text{div}})$. If $A_{\text{div}}$ is a torsion étale group, we denote $A_{\text{div}}^{\text{PD}}=(A_{\text{div}})^{\text{PD}}$. The notation $A_{/\text{div}}^{\wedge }$ means $(A_{/\text{div}})^{\wedge }$. For $A\in \text{FGEt}/k$, we denote $A_{/\text{tor}}^{\text{LD}\wedge }=(A_{/\text{tor}}^{\text{LD}})^{\wedge }$. (The general rule here is that we read the subscript first and then the superscript, from the inside to the outside. Sticking to one such convention reduces excessive usage of parentheses. But we are not strictly consistent with this rule, as the ind-rational pro-étale site $\operatorname{Spec}k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}$ or the open subscheme ${\mathcal{A}}_{0}^{\vee }\subset {\mathcal{A}}^{\vee }$ of the Néron model shows.)

We denote $\mathbb{A}^{\infty }=\hat{\mathbb{Z}}\otimes _{\mathbb{Z}}\mathbb{Q}\in \text{I}\text{P}\text{FEt}/k$ (which is the adele ring of $\mathbb{Q}$ without components of infinite places). We have $\mathbb{A}^{\infty }=V(\mathbb{Q}/\mathbb{Z})$. The assumption that $A$ is divisibly ML in the above proposition cannot be dropped. For instance, applying $R\,\mathop{\varprojlim }\nolimits_{n}$ to the exact sequence $0\rightarrow \bigoplus _{p}\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 0$, we know that $R\,\mathop{\varprojlim }\nolimits_{n}\bigoplus _{p}\mathbb{Z}/p\mathbb{Z}$ is isomorphic to $[\mathbb{A}^{\infty }\rightarrow \mathbb{A}^{\infty }][-1]$, or $[\hat{\mathbb{Z}}\rightarrow \hat{\mathbb{Z}}][-1]\otimes \mathbb{Q}$, or $(\prod _{p}\mathbb{Z}/p\mathbb{Z})/(\bigoplus _{p}\mathbb{Z}/p\mathbb{Z})[-1]$.

For $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, there is a canonical choice of a mapping cone of the above defined morphism . We denote this cone by . It is an object of $D(\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}))$. (This construction can also be understood more conceptually using the derived functor of the procategory extension of the two variable functor $\otimes$, at least when $A$ is assumed to be bounded above.)

Proposition 2.3.5. Let $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. The distinguished triangle

induces an exact sequence

in $\text{P}\operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ for any $i\in \mathbb{Z}$.

Proposition 2.3.6. If a morphism $A\rightarrow B$ in $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ induces an isomorphism

$$\begin{eqnarray}A\otimes ^{L}\mathbb{Z}/n\mathbb{Z}\overset{{\sim}}{\rightarrow }B\otimes ^{L}\mathbb{Z}/n\mathbb{Z}\end{eqnarray}$$

for any $n\geqslant 1$, then it also induces an isomorphism

If $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, we can apply $R\varprojlim$ to . We denote the result by

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}).\end{eqnarray}$$

Proposition 2.3.7. For $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, we have a distinguished triangle

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,A\rightarrow A\rightarrow R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z}).\end{eqnarray}$$

If a morphism $A\rightarrow B$ in $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ induces an isomorphism

$$\begin{eqnarray}A\otimes ^{L}\mathbb{Z}/n\mathbb{Z}\overset{{\sim}}{\rightarrow }B\otimes ^{L}\mathbb{Z}/n\mathbb{Z}\end{eqnarray}$$

for any $n\geqslant 1$, then it also induces an isomorphism

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\overset{{\sim}}{\rightarrow }R\,\underset{n}{\varprojlim }\,(B\otimes ^{L}\mathbb{Z}/n\mathbb{Z}).\end{eqnarray}$$

Proposition 2.3.8. Let $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. Assume that

$$\begin{eqnarray}R^{i}\,\underset{n}{\varprojlim }\,(H^{j}(A)\otimes \mathbb{Z}/n\mathbb{Z})=R^{i}\,\underset{n}{\varprojlim }\,(H^{j}(A)[n])=0\end{eqnarray}$$

for any $i\geqslant 1$ and $j\in \mathbb{Z}$. Then we have an exact sequence

$$\begin{eqnarray}0\rightarrow H^{i}(A)^{\wedge }\rightarrow H^{i}R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\rightarrow TH^{i+1}(A)\rightarrow 0.\end{eqnarray}$$

for all $i\in \mathbb{Z}$.

Proposition 2.3.9. For any $A\in D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, the natural morphism

$$\begin{eqnarray}\Bigl(R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\Bigr)\otimes \mathbb{Q}\rightarrow R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Q}/\mathbb{Z})\end{eqnarray}$$

is an isomorphism. Denote these isomorphic objects by $A\,\hat{\otimes }\,\mathbb{A}^{\infty }$. We have a natural distinguished triangle

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,A\rightarrow A\otimes \mathbb{Q}\rightarrow A\,\hat{\otimes }\,\mathbb{A}^{\infty }.\end{eqnarray}$$

The assignment $A\mapsto A\,\hat{\otimes }\,\mathbb{A}^{\infty }$ is a triangulated endofunctor on $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ that sends $\mathbb{Z}$ to $\mathbb{A}^{\infty }$.

Proof. We have a distinguished triangle

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,A\rightarrow A\rightarrow R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\end{eqnarray}$$

and hence a distinguished triangle

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,A\rightarrow A\otimes \mathbb{Q}\rightarrow \Bigl(R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})\Bigr)\otimes \mathbb{Q}.\end{eqnarray}$$

Also we have a distinguished triangle

$$\begin{eqnarray}A\rightarrow A\otimes \mathbb{Q}\rightarrow A\otimes ^{L}\mathbb{Q}/\mathbb{Z}\end{eqnarray}$$

and hence a distinguished triangle

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,A\rightarrow A\otimes \mathbb{Q}\rightarrow R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Q}/\mathbb{Z}).\end{eqnarray}$$

Comparing the two triangles, we get the result. ◻

The boundedness conditions in the following two propositions might be unnecessary.

Proposition 2.3.10. For $A\in D^{+}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, there is a canonical isomorphism

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z})=R\,\mathbf{Hom}_{k}(\mathbb{Q}/\mathbb{Z},A)[1].\end{eqnarray}$$

Proof. We have

$$\begin{eqnarray}R\,\mathbf{Hom}_{k}(\mathbb{Q},A)=R\,\underset{n}{\varprojlim }\,R\,\mathbf{Hom}_{k}(\mathbb{Z},A)=R\,\underset{n}{\varprojlim }\,A\end{eqnarray}$$

by the proof of [Reference SuzukiSuz14, Proposition (2.2.3)] (which needs the bounded below condition). Comparing the two distinguished triangles

$$\begin{eqnarray}\displaystyle & \displaystyle R\,\underset{n}{\varprojlim }\,A\rightarrow A\rightarrow R\,\underset{n}{\varprojlim }\,(A\otimes ^{L}\mathbb{Z}/n\mathbb{Z}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\,\mathbf{Hom}_{k}(\mathbb{Q},A)\rightarrow R\,\mathbf{Hom}_{k}(\mathbb{Z},A)\rightarrow R\,\mathbf{Hom}_{k}(\mathbb{Q}/\mathbb{Z},A)[1], & \displaystyle \nonumber\end{eqnarray}$$

we get the result. ◻

Proposition 2.3.11. For $A\in D^{-}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$ and $B\in D^{+}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$, there is a canonical isomorphism

$$\begin{eqnarray}R\,\underset{n}{\varprojlim }(R\,\mathbf{Hom}_{k}(A,B)\otimes ^{L}\mathbb{Z}/n\mathbb{Z})=R\,\mathbf{Hom}_{k}(A\otimes ^{L}\mathbb{Q}/\mathbb{Z},B)[1].\end{eqnarray}$$

Proof. We know that $R\,\mathbf{Hom}_{k}(A,B)$ is bounded below. By the previous proposition, the left-hand side can be written as

$$\begin{eqnarray}R\,\mathbf{Hom}_{k}(\mathbb{Q}/\mathbb{Z},R\,\mathbf{Hom}_{k}(A,B))[1].\end{eqnarray}$$

This is also the right-hand side by the derived tensor-Hom adjunction. ◻

2.4 Premorphisms of sites

For a site $S$, we denote its category of sheaves of sets (resp. abelian groups) by $\operatorname{Set}(S)$ (resp. $\operatorname{Ab}(S)$). The category of complexes in $\operatorname{Ab}(S)$ is $\operatorname{Ch}(S)$, its homotopy category $K(S)$ and derived category $D(S)$. We say $A\in \operatorname{Ch}(S)$ is K-limp if $R\unicode[STIX]{x1D6E4}(X,A)=\unicode[STIX]{x1D6E4}(X,A)$ for any object $X\in S$, where $\unicode[STIX]{x1D6E4}$ in the right-hand side is applied termwise. See also [Reference SpaltensteinSpa88, Corollary 5.17] and [Reference SchnürerSch17, Appendix A]. This is not a “K”-version of limp sheaf as defined in [Sta18, Tag 072Y] since $X$ is not allowed to be an arbitrary sheaf of sets but only a representable one. K-injectives are K-limp.

The following three propositions are essentially well known, at least for bounded below complexes of sheaves or just sheaves; see [Reference MilneMil80, III, Sections 1, 2].

Proof. Note that $\mathbf{Hom}_{S}(A,J)=R\,\mathbf{Hom}_{S}(A,J)$. By [Reference Kashiwara and SchapiraKS06, Proposition 18.6.6], we have $R\unicode[STIX]{x1D6E4}(X,R\,\mathbf{Hom}_{S}(A,J))=R\operatorname{Hom}_{S/X}(A,J)$ for any $X\in S$, where $S/X$ is the localization of $S$ at $X$. Hence

$$\begin{eqnarray}\displaystyle & & \displaystyle R\unicode[STIX]{x1D6E4}(X,\mathbf{Hom}_{S}(A,J))=R\operatorname{Hom}_{S/X}(A,J)\nonumber\\ \displaystyle \hspace{74.39996pt} & & \displaystyle \quad =\operatorname{Hom}_{S/X}(A,J)=\unicode[STIX]{x1D6E4}(X,\mathbf{Hom}_{S}(A,J)).\hspace{48.0pt}\square\nonumber\end{eqnarray}$$

Let $f^{-1}$ be a functor from the underlying category of a site $S$ to the underlying category of another site $S^{\prime }$. The right composition with $f^{-1}$ defines a functor from the category of presheaves of sets on $S^{\prime }$ to the category of presheaves of sets on $S$, which is the pushforward functor for presheaves. If this functor sends sheaves to sheaves, $f^{-1}$ is called a continuous functor in the terminology of [AGV72, III, Definition 1.1]. We say that $f^{-1}$ then defines a continuous map of sites $f:S^{\prime }\rightarrow S$. The pushforward functor $\operatorname{Set}(S^{\prime })\rightarrow \operatorname{Set}(S)$ or $\operatorname{Ab}(S^{\prime })\rightarrow \operatorname{Ab}(S)$ is denoted by $f_{\ast }$. The left adjoint to $f_{\ast }:\operatorname{Set}(S^{\prime })\rightarrow \operatorname{Set}(S)$ exists by [AGV72, III, Proposition 1.2], which we denote by $f^{\ast \text{set}}:\operatorname{Set}(S)\rightarrow \operatorname{Set}(S^{\prime })$. The left adjoint to $f_{\ast }:\operatorname{Ab}(S^{\prime })\rightarrow \operatorname{Ab}(S)$ exists by [AGV72, III, Proposition 1.7], which we denote by $f^{\ast }:\operatorname{Set}(S)\rightarrow \operatorname{Set}(S^{\prime })$. They are called pullback functors. The continuous map $f:S^{\prime }\rightarrow S$ is called a morphism of sites if the pullback $f^{\ast \text{set}}:\operatorname{Set}(S)\rightarrow \operatorname{Set}(S^{\prime })$ for sheaves of sets is exact. In this case, the pullback for sheaves of abelian groups $f^{\ast }:\operatorname{Ab}(S)\rightarrow \operatorname{Ab}(S^{\prime })$ and $f^{\ast \text{set}}$ are compatible with forgetting group structures by [AGV72, III, Proposition 1.7, 4)], so we do not have to distinguish $f^{\ast }$ and $f^{\ast \text{set}}$. In general, we use $f^{\ast }$ to mean the pullback functor for sheaves of abelian groups.

If $S$ and $S^{\prime }$ are sites defined by pretopologies, and if $f^{-1}$ is a functor from the underlying category of $S$ to that of $S^{\prime }$ that sends coverings to coverings and $f^{-1}(Y\times _{X}Z)=f^{-1}(Y)\times _{f^{-1}(X)}f^{-1}(Z)$ whenever $Y\rightarrow X$ appears in a covering family, then $f^{-1}$ is called a morphism of topologies in the terminology of [Reference ArtinArt62, Definition 2.4.2] and defines a continuous map $f:S^{\prime }\rightarrow S$. We call such a continuous map a premorphism of sites. In this case, $f_{\ast }$ sends acyclic sheaves (i.e., those $A^{\prime }$ with $H^{n}(X^{\prime },A^{\prime })=0$ for any object $X^{\prime }$ of $S^{\prime }$ and $n\geqslant 1$) to acyclic sheaves and hence induces the Leray spectral sequence $R\unicode[STIX]{x1D6E4}(X,Rf_{\ast }A^{\prime })=R\unicode[STIX]{x1D6E4}(f^{-1}(X),A^{\prime })$ for any $X\in S$ and $A^{\prime }\in \operatorname{Ab}(S^{\prime })$ [Reference ArtinArt62, Section II.4]. Here is a slight generalization.

Proposition 2.4.3. Let $f:S^{\prime }\rightarrow S$ be a premorphism of sites defined by pretopologies. Then $f_{\ast }$ sends K-limp complexes to K-limp complexes. For any $X\in S$ and $A^{\prime }\in D(S^{\prime })$, we have an isomorphism

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}(X,Rf_{\ast }A^{\prime })=R\unicode[STIX]{x1D6E4}(f^{-1}X,A^{\prime }),\end{eqnarray}$$

where $f^{-1}:S\rightarrow S^{\prime }$ is the underlying functor of $f$.

Proof. By [Reference SuzukiSuz13, Lemma 3.7.2], we know that $f^{\ast }:\operatorname{Ab}(S)\rightarrow \operatorname{Ab}(S^{\prime })$ admits a left derived functor $Lf^{\ast }:D(S)\rightarrow D(S^{\prime })$, which is left adjoint to $Rf_{\ast }:D(S^{\prime })\rightarrow D(S)$ and sends the sheaf $\mathbb{Z}[X]$ of free abelian groups generated by any $X\in S$ to $\mathbb{Z}[f^{-1}X]$. For any $A^{\prime }\in \operatorname{Ch}(S^{\prime })$ and $X\in S$, we have

$$\begin{eqnarray}\displaystyle R\unicode[STIX]{x1D6E4}(X,Rf_{\ast }A^{\prime }) & = & \displaystyle R\operatorname{Hom}_{S}(\mathbb{Z}[X],Rf_{\ast }A^{\prime })=R\operatorname{Hom}_{S^{\prime }}(Lf^{\ast }\mathbb{Z}[X],A^{\prime })\nonumber\\ \displaystyle & = & \displaystyle R\operatorname{Hom}_{S^{\prime }}(\mathbb{Z}[f^{-1}X],A^{\prime })=R\unicode[STIX]{x1D6E4}(f^{-1}X,A^{\prime }).\nonumber\end{eqnarray}$$

If $A^{\prime }$ is K-limp, then $Rf_{\ast }A^{\prime }=f_{\ast }A^{\prime }$ by Proposition 2.4.2 and $R\unicode[STIX]{x1D6E4}(f^{-1}X,A^{\prime })=\unicode[STIX]{x1D6E4}(f^{-1}X,A^{\prime })$. Hence $R\unicode[STIX]{x1D6E4}(X,f_{\ast }A^{\prime })=\unicode[STIX]{x1D6E4}(f^{-1}X,A^{\prime })=\unicode[STIX]{x1D6E4}(X,f_{\ast }A^{\prime })$, and so $f_{\ast }A^{\prime }$ is K-limp.◻

2.5 Local duality without relative sites

Let $\hat{K}_{x}$ be a complete discrete valuation field with perfect residue field $k_{x}$ ($\in {\mathcal{U}}_{0}$) of characteristic $p>0$. The ring of integers is denoted by $\hat{{\mathcal{O}}}_{x}$ with maximal ideal $\hat{\mathfrak{p}}_{x}$. By the fppf site $\operatorname{Spec}\hat{{\mathcal{O}}}_{x,\text{fppf}}$ of $\hat{{\mathcal{O}}}_{x}$, we mean the category of (${\mathcal{U}}_{0}$-small) $\hat{{\mathcal{O}}}_{x}$-algebras endowed with the fppf topology. The same applies to the fppf site $\operatorname{Spec}\hat{K}_{x,\text{fppf}}$ of $\hat{K}_{x}$. Sheaves take values in $\operatorname{Set}$ or $\operatorname{Ab}$ (of ${\mathcal{U}}$-small sets or abelian groups). The Hom functor and the sheaf-Hom functor for $\operatorname{Spec}\hat{{\mathcal{O}}}_{x,\text{fppf}}$ are denoted by $\operatorname{Hom}_{\hat{{\mathcal{O}}}_{x}}$ and $\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}$, respectively. Their derived functors are denoted by $\operatorname{Ext}_{\hat{{\mathcal{O}}}_{x}}^{n}$, $\mathbf{Ext}_{\hat{{\mathcal{O}}}_{x}}^{n}$, $R\operatorname{Hom}_{\hat{{\mathcal{O}}}_{x}}$, $R\,\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}$. Similar notation applies to $\operatorname{Spec}\hat{K}_{x,\text{fppf}}$. We denote $k_{x}^{\text{ind}\text{rat}}=(k_{x})^{\text{ind}\text{rat}}$, $\operatorname{Spec}k_{x,\text{et}}^{\text{ind}\text{rat}}=\operatorname{Spec}(k_{x})_{\text{et}}^{\text{ind}\text{rat}}$ and $\operatorname{Spec}k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}=\operatorname{Spec}(k_{x})_{\text{pro}\text{et}}^{\text{ind}\text{rat}}$.

The ring $\hat{{\mathcal{O}}}_{x}$ has a natural structure of a $W(k_{x})$-algebra of profinite length, where $W$ denotes the ring of $p$-typical Witt vectors of infinite length. In equal characteristic, this structure factors through $k_{x}$ so that $\hat{{\mathcal{O}}}_{x}$ is profinite length over $k_{x}$. In mixed characteristic, it is finite free over $W(k_{x})$. For $k_{x}^{\prime }\in k_{x}^{\text{ind}\text{rat}}$, we define

$$\begin{eqnarray}\displaystyle & \displaystyle \hat{\mathbf{O}}_{x}(k_{x}^{\prime })=W(k_{x}^{\prime })\hat{\otimes }_{W(k_{x})}\hat{{\mathcal{O}}}_{x}=\underset{n}{\varprojlim }(W_{n}(k_{x}^{\prime })\otimes _{W_{n}(k_{x})}\hat{{\mathcal{O}}}_{x}/\hat{\mathfrak{p}}_{x}^{n}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hat{\mathbf{K}}_{x}(k_{x}^{\prime })=\hat{\mathbf{O}}_{x}(k_{x}^{\prime })\otimes _{\hat{{\mathcal{O}}}_{x}}\hat{K}_{x}. & \displaystyle \nonumber\end{eqnarray}$$

Proposition 2.5.1. The functors $k_{x}^{\prime }\mapsto \hat{\mathbf{O}}_{x}(k_{x}^{\prime }),\hat{\mathbf{K}}_{x}(k_{x}^{\prime })$ define premorphisms of sites

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}}:\operatorname{Spec}\hat{{\mathcal{O}}}_{x,\text{fppf}}\rightarrow \operatorname{Spec}k_{x,\text{et}}^{\text{ind}\text{rat}}, & & \displaystyle \nonumber\\ \displaystyle \unicode[STIX]{x1D70B}_{\hat{K}_{x}}:\operatorname{Spec}\hat{K}_{x,\text{fppf}}\rightarrow \operatorname{Spec}k_{x,\text{et}}^{\text{ind}\text{rat}}. & & \displaystyle \nonumber\end{eqnarray}$$

The scheme morphism $j:\operatorname{Spec}\hat{K}_{x}{\hookrightarrow}\operatorname{Spec}\hat{{\mathcal{O}}}_{x}$ defines a morphism of sites

$$\begin{eqnarray}j:\operatorname{Spec}\hat{K}_{x,\text{fppf}}\rightarrow \operatorname{Spec}\hat{{\mathcal{O}}}_{x,\text{fppf}}.\end{eqnarray}$$

Its pullback functor $j^{\ast }$ is the restriction functor for sheaves from $\hat{{\mathcal{O}}}_{x}$ to $\hat{K}_{x}$. This restriction functor is frequently omitted by abuse of notation, so a sheaf (or a complex of sheaves) $F$ on $\operatorname{Spec}\hat{{\mathcal{O}}}_{x,\text{fppf}}$ restricted to $\operatorname{Spec}\hat{K}_{x,\text{fppf}}$ will frequently be written by just $F$. More generally, if $f:Z\rightarrow Y$ is a morphism of schemes, then the restriction $f^{\ast }F$ of a sheaf (or a complex of sheaves) $F$ on $Y_{\text{fppf}}$ will frequently be written by just $F$. But the important point is that $f$ is a localization morphism [AGV72, III, Section 5]. Recall from [AGV72, III, Section 5] that for a site $S$ and its object $V$, the category $S/V$ of pairs $(W,g)$ ($W$ an object of $S$ and $g:W\rightarrow V$ a morphism in $S$) is equipped with the natural induced topology. The functor $j_{V}:(W,g)\mapsto W$ defines a continuous map of sites $S\rightarrow S/V$ whose pushforward functor $\operatorname{Set}(S)\rightarrow \operatorname{Set}(S/V)$ is the restriction functor. In our situation, by $f$ being a localization morphism, we mean that $f:Z\rightarrow Y$ belongs to the underlying category of $Y_{\text{fppf}}$ and $Z_{\text{fppf}}$ can be identified with $Y_{\text{fppf}}/Z$. Hence $f^{\ast }$ admits an exact left adjoint $f_{!}:\operatorname{Ab}(Z_{\text{fppf}})\rightarrow \operatorname{Ab}(Y_{\text{fppf}})$ by [AGV72, IV, Proposition 11.3.1]. Hence $f^{\ast }$ send (K-)injectives to (K-)injectives. This will be needed to pass from sheaf or complex level statements to derived categorical statements.

We have the pushforward functors

$$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}})_{\ast }:\operatorname{Ab}(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D70B}_{\hat{K}_{x}})_{\ast }:\operatorname{Ab}(\hat{K}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}}) & \displaystyle \nonumber\end{eqnarray}$$

by $\unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}}$ and $\unicode[STIX]{x1D70B}_{\hat{K}_{x}}$. Let $\operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}})\rightarrow \operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$ be the pro-étale sheafification functor. The composite functors of these functors are denoted by

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;):\operatorname{Ab}(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\;\cdot \;):\operatorname{Ab}(\hat{K}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}). & \displaystyle \nonumber\end{eqnarray}$$

We have the right derived functors

$$\begin{eqnarray}\displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;):D(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\;\cdot \;):D(\hat{K}_{x,\text{fppf}})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}). & \displaystyle \nonumber\end{eqnarray}$$

Since sheafification is exact, $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ is the composite of $R(\unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}})_{\ast }$ and the pro-étale sheafification functor $D(k_{x,\text{et}}^{\text{ind}\text{rat}})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$. The same is true for $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\;\cdot \;)$. We denote $\mathbf{H}^{n}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)=H^{n}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ and similarly $\mathbf{H}^{n}(\hat{K}_{x},\;\cdot \;)$. By Proposition A.1, these functors are compatible with the functors $R\tilde{\boldsymbol{\unicode[STIX]{x1D6E4}}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ and $R\tilde{\boldsymbol{\unicode[STIX]{x1D6E4}}}(\hat{K}_{x},\;\cdot \;)$ in the notation of [Reference SuzukiSuz14, paragraph before Proposition (3.3.8)].

The pro-étale sheafification in general makes it hard to calculate the derived global section $R\unicode[STIX]{x1D6E4}(k_{x,\text{pro}\text{et}}^{\prime },\;\cdot \;)$ of the objects $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},G)$ and $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},G)$ at each $k_{x}^{\prime }\in k_{x}^{\text{ind}\text{rat}}$ for a general complex $G$. In general, it is not clear whether the natural morphisms

$$\begin{eqnarray}\displaystyle & \displaystyle R\unicode[STIX]{x1D6E4}(\hat{{\mathcal{O}}}_{x},G)\rightarrow R\unicode[STIX]{x1D6E4}(k_{x,\text{pro}\text{et}},R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},G)) & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\unicode[STIX]{x1D6E4}(\hat{K}_{x},G)\rightarrow R\unicode[STIX]{x1D6E4}(k_{x,\text{pro}\text{et}},R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},G)) & \displaystyle \nonumber\end{eqnarray}$$

are isomorphisms or not. The first (resp. second) morphism is an isomorphism if $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},G)$ (resp. $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},G)$) is P-acyclic in the sense of [Reference SuzukiSuz14, Section 2.4]. This condition is satisfied for $G$ a smooth group scheme or a finite flat group scheme by [Reference SuzukiSuz14, Propositions (3.4.2), (3.4.3)]. This is why the notion of P-acyclicity is introduced in [Reference SuzukiSuz14, Section 2.4]. The following proposition shows we can eliminate P-acyclicity from [Reference SuzukiSuz14] to a certain extent.

Proposition 2.5.2. Let $G\in D(\hat{{\mathcal{O}}}_{x,\text{fppf}})$. For any w-contractible $k_{x}^{\prime }\in k_{x}^{\text{ind}\text{rat}}$ ([Reference Bhatt and ScholzeBS15, Definition 2.4.1], which includes any algebraically closed field over $k_{x}$), we have

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}(k_{x,\text{pro}\text{et}}^{\prime },R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},G))=R\unicode[STIX]{x1D6E4}(\hat{\mathbf{O}}_{x}(k_{x}^{\prime }),G).\end{eqnarray}$$

A similar relation holds with $\hat{{\mathcal{O}}}_{x}$ replaced by $\hat{K}_{x}$.

Proof. We have

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}(k_{x,\text{et}}^{\prime },R(\unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}})_{\ast }G)=R\unicode[STIX]{x1D6E4}(\hat{\mathbf{O}}_{x}(k_{x}^{\prime }),G)\end{eqnarray}$$

by Proposition 2.4.3. Hence it is enough to show that $R\unicode[STIX]{x1D6E4}(k_{x,\text{pro}\text{et}}^{\prime },\tilde{F})=R\unicode[STIX]{x1D6E4}(k_{x,\text{et}}^{\prime },F)$ for any $F\in D(k_{\text{et}}^{\text{ind}\text{rat}})$, where $\tilde{F}$ is the pro-étale sheafification of $F$. The section functor $\unicode[STIX]{x1D6E4}(k_{x}^{\prime },\;\cdot \;)$ is exact on $\operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$ and $\operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}})$ since $k_{x}^{\prime }$ is w-contractible. Hence it is enough to show that $\unicode[STIX]{x1D6E4}(k_{x}^{\prime },\tilde{F})=\unicode[STIX]{x1D6E4}(k_{x}^{\prime },F)$ for any $F\in \operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}})$. Sheafification is given by applying the zeroth Čech cohomology presheaf functor $\text{}\underline{\check{H}}^{0}$ twice [Reference MilneMil80, III, Remark 2.2(c)]. Any pro-étale covering of a w-contractible scheme has a section by definition and the Čech complex for a covering with section is null-homotopic. Hence for any presheaf $F$ on $\operatorname{Spec}k_{x,\text{pro}\text{et}}^{\prime }$, we have

$$\begin{eqnarray}\hspace{72.60004pt}\unicode[STIX]{x1D6E4}(k_{x}^{\prime },\text{}\underline{\check{H}}^{0}\text{}\underline{\check{H}}^{0}(F))=\unicode[STIX]{x1D6E4}(k_{x}^{\prime },\text{}\underline{\check{H}}^{0}(F))=\unicode[STIX]{x1D6E4}(k_{x}^{\prime },F).\hspace{44.64001pt}\square\end{eqnarray}$$

We define a functor by the mapping fiber construction

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\;\cdot \;):=[\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)\rightarrow \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},j^{\ast }\;\cdot \;)][-1]:\operatorname{Ch}(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow \operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})\end{eqnarray}$$

on the category of complexes of sheaves. This is an additive functor that commutes with the translation functors, that is, a functor of additive categories with translation in the terminology of [Reference Kashiwara and SchapiraKS06, Definition 10.1.1]. It induces a functor on the homotopy categories by [Reference Kashiwara and SchapiraKS06, Proposition 11.2.9]. We have its right derived functor

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)=[R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},j^{\ast }\;\cdot \;)][-1]:D(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})\end{eqnarray}$$

by [Reference Kashiwara and SchapiraKS06, Theorem 14.3.1(vi)]. We set $\mathbf{H}_{x}^{n}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)=H^{n}R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$. Note that $\mathbf{H}_{x}^{0}\neq \boldsymbol{\unicode[STIX]{x1D6E4}}_{x}$ in this definition. (It can be shown that they define the same right derived functors by the same method as [Reference SuzukiSuz14, Proposition (3.3.3)].) By Proposition A.2, this is compatible with the functor denoted by $R\tilde{\boldsymbol{\unicode[STIX]{x1D6E4}}}_{x}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ in [Reference SuzukiSuz14, paragraph before Proposition (3.3.8)]. We frequently omit the restriction functor $j^{\ast }$ from this type of formulas by abuse of notation.

Let $A,B\in \operatorname{Ch}(\hat{{\mathcal{O}}}_{x,\text{fppf}})$. For any $k_{x}^{\prime }\in k_{x}^{\text{ind}\text{rat}}$, the functoriality of $\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ gives a morphism

$$\begin{eqnarray}\operatorname{Hom}_{\hat{\mathbf{O}}_{x}(k_{x}^{\prime })}(A,B)\rightarrow \operatorname{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}/k_{x}^{\prime }}(\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},A),\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},B))\end{eqnarray}$$

in $\operatorname{Ch}(\operatorname{Ab})$, where $\operatorname{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}/k_{x}^{\prime }}$ is the Hom functor for the category of sheaves on the localization of $\operatorname{Spec}k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}$ at $k_{x}^{\prime }$. This is functorial on $k_{x}^{\prime }$, so we have a morphism

(2.5.1)$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}(A,B))\rightarrow \mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},A),\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},B))\end{eqnarray}$$

in $\operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$. The right derived functor of the left-hand side (as a functor on $A,B$) is

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},R\,\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}(A,B))\end{eqnarray}$$

by Propositions 2.4.1, 2.4.2 and the theorem on derived functors of composition [Reference Kashiwara and SchapiraKS06, Proposition 10.3.5(ii)]. Hence we have a morphism

(2.5.2)$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},R\,\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}(A,B))\rightarrow R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},A),R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},B))\end{eqnarray}$$

in $D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$, functorial on $A,B\in D(\hat{{\mathcal{O}}}_{x,\text{fppf}})$, by the universal property of the right derived functor. (Note that the right-hand side of this is not the right derived functor of the right-hand side of (2.5.1). The problem is that $\unicode[STIX]{x1D70B}_{\hat{{\mathcal{O}}}_{x}}$ is only a premorphism of sites and hence its pushforward functor might not send (K-)injectives to (K-)injectives. Sheafification also might not send (K-)injectives to (K-)injectives.) Similarly, we have a morphism

(2.5.3)$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}(A,B))\rightarrow \mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},A),\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},B))\end{eqnarray}$$

in $\operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$, functorial on $A,B\in \operatorname{Ch}(\hat{{\mathcal{O}}}_{x,\text{fppf}})$. Deriving, we have a morphism

(2.5.4)$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},R\,\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}(A,B))\rightarrow R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},A),R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},B))\end{eqnarray}$$

in $D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$, functorial on $A,B\in D(\hat{{\mathcal{O}}}_{x,\text{fppf}})$. Also we have a morphism

(2.5.5)$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\mathbf{Hom}_{\hat{K}_{x}}(A,B))\rightarrow \mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A),\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},B))\end{eqnarray}$$

in $\operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$, functorial on $A,B\in \operatorname{Ch}(\hat{K}_{x,\text{fppf}})$. Deriving, we have a morphism

(2.5.6)$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},R\,\mathbf{Hom}_{\hat{K}_{x}}(A,B))\rightarrow R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A),R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},B))\end{eqnarray}$$

in $D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$, functorial on $A,B\in D(\hat{K}_{x,\text{fppf}})$. These morphisms are compatible with the morphisms in [Reference SuzukiSuz14, Proposition (3.3.8)] by Proposition A.3.

With the comparison results in Appendix A, we can translate the results of [Reference SuzukiSuz14] to our setting. In particular, by [Reference SuzukiSuz14, Propositions (3.4.2)(a), (3.4.3)(e); Section 4.1], any term in the localization triangle

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\mathbf{G}_{m})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\mathbf{G}_{m})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\mathbf{G}_{m})[1]\end{eqnarray}$$

is concentrated in degree zero, where we have an exact sequence

$$\begin{eqnarray}0\rightarrow \hat{\mathbf{O}}_{x}^{\times }\rightarrow \hat{\mathbf{K}}_{x}^{\times }\rightarrow \mathbb{Z}\rightarrow 0.\end{eqnarray}$$

We call the (iso)morphisms

(2.5.7)$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\mathbf{G}_{m})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\mathbf{G}_{m})[1]=\mathbb{Z}\end{eqnarray}$$

the trace (iso)morphisms (at $x$).

Let $A$ be an abelian variety over $\hat{K}_{x}$ with dual $A^{\vee }$. By the Barsotti–Weil formula [Reference OortOor66, Chapter III, Theorem (18.1)], we have a canonical isomorphism $A^{\vee }\cong \mathbf{Ext}_{\hat{K}_{x}}^{1}(A,\mathbf{G}_{m})$ defined by the Poincaré bundle. With $\mathbf{Hom}_{\hat{K}_{x}}(A,\mathbf{G}_{m})=0$, we have a canonical morphism $A^{\vee }\rightarrow R\,\mathbf{Hom}_{\hat{K}_{x}}(A,\mathbf{G}_{m})[1]$ in $D(\hat{K}_{x,\text{fppf}})$. Hence we have a morphism

$$\begin{eqnarray}\displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A^{\vee }) & \rightarrow & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},R\,\mathbf{Hom}_{\hat{K}_{x}}(A,\mathbf{G}_{m}))[1]\nonumber\\ \displaystyle & \rightarrow & \displaystyle R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A),R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},\mathbf{G}_{m}))[1]\nonumber\\ \displaystyle & \rightarrow & \displaystyle R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A),\mathbb{Z})[1]=R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)^{\text{SD}_{x}}[1]\nonumber\end{eqnarray}$$

by (2.5.6) and (2.5.7), where $\text{SD}_{x}=R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\;\cdot \;,\mathbb{Z})$. Its Serre dual, when $A$ is replaced by $A^{\vee }$, is denoted by

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{A}:R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A^{\vee })^{\text{SD}_{x}\text{SD}_{x}}\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)^{\text{SD}_{x}}[1].\end{eqnarray}$$

These morphisms agree with the morphisms defined in [Reference SuzukiSuz14, Section 4.1]. Hence [Reference SuzukiSuz14, Theorem (4.1.2)] implies that $\unicode[STIX]{x1D703}_{A}$ is an isomorphism.

Proposition 2.5.3. Let $A$ be an abelian variety over $\hat{K}_{x}$ with Néron model ${\mathcal{A}}$. Let ${\mathcal{A}}_{0}$ be the maximal open subgroup scheme of ${\mathcal{A}}$ with connected fibers and ${\mathcal{A}}_{x}$ the special fiber of ${\mathcal{A}}$ over $x=\operatorname{Spec}k_{x}$. Then $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})$ is concentrated in degree $0$; $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})$ in degree $0$; $R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})$ in degree $2$; and $R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})$ in degrees $1,2$. We have

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})=\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)\in \text{P}\text{Alg}/k_{x},\qquad \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})=\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)_{0}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbf{H}_{x}^{1}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})=\unicode[STIX]{x1D70B}_{0}({\mathcal{A}}_{x})\in \text{FEt}/k_{x}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbf{H}_{x}^{2}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})=\mathbf{H}_{x}^{2}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})=\mathbf{H}^{1}(\hat{K}_{x},A)\in \text{I}\text{Alg}_{\text{uc}}/k_{x}. & \displaystyle \nonumber\end{eqnarray}$$

The isomorphic groups in the third line are divisible.

Proof. The group $\mathbf{H}^{1}(\hat{K}_{x},A)$ is divisible since $\mathbf{H}^{2}(\hat{K}_{x},A[n])=0$ for any $n\geqslant 1$ by [Reference SuzukiSuz14, Proposition (3.4.3)(b)]. The statements about $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})$, $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0})$ follow from [Reference SuzukiSuz14, Propositions (3.4.2)(a), (3.4.3)(d)]. These propositions, the localization triangle

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)\end{eqnarray}$$

and the similar triangle for ${\mathcal{A}}_{0}$ imply the rest of the statements.◻

We recall [Reference SuzukiSuz14, Remark (4.2.10)]. Let ${\mathcal{A}}$ and ${\mathcal{A}}^{\vee }$ be the Néron models of $A$ and $A^{\vee }$, respectively. Let ${\mathcal{A}}_{0}^{\vee }$ be the maximal open subgroup scheme of ${\mathcal{A}}^{\vee }$ with connected fibers. The Poincaré biextension $A^{\vee }\otimes ^{L}A\rightarrow \mathbf{G}_{m}[1]$ as a morphism in $D(\hat{K}_{x,\text{fppf}})$ canonically extends to a biextension ${\mathcal{A}}_{0}^{\vee }\otimes ^{L}{\mathcal{A}}\rightarrow \mathbf{G}_{m}[1]$ as a morphism in $D(\hat{{\mathcal{O}}}_{x,\text{fppf}})$ by [Gro72, IX, 1.4.3]. Hence we have a morphism ${\mathcal{A}}_{0}^{\vee }\rightarrow R\,\mathbf{Hom}_{\hat{{\mathcal{O}}}_{x}}({\mathcal{A}},\mathbf{G}_{m}[1])$. With the functoriality morphism (2.5.3) and the trace isomorphism (2.5.7), we have morphisms

$$\begin{eqnarray}\displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}_{0}^{\vee }) & \rightarrow & \displaystyle R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}),R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},\mathbf{G}_{m}[1]))\nonumber\\ \displaystyle & = & \displaystyle R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}}),\mathbb{Z})=R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},{\mathcal{A}})^{\text{SD}_{x}}.\nonumber\end{eqnarray}$$

Proposition 2.5.4. The diagram

is a morphism of distinguished triangles. Applying $\text{SD}_{x}\text{SD}_{x}$, the induced diagram

is an isomorphism of distinguished triangles.

Assume that $k_{x}$ is a finite extension of another perfect field $k$. We have a finite étale morphism $f_{x}:\operatorname{Spec}k_{x}\rightarrow \operatorname{Spec}k$. This induces a morphism of sites

$$\begin{eqnarray}f_{x}:\operatorname{Spec}k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}\rightarrow \operatorname{Spec}k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}\end{eqnarray}$$

Its pushforward functor is the Weil restriction functor $\operatorname{Res}_{k_{x}/k}$, which is exact [Reference Bhatt and ScholzeBS15, Lemma 6.1.17]. We denote the composites of $\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$, $R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},\;\cdot \;)$ and $(f_{x})_{\ast }=\operatorname{Res}_{k_{x}/k}$ by

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x}/k,\;\cdot \;):\operatorname{Ab}(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x}/k,\;\cdot \;):D(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\end{eqnarray}$$

respectively, with cohomology $\mathbf{H}^{n}(\hat{{\mathcal{O}}}_{x}/k,\;\cdot \;)=H^{n}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x}/k,\;\cdot \;)$. Similar functors

$$\begin{eqnarray}\displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,\;\cdot \;):\operatorname{Ab}(\hat{K}_{x,\text{fppf}})\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}), & & \displaystyle \nonumber\\ \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x}/k,\;\cdot \;):\operatorname{Ch}(\hat{{\mathcal{O}}}_{x,\text{fppf}})\rightarrow \operatorname{Ch}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}) & & \displaystyle \nonumber\end{eqnarray}$$

and their derived functor are defined. We have

$$\begin{eqnarray}(f_{x})_{\ast }R\,\mathbf{Hom}_{k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\;\cdot \;,\mathbb{Z})=R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}((f_{x})_{\ast }(\;\cdot \;),\mathbb{Z})\end{eqnarray}$$

by the duality for finite étale morphisms [Reference MilneMil80, V, Proposition 1.13] (which is for the étale topology; the same proof works for the pro-étale topology). In other words, we have

$$\begin{eqnarray}\operatorname{Res}_{k_{x}/k}\,\circ \,\text{SD}_{x}=\text{SD}\,\circ \,\operatorname{Res}_{k_{x}/k},\end{eqnarray}$$

where $\text{SD}=R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(\;\cdot \;,\mathbb{Z})$ as before. Hence applying $\operatorname{Res}_{k_{x}/k}$ to the duality statements above over $k_{x}$ defines some new duality statements over $k$. Doing this for $\unicode[STIX]{x1D703}_{A}$ for instance defines a new isomorphism

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A^{\vee })^{\text{SD}\text{SD}}\overset{{\sim}}{\rightarrow }R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A)^{\text{SD}}\end{eqnarray}$$

in $D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})$. (Here we are not using any specific property of our duality isomorphism. We are using nothing but the obvious fact that if a morphism $C\rightarrow D$ in $D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$ is an isomorphism, then the induced morphism $\operatorname{Res}_{k_{x}/k}C\rightarrow \operatorname{Res}_{k_{x}/k}D$ is an isomorphism.) More explicitly, this morphism comes from the morphisms

$$\begin{eqnarray}\displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A^{\vee }) & \rightarrow & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,R\,\mathbf{Hom}_{\hat{K}_{x}}(A,\mathbf{G}_{m}))[1]\nonumber\\ \displaystyle & \rightarrow & \displaystyle R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A),R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,\mathbf{G}_{m}))[1]\nonumber\\ \displaystyle & \rightarrow & \displaystyle R\,\mathbf{Hom}_{k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}}(R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A),\mathbb{Z})[1]=R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,A)^{\text{SD}}[1]\nonumber\end{eqnarray}$$

The trace morphism in this situation used here is

(2.5.8)$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x}/k,\mathbf{G}_{m})\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x}/k,\mathbf{G}_{m})[1]=\operatorname{Res}_{k_{x}/k}\hat{\mathbf{K}}_{x}^{\times }/\hat{\mathbf{O}}_{x}^{\times }=\operatorname{Res}_{k_{x}/k}\mathbb{Z}\rightarrow \mathbb{Z},\end{eqnarray}$$

where the last morphism is

$$\begin{eqnarray}\mathbb{Z}[\operatorname{Hom}_{k}(k_{x},\overline{k})]\cong \mathbb{Z}^{[k_{x}:k]}\overset{\text{sum}}{{\twoheadrightarrow}}\mathbb{Z}\end{eqnarray}$$

on geometric points.

2.6 Henselizations and completions

Let ${\mathcal{O}}_{x}^{h}$ be an excellent Henselian discrete valuation ring of equal characteristic $p>0$ with perfect residue field $k_{x}$ ($\in {\mathcal{U}}_{0}$). We denote the maximal ideal by $\mathfrak{p}_{x}^{h}$ and the fraction field by $K_{x}^{h}$. The corresponding objects after completion are denoted by $\hat{{\mathcal{O}}}_{x}$, $\hat{\mathfrak{p}}_{x}$ and $\hat{K}_{x}$.

We make a variant of the constructions in the previous subsection using Henselizations instead of completions. For $k_{x}^{\prime }\in k_{x}^{\text{ind}\text{rat}}$, the Henselization of the (nonlocal) ring $k_{x}^{\prime }\otimes _{k_{x}}{\mathcal{O}}_{x}^{h}$ at the ideal $k_{x}^{\prime }\otimes _{k_{x}}\mathfrak{p}_{x}^{h}$ [Reference RaynaudRay70b, Chapter XI, Section 2] is denoted by $(k_{x}^{\prime }\otimes _{k_{x}}{\mathcal{O}}_{x}^{h})^{h}$. We define

$$\begin{eqnarray}\mathbf{O}_{x}^{h}(k_{x}^{\prime })=(k_{x}^{\prime }\otimes _{k_{x}}{\mathcal{O}}_{x}^{h})^{h},\qquad \mathbf{K}_{x}^{h}(k_{x}^{\prime })=\mathbf{O}_{x}^{h}(k_{x}^{\prime })\otimes _{{\mathcal{O}}_{x}^{h}}K_{x}^{h}.\end{eqnarray}$$

The functors $\mathbf{O}_{x}^{h}$ and $\mathbf{K}_{x}^{h}$ define premorphisms of sites

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{{\mathcal{O}}_{x}^{h}}:\operatorname{Spec}{\mathcal{O}}_{x,\text{fppf}}^{h}\rightarrow \operatorname{Spec}k_{x,\text{et}}^{\text{ind}\text{rat}},\qquad \unicode[STIX]{x1D70B}_{K_{x}^{h}}:\operatorname{Spec}K_{x,\text{fppf}}^{h}\rightarrow \operatorname{Spec}k_{x,\text{et}}^{\text{ind}\text{rat}}.\end{eqnarray}$$

The pro-étale sheafifications of their pushforward functors are denoted by

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h},\;\cdot \;),:\operatorname{Ab}({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow \operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}(K_{x}^{h},\;\cdot \;):\operatorname{Ab}(K_{x,\text{fppf}}^{h})\rightarrow \operatorname{Ab}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}). & \displaystyle \nonumber\end{eqnarray}$$

We set

$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)=[\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h},\;\cdot \;),\rightarrow \boldsymbol{\unicode[STIX]{x1D6E4}}(K_{x}^{h},j^{\ast }\;\cdot \;)][-1]: & \displaystyle \nonumber\\ \displaystyle & \displaystyle \operatorname{Ch}({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow \operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\end{eqnarray}$$

where $j:\operatorname{Spec}K_{x}^{h}\rightarrow \operatorname{Spec}{\mathcal{O}}_{x}^{h}$ is the natural morphism inducing a morphism on the fppf sites. We have their right derived functors

$$\begin{eqnarray}\displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h},\;\cdot \;),:D({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}(K_{x}^{h},\;\cdot \;),:D(K_{x,\text{fppf}}^{h})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)=[R\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h},\;\cdot \;)\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}(K_{x}^{h},j^{\ast }\;\cdot \;)][-1]: & \displaystyle \nonumber\\ \displaystyle & \displaystyle D({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow D(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}}). & \displaystyle \nonumber\end{eqnarray}$$

Again, we frequently omit the restriction functor $j^{\ast }$ by abuse of notation.

We say that a sheaf $F\in \operatorname{Set}({\mathcal{O}}_{x,\text{fppf}}^{h})$ is locally of finite presentation if it commutes with filtered direct limits as a functor on the category of ${\mathcal{O}}_{x}^{h}$-algebras.

Proposition 2.6.1. For any sheaf $A\in \operatorname{Ab}({\mathcal{O}}_{x,\text{fppf}}^{h})$ locally of finite presentation and $n\geqslant 0$, the sheaf $R^{n}(\unicode[STIX]{x1D70B}_{{\mathcal{O}}_{x}^{h}})_{\ast }A\in \operatorname{Ab}(k_{x,\text{et}}^{\text{ind}\text{rat}})$ is locally of finite presentation. In particular, we have $\mathbf{H}^{n}({\mathcal{O}}_{x}^{h},A)=R^{n}(\unicode[STIX]{x1D70B}_{{\mathcal{O}}_{x}^{h}})_{\ast }A$. That is, $\mathbf{H}^{n}({\mathcal{O}}_{x}^{h},A)$ is the étale sheafification of the presheaf

$$\begin{eqnarray}k_{x}^{\prime }\mapsto H^{n}(\mathbf{O}_{x}^{h}(k_{x}^{\prime }),A).\end{eqnarray}$$

A similar statement holds when ${\mathcal{O}}_{x}^{h}$ is replaced by $K_{x}^{h}$.

Proof. The sheaf $\mathbf{O}_{x}^{h}$ is locally of finite presentation by the construction of Henselization. Let $A\in \operatorname{Ab}({\mathcal{O}}_{x,\text{fppf}}^{h})$ be locally of finite presentation. Let $k_{x}^{\prime }=\bigcup k_{x,\unicode[STIX]{x1D706}}^{\prime }\in k^{\text{ind}\text{rat}}$ with $k_{x,\unicode[STIX]{x1D706}}^{\prime }\in k_{x}^{\text{rat}}$. Then for any $n\geqslant 0$, we have

$$\begin{eqnarray}H^{n}(\mathbf{O}_{x}^{h}(k_{x}^{\prime }),A)=\underset{\unicode[STIX]{x1D706}}{\varinjlim }\,H^{n}(\mathbf{O}_{x}^{h}(k_{x,\unicode[STIX]{x1D706}}^{\prime }),A).\end{eqnarray}$$

By sheafification, we know that $R^{n}(\unicode[STIX]{x1D70B}_{{\mathcal{O}}_{x}^{h}})_{\ast }A$ is locally of finite presentation. The same proof works for $K_{x}^{h}$.◻

Define a functor $\operatorname{Ch}({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow \operatorname{Ch}(k_{x,\text{pro}\text{et}}^{\text{ind}\text{rat}})$ of additive categories with translation by

$$\begin{eqnarray}o_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)=[\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)\rightarrow \boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},f^{\ast }\;\cdot \;)],\end{eqnarray}$$

where $f:\operatorname{Spec}\hat{{\mathcal{O}}}_{x}\rightarrow \operatorname{Spec}{\mathcal{O}}_{x}^{h}$ is the natural scheme morphism inducing a morphism on the fppf sites. We call its right derived functor

$$\begin{eqnarray}Ro_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)=[R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)\rightarrow R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},f^{\ast }\;\cdot \;)]\end{eqnarray}$$

the obstruction for cohomological approximation (at $x$). Define $D({\mathcal{O}}_{x,\text{fppf}}^{h})_{\text{ca}}$ to be the kernel of the functor $Ro_{x}({\mathcal{O}}_{x}^{h},\;\cdot \;)$, that is, the full subcategory of $D({\mathcal{O}}_{x,\text{fppf}}^{h})$ consisting of objects $A$ with $Ro_{x}({\mathcal{O}}_{x}^{h},A)=0$, or

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},A)=R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},f^{\ast }A).\end{eqnarray}$$

Such an object $A$ is said to satisfy cohomological approximation.

Proof. Any finite flat group scheme $N$ is a closed subgroup scheme of some smooth affine group scheme $G$ by [Reference BégueriBég81, Proposition 2.2.1]. The fppf quotient $H=G/N$ is a smooth affine group scheme by descent. We have a distinguished triangle $Ro_{x}({\mathcal{O}}_{x}^{h},N)\rightarrow Ro_{x}({\mathcal{O}}_{x}^{h},G)\rightarrow Ro_{x}({\mathcal{O}}_{x}^{h},H)$. If two of the terms are zero, then so is the other. Hence the finite flat case is reduced to the smooth case.

Assume that $A$ is a smooth group scheme over ${\mathcal{O}}_{x}^{h}$. Let $N\subset A$ be the schematic closure of the identity section of $A\times _{{\mathcal{O}}_{x}^{h}}K_{x}^{h}$. Then $N$ is an étale group scheme over ${\mathcal{O}}_{x}^{h}$ with trivial generic fiber and $A/N$ is a separated smooth group scheme over ${\mathcal{O}}_{x}^{h}$ by [Reference RaynaudRay70a, Proposition 3.3.5]. The both objects $R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h},N)$ and $R\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}(\hat{{\mathcal{O}}}_{x},N)$ are isomorphic to $N\times _{{\mathcal{O}}_{x}^{h}}k_{x}$.

Hence we may assume that $A$ is smooth separated. Then we have $\mathbf{H}_{x}^{0}(\hat{{\mathcal{O}}}_{x},A)=0$. Also $\mathbf{H}^{n}(\hat{{\mathcal{O}}}_{x},A)=0$ for $n\geqslant 1$ by [Reference SuzukiSuz14, Proposition (3.4.2)(a)]. Hence

$$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{H}_{x}^{1}(\hat{{\mathcal{O}}}_{x},A)=\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{K}_{x},A)/\boldsymbol{\unicode[STIX]{x1D6E4}}(\hat{{\mathcal{O}}}_{x},A), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \mathbf{H}_{x}^{n}(\hat{{\mathcal{O}}}_{x},A)=\mathbf{H}^{n-1}(\hat{K}_{x},A) & \displaystyle \nonumber\end{eqnarray}$$

for $n\geqslant 2$. These sheaves are locally of finite presentation (even before pro-étale sheafification) by the Greenberg approximation argument [Reference SuzukiSuz14, Propositions (3.2.8), (3.2.9)]. On the other hand, the sheaves $\mathbf{H}^{n}({\mathcal{O}}_{x}^{h},A)$, $\mathbf{H}^{n}(K_{x}^{h},A)$ and thus $\mathbf{H}_{x}^{n}({\mathcal{O}}_{x}^{h},A)$ are locally of finite presentation for $n\geqslant 0$ by the previous proposition. Therefore it is enough to show that the morphism

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}_{x}({\mathcal{O}}_{x}^{h},A)\rightarrow R\unicode[STIX]{x1D6E4}_{x}(\hat{{\mathcal{O}}}_{x},A)\end{eqnarray}$$

is an isomorphism when $k$ is algebraically closed by [Reference SuzukiSuz14, the second paragraph after Proposition (2.4.1)]. The statements to prove are

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}(K_{x}^{h},A)/\unicode[STIX]{x1D6E4}({\mathcal{O}}_{x}^{h},A)=\unicode[STIX]{x1D6E4}(\hat{K}_{x},A)/\unicode[STIX]{x1D6E4}(\hat{{\mathcal{O}}}_{x},A), & \displaystyle \nonumber\\ \displaystyle & \displaystyle H^{n}(K_{x}^{h},A)=H^{n}(\hat{K}_{x},A),\quad n\geqslant 1. & \displaystyle \nonumber\end{eqnarray}$$

They can be proven in the same way as the Greenberg approximation argument [Reference SuzukiSuz14, Propositions (3.2.8), (3.2.9)] (or equivalently, by the same argument as [Reference MilneMil06, I, Remark 3.10] combined with [Reference ČesnavičiusČes15, Proposition 3.5(b)]). ◻

Perhaps any group scheme locally of finite type over ${\mathcal{O}}_{x}^{h}$ might satisfy cohomological approximation since the Greenberg approximation itself holds in this generality. We do not pursue this point. But see [Reference Demarche and HarariDH18, Lemma 2.6, Remark 2.7] for a related result.

If $k_{x}$ is a finite extension of another perfect field $k$ and $f:\operatorname{Spec}k_{x}\rightarrow \operatorname{Spec}k$ is the natural morphism, then the composite of $\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h},\;\cdot \;)$ and $f_{\ast }$ is denoted by

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h}/k,\;\cdot \;):\operatorname{Ch}({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow \operatorname{Ch}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})\end{eqnarray}$$

with right derived functor

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}({\mathcal{O}}_{x}^{h}/k,\;\cdot \;):D({\mathcal{O}}_{x,\text{fppf}}^{h})\rightarrow D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}).\end{eqnarray}$$

Similar notation applies to other objects, defining $\boldsymbol{\unicode[STIX]{x1D6E4}}(K_{x}^{h}/k,\;\cdot \;)$, $\boldsymbol{\unicode[STIX]{x1D6E4}}_{x}({\mathcal{O}}_{x}^{h}/k,\;\cdot \;)$, $o_{x}({\mathcal{O}}_{x}^{h}/k,\;\cdot \;)$ and their derived functors.

2.7 The fppf site of a curve over the rational étale site of the base

Let $U$ be a smooth geometrically connected curve over a perfect field $k$ ($\in {\mathcal{U}}_{0}$) of characteristic $p>0$, with smooth compactification $X$ and function field $K$. By the fppf site $U_{\text{fppf}}$ of $U$, we mean the category of (${\mathcal{U}}_{0}$-small) $U$-schemes endowed with the fppf topology. For $k^{\prime }\in k^{\text{ind}\text{rat}}$, we denote $U_{k^{\prime }}=U\times _{k}k^{\prime }$. The functor sending $k^{\prime }\in k^{\text{ind}\text{rat}}$ to $U_{k^{\prime }}$ defines a premorphism of sites

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{U}:U_{\text{fppf}}\rightarrow \operatorname{Spec}k_{\text{et}}^{\text{ind}\text{rat}}.\end{eqnarray}$$

We define a left exact functor

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D6E4}}(U,\;\cdot \;):\operatorname{Ab}(U_{\text{fppf}})\rightarrow \operatorname{Ab}(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}})\end{eqnarray}$$

by the composite of the pushforward functor $\unicode[STIX]{x1D70B}_{U\ast }$ and the pro-étale sheafification. We have its right derived functor

$$\begin{eqnarray}R\boldsymbol{\unicode[STIX]{x1D6E4}}(U,\;\cdot \;):D(U_{\text{fppf}})\rightarrow D(k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}),\end{eqnarray}$$

with cohomologies $\mathbf{H}^{n}=H^{n}R\boldsymbol{\unicode[STIX]{x1D6E4}}$. For any $A\in \operatorname{Ab}(U_{\text{fppf}})$ and $n\geqslant 0$, the sheaf $\mathbf{H}^{n}(U,A)$ on $\operatorname{Spec}k_{\text{pro}\text{et}}^{\text{ind}\text{rat}}$ is the pro-étale sheafification of the presheaf