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Lemma 41.49.2. Let $X$ be a scheme. Let $\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$. Let $\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\textit{Sch}/X)_\tau$ be full subcategories with the following properties:
  1. For an object $U/X$ of $\mathcal{C}_t$,
    1. if $\{U_i \to U\}$ is a covering of $(\textit{Sch}/X)_\tau$, then $U_i/X$ is an object of $\mathcal{C}_t$,
    2. $U \times \mathbf{A}^1/X$ is an object of $\mathcal{C}_t$.
  2. $X/X$ is an object of $\mathcal{C}_t$.
We endow $\mathcal{C}_t$ with the structure of a site whose coverings are exactly those coverings $\{U_i \to U\}$ of $(\textit{Sch}/X)_\tau$ with $U \in \mathop{\rm Ob}\nolimits(\mathcal{C}_t)$. Then
  1. The functor $\mathcal{C}_1 \to \mathcal{C}_2$ is fully faithful, continuous, and cocontinuous.
Denote $g : \mathop{\textit{Sh}}\nolimits(\mathcal{C}_1) \to \mathop{\textit{Sh}}\nolimits(\mathcal{C}_2)$ the corresponding morphism of topoi. Denote $\mathcal{O}_t$ the restriction of $\mathcal{O}$ to $\mathcal{C}_t$. Denote $g_!$ the functor of Modules on Sites, Definition 17.16.1.
  1. The canonical map $g_!\mathcal{O}_1 \to \mathcal{O}_2$ is an isomorphism.

Proof. Assertion (\romannumeral1) is immediate from the definitions. In this proof all schemes are schemes over $X$ and all morphisms of schemes are morphisms of schemes over $X$. Note that $g^{-1}$ is given by restriction, so that for an object $U$ of $\mathcal{C}_1$ we have $\mathcal{O}_1(U) = \mathcal{O}_2(U) = \mathcal{O}(U)$. Recall that $g_!\mathcal{O}_1$ is the sheaf associated to the presheaf $g_{p!}\mathcal{O}_1$ which associates to $V$ in $\mathcal{C}_2$ the group $$ \mathop{\rm colim}\nolimits_{V \to U} \mathcal{O}(U) $$ where $U$ runs over the objects of $\mathcal{C}_1$ and the colimit is taken in the category of abelian groups. Below we will use frequently that if $$ V \to U \to U' $$ are morphisms with $U, U' \in \mathop{\rm Ob}\nolimits(\mathcal{C}_1)$ and if $f' \in \mathcal{O}(U')$ restricts to $f \in \mathcal{O}(U)$, then $(V \to U, f)$ and $(V \to U', f')$ define the same element of the colimit. Also, $g_!\mathcal{O}_1 \to \mathcal{O}_2$ maps the element $(V \to U, f)$ simply to the pullback of $f$ to $V$.

Surjectivity. Let $V$ be a scheme and let $h \in \mathcal{O}(V)$. Then we obtain a morphism $V \to X \times \mathbf{A}^1$ induced by $h$ and the structure morphism $V \to X$. Writing $\mathbf{A}^1 = \mathop{\rm Spec}(\mathbf{Z}[x])$ we see the element $x \in \mathcal{O}(X \times \mathbf{A}^1)$ pulls back to $h$. Since $X \times \mathbf{A}^1$ is an object of $\mathcal{C}_1$ by assumptions (1)(b) and (2) we obtain the desired surjectivity.

Injectivity. Let $V$ be a scheme. Let $s = \sum_{i = 1, \ldots, n} (V \to U_i, f_i)$ be an element of the colimit displayed above. For any $i$ we can use the morphism $f_i : U_i \to X \times \mathbf{A}^1$ to see that $(V \to U_i, f_i)$ defines the same element of the colimit as $(f_i : V \to X \times \mathbf{A}^1, x)$. Then we can consider $$ f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n $$ and we see that $s$ is equivalent in the colimit to $$ \sum\nolimits_{i = 1, \ldots, n} (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_i) = (f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_1 + \ldots + x_n) $$ Now, if $x_1 + \ldots + x_n$ restricts to zero on $V$, then we see that $f_1 \times \ldots \times f_n$ factors through $X \times \mathbf{A}^{n - 1} = V(x_1 + \ldots + x_n)$. Hence we see that $s$ is equivalent to zero in the colimit. $\square$

\begin{lemma}
\label{lemma-compare-structure-sheaves}
Let $X$ be a scheme. Let
$\tau \in \{Zariski, \acute{e}tale, smooth, syntomic, fppf\}$.
Let $\mathcal{C}_1 \subset \mathcal{C}_2 \subset (\Sch/X)_\tau$ be full
subcategories with the following properties:
\begin{enumerate}
\item For an object $U/X$ of $\mathcal{C}_t$,
\begin{enumerate}
\item if $\{U_i \to U\}$ is a covering of $(\Sch/X)_\tau$, then
$U_i/X$ is an object of $\mathcal{C}_t$,
\item $U \times \mathbf{A}^1/X$ is an object of $\mathcal{C}_t$.
\end{enumerate}
\item $X/X$ is an object of $\mathcal{C}_t$.
\end{enumerate}
We endow $\mathcal{C}_t$ with the structure of a site whose coverings are
exactly those coverings $\{U_i \to U\}$ of $(\Sch/X)_\tau$ with
$U \in \Ob(\mathcal{C}_t)$. Then
\begin{enumerate}
\item[(\romannumeral1)] The functor $\mathcal{C}_1 \to \mathcal{C}_2$
is fully faithful, continuous, and cocontinuous.
\end{enumerate}
Denote $g : \Sh(\mathcal{C}_1) \to \Sh(\mathcal{C}_2)$ the corresponding
morphism of topoi. Denote $\mathcal{O}_t$ the restriction of $\mathcal{O}$
to $\mathcal{C}_t$. Denote $g_!$ the functor of
Modules on Sites, Definition \ref{sites-modules-definition-g-shriek}.
\begin{enumerate}
\item[(\romannumeral2)] The canonical map $g_!\mathcal{O}_1 \to \mathcal{O}_2$
is an isomorphism.
\end{enumerate}
\end{lemma}

\begin{proof}
Assertion (\romannumeral1) is immediate from the definitions.
In this proof all schemes are schemes over $X$ and all morphisms of
schemes are morphisms of schemes over $X$. Note that $g^{-1}$ is
given by restriction, so that for an object $U$ of $\mathcal{C}_1$
we have $\mathcal{O}_1(U) = \mathcal{O}_2(U) = \mathcal{O}(U)$.
Recall that $g_!\mathcal{O}_1$ is the sheaf associated to the presheaf
$g_{p!}\mathcal{O}_1$ which associates to $V$ in $\mathcal{C}_2$ the group
$$
\colim_{V \to U} \mathcal{O}(U)
$$
where $U$ runs over the objects of $\mathcal{C}_1$ and the colimit is
taken in the category of abelian groups. Below we will use frequently
that if
$$
V \to U \to U'
$$
are morphisms with $U, U' \in \Ob(\mathcal{C}_1)$
and if $f' \in \mathcal{O}(U')$ restricts to $f \in \mathcal{O}(U)$,
then $(V \to U, f)$ and $(V \to U', f')$ define the same element of the
colimit. Also, $g_!\mathcal{O}_1 \to \mathcal{O}_2$ maps the element
$(V \to U, f)$ simply to the pullback of $f$ to $V$.

\medskip\noindent
Surjectivity. Let $V$ be a scheme and let $h \in \mathcal{O}(V)$.
Then we obtain a morphism $V \to X \times \mathbf{A}^1$ induced by $h$
and the structure morphism $V \to X$. Writing
$\mathbf{A}^1 = \Spec(\mathbf{Z}[x])$ we see the element
$x \in \mathcal{O}(X \times \mathbf{A}^1)$ pulls
back to $h$. Since $X \times \mathbf{A}^1$ is an object of $\mathcal{C}_1$
by assumptions (1)(b) and (2) we obtain the desired surjectivity.

\medskip\noindent
Injectivity. Let $V$ be a scheme. Let
$s = \sum_{i = 1, \ldots, n} (V \to U_i, f_i)$ be an element of the colimit
displayed above. For any $i$ we can use the morphism
$f_i : U_i \to X \times \mathbf{A}^1$
to see that $(V \to U_i, f_i)$ defines the same element of the colimit as
$(f_i : V \to X \times \mathbf{A}^1, x)$. Then we can consider
$$
f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n
$$
and we see that $s$ is equivalent in the colimit to
$$
\sum\nolimits_{i = 1, \ldots, n}
(f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n, x_i) =
(f_1 \times \ldots \times f_n : V \to X \times \mathbf{A}^n,
x_1 + \ldots + x_n)
$$
Now, if $x_1 + \ldots + x_n$ restricts to zero on $V$, then we see
that $f_1 \times \ldots \times f_n$ factors through
$X \times \mathbf{A}^{n - 1} = V(x_1 + \ldots + x_n)$. Hence we see
that $s$ is equivalent to zero in the colimit.
\end{proof}

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