# The Stacks Project

## Tag 05VD

Lemma 18.38.1. Let $(f, f^\sharp) : (\mathop{\mathit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\mathop{\mathit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi or ringed sites. Then $f^*\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{C}$-module whenever $\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{D}$-module.

Proof. Choose a diagram as in Lemma 18.7.2. Recall that being a flat module is intrinsic (see Section 18.18 and Definition 18.28.1). Hence it suffices to prove the lemma for the morphism $(h, h^\sharp) : (\mathop{\mathit{Sh}}\nolimits(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \to (\mathop{\mathit{Sh}}\nolimits(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'})$. In other words, we may assume that our sites $\mathcal{C}$ and $\mathcal{D}$ have all finite limits and that $f$ is a morphism of sites induced by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ which commutes with finite limits.

Recall that $f^*\mathcal{F} = \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{F}$ (Definition 18.13.1). By Lemma 18.28.11 it suffices to prove that $f^{-1}\mathcal{F}$ is a flat $f^{-1}\mathcal{O}_\mathcal{D}$-module. Combined with the previous paragraph this reduces us to the situation of the next paragraph.

Assume $\mathcal{C}$ and $\mathcal{D}$ are sites which have all finite limits and that $u : \mathcal{D} \to \mathcal{C}$ is a continuous functor which commutes with finite limits. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$ and let $\mathcal{F}$ be a flat $\mathcal{O}$-module. Then $u$ defines a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ (Sites, Proposition 7.14.6). To show: $f^{-1}\mathcal{F}$ is a flat $f^{-1}\mathcal{O}$-module. Let $U$ be an object of $\mathcal{C}$ and let $$f^{-1}\mathcal{O}|_U \xrightarrow{(f_1, \ldots, f_n)} f^{-1}\mathcal{O}|_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} f^{-1}\mathcal{F}|_U$$ be a complex of $f^{-1}\mathcal{O}|_U$-modules. Our goal is to construct a factorization of $(s_1, \ldots, s_n)$ on the members of a covering of $U$ as in Lemma 18.28.12 part (2). Consider the elements $s_a \in f^{-1}\mathcal{F}(U)$ and $f_a \in f^{-1}\mathcal{O}(U)$. Since $f^{-1}\mathcal{F}$, resp. $f^{-1}\mathcal{O}$ is the sheafification of $u_p\mathcal{F}$ we may, after replacing $U$ by the members of a covering, assume that $s_a$ is the image of an element $s'_a \in u_p\mathcal{F}(U)$ and $f_a$ is the image of an element $f'_a \in u_p\mathcal{O}(U)$. Then after another replacement of $U$ by the members of a covering we may assume that $\sum f'_as'_a$ is zero in $u_p\mathcal{F}(U)$. Recall that the category $(\mathcal{I}_U^u)^{opp}$ is directed (Sites, Lemma 7.5.2) and that $u_p\mathcal{F}(U) = \mathop{\mathrm{colim}}\nolimits_{(\mathcal{I}_U^u)^{opp}} \mathcal{F}(V)$ and $u_p\mathcal{O}(U) = \mathop{\mathrm{colim}}\nolimits_{(\mathcal{I}_U^u)^{opp}} \mathcal{O}(V)$. Hence we may assume there is a pair $(V, \phi) \in \mathop{\mathrm{Ob}}\nolimits(\mathcal{I}_U^u)$ where $V$ is an object of $\mathcal{D}$ and $\phi$ is a morphism $\phi : U \to u(V)$ of $\mathcal{D}$ and elements $s''_a \in \mathcal{F}(V)$ and $f''_a \in \mathcal{O}(V)$ whose images in $u_p\mathcal{F}(U)$ and $u_p\mathcal{O}(U)$ are equal to $s'_a$ and $f'_a$ and such that $\sum f''_a s''_a = 0$ in $\mathcal{F}(V)$. Then we obtain a complex $$\mathcal{O}|_V \xrightarrow{(f''_1, \ldots, f''_n)} \mathcal{O}|_V^{\oplus n} \xrightarrow{(s''_1, \ldots, s''_n)} \mathcal{F}|_V$$ and we can apply the other direction of Lemma 18.28.12 to see there exists a covering $\{V_i \to V\}$ of $\mathcal{D}$ and for each $i$ a factorization $$\mathcal{O}|_{V_i}^{\oplus n} \xrightarrow{B''_i} \mathcal{O}|_{V_i}^{\oplus l_i} \xrightarrow{(t''_{i1}, \ldots, t''_{il_i})} \mathcal{F}|_{V_i}$$ of $(s''_1, \ldots, s''_n)|_{V_i}$ such that $B_i \circ (f''_1, \ldots, f''_n)|_{V_i} = 0$. Set $U_i = U \times_{\phi, u(V)} u(V_i)$, denote $B_i \in \text{Mat}(l_i \times n, f^{-1}\mathcal{O}(U_i))$ the image of $B''_i$, and denote $t_{ij} \in f^{-1}\mathcal{F}(U_i)$ the image of $t''_{ij}$. Then we get a factorization $$f^{-1}\mathcal{O}|_{U_i}^{\oplus n} \xrightarrow{B_i} f^{-1}\mathcal{O}|_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}$$ of $(s_1, \ldots, s_n)|_{U_i}$ such that $B_i \circ (f_1, \ldots, f_n)|_{U_i} = 0$. This finishes the proof. $\square$

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 6475–6488 (see updates for more information).

\begin{lemma}
\label{lemma-pullback-flat}
\begin{reference}
\cite[Expos\'e V, Corollary 1.7.1]{SGA4}
\end{reference}
Let
$(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi or ringed sites.
Then $f^*\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{C}$-module
whenever $\mathcal{F}$ is a flat $\mathcal{O}_\mathcal{D}$-module.
\end{lemma}

\begin{proof}
Choose a diagram as in
Lemma \ref{lemma-morphism-ringed-topoi-comes-from-morphism-ringed-sites}.
Recall that being a flat module is intrinsic
(see Section \ref{section-intrinsic} and
Definition \ref{definition-flat}).
Hence it suffices to prove the lemma for
the morphism $(h, h^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \to (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'})$.
In other words, we may assume that
our sites $\mathcal{C}$ and $\mathcal{D}$
have all finite limits and that $f$ is a morphism
of sites induced by a continuous functor $u : \mathcal{D} \to \mathcal{C}$
which commutes with finite limits.

\medskip\noindent
Recall that $f^*\mathcal{F} = \mathcal{O}_\mathcal{C} \otimes_{f^{-1}\mathcal{O}_\mathcal{D}} f^{-1}\mathcal{F}$ (Definition \ref{definition-pushforward}).
By Lemma \ref{lemma-flat-change-of-rings} it suffices to
prove that $f^{-1}\mathcal{F}$ is a flat
$f^{-1}\mathcal{O}_\mathcal{D}$-module. Combined with
the previous paragraph this reduces us to the situation
of the next paragraph.

\medskip\noindent
Assume $\mathcal{C}$ and $\mathcal{D}$ are sites which
have all finite limits and that $u : \mathcal{D} \to \mathcal{C}$
is a continuous functor which commutes with finite limits.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{D}$
and let $\mathcal{F}$ be a flat $\mathcal{O}$-module.
Then $u$ defines a morphism of sites $f : \mathcal{C} \to \mathcal{D}$
(Sites, Proposition \ref{sites-proposition-get-morphism}).
To show: $f^{-1}\mathcal{F}$ is a flat $f^{-1}\mathcal{O}$-module.
Let $U$ be an object of $\mathcal{C}$ and let
$$f^{-1}\mathcal{O}|_U \xrightarrow{(f_1, \ldots, f_n)} f^{-1}\mathcal{O}|_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} f^{-1}\mathcal{F}|_U$$
be a complex of $f^{-1}\mathcal{O}|_U$-modules.
Our goal is to construct a factorization of
$(s_1, \ldots, s_n)$ on the members of a covering of $U$
as in Lemma \ref{lemma-flat-eq} part (2).
Consider the elements $s_a \in f^{-1}\mathcal{F}(U)$
and $f_a \in f^{-1}\mathcal{O}(U)$.
Since $f^{-1}\mathcal{F}$, resp.\ $f^{-1}\mathcal{O}$
is the sheafification of $u_p\mathcal{F}$ we may,
after replacing $U$ by the members of a covering,
assume that $s_a$ is the image of an element $s'_a \in u_p\mathcal{F}(U)$ and
$f_a$ is the image of an element $f'_a \in u_p\mathcal{O}(U)$.
Then after another replacement of $U$ by the members of a covering
we may assume that $\sum f'_as'_a$ is zero in $u_p\mathcal{F}(U)$.
Recall that the category $(\mathcal{I}_U^u)^{opp}$ is directed
(Sites, Lemma \ref{sites-lemma-directed})
and that $u_p\mathcal{F}(U) = \colim_{(\mathcal{I}_U^u)^{opp}} \mathcal{F}(V)$
and $u_p\mathcal{O}(U) = \colim_{(\mathcal{I}_U^u)^{opp}} \mathcal{O}(V)$.
Hence we may assume there is a pair $(V, \phi) \in \Ob(\mathcal{I}_U^u)$
where $V$ is an object of $\mathcal{D}$
and $\phi$ is a morphism $\phi : U \to u(V)$ of $\mathcal{D}$
and elements $s''_a \in \mathcal{F}(V)$ and $f''_a \in \mathcal{O}(V)$
whose images in $u_p\mathcal{F}(U)$ and $u_p\mathcal{O}(U)$
are equal to $s'_a$ and $f'_a$ and such that
$\sum f''_a s''_a = 0$ in $\mathcal{F}(V)$.
Then we obtain a complex
$$\mathcal{O}|_V \xrightarrow{(f''_1, \ldots, f''_n)} \mathcal{O}|_V^{\oplus n} \xrightarrow{(s''_1, \ldots, s''_n)} \mathcal{F}|_V$$
and we can apply the other direction of Lemma \ref{lemma-flat-eq}
to see there exists a covering $\{V_i \to V\}$ of $\mathcal{D}$
and for each $i$ a factorization
$$\mathcal{O}|_{V_i}^{\oplus n} \xrightarrow{B''_i} \mathcal{O}|_{V_i}^{\oplus l_i} \xrightarrow{(t''_{i1}, \ldots, t''_{il_i})} \mathcal{F}|_{V_i}$$
of $(s''_1, \ldots, s''_n)|_{V_i}$ such that
$B_i \circ (f''_1, \ldots, f''_n)|_{V_i} = 0$.
Set $U_i = U \times_{\phi, u(V)} u(V_i)$, denote
$B_i \in \text{Mat}(l_i \times n, f^{-1}\mathcal{O}(U_i))$
the image of $B''_i$, and denote
$t_{ij} \in f^{-1}\mathcal{F}(U_i)$ the image of
$t''_{ij}$. Then we get a factorization
$$f^{-1}\mathcal{O}|_{U_i}^{\oplus n} \xrightarrow{B_i} f^{-1}\mathcal{O}|_{U_i}^{\oplus l_i} \xrightarrow{(t_{i1}, \ldots, t_{il_i})} \mathcal{F}|_{U_i}$$
of $(s_1, \ldots, s_n)|_{U_i}$ such that
$B_i \circ (f_1, \ldots, f_n)|_{U_i} = 0$.
This finishes the proof.
\end{proof}

## References

[SGA4, Exposé V, Corollary 1.7.1]

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