The Stacks Project


Tag 05VD

Chapter 18: Modules on Sites > Section 18.38: Pullbacks of flat modules

Lemma 18.38.1. Let $(f, f^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}_\mathcal{D})$ 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 $(h, h^\sharp) : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'})$. 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 $u_p\mathcal{F}(U) = \mathop{\rm colim}\nolimits_{(\mathcal{I}_U^u)^{opp}} \mathcal{F}(V)$ and $u_p\mathcal{O}(U) = \mathop{\rm colim}\nolimits_{(\mathcal{I}_U^u)^{opp}} \mathcal{O}(V)$. Hence we may assume there is a pair $(V, \phi) \in \mathop{\rm Ob}\nolimits(\mathcal{I}_U^u)$ 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]

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 05VD

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?