The Stacks project

18.31 Flat morphisms

Definition 18.31.1. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. We say $(f, f^\sharp )$ is flat if the ring map $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$ is flat. We say a morphism of ringed sites is flat if the associated morphism of ringed topoi is flat.

Lemma 18.31.2. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ be a morphism of ringed topoi. Then

\[ f^{-1} : \textit{Ab}(\mathcal{C}') \longrightarrow \textit{Ab}(\mathcal{C}), \quad \mathcal{F} \longmapsto f^{-1}\mathcal{F} \]

is exact. If $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ is a flat morphism of ringed topoi then

\[ f^* : \textit{Mod}(\mathcal{O}') \longrightarrow \textit{Mod}(\mathcal{O}), \quad \mathcal{F} \longmapsto f^*\mathcal{F} \]

is exact.

Proof. Given an abelian sheaf $\mathcal{G}$ on $\mathcal{C}'$ the underlying sheaf of sets of $f^{-1}\mathcal{G}$ is the same as $f^{-1}$ of the underlying sheaf of sets of $\mathcal{G}$, see Sites, Section 7.44. Hence the exactness of $f^{-1}$ for sheaves of sets (required in the definition of a morphism of topoi, see Sites, Definition 7.15.1) implies the exactness of $f^{-1}$ as a functor on abelian sheaves.

To see the statement on modules recall that $f^*\mathcal{F}$ is defined as the tensor product $f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}', f^\sharp } \mathcal{O}$. Hence $f^*$ is a composition of functors both of which are exact. $\square$

Definition 18.31.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. We say that $\mathcal{F}$ is flat over $(\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ if $\mathcal{F}$ is flat as an $f^{-1}\mathcal{O}'$-module.

This is compatible with the notion as defined for morphisms of ringed spaces, see Modules, Definition 17.20.3 and the discussion following.

Lemma 18.31.4. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_\mathcal {D}$-modules. If $\mathcal{F}$ is finitely presented and $f$ is flat, then the canonical map

\[ f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G}) \]

of Remark 18.27.3 is an isomorphism.

Proof. Say $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. We have to show that the restriction of the map to $\mathcal{C}/U$ for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is an isomorphism. We may replace $U$ by the members of a covering of $U$. Hence by Sites, Lemma 7.14.10 we may assume there exists a morphism $U \to u(V)$ for some $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Of course, then we may replace $U$ by $u(V)$. Then since $u$ is continuous, we may replace $V$ by a covering and assume there is a presentation $\mathcal{O}_ V^{\oplus m} \to \mathcal{O}_ V^{\oplus n} \to \mathcal{F}|_ V \to 0$ over $\mathcal{D}/V$. Since formation of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with localization (Lemma 18.27.2) we may replace $f$ by the morphism $(\mathcal{C}/u(V), \mathcal{O}_{u(V)}) \to (\mathcal{D}/V, \mathcal{O}_ V)$ induced by $f$. Hence we reduce to the case where $\mathcal{F}$ has a global presentation $\mathcal{O}_\mathcal {D}^{\oplus m} \to \mathcal{O}_\mathcal {D}^{\oplus n} \to \mathcal{F} \to 0$. Since $f$ is flat and $f^*\mathcal{O}_\mathcal {D} = \mathcal{O}_\mathcal {C}$ we obtain a corresponding presentation $\mathcal{O}_\mathcal {C}^{\oplus m} \to \mathcal{O}_\mathcal {C}^{\oplus n} \to f^*\mathcal{F} \to 0$, see Lemma 18.31.2. Using that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with finite direct sums in the first variable, using that both $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(\mathcal{O}_\mathcal {C}, -)$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{O}_\mathcal {D}, -)$ are the identity functor, and using the functoriality of the construction of Remark 18.27.3 we obtain a commutative diagram

\[ \xymatrix{ 0 \ar[r] & f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{F}, \mathcal{G}) \ar[d] \ar[r] & f^*\mathcal{G}^{\oplus n} \ar[d] \ar[r] & f^*\mathcal{G}^{\oplus n} \ar[d] \\ 0 \ar[r] & \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(f^*\mathcal{F}, f^*\mathcal{G}) \ar[r] & f^*\mathcal{G}^{\oplus n} \ar[r] & f^*\mathcal{G}^{\oplus n} } \]

where the right two vertical arrows are isomorphisms. By Lemma 18.27.5 the rows are exact. We conclude by the 5 lemma. $\square$

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