The Stacks project

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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