The Stacks project

Lemma 30.23.9. Let $f : X \to Y$ be a morphism of Noetherian schemes. Let $\mathcal{J} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals and set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X$. Then there is a right exact functor

\[ f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]

which sends $(\mathcal{G}_ n)$ to $(f^*\mathcal{G}_ n)$. If $f$ is flat, then $f^*$ is an exact functor.

Proof. Since $f^* : \textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(\mathcal{O}_ X)$ is right exact we have

\[ f^*\mathcal{G}_ n = f^*(\mathcal{G}_{n + 1}/\mathcal{I}^ n\mathcal{G}_{n + 1}) = f^*\mathcal{G}_{n + 1}/f^{-1}\mathcal{I}^ nf^*\mathcal{G}_{n + 1} = f^*\mathcal{G}_{n + 1}/\mathcal{J}^ nf^*\mathcal{G}_{n + 1} \]

hence the pullback of a system is a system. The construction of cokernels in the proof of Lemma 30.23.2 shows that $f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})$ is always right exact. If $f$ is flat, then $f^* : \textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(\mathcal{O}_ X)$ is an exact functor. It follows from the construction of kernels in the proof of Lemma 30.23.2 that in this case $f^* : \textit{Coh}(Y, \mathcal{J}) \to \textit{Coh}(X, \mathcal{I})$ also transforms kernels into kernels. $\square$

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