Lemma 76.42.10. Let S be a scheme. Let f : X \to Y be a representable proper morphism of Noetherian algebraic spaces over S. Let \mathcal{J}, \mathcal{K} \subset \mathcal{O}_ Y be quasi-coherent sheaves of ideals. Assume f is an isomorphism over V = Y \setminus V(\mathcal{K}). Set \mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X. Let (\mathcal{G}_ n) be an object of \textit{Coh}(Y, \mathcal{J}), let \mathcal{F} be a coherent \mathcal{O}_ X-module, and let \beta : (f^*\mathcal{G}_ n) \to \mathcal{F}^\wedge be an isomorphism in \textit{Coh}(X, \mathcal{I}). Then there exists a map
\alpha : (\mathcal{G}_ n) \longrightarrow (f_*\mathcal{F})^\wedge
in \textit{Coh}(Y, \mathcal{J}) whose kernel and cokernel are annihilated by a power of \mathcal{K}.
Proof.
Since f is a proper morphism we see that f_*\mathcal{F} is a coherent \mathcal{O}_ Y-module (Cohomology of Spaces, Lemma 69.20.2). Thus the statement of the lemma makes sense. Consider the compositions
\gamma _ n : \mathcal{G}_ n \to f_*f^*\mathcal{G}_ n \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}).
Here the first map is the adjunction map and the second is f_*\beta _ n. We claim that there exists a unique \alpha as in the lemma such that the compositions
\mathcal{G}_ n \xrightarrow {\alpha _ n} f_*\mathcal{F}/\mathcal{J}^ nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F})
equal \gamma _ n for all n. Because of the uniqueness and étale descent for \textit{Coh}(Y, \mathcal{J}) it suffices to prove this étale locally on Y. Thus we may assume Y is the spectrum of a Noetherian ring. As f is representable we see that X is a scheme as well. Thus we reduce to the case of schemes, see proof of Cohomology of Schemes, Lemma 30.25.3.
\square
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