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Let $X$ be a noetherian, affine, normal, isolated singularity and $\pi:\widetilde{X} \to X$ be a resolution of singularities. Suppose, we have an exact sequence (not necessarily short exact): $$\mathcal{O}_X^{a_1} \xrightarrow{\phi_1} \mathcal{O}_X^{a_2} \xrightarrow{\phi_2} \mathcal{O}_X^{a_3}$$ with $\phi_1$ and $\phi_2$ defined by an $a_1 \times a_2$ and $a_2 \times a_3$-matrices respectively, with coefficients in $\Gamma(\mathcal{O}_X)$. Using the natural inclusion $\Gamma(\mathcal{O}_X) \subset \Gamma(\mathcal{O}_{\widetilde{X}})$, the matrices corresponding to $\phi_1$ and $\phi_2$ define natural morphisms $\phi_1: \mathcal{O}_{\widetilde{X}}^{a_1} \to \mathcal{O}_{\widetilde{X}}^{a_2}$ and $\phi_2: \mathcal{O}_{\widetilde{X}}^{a_2} \to \mathcal{O}_{\widetilde{X}}^{a_3}$. Is the resulting complex $$\mathcal{O}_{\widetilde{X}}^{a_1} \xrightarrow{\phi_1} \mathcal{O}_{\widetilde{X}}^{a_2} \xrightarrow{\phi_2} \mathcal{O}_{\widetilde{X}}^{a_3}$$ also exact?

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    $\begingroup$ In general, it is not exact. $\endgroup$ – Sasha Nov 8 at 18:30

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