# Exactness of sequences preserved under resolution of singularities

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?

• In general, it is not exact. – Sasha Nov 8 at 18:30