# ag.algebraic geometry – Accuracy of sequences preserved under singularity resolution

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