# Pushforward of locally free sheaves and resolution of singularities

Let $$X$$ be an noetherian, affine, isolated of dimension at least $$2$$ and $$\pi:\widetilde{X} \to X$$ a resolution of singularities. Let $$\mathcal{E}$$ be a locally-free sheaf on $$\widetilde{X}$$ such that $$\pi_*\mathcal{E}$$ is a free $$\mathcal{O}_X$$-module. Is $$\mathcal{E}$$ necessarily trivial i.e., a direct sum of copies of $$\mathcal{O}_{\widetilde{X}}$$?

No. If $$X$$ is the cone over a normal rational curve of degree $$d > 1$$ and $$\tilde{X}$$ is its blowup at the vertex with the exceptional curve $$E$$ then $$\pi_*\mathcal{O}(nE) = \mathcal{O}$$ for any $$n \ge 0$$.