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ag.algebraic geometry – Is the pushforward of a regionally free sheaf by an launch immersion coherent?

By the route, assuming by varieties, you denote irreducible varieties, then for the second query, the respond is sure.

For the torsion free-ness, suppose that $r in H^0(U, O_X)$ kills some non-zero component $z in H^0(U, f_* mathcal{F}) = H^0(U cap X, mathcal{F})$. By restriction, $r$ is a non-zero component of $H^0(X cap U, mathcal{O}_Z)$. We quiet have $rz = 0$ plane on this setting, and so by limiting to an affine mask of $X$, it quiet occurs. This will contradict the torsion-freeness (and thus specifically the locally-freeness) of $mathcal{F}$.

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