Let $f:E \to S$ be a (family of) elliptic curve. Let $[0]:S \to E$ be the zero section and
$I:=I([0])$ its ideal sheaf. Why is $f_*(I^{-n})$ locally free, as claimed by Hida in the following excerpt from his book Geometric Modular forms and Elliptic Curves?
2026-03-24 23:56:08.1774396568
Weierstrass equation for a family of elliptic curves, pushforward of sheaf
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$f:E\rightarrow S$ is proper and flat. By Cohomology and Base Change, $R^1f_*(I([0])^{-n})= 0$ implies that $f_*I([0])^{-n}$ is locally free. You can check on a fibre to see that the rank is $n$.