In the higher direct image sheaves section of Vakil (18.8), he makes the assumptions that $\mathscr{O}_Y$ is coherent over itself several times. How could $\mathscr{O}_Y$ not be Noetherian over itself? Over each affine open $U_i = $Spec $A_i$, isn't $\mathscr{O}_Y|_{U_i} = \widetilde{A_i}$, which is generated by 1?
2026-03-24 23:43:14.1774395794
When is the structure sheaf not coherent over itself?
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