# When is the structure sheaf not coherent over itself?

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?

• Note that when the scheme is not locally Noetherian, then a coherent sheaf is not necessarily one that is locally a finitely generated O_X-module. See 13.6.1 in Vakil. So what you're looking here for is an example of a non-coherent ring, which Vakil also gives an example here math.stanford.edu/~vakil/216blog/incoherent.pdf . – loch Feb 4 '18 at 5:23