When reading the proof of Theorem II, 7.11 in "Residues and Duality" by R. Hartshorne, I have encountered some steps that are unclear to me.
1.Firstly, I am unsure about the notation $\mathscr{O}_Z$ mentioned on the third line of page 129. Could you please clarify its meaning? (As Aphelli commented, it is the extension by zero of the structure sheaf of the integral subscheme $Z\subset X$.)
2.Additionally, I would appreciate further explanation on how to deduce that the image of $\phi$ has support in $Z'$ as stated on the seventh line of page 129.
- How can we shrink $U$ so that $\psi$ is injective on $U$?
Any comments or insights on these matters would be greatly appreciated. Thank you in advance for your assistance.
