I have a question about the 1 to 1 correspondence between quasi-coherent sheaf of ideals and the closed subschemes.
suppose $(i,i^×):Y\subset X$ is a closed subscheme,where $i^×:O_X\rightarrow i_*O_Y$.denote $I$ the kernel of $i^×$ denote $Z=Supp(O_X/I)$,we need to check $Y=Z$.He says the unicity is clear.but I don't know how to prove $Y=Z$,it is easy to see $Z$ is closed and $Z\subset Y$.How to show is equals?
I also find that in QingLiu's book,proposition2.24.the proof is :using $0\rightarrow I\rightarrow O_X\rightarrow i_*O_Y\rightarrow 0$ we deduce that $I_x=O_{X,x}$ iff $x\notin Y$,where we know $(i_*O_Y)_x=O_{Y,x}$ if $x\in Y$,and 0,otherwise. I can't understand this if $x\in Y$,and $O_{Y,x}=0$,we can also get $I_x=O_{X,x}$.Of course,I know this can't be true,but how to prove this material.
Thanks for your help.