I'm studying algebraic geometry, now specifically sheaf of modules.
I'm trying to figure out how sheaves of $\cal{O}_\textrm{X}$-module change in a closed subscheme.
Let $X$ be a scheme and $Y$ be a closed subscheme with immersion $i:Y\rightarrow X$.
Let $\cal{F}$ be a (maybe quasi-)coherent sheaves of $\mathcal{O}_X$-module.
Let $U=Spec A$ be an open set in $X$, then $\mathcal F|_U\simeq \widetilde{M}$.
I know that there is some ideal $I$ of $A$ such that $\mathcal{O} _ Y|_{Y\cap U}\simeq Spec A/I$.
Now, I want to know that $i_*i^*\mathcal F|_U\simeq\widetilde{M/IM}$?
More generally, is it reasonable to say that $i_*i^*\mathcal F\simeq \cal F/I$ for an ideal sheaf $\cal I$ corresponding closed subscheme $Y$?