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 $Y$?

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    $\begingroup$ Yes, it is true that $i_*i^*\mathcal F\simeq \cal F/\mathcal I\mathcal F$ for every sheaf $\mathcal F$ of $\mathcal O_X$-Modules, quasi-coherent or not. $\endgroup$ – Georges Elencwajg Nov 15 '16 at 7:53

First we can look at $i^*\mathcal{F}$, by the properties of pull back map, locally this is the coherent sheaf defined by $M\otimes A/IA=M/IM$ (as an $A/I$ module).

Second, locally $i_*i^*\mathcal{F}$ is the coherent sheaf on $X|_U$ defined by the module $M/IM$ (but this time as an $A$ module!), since you have the natural map $A\to A/I$.

  • $\begingroup$ Can you give me some more detail like; $i^{-1}\mathcal F\otimes _{i^{-1}\mathcal O_X}\mathcal O_Y$ $\endgroup$ – wooa0923 Nov 15 '16 at 3:46
  • $\begingroup$ Those properties can be found at Hartshorne Chaper 2 Prop 5.2. $\endgroup$ – Xuqiang QIN Nov 15 '16 at 4:20
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    $\begingroup$ Btw, to get around with the unpleasant complexity caused by $i^{-1}$, use adjunction and show results about $i_*$ instead. $\endgroup$ – Xuqiang QIN Nov 15 '16 at 4:27
  • $\begingroup$ Right. I missed it. Thanks. $\endgroup$ – wooa0923 Nov 16 '16 at 14:05

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