# Pull back of the associated sheaf of Rees algebra

Let $$A$$ be a Noetherian ring. Let $$I\subset A$$ be an ideal. Set B=$$\oplus_{n\ge 0}I^n$$ to be the Rees algebra associated to $$I$$, $$\widetilde B$$ to be the associated sheaf of $$B$$ on $$\operatorname{Spec} A$$. Let $$f:X\to \operatorname{Spec} A$$ be a proper morphism.

Definition. For a quasicoherent sheaf $$\mathcal F$$on $$X$$, define $$I^n\mathcal F=\operatorname{Im}(f^*I^n\otimes_{\mathcal O_X}\mathcal F\to \mathcal F)$$.

In this Stacks tag the authors write $$f^*\widetilde B=\oplus_{n\ge 0}I^n\mathcal O_X$$ without any explanation:

Suppose $$\operatorname{Spec} C\subset X$$ is an open affine subset, the section of $$f^*\widetilde B$$ over $$\operatorname{Spec} C$$ is $$\oplus_{n\ge0} C\otimes_A I^n$$, and with $$f^*\widetilde B$$ identifying to $$\oplus_{n\ge 0}I^n\mathcal O_X$$ we have $$\oplus_{n\ge0} C\otimes_A I^n=\operatorname{Im}(\oplus _{n\ge 0}C\otimes_AI^n\to C)$$, but this doesn't look correct. Anyone can help?

• Please use \operatorname{Spec} to format $\operatorname{Spec}$: this produces better spacing, and I've made the upgrade in your post this time. Do also note that pictures of text are discouraged on this network - please consider replacing your images with text. Commented Sep 14, 2020 at 19:27
• @ KReiser thank you to point it out,I have made some edits. Commented Sep 15, 2020 at 3:29

Note that $$\oplus_{n\ge0} C\otimes_A I^n=\oplus _{n\ge 0}\operatorname{Im}(C\otimes_AI^n\to C)$$ rather than $$\operatorname{Im}(\oplus _{n\ge 0}C\otimes_AI^n\to C)$$. The latter makes little sense.
The notation $$\operatorname{Im}(C\otimes_AI^n\to C)$$ is nothing but the ideal of $$C$$ generated by the image of $$I^n$$ via the obvious map $$A\to C$$. The latter coincides with $$C\otimes_A I^n$$, as predicted.