# When is $f^{-1}(\tilde{M})$ a quasi-coherent sheaf?

To be precise, I want to know whether the following statement is true or false:

Let $A$ be a ring (it can be reduced), $f:\rm{Spec}(A/I) \to \rm{Spec}(A)$ is a closed immersion, $\tilde{M}$ is a quasi-coherent sheaf of $\mathcal{O}_{\rm{Spec}(A)}$-module, is it true $f^{-1}(\tilde{M})$ is also a quasi-coherent sheaf of $\mathcal{O}_{\rm{Spec}(A/I)}$-module?

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No, it is even not a module over $O_{\mathrm{Spec}(A/I)}$ in general. Actually let $F=\tilde{M}$. The stalk $f^{-1}(F)_x=F_{f(x)}$ and the RHS is not a $A/I$-module (not killed by $I$) in general. –  user18119 Jan 10 '12 at 20:43
Yes, I see. Thank you! –  Li Zhan Jan 11 '12 at 17:58