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Let $X = \operatorname{Spec}(A)$ be the spectrum of a local ring A with maximal ideal $\mathfrak{m}$, closed immersion $i: \{\mathfrak{m}\} \hookrightarrow X$ and $\mathcal{F}$ a sheaf on $X$ with supports $\mathfrak{m}$. Then, we should have an isomorphism $i_*i^*\mathcal{F} \cong \mathcal{F}$.

But this can't be true: Take $A = \mathbf{Z}_{(p)}, \mathcal{F} = \mathbf{Z}/p^2$, then $i_*i^*\mathcal{F} = \mathbf{Z}/p$.

Where is the mistake in this?

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Isn't $i_* i^* \mathcal{F}$ just the skyscraper sheaf of $\mathbf{Z}/p^2$ concentrated at $\mathfrak{m}$? –  Zhen Lin Nov 24 '11 at 22:46
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