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Let $f \colon X \to Y$ be a continuous map and $\mathcal{O}_X$ a sheaf (of sets) on $X$.

Question: Is the stalk $(f_*\mathcal{O}_X)_{f(p)}$ for $P \in X$ isomorphic to the stalk $\mathcal{O}_{X,P}$ ? If it is, how exactly does the isomorphism map elements of these stalks (which are equivalence classes of pairs $(U,g)$) to one another ? Is the isomorphism unique/universal/canonical in some sense?

Remark: I know (just) a little bit about adjunctions, but if the answer is related to the inverse image functor (which I don't feel familiar with) and the adjunction to the pushforward, please give me a detailed explanation. Thank you.

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No, there is no such isomorphism in general. For instance, let $X$ be a proper $k$-variety of positive dimension and $f:X\to Y=\textrm{pt}$ its structural morphism. Let $\mathcal O_X$ be the sheaf of regular functions on $X$. Then for any regular point (say) $P\in X$ the stalk $(f_\ast\mathcal O_X)_{f(P)}=(f_\ast\mathcal O_X)_\textrm{pt}=\mathcal O_X(X)=k$, but $\mathcal O_{X,P}$ is a ring of dimension $\dim X>0$.

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  • $\begingroup$ Interesting, I was hoping the answer would be yes. Because then, I could define the induced morphism on the stalks of a morphism of ringed spaces $f \colon X \to Y$ with comorphism $f^\# \colon \mathcal{O}_Y \to f_*\mathcal{O}_X$ to be the composition of $f^\#_{f(p)} \colon \mathcal{O}_{Y,f(p)} \to (f_*\mathcal{O}_X)_{f(p)}$ with the isomorphism to $\mathcal{O}_{X,p}$. Maybe in this situation there is such an isomorphism? $\endgroup$ – legacytron Jan 1 '15 at 14:26
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    $\begingroup$ @legacytron There is, at least, always a map $(f_*\mathcal{O}_X)_{f(p)} \to \mathcal{O}_{X, p}$ and you can compose this with $f^\sharp_{f(p)}$ to get the desired map. $\endgroup$ – Hoot Jan 1 '15 at 15:24
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    $\begingroup$ Yeah, you probably mean the map $(f_*\mathcal{O}_X)_{f(p)} \to \mathcal{O}_{X,p}$ given by $[(U,g)] \mapsto [(f^{-1}(U),g)]$. Is the finally resulting composition $\mathcal{O}_{Y,f(p)} \to \mathcal{O}_{X,p}$ then the one that is required to be a local homomorphism for $f$ to be a morphism of locally ringed spaces? $\endgroup$ – legacytron Jan 1 '15 at 15:48
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    $\begingroup$ @legacytron Yes. Very check-able! $\endgroup$ – Hoot Jan 4 '15 at 9:09

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