Pullback morphism of structure sheaves induced by a morphism of schemes

Question

I'm led to believe there is a morphism of structure sheaves $f': f^{-1}\mathcal O_Y \rightarrow \mathcal O_X$ induced by a morphism of schemes $f:X \rightarrow Y$. Could someone describe this morphism to me or perhaps provide a reference?

Answer 1

A hint was provided in the comments.

Recall that a morphism of schemes is a morphism of locally ringed spaces, i.e. it is a pair $(\phi, \phi')$ where $\phi : |X| \to |Y|$ is continuous and $\phi' : O_Y \to f_*O_X$ is a morphism of sheaves of rings on $Y$.

The functor $f^{-1}$ is a left adjoint to $f_*$ (see exercise II.1.18 in Hartshorne's Algebraic geometry), i.e. we have natural bijections $$\mathrm{Hom}(f^{-1}G, F) \cong \mathrm{Hom}(G, f_*F)$$ where $F$ is a sheaf on $X$ anf $G$ is a sheaf on $Y$.

In particular, $\phi' \in \mathrm{Hom}(O_Y, f_*O_X)$ corresponds to a unique morphism of sheaves $\psi' : f^{-1}O_Y \to O_X$.

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— If $f : X \to Y$ is an open immersion, then for any open $U \subset X$, we have the following description : $$\psi'_U : (f^{-1}O_Y)(U) = O_Y(f(U)) \to O_X(U)$$ is equal to $$\phi'_{f(U)} : O_Y(f(U)) \to O_X[f^{-1}(f(U))] = O_X(U).$$

— More generally, the map $$\psi'_U : (f^{-1}O_Y)(U) = \varinjlim_{V \supseteq f(U), V \text{ open in } Y} O_Y(V) \longrightarrow O_X(U)$$ is induced by $$\mathrm{Res} \circ \phi'_V : O_Y(V) \to (f_*O_X)(V) = O_X(f^{-1}(V)) \to O_X(U)$$ whenever $V \supseteq f(U)$ (which implies $f^{-1}(V) \supseteq f^{-1}(f(U)) \supseteq U$).