# Pullback morphism of structure sheaves induced by a morphism of schemes

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?

• To give a morphism of schemes, is giving among other things a morphism $O_Y \to f_* O_X$ as $f^{-1}$ is adjoint do $f_*$ in the category of sheaves, you have such a morphism.
– Ahr
Jun 4, 2018 at 11:27

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 $\psi' : f^{-1}O_Y \to O_X$.

— 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 $f^{-1}(V) \supseteq f^{-1}(f(U)) \supseteq U$).

• Ah I see. I was just gonna ask about what the map was explicitly. Can you perhaps describe the map in the case $f$ isn't an open immersion as I don't think mine is? Jun 4, 2018 at 12:13
• Take $V$ an open set of $Y$ et $U \subset X$ an open set mapped in $V\subset Y$ by $f$, then you have a map $O_Y(V)\to O_X( f^{-1} (V))\to O_X(U)$ the last map is just the restriction, the first one is given by the morphism of schemes it self. As the presheaf inverse image is defined by taking the colimit of such $O_Y(V)$ you get a map of presheaves and thus a map of sheaves in the sheafification.
– Ahr
Jun 4, 2018 at 12:38
• @Fromage : the above comment seems to be relevant. Jun 4, 2018 at 12:39
• @A.Rod thanks very much both, this is exactly what I was looking for. Jun 4, 2018 at 13:51