The <tensor product of sheaves> $\mathcal F\otimes_{\mathcal O_X}\mathcal G$ is the sheafification of the presheaf
$$
U\longmapsto\mathcal F(U)\otimes_{\mathcal O_X(U)}\mathcal G(U).
$$
Equivalently, its stalk at $x$ is $\mathcal F_x\otimes_{\mathcal O_{X,x}}\mathcal G_x$.

The <direct image sheaf> is defined on each open set $V\subseteq Y$ by
$$
(f_*\mathcal F)(V)=\mathcal F(f^{-1}V).
$$
The <pullback of a sheaf of modules> is
$$
f^*\mathcal E=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal E,
$$
where $f^{-1}\mathcal E$ is the <inverse image sheaf> and the tensor product uses the structural morphism $f^{-1}\mathcal O_Y\to\mathcal O_X$ of the given <morphism of ringed spaces>[ringed-space morphism].

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 113 / 2 / a / Solution

{wiki=Ringed_space#Morphisms_of_ringed_spaces}

A morphism of ringed spaces $f:(X,\mathcal O_X)\to(Y,\mathcal O_Y)$ consists of a continuous map $f:X\to Y$ and a morphism of <sheaf of rings>[sheaves of rings] $f^{-1}\mathcal O_Y\to\mathcal O_X$.

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 Incoming links: Morphism of ringed spaces