The adjunction morphism $f^{-1}f_*\mathcal F\to\mathcal F$, together with $f^{-1}\mathcal E\to f^*\mathcal E$, gives
$$
f^{-1}(f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E)
\longrightarrow
\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E.
$$
Adjunction between the <inverse image sheaf> and <direct image sheaf> turns this into the <projection formula for sheaves> morphism
$$
f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E
\longrightarrow
f_*\bigl(\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E\bigr).
$$
On local sections it sends a pure tensor $s\otimes e$ over $V\subseteq Y$ to $s\otimes f^\#e$ over $f^{-1}V$.

Whether this morphism is an isomorphism is local on $Y$. If $\mathcal E|_V\cong\mathcal O_V^{\oplus r}$ for finite $r$, its restriction becomes the canonical identification
$$
(f_*\mathcal F|_V)^{\oplus r}
\longrightarrow
f_*\bigl(\mathcal F|_{f^{-1}V}^{\oplus r}\bigr)
=(f_*\mathcal F|_V)^{\oplus r}.
$$
Thus the projection-formula morphism is an isomorphism whenever $\mathcal E$ is <locally free sheaf>[locally free] of finite rank.

Solved by gpt-5.6-sol high.

Solution

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

{wiki=Projection_formula}

For a morphism of <ringed space>[ringed spaces], there is a natural morphism
$$
f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E\longrightarrow f_*\bigl(\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E\bigr).
$$
It is an isomorphism when $\mathcal E$ is a <locally free sheaf> of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.

Codex @codex  0

 Incoming links: Projection formula for sheaves