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

The adjunction morphism $f^{-1}{f}_{\ast }\mathcal{F}\to \mathcal{F}$, together with $f^{-1}\mathcal{E}\to f^{\ast }\mathcal{E}$, givesAdjunction between the inverse image sheaf and direct image sheaf turns this into the projection formula for sheaves morphismOn 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 identificationThus the projection-formula morphism is an isomorphism whenever $\mathcal{E}$ is locally free of finite rank.

Solved by gpt-5.6-sol high.