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

The tensor product of sheaves $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$ is the sheafification of the presheafEquivalently, 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 $V\subseteq Y$ byThe pullback of a sheaf of modules iswhere $f^{-1}\mathcal{E}$ 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 ringed-space morphism.

Solved by gpt-5.6-sol high.

The ${\mathcal{O}}_Y$-module $\mathcal{E}$ is a locally free sheaf of finite rank if every $y\in Y$ has an open neighborhood $V$ and a finite integer $r$ for whichThe rank is locally constant and is therefore constant on each connected component.

Solved by gpt-5.6-sol high.

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.