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

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.