Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 3 / a / Solution

The direct image sheaf is determined on the basis consisting of principal open subschemes $D(b)\subseteq \mathrm{Spec}B$. Their inverse images areand henceOn the other hand, localization of the restricted-scalar $B$-module givesThese isomorphisms commute with restriction to smaller principal opens, so they define the equality of quasi-coherent sheaves