The <direct image sheaf> is determined on the basis consisting of <principal open subschemes> $D(b)\subseteq\operatorname{Spec}B$. Their inverse images are
$$
f^{-1}D(b)=D(\varphi(b))\subseteq\operatorname{Spec}A,
$$
and hence
$$
f_*\widetilde M(D(b))
=\widetilde M(D(\varphi(b)))
=M_{\varphi(b)}.
$$
On the other hand, localization of the restricted-scalar $B$-module gives
$$
(M_B)_b\cong M_{\varphi(b)}.
$$
These isomorphisms commute with restriction to smaller principal opens, so they define the equality of <quasi-coherent sheaves>
$$
f_*\widetilde M=\widetilde{M_B}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-113/3/a/solution