ev}}_{y_0}\left(H^0({\alpha})\tau(1{\otimes}e)\right)=1{\otimes}e$, which means that $H^0({\alpha})\tau$ sends $1{\otimes}E$ into the conservative sections, so that we only need to show that *any* conservative section $s$ is of the form $H^0({\alpha})\tau(1{\otimes}e)$. We write $s=H^0({\alpha})\tau(v)$ with $v\in B{\otimes}_AE$. As $s$ is conservative $\iota{{\rm ev}}_{y_0}(s)= 1{\otimes}e$ for some $e\in E$; as ${\alpha}$ is adapted, $\iota{{\rm ev}}_{y_0}\big(H^0({\alpha})\tau(v)\big)=v$, and hence $v=1{\otimes}e$.

Granted these preparations, we can now have a better control on kernels and cokernels in ${\mathfrak{T}}_{\varphi}$.

\[05.04.2018–3\] Let ${\overline{{{\mathscr{E}}}}}$ be another object of ${\mathfrak{T}}_{\varphi}$ and $u:{{\mathscr{E}}}\to{\overline{{{\mathscr{E}}}}}$ a morphism of ${{\mathscr{O}}}_X$-modules. Write ${\overline{E}}$ for the $A$-module ${\overline{{{\mathscr{E}}}}}|_{x_0}$, ${\overline{\tau}}$ and ${\overline{\iota}}$ for the the canonical morphisms $B{\otimes}_A{\overline{E}}\to H^0({{\mathscr{O}}}_Y{\otimes}_A{\overline{E}})$ and ${\varphi}^*({\overline{{{\mathscr{E}}}}})|_{y_0}\to B{\otimes}_A{\overline{E}}$. Let ${\alpha}:{{\mathscr{O}}}_{Y}{\otimes}_AE\stackrel\sim\to{\varphi}^*({{\mathscr{E}}})$ and ${\overline{{\alpha}}}:{{\mathscr{O}}}_{Y}{\otimes}_A{\overline{E}}\stackrel\sim\to{\varphi}^*({\overline{{{\mathscr{E}}}}})$ be adapted to the point $y_0$. The following claims hold true.

1.  The arrow $H^0({\varphi}^*u)$ takes conservative sections to conservative sections.

2.  There exists a morphism of $A$-modules $u_0:E\to {\overline{E}}$ such that $$\xymatrix{{\varphi}^*{{\mathscr{E}}}\ar[rr]^{{\varphi}^*u}&& {\varphi}^*{\overline{{{\mathscr{E}}}}} 
    \\
    \ar[u]^{\alpha}{{\mathscr{O}}}_{Y}{\otimes}_AE\ar[rr]_-{{\mathrm{id}}{\underset{A}{\otimes}} u_0}&&{{\mathscr{O}}}_{Y}{\otimes}_A{\overline{E}}\ar[u]_{{\overline{{\alpha}}}}.
    }$$ commutes. Put differently, ${\varphi}^*(u)$ “descends” to $A$.

3.  The coherent sheaves ${\rm Ker}({\varphi}^*u)$ and ${\rm Coker}({\varphi}^*u)$ are trivial relatively to $A$.

\(1) Let $s\in H^0({\varphi}^*({{\mathscr{E}}}))$ be conservative at $y_0$, so that $\iota{\rm ev}_{y_0}(s)=1{\otimes}e$ for some $e\in E$. Now $${{\rm ev}}_{y_0}\left\{H^0({\varphi}^*u)(s)\right\}=({\varphi}^*u)|_{y_0}\left\{{{\rm ev}}_{y_0}(s)\right\}.$$ But, $${\overline{\iota}}\circ \left(({\varphi}^*u)|_{y_0}\right)=\left({\mathrm{id}}_B{\otimes}_Au|_{x_0}\right)\circ\iota,$$ which shows that ${\overline{\iota}}{{\rm ev}}_{y_0}\left\{H^0({\varphi}^*(u))(s)\right\}=\left({\mathrm{id}}_B{\otimes}_Au|_{x_0}\right)\circ\iota({{\rm ev}}_{y_0}(s))=\left({\mathrm{id}}_B{\otimes}_Au|_{x_0}\right)(1{\otimes}e)$.

\(2) Let $e\in E$. We 