know that $s_e:=H^0({\alpha})\tau(1{\otimes}e)$ is conservative at $y_0$ (Lemma \[26.03.2018–1\]), so that $
H^0\left( {\varphi}^*(u)\right) (s_e)$ is equally conservative (by part (1)). Employing again Lemma \[26.03.2018–1\], we guarantee that $$\label{05.09.2018--1}
H^0({\varphi}^*(u))(s_e)=H^0({\overline{{\alpha}}}){\overline{\tau}}(1{\otimes}u_0(e))$$ for some $u_0(e)\in {\overline{E}}$. Since ${\overline{E}}\to B{\otimes}_A{\overline{E}}$ is injective, $u_0(e)$ is uniquely determined. This allows us to define a map $u_0:E\to {\overline{E}}$ which is easily seen to be $A$-linear. Now, $$\xymatrix{{{\mathscr{O}}}_{Y}{\otimes}_AE\ar[rr]^{{\mathrm{id}}{\otimes}_Au_0}&&{{\mathscr{O}}}_{Y}{\otimes}_A{\overline{E}}}$$ sends $\tau(1{\otimes}e)$ to ${\overline{\tau}}(1{\otimes}u_0(e))$, and hence $$\xymatrix{ {\varphi}^*{{\mathscr{E}}}\ar[rrr]^{{\overline{{\alpha}}}\circ ({\mathrm{id}}{\otimes}_Au_0)\circ  {\alpha}^{-1}}&&& {\varphi}^*{\overline{{{\mathscr{E}}}}}}$$ sends $s_e=H^0({\alpha})\tau(1{\otimes}e)$ to $H^0({\overline{{\alpha}}}){\overline{\tau}}(1{\otimes}u_0(e))$. In conclusion, replacing ${\varphi}^*(u)$ by ${\overline{{\alpha}}}\circ({\rm id}{\otimes}_Au_0)\circ{\alpha}^{-1}$ in eq. still produces a true statement. Therefore ${\overline{{\alpha}}}\circ ({\mathrm{id}}{\otimes}_Au_0)\circ  {\alpha}^{-1}={\varphi}^*(u)$.

\(3) This follows from the fact that both kernel and cokernel of ${\mathrm{id}}{\otimes}_Au_0$ are trivial relative to $A$ since $Y$ is flat over $A$.

\[01.02.2019–1\]Let $X={\mathrm{Spec}\,}{{{\mathds{C}}\llbrackett\rrbracket}}$ and $Y={\mathrm{Spec}\,}B$, where $B={{\mathds{C}}\llbracket\sqrt t\rrbracket}$. Let ${{\mathscr{E}}}={{\mathscr{O}}}_X$ and define $v:{{\mathscr{O}}}_Y\to{{\mathscr{O}}}_Y$ as multiplication by $\sqrt t$. Clearly ${\rm Coker}(v)$ cannot be trivial relatively to $A$; of course this does not contradict Proposition \[05.04.2018–3\] since $v$ is not induced by any morphism ${{\mathscr{O}}}_X\to{{\mathscr{O}}}_X$.

Further properties of the category of coherent modules trivialized by a proper morphism {#04.07.2018--1}
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In this section, we let $X$ be a proper and flat $A$-scheme with [*reduced fibres*]{}, and $x_0$ an $A$-point of $X$. Let ${\varphi}:Y\to X$ 