$B$-point $y_0$, we can say that $B{\otimes}_A
{\overline{E}}\simeq B{\otimes}_AE$ as $B$-modules. Now, agreeing to write $A(\ell)=A/(\pi^\ell)$, we have $$E\simeq A^r{\oplus}A({\delta}_1){\oplus}\cdots{\oplus}A({\delta}_m)\quad\text{and}\quad {\overline{E}}\simeq A^{{\overline{r}}}{\oplus}A({\overline{{\delta}}}_1){\oplus}\cdots{\oplus}A({\overline{{\delta}}}_{{\overline{m}}});$$ here $m$ is either a positive integer, in which case ${\delta}_1\ge\cdots\ge{\delta}_m$ are also positive integers, or $m=0$ and the factors $A({\delta}_i)$ are to be dropped, and analogous considerations are in force for ${\overline{E}}$. As $B$ is free over $A$, of rank $s\ge1$ say, we have, as $A$-modules, $$B{\otimes}_AE\simeq A^{rs}{\oplus}\underbrace{A({\delta}_1){\oplus}\cdots{\oplus}A({\delta}_1)}_{s}{\oplus}\cdots{\oplus}\underbrace{A({\delta}_m){\oplus}\cdots{\oplus}A({\delta}_m)}_{s}$$ and $$B{\otimes}_A{\overline{E}}\simeq A^{{\overline{r}}s}{\oplus}\underbrace{A({\overline{{\delta}}}_1){\oplus}\cdots{\oplus}A({\overline{{\delta}}}_1)}_{s}{\oplus}\cdots{\oplus}\underbrace{A({\overline{{\delta}}}_{{\overline{m}}}){\oplus}\cdots{\oplus}A({\overline{{\delta}}}_{{\overline{m}}})}_{s},$$ so that the isomorphism $B{\otimes}_AE\simeq B{\otimes}_A{\overline{E}}$ implies that $r={\overline{r}}$, $m={{\overline{m}}}$ and ${\delta}_i={\overline{{\delta}}}_i$.

From now on we write $$\boxed{E:={{\mathscr{E}}}|_{x_0}}$$ and let $$\boxed{
\tau:B{\otimes}_AE{\stackrel{\sim}{\longrightarrow}} H^0({{\mathscr{O}}}_Y{\otimes}_AE)
}$$ be the canonical isomorphism of $B$-modules.

Of course one can find many isomorphisms ${{\mathscr{O}}}_Y{\otimes}_AE\simeq{\varphi}^*({{\mathscr{E}}})$ and we now single out a special class of such. Using $y_0$ and the fact that it “extends” $x_0$, we arrive at a canonical isomorphism $${\varphi}^*({{\mathscr{E}}})|_{y_0} \xymatrix{\ar[rr]^{\iota}_\sim&&}B{\otimes}_AE$$ which, denoting by ${\rm ev}_{y_0}$ the “evaluation” $H^0({\varphi}^*({{\mathscr{E}}}))\to{\varphi}^*({{\mathscr{E}}})|_{y_0}$, allows us to introduce the arrow of $B$-modules $$\xymatrix{ H^0({\varphi}^*({{\mathscr{E}}})) \ar[rr]^{\iota{{\rm ev}}_{y_0}}&& B{\otimes}_AE.}$$ Because $B{\otimes}_AE\simeq H^0({{\mathscr{O}}}_Y{\otimes}_AE)$, the reader should note that *${{\rm ev}}_{y_0}$ is an isomorphism*. Hence, if ${\alpha}:{{\mathscr{O}}}_Y{\otimes}_AE\to{\varphi}^*({{\mathscr{E}}})$ is an isomorphism, we arrive at a commutative diagram $$\label{22.03.2018--1}
\xymatrix{H^0({{\mathscr{O}}}_Y{\otimes}_AE)\ar[rr]^-{H^0({\alpha})}&& H^0({\varphi}^*({{\mathscr{E}}}))\ar[d]^{\iota  {\rm ev}_{y_0}} \\ 
B{\otimes}_AE\ar[u]^\tau\ar[rr] && B{\otimes}_AE.
}$$

\[06.09.2018–1\]

1.  A global section 