$s\in H^0({\varphi}^*({{\mathscr{E}}}))$ is said to be *conservative at $y_0$* if it is taken to $1{\otimes}E$ under $$\xymatrix{   H^0({\varphi}^*({{\mathscr{E}}}))        \ar[rr]^{\iota{\rm ev}_{y_0}}  && B{\otimes}_AE. }$$

2.  An isomorphism ${\alpha}:{{\mathscr{O}}}_Y{\otimes}_AE\to{\varphi}^*({{\mathscr{E}}})$ is called [*adapted to $y_0$*]{} if the lower horizontal arrow in diagram is the identity.

Note that the canonical arrow $E\to 1{\otimes}E$ is injective by faithful flatness of $A\to B$ and [@matsumura 7.5(i), p.49].

\[23.03.2018–1\]Isomorphisms adapted to $y_0$ always exist.

We first show that for an automorphism of $B$-modules $$\xymatrix{H^0({{\mathscr{O}}}_{Y}{\otimes}_AE)\ar[rr]^c_\sim&&H^0({{\mathscr{O}}}_{Y}{\otimes}_AE)},$$ it is possible to find an automorphism of ${{\mathscr{O}}}_Y$-modules $$\xymatrix{{{\mathscr{O}}}_Y{\otimes}_AE\ar[rr]^{{\gamma}}_\sim&& {{\mathscr{O}}}_{Y}{\otimes}_AE}$$ such that $H^0({\gamma})=c$. Write $\tilde c=\tau^{-1}c\tau$; this is an automorphism of $B{\otimes}_AE$. Pulling $\tilde c$ back to $Y$ via the structure morphism, we find an automorphism ${\gamma}:{{\mathscr{O}}}_{Y}{\otimes}_AE\stackrel\sim\to{{\mathscr{O}}}_{Y}{\otimes}_AE$ such that $H^0({\gamma})(\tau(1{\otimes}e))=\tau(\tilde c(1{\otimes}e))$. This implies that $H^0({\gamma})(\tau(1{\otimes}e))= c\tau(1{\otimes}e)$. Since $\{\tau(1{\otimes}e)\,:\,e\in E\}$ is a set of generators of $H^0({{\mathscr{O}}}_{Y}{\otimes}_AE)$, we see that $H^0({\gamma})=c$.

We now choose any isomorphism ${\alpha}:{{\mathscr{O}}}_Y{\otimes}_AE\stackrel\sim\to{\varphi}^*({{\mathscr{E}}})$ and let $b$ be the composition $$\xymatrix{H^0({{\mathscr{O}}}_{Y}{\otimes}_AE)\ar[rrr]^{\tau\circ \iota{\rm ev}_{y_0}\circ H^0({\alpha})}&&& H^0({{\mathscr{O}}}_{Y}{\otimes}_AE)}.$$ Let ${\beta}:{{\mathscr{O}}}_{Y}{\otimes}_AE\stackrel\sim\to{{\mathscr{O}}}_{Y}{\otimes}_AE$ induce $b^{-1}$ on global sections; it follows that ${\alpha}{\beta}$ is adapted to $y_0$.

\[26.03.2018–1\] If ${\alpha}:{{\mathscr{O}}}_{Y}{\otimes}_AE\stackrel\sim\to{\varphi}^*({{\mathscr{E}}})$ is adapted to $y_0$, then $H^0({\alpha})\tau$ defines a bijection between $1{\otimes}_AE$ and the conservative sections of $H^0({\varphi}^*({{\mathscr{E}}}))$.

Because ${\alpha}$ is adapted to $y_0$, we see that $\iota{{\rm 