free. The following will be useful further ahead.

\[21.02.2019–1\] In the above notation, a coherent module ${{\mathscr{F}}}\in{\mathbf{coh}}(X)$ belongs to ${\mathfrak{T}}_{\varphi}$ if and only if ${\varphi}^*({{\mathscr{E}}})$ is trivial relatively to $B$.

We only show the “if” clause. In this case, ${\varphi}^*({{\mathscr{F}}})\simeq{{\mathscr{O}}}_Y{\otimes}_BF$ and using the point $y_0$, we conclude that $F\simeq {\varphi}^*({{\mathscr{F}}})|_{y_0}$. Now, since $y_0$ is taken to the $A$-point $x_0$, we have that ${\varphi}^*({{\mathscr{F}}})|_{y_0}\simeq B{\otimes}_A({{\mathscr{F}}}|_{x_0})$ so that ${\varphi}^*({{\mathscr{F}}})\simeq {{\mathscr{O}}}_Y{\otimes}_A({{\mathscr{F}}}|_{x_0})$.

We now give ourselves an object ${{\mathscr{E}}}$ of ${\mathfrak{T}}_{\varphi}$: this means that ${\varphi}^*({{\mathscr{E}}})$ is isomorphic to the pull-back of a certain $A$-module to $Y$. In what follows we wish to find more propitious candidates for the latter isomorphism and module so to achieve a proof of Proposition \[05.04.2018–3\] below. This Proposition is a key step in endowing the category ${\mathfrak{T}}_{\varphi}$ with kernels and cokernels (see Section \[04.07.2018–1\]). To have a better hold of the subtlety behind Proposition \[05.04.2018–3\], the reader is asked to read Remark \[01.02.2019–1\] below.

As the category of finite $A$-modules is rather simple, we have:

\[20.02.2019–4\] Let $E={{\mathscr{E}}}|_{x_0}$. Then the ${{\mathscr{O}}}_{Y}$-modules ${{\mathscr{O}}}_{Y}{\otimes}_AE$ and ${\varphi}^*({{\mathscr{E}}})$ are (non-canonically) isomorphic.

We already know that ${\varphi}^*({{\mathscr{E}}})\simeq{{\mathscr{O}}}_{Y}{\otimes}_A{\overline{E}}$ for [*some*]{} $A$-module ${\overline{E}}$, and all we are required to show is that ${\overline{E}}\simeq E$. Using the 