trivial relatively to $A$ if and only if it is trivial in according to our conventions in Section \[notations\_conventions\].

In the above definition, it is to be expected that several different choices concerning the “descended” module are possible. The next lemma, which we state here for future use, explains how to be more canonical.

\[29.03.2018–1\]Let $Y$ be proper and $H^0$-flat over $A$. Write $B=H^0({{\mathscr{O}}}_Y)$ and let $F$ be a finite $A$-module. Then the canonical morphism $${{\mathscr{O}}}_Y{\underset{B}{\otimes}}H^0({{\mathscr{O}}}_Y{\otimes}_AF){\longrightarrow}{{\mathscr{O}}}_Y{\otimes}_AF$$ is an isomorphism.

By definition of $H^0$-flatness, the canonical arrow $${\sigma}:B{\otimes}_AF{\longrightarrow}H^0({{\mathscr{O}}}_Y{\otimes}_AF)$$ is bijective. We then consider the commutative diagram $$\xymatrix{{{\mathscr{O}}}_Y{\underset{B}{\otimes}}H^0({{\mathscr{O}}}_Y{\otimes}_AF)\ar[rr]&&{{\mathscr{O}}}_Y{\otimes}_AF\\{{\mathscr{O}}}_Y{\underset{B}{\otimes}}(B{\otimes}_AF)\ar[rru]_\sim\ar[u]^-{\sim}_-{{\rm id}{\otimes}{\sigma}}}$$ where the horizontal arrow is the one of the statement.

Here is the central definition of this section.

\[05.09.2018–3\]If ${\varphi}:Y\to X$ is a morphism of $A$-schemes, we let ${\mathfrak{T}}_{\varphi}$ stand for the *full* subcategory of ${\mathbf{coh}}(X)$ consisting of those coherent sheaves ${{\mathscr{E}}}$ such that the coherent ${{\mathscr{O}}}_Y$-module ${\varphi}^*({{\mathscr{E}}})$ is trivial relatively to $A$.

Let $X$ be a connected, proper and flat $A$-scheme, and $x_0$ be an $A$-point of $X$. Let ${\varphi}:Y\to X$ an object of ${\mathfrak{S}}(X,x_0)$ (see Definition \[26.06.2018–1\]). If $B=H^0({{\mathscr{O}}}_Y)$, we let $y_0$ stand for the $B$-point of $Y$ above $x_0$. We note that the $A$-module $B$ is finite and torsion-free, hence 