$$\left(K[G]_{\rm right} \right)^{H_K} = K\quad\text{and}\quad \left(k[G]_{\rm right} \right)^{H_k}=k.$$ By Lemma \[29.11.2018–1\] we conclude that $u_K:H_K\to G_K$ and $u_k:H_k\to G_k$ are faithfully flat. Because of [@duong-hai18 Theorem 4.1.1], $u$ is faithfully flat.

(2)$\Rightarrow$(3). This is trivial.

(3)$\Rightarrow$(1). Any $a\in  A[G]_{\rm right}$, respectively $b\in k[G]_{\rm right}$, belongs to a certain $V\in{\mathfrak{s}}(G)$, respectively $M\in{\mathfrak{s}}_0(G)$ because of “local finiteness” [@serre68 1.5, Corollary, p.40]. The conclusion then follows.

Modules trivialized by a proper and $H^0$-flat scheme {#14.04.2018--3}
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In this section we introduce the category of coherent sheaves on which all further developments hinge: the category of sheaves which became “trivial” after a pull-back by a proper morphisms. Since we wish to work with schemes over a d.v.r., the notion of triviality of a coherent module becomes more subtle than the one over a field, and we need to account for modules coming from the base-ring. This is a source of difficulty specially because the “base” of the scheme effectuating the trivialisation might grow.

\[20.02.2019–3\]Let $Y$ be any $A$-scheme. We say that ${{\mathscr{F}}}\in{\mathbf{coh}}(Y)$ is trivial relatively to $A$ if there exists a coherent sheaf (a finite $A$-module) $F$ such that ${{\mathscr{F}}}={{\mathscr{O}}}_Y{\otimes}_AF$.

Note that, if $A$ is a field, then a coherent sheaf on $Y$ is 