is denoted by $A_n$.

(2) Given an object $W$ over $A$ (a scheme, a module, etc) and an $A$-algebra $B$, we find useful to write $W_B$ instead of $W{\otimes}_AB$. If context prevents any misunderstanding, we also employ $W_n$ instead of $W_{A_n}$.

(3) The characteristic of a discrete valuation ring is the couple $(r,p)$, where $r$ is the characteristic of the field of fractions and $p$ that of the residue field.

 On general algebraic geometry {#on-general-algebraic-geometry .unnumbered}
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(1) A [*vector bundle*]{} over a scheme is a locally free sheaf of finite rank. A vector bundle is said to be trivial if it is isomorphic to a direct sum of a copies of the structure sheaf (in particular the rank is constant).

(2) Let $R$ be a noetherian ring and $Y$ a proper $R$-scheme. We shall say that $Y$ is [*$H^0$-flat over $R$*]{} if it is flat and cohomologically flat [@bosch-lutkebohmert-raynaud90 p.206] in degree zero over ${\mathrm{Spec}\,}R$. In this work we shall employ constantly that if $Y$ is $R$-flat, then (a) $H^0$-flatness amounts to exactness of $M\mapsto H^0({{\mathscr{O}}}_Y{\otimes}_AM)$ [@hartshorne77 III.12.5 and III.12.6] and (b) if $Y$ the fibres of $Y$ are geometrically reduced [\[EGA $\textsc{IV}_{2}$, 4.6.2, p.68\]]{}, then $Y$ is $H^0$-flat 