$\Pi(X,x_0)$ is [*strictly*]{} pro-finite (Theorem \[15.03.2019–1\]). Finally, an indirect argument allows us to deduce from this that $\Pi(X,x_0)$ is [*strictly*]{} pro-finite even if $A$ fails to be complete: see Theorem \[25.03.2019–1\]. As a consequence, the ring of functions of $\Pi(X,x_0)$ is a Mittag-Leffler $A$-module (Corollary \[16.04.2019–2\]).

In Section \[20.03.2019–1\] we make a brief digression to exhibit examples and applications of the preceding theory to the case of invertible sheaves, see Proposition \[11.01.2019–1\]. These examples were a constant source of guidance while elaborating our rhetoric and we believe they shall be useful to the reader.

Section \[15.03.2019–2\] sets out to give conditions for a finite group scheme—appearing as the structure group of a torsor over $X$—to be a quotient of $\Pi(X,x_0)$, see Proposition \[29.11.2018–2\] and Corollary \[27.03.2019–1\].

In Section \[06.03.2019–1\] we review parts of [@mehta-subramanian13]. Let us explain the reason for offering such a revision before summarising the contents of Section \[06.03.2019–1\]. If $X_0$ stands for the special fibre of $X$, one of the main points of [@mehta-subramanian13] is to use a technique in [@deninger-werner05], see pages 574 to 576 there, to show, speaking colloquially, how to trivialise a vector bundle $E\in{\mathbf{VB}}(X_0)$ on the special fibre of a [*flat*]{} $A$-scheme $Y$ 