Proposition 6

Let

${E'}^{f'}$ E

$p'p$

${B'}^f$ B

be a cartesian square. Suppose that the mapping $f$ is proper, separated and surjective, and make one of the following assumptions:

(i) The fibres of $f$ are locally connected and the fibre square $B'\times_BB'$ is locally connected.

(ii) The fibres of $f$ are finite, the diagonal $\Delta_{B'}$ of $B'\times_BB'$ is open in $B'\times_BB'$ and $B'\times_BB'-\Delta_{B'}$ is a locally connected space.

Then, if $(E', p')$ is a covering, $(E, p)$ is a covering.

Since the mapping $f$ is universally strict (I, p. 20, corollary), the mapping $p$ is étale (I, p. 31, prop. 8) and separated (I, p. 27, prop. 4). We shall suppose, which is permissible, that $E'= B'\times_BE$.

Let $a$ be a point of B; it is a question of proving that the point $a$ has a neighbourhood W such that the W-space $(E_W, p_W)$ is a covering

which is trivializable. Put $B'_a$ = $\overset{-1}{f}(a)$ and denote by $E'_a$ = $(^-{p'}^1)(E_a)$. The mapping $t_a: E'_a\rightarrow B'_a\times E_a$ defined by $t_a(y) = (p'(y), f'(y))$ is a $B'_a$-isomorphism (I, p. 9, prop. 4), hence $E'_a$ is a trivializable covering of $B'_a$ and $t_a$ is a trivialization of this covering.

Let us prove that there exists a neighbourhood $V'$ of $B'_a$ in $B'$ and a continuous trivialization $t$ of the covering $(E'_{V'}, p_{V'})$ extending $t_a$. Under assumption (ii), $B'_a$ is finite and its points have pairwise disjoint open neighbourhoods over which the covering $E'$ is trivializable, whence the assertion in this case. Under assumption (i), $B'_a$ is locally connected, and so is $B'($IV, p. 390, lemma 1); since the mapping $f$ is proper and separated, $B'_a$ is compact and two distinct points have disjoint neighbourhoods in $B'$, so that the pair $(B',B'_a)$ satisfies property (PCV) ( I, p. 37, lemma 1). The assertion therefore follows from Cor. 2 of I, p. 90.

As $f$ is proper, there exists a neighbourhood V of $a$ in B such that $V'$ contains $\overset{-1}{f}(V)$ (lemma, I, p. 75). We may thus suppose that

Let us endow the $V'$-spaces $V'\times E_a$ and $E'_{V'}= V'\times_BE$ with their canonical descent data relative to $f_V: V'\rightarrow V$. We shall now prove that, after shrinking V and $V'$, the isomorphism of $V'$-spaces $t: E'_{V'}\rightarrow$ $V'\times E_a$ which we have just defined is compatible with the descent data, that is to say that one has $t(b'_1, x) =t(b'_2, x)$ if $(b'_1, b'_2)\in V'\times_VV'$ and $x\in E_{f(b'_1)}$. Let $\widetilde{t}$ denote the mapping pr$_2\circ t: E'_{V'}\rightarrow E_a$.

Put $V''= V'\times_VV'$; let us consider it as a $V'$-space by means of the first projection. The mapping $((b'_1, b'_2), x)\mapsto ((b'_1, b'_2),(b'_1, x))$ of $V''\times_VE$ into $V''\times_{V'}E'$ is an isomorphism of $V''$-spaces; this shows that $V''\times_VE$ is a covering of $V''$.

For $i= 1$, 2, define a mapping $u_i: V''\times_VE\rightarrow V''\times E_a$ by setting $u_i(b'_1, b'_2, x) = (b'_1, b'_2,\widetilde{t}(b'_i, x))$; these are trivializations of the covering $V''\times_VE$. Let $W''$ be the set of points $w\in V''$ over which these trivializations $u_1$ and $u_2$ coincide; it contains $B'_a\times B'_a$, as well as the diagonal $\Delta_{V'}$. Let us prove that $W''$ is a neighbourhood of $B'_a\times B'_a$. Under assumption (i), this follows from Cor. 2 of I, p. 80, since $B'\times B'$ is locally connected. Under assumption (ii), $W''$ contains a neighbourhood of $(B'_a\times B'_a)-\Delta_{B'_a}$ in $(B'\times_BB')-\Delta_{B'}$, for this set is locally connected (loc. cit.). As $\Delta_{V'}$ is open in $B'\times_BB',W''$ is a neighbourhood of $B'_a\times B'_a$ in $V''$.

As $f$ is proper, the canonical mapping $f''$ of $B'\times_BB'$ into B is proper, for it is the composite of the projection pr$_1: B'\times_BB'\rightarrow B'$ and the mapping $f$.

By the lemma of I, p. 75, there exists a neighbourhood W of $a$ in V

such that $(f\overset{-1}{''})(W)$ is contained in $W''$; let us put $W'=(\overset{-1}{f}')(W)$; it is a subset of $V'$ and the isomorphism of étale spaces $t: E'_{W'}\rightarrow W'\times E_a$ is compatible with the canonical descent data relative to the mapping $f_W: W'\rightarrow W$. It follows from corollary (IV, p. 388) that the W-étale spaces $E_W$ and $W\times E_a$ are isomorphic. In particular, $E_W$ is a trivializable covering, whence the proposition.