Proposition 5

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous mapping. Suppose that the mapping $f$ is proper and separated, or else open. Every descent datum relative to $f$ on a spread X-space is effective. Moreover, if $f$ is surjective, the quotient space is a spread Y-space.

Let $(Z, p)$ be an X-étalé space and let $\tau$ be a descent datum relative to $f$ on $(Z, p)$. Let us denote by $R_{\tau}$ the equivalence relation in Z associated with $\tau$, let $g: Z\rightarrow Z/R_{\tau}$ be the canonical surjection and let $h$ denote the mapping of Z into $X\times_Y(Z/R_{\tau})$ defined by $z\mapsto (p(z), g(z))$. It is continuous and bijective; let us prove that it is a homeomorphism.

a) Suppose first that the mapping $f$ is open.

Let $z_0$ be a point of Z; put $x_0=p(z_0)$. Since $p$ is étale, there exists a neighbourhood U of $x_0$ in X and a continuous section $s: U\rightarrow Z$ of $p$ over U such that $s(x_0) =z_0$. The mapping $p\times p: Z\times_YZ\rightarrow$ $X\times_YX$ is étale, and the mappings of $U\times_YU$ into $Z\times_YZ$ defined by $(x, x')\mapsto (s(x), s(x'))$ and $(x, x')\mapsto (s(x), \tau (s(x), x'))$ are continuous sections of it over $U\times_YY$. Since they coincide at every point of $U\times_YU$ of the form $(x, x)$, they therefore coincide on an open neighbourhood V of $\Delta_U$ contained in $U\times_YU$. Let $U_0$ be a neighbourhood of $x_0$ in X such that $U_0\times_YU_0$ is contained in V. Let $(x, u)$ be a point of $U_0\times_Yg(s(U_0))$; let $x'\in U_0$ be such that $u=g(s(x'))$. We have therefore $s(x) =\tau (s(x'), x)$, whence $R_{\tau}\{s(x), s(x')\}$ and $(x, u) = (x, g(s(x'))) =$ $(x, g(s(x))) =h(s(x))$.

Since the mapping $g$ is open (IV, p. 386, prop. 2), the set $U_0\times_Yg(s(U_0))$ is an open subset of $X\times_Y(Z/R_{\tau})$ on which the mapping $h^{-1}$ is equal to the continuous mapping $s\circ$ pr$_1$. The mapping $h$ is therefore a homeomorphism.

b) Suppose now that the mapping $f$ is proper and separated.

Let $z_0$ be a point of Z; put $x_0=p(z_0)$ and $y_0=f(x_0)$. The mapping $s$ given by $x\mapsto \tau (z_0, x)$ is a section of $p$ over $\overset{-1}{f}(y_0)$.

The set $\overset{-1}{f}(y_0)$ is compact (TG, I, p. 75, Theorem 1 and I, p. 26, Remark 2), and two distinct points of $\overset{-1}{f}(y_0)$ have disjoint neighbourhoods in X, since $f$ is separated (I, p. 25, Proposition 1). By Theorem 2 of I, p. 37, there therefore exists a neighbourhood $U_0$ of $\overset{-1}{f}(y_0)$ in X and a section $s_0$ of $p$ over $U_0$ extending $s$. The set $s_0(U_0)$ is an open set of Z, since $p$ is étale (I, p. 30, Corollary 3), and contains the saturated set of $\{z_0\}$ for the relation $R_{\tau}$. The mapping $g$ is closed (IV, p. 386, Proposition 2). There then exists an open set V of $Z/R_{\tau}$ such that $W =\overset{-1}{g}(V)\cap \overset{-1}{p}(U_0)$ is a neighbourhood of $z_0$ contained in $s_0(U_0)$ (I, p. 75, Lemma).

Let now $(x, u)\in U_0\times_YV$, and let $z$ be the unique point of Z such that $h(z) = (x, u)$; by definition, one has $z\in W$. Since $W\subset s_0(U_0)$, one has $z=s_0(p(x))$. This shows that the restriction of $h^{-1}$ to the open set $U_0\times_YV$ of $X\times_Y(Z/R_{\tau})$ is equal to $s_0\circ p\circ$ pr$_1$. Consequently, $h$ is a homeomorphism.

Thus we have shown that the descent datum $\tau$ is effective. The mapping $f$ is universally strict (I, p. 20, corollary). Under the assumption that $f$ is surjective, it then follows from Proposition 8 of I, p. 31 that $q: Z/R_{\tau}\rightarrow Y$ is étale.