Proposition 4

Let A and B be topological spaces, $u: A\rightarrow B$ a continuous mapping, $\mathscr{G}$ a presheaf on B$,\mathscr{F}$ a sheaf on A.

For every presheaf morphism $\varphi :\mathscr{G}\rightarrow u_*\mathscr{F}$, there exists a unique sheaf morphism $\psi :u^*(\mathscr{G})\rightarrow \mathscr{F}$ such that $\varphi =u_*(\psi )\circ \beta_{\mathscr{G}}$.

In other words, the canonical mapping

Mor($u^*(\mathscr{G}),\mathscr{F}$)$\rightarrow$ Mor($\mathscr{G}, u_*(\mathscr{F})$)$,\psi \mapsto u_*(\psi )\circ \beta_{\mathscr{G}}$

is a bijection.

With the notation of proposition 4, one will sometimes denote $\psi =\varphi^{\sharp}$ and $\varphi =\psi^{\flat}$.

Let us prove proposition 4. By remark 2 applied to the morphism $\sigma_{\mathscr{G}}:\mathscr{G}\rightarrow \widetilde{\mathscr{G}}$, the morphism $\beta_{\mathscr{G}}$ is equal to the composition

where $u^*(\sigma_{\mathscr{G}}):u^*(\mathscr{G})\rightarrow u^*(\widetilde{\mathscr{G}})$ is the canonical isomorphism of the remark in I, p. 58. Let $\widetilde{\varphi}:\widetilde{\mathscr{G}}\rightarrow u_*\mathscr{F}$ be the unique sheaf morphism such that $\widetilde{\varphi}\circ \sigma_{\mathscr{G}}=\varphi ($I, p. 54, prop. 3). It is then enough to prove that there exists a unique sheaf morphism $\widetilde{\psi}:u^*(\mathscr{G})\rightarrow \mathscr{F}$ such that $u_*(\widetilde{\psi})\circ \beta_{\mathscr{G}}=\widetilde{\varphi}$.

We p$\widetilde{o}$ssibly may therefore suppose that $\mathscr{G}$ is a sheaf. In order that a sheaf morphism $\psi :u^*(\mathscr{G})\rightarrow \mathscr{F}$ satisfy the conclusion of proposition 4, it is necessary and sufficient that for every open set V of B and every section $t$ of $E_{\mathscr{G}}$ over V, one have

Let $U_0$ be an open set of A and $s_0$ an element of $u^*(\mathscr{G})(U_0)$, in other words a B-morphism from $U_0$ into $E_{\mathscr{G}}$. Let $S(U_0, s_0)$ be the set of triples $(U,V, t)$ where U is an open set of A contained in $U_0$, V is an open set of B such that $u(U)\subset V$, and $t$ a section of $E_{\mathscr{G}}$ over V such that one has

If $U_1$ and $U_2$ are open sets of A with $U_1\subset U_2$, let us denote by $f_{U_1U_2}$ the restriction mapping $\mathscr{F}(U_2)\rightarrow \mathscr{F}(U_1)$. For every $(U,V, t)\in S(U_0, s_0)$, one then has the relation

Consequently, if $\psi :u^*(\mathscr{G})\rightarrow \mathscr{F}$ satisfies (6), one has

Let us prove that, for every point $a$ of $U_0$, there exists a triple $(U,V, t)\in S(U_0, s_0)$ such that $a\in U$. For indeed, let $a$ be a point of $U_0$. There exists an open neighbourhood V of B containing $u(a)$ and a section $t$ of the étale space $E_{\mathscr{G}}$ over V such that $t(u(a)) =s_0(a)$ (I, p. 33, prop. 9). Let $U_1=\overset{-1}{u}(V)\cap U_0$. The sections $s_0|U_1$ and $t\circ u|U_1$ of the $U_1$-étale space $E_{\mathscr{G}}\times_BU_1$ coincide at the point $a$. By proposition 11, b) of I, p. 34, the set of points where they coincide is an open set U of $U_1$ containing $a$. The triple $(U,V, t)$ then belongs to $S(U_0, s_0)$.

Formula (8) and property $(F_1)$ of sheaves (I, p. 43) then imply the uniqueness of $\psi$.

Let $(U,V, t)$ and $(U',V', t')$ be elements of $S(U_0, s_0)$. By relation (7), the restrictions of $t$ and $t'$ to $u(U\cap U')$ coincide. By Prop. 11, b) of I, p. 34, there exists an open set W of B such that $u(U\cap U')\subset$ $W\subset V\cap V'$ and $t|W =t'|W$. Hence

whence

By properties $(F_1)$ and $(F_2)$ for the sheaf $\mathscr{F}$, there exists a unique element $s'$ of $\mathscr{F}(U_0)$ such that for every triple $(U,V, t)$ of $S(U_0, s_0)$, one has:

Let us denote this element by $\psi_{U_0}(s_0)$.

Let $U_1$ be an open set contained in $U_0$ and let $s_1=s_0|U_1$. If $(U,V, t)\in$ $S(U_1, s_1)$, U is an open set contained in $U_0$ and $t\circ u|U =s_1|U =s_0|U$, hence $(U, v, t)\in S(U_0, s_0)$ and relation (10) then implies that

By definition of $\psi_{U_1}(s_1)$, one therefore has $\psi_{U_1}(s_1) =f_{U_1U_0}(\psi_{U_0}(s_0))$. This proves that the family $\psi = (\psi_U)$ is a morphism of sheaves from $u^*(\mathscr{G})$ into $\mathscr{F}$.

Let us prove that $\psi$ satisfies relation (6). Let V thus be an open set of B and let $t$ be a section of $E_{\mathscr{G}}$ over V. If U = $\overset{-1}{u}(V)$ and if $s=t\circ u|U$, the triple $(U,V, t)$ belongs to $S(U, s)$ and relation (6) is an immediate consequence of relation (10), applied to $U = U_0$.