# Aggressive geometry – "Basic change" on the map étale

Let $$f colon Y to X$$ a morphism of schemas. I note by $$f _ *$$ and $$f ^ *$$ pushing forward and removing sheaves of modules. Let $$U subseteq X$$ to be an open set (Zariski), and denote by $$i colon U to X$$ l & # 39; inclusion. Also, note by
$$begin {equation} f_U colon f ^ {- 1} (U) to U end {equation}$$
the map induced by $$f$$and denote by $$j colon f ^ {- 1} (U) to Y$$ l & # 39; inclusion. I think it's pretty immediate to show that, if $$mathscr F$$ is a sheaf of $$mathscr O_Y$$-modules, then we have an isomorphism of "base change"
$$begin {equation} i ^ * f_ * mathscr F cong {f_U} _ * j ^ * mathscr F end {equation}$$
of wreaths $$mathscr O_U$$-modules. After all, $$i$$ and $$j$$ are just inclusions of open sets, so LHS and RHS above are given by $$V mapsto mathscr F (f ^ {- 1} (V))$$, does not matter when $$V subseteq U$$.

Now, I wonder: and if we worked with a different topology, let's say the topology spreads? Assume everything as above but $$i colon U to X$$ a map spread, and interpreter $$f ^ {- 1} (U)$$ as the withdrawal $$Y times_ {X} U$$. Is the "base change" isomorphism above still valid? I would say "yes": if so, is there a reference for that, or even something more general?