Q: Specific Example of a Morphism of Schemes for which the Push-Pull Morphism is not an Isomorphism Consider a Cartesian diagram of schemes as follows:
$\require{AMScd}
\begin{CD}
X \times_Z Y @>{\tilde{\pi}}>> Y\\
@VV{\tilde{\phi}}V @VV{\phi}V\\
X @>{\pi}>> Z
\end{CD}$
From the isomorphism $\phi_*\tilde{\pi}_*\cong \pi_*\tilde{\phi}_*$, for any quasicoherent sheaf $F \in QCoh(Y)$, one obtains by adjunction the push-pull map $\pi^*\phi_{*}(F) \to \tilde{\phi}_*\tilde{\pi}^*(F)$. There are many sources which claim that this map is not an isomorphism, thus motivating the need for derived algebraic geometry (for example, the fourth page in this section of Gaitsgory-Rozenblyum). However, is there a specific example where $\phi$ is quasicompact and this morphism fails to be an isomorphism?
I have read through the example where $Y$ is the countable disjoint union of points and $X = \mathbb{A}^1$, but because pushforward doesn't send quasicoherent sheaves to quasicoherent sheaves in this case, this doesn't seem (to me) to be one of the problems that derived algebraic geometry fixes.
