$\newcommand{\Zar}{\mathrm{Zar}}\newcommand{\Sh}{\mathrm{Sh}}$The category of schemes is a full subcategory of the big Zariski topos $\Zar$. For each scheme $X$, there is a local geometric morphism $\pi : \Zar/X\to \Sh(X)$. I denote the associated functors by $\pi^{-1} \dashv \pi_\ast \dashv \pi^!$. The pushforward (which sends an $X$-space to its sheaf of sections) sends $\mathbb A_X$-modules to $\mathcal O_X$-modules. We may define a left adjoint $\pi^\ast$ by setting $$\pi^\ast \mathcal F = \pi^{-1} \mathcal F\otimes_{\pi^{-1}\mathcal O_X}\mathbb A_X$$

The same set-up should also work for other topologies (and the associated big and small topoi).

Given a morphism $f:X\to Y$ of schemes, we get a commutative square of geometric morphisms as shown below. $\require{AMScd}$ \begin{CD} \Zar/X @>f>> \Zar/Y\\ @V \pi V V @VV \pi V\\ \Sh(X) @>>f> \Sh(Y) \end{CD} Associated to that diagram are two Beck-Chevalley conditions. There are natural comparison cells $f^{-1}\pi_\ast \Rightarrow \pi_\ast f^{-1}$ and $\pi^{-1}f_\ast \Rightarrow f_\ast \pi^{-1}$. Under which conditions on $f$ are those comparison cells natural isomorphisms? Which components of the comparison cells are isomorphisms? Similarly, under which conditions do the two Beck-Chevalley conditions for the functors $\pi^\ast \dashv \pi_\ast$ and $f^\ast \dashv f_\ast$ between the categories of internal modules hold?

I am looking in fo a reference, I have not been able to find anything useful yet. The stacks-project contains all the definition, but nothing about the Beck-Chevalley conditions as far as I can tell. I already know that they will sometimes fail, and I am especially interested in positive results.

Edit. Here are some special cases, which I have already solved. For some examples I will need the following construction. If $\mathcal F$ is a quasicoherent $\mathcal O_X$-module, then I write $V(\mathcal F)$ to denote the associated $\mathbb A_X$-module $Spec_X(Sym\,\mathcal F)\to X$. The functor $V$ is a contravariant fully faithful embedding $QCoh(X)\to Mod(\mathbb A_X)$ and there is an adjoint $\mathcal L$ which sends an $\mathbb A_X$-module $W$ to its sheaf $\mathcal L(W)$ of linear functions. It is the case that $\mathcal L(W) = \pi_\ast(W^\vee)$.

  • The Beck-Chevalley morphism $f^\ast \pi_\ast E\to \pi_\ast f^\ast E$ is an isomorphism for all $\mathbb A_Y$-modules of the form $E = \pi^\ast \mathcal E$. This is simply because $f^\ast \pi_\ast \pi^\ast = f^\ast = \pi_\ast \pi^\ast f^\ast = \pi_\ast f^\ast \pi^\ast$.

  • The Beck-Chevalley morphism $f^\ast \pi_\ast E \to \pi_\ast f^\ast E$ need not be an isomorphism for all $\mathbb A_Y$-modules of the form $E = V(\mathcal E)$. To see this, look at the example $Z=V(xy) \subset \mathbb A^2_\mathbb k$ where $k$ is a field. Let $p: Spec(k)\to Z$ be the inclusion of the origin. Then we may compute that $\pi_\ast p^\ast V(\Omega_{Z/k})$ is $((\Omega_{Z/k})_p\otimes_{\mathcal O_{Z,p}}{\kappa(p)})^\vee$, while $p^\ast \pi_\ast V(\Omega_{Z/k})$ is equal to $\Omega_{Z/k}^\vee \otimes_{\mathcal O_{X,p}}\kappa(p)$ and those won't be the same.

  • The Beck-Chevalley condition $i_U^\ast\pi_\ast= \pi_\ast i_U^\ast$ holds when $i_U$ is the embedding of an open subscheme.

  • If $g_p : Spec(\mathcal O_{X,p})\to X$ is the inclusion of a germ into $X$ and $E = V(\mathcal E)$ for a coherent sheaf $\mathcal E$, then the Beck-Chevalley condition $g_p^\ast \pi_\ast = \pi_\ast g_p^\ast$ also holds. This is a consequence of the fact that $(\mathcal E^\vee)_p = (\mathcal E_p)^\vee$ when $\mathcal E$ is coherent.

  • When $\mathcal E$ is locally free on $X$, then $V(\mathcal E) = \pi^\ast(\mathcal E^\vee)$ and the Beck-Chevalley condition holds for any morphism into $X$.

I'd like to learn about more (and less elementary) examples, if someone happes to know some. :)

  • 2
    $\begingroup$ There are positive results regarding Beck–Chevalley conditions in Johnstone's Elephant. If your diagram would be a pullback diagram, then you could use that $\pi$ is local (in particular tidy) to show that the BC condition $f^{-1}\pi_* \simeq \pi_* f^{-1}$ holds (Corollary C3.4.11 in the Elephant). $\endgroup$ Feb 16 at 14:11
  • 3
    $\begingroup$ Unfortunately, the diagram you mention is not typically a pullback diagram. Take for example $Y=\mathrm{Spec}(K)$ and $X = \mathrm{Spec}(L)$ for $L/K$ a field extension. Then you get an equivalence at the level of small toposes but not at the level of big toposes. The BC condition $f^{-1}\pi_* \simeq \pi_*f^{-1}$ also fails in this case. $\endgroup$ Feb 16 at 14:11


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