All of the following comes from these notes by Bhatt. I apologize for my inexperience with topoi.
Fix flat ideal $I\subset R$ such that $I^2=I$. Define the category $\operatorname{Mod}^{a}_{R,I}$ of almost modules to be the Serre quotient of $\operatorname{Mod}_R$ by those $I$-torsion $\operatorname{Mod}_{R/I}$.
Define
- the subcategory inclusion functor $j_!:\operatorname{Mod}^{a}_{R,I}\to\operatorname{Mod}_R$;
- the Serre quotient functor $j^*:\operatorname{Mod}_R\to\operatorname{Mod}^{a}_{R,I}$; and
- the coëxtension functor $j_*:\operatorname{Mod}^{a}_{R,I}\to\operatorname{Mod}_R:M\mapsto\operatorname{Hom}_R(I,M)$.
One notices two significant facts:
- There's an adjunction triple $j_!\dashv j^*\dashv j_*$; and
- The counit $f^* f_* \Rightarrow\operatorname{id}$ is an isomorphism;
which induces one's wonder on if there's an open geometric embedding of (ringed) topoi for which objects of $\operatorname{Mod}_R$ and $\operatorname{Mod}^{a}_{R,I}$ are quasi-coherent modules.
Also, recall (from SGA 4 or something, if someone can recall where it is please comment) that if one decomposes say a scheme $X$ into a pair of mutually complementary components $U,Z$, one open, open closed, one has
where the composition of any two consecutive morphisms in the same direction results in the zero functor. Surprisingly, one can "decompose" a topos over $\operatorname{Spec}R$ in a similar fashion:
where the three functors on the last are given by the usual functors for change of rings.
Moreover, Bhatt (pp. 21-22) and Gabber-Ramero (at p. 11) both mentioned that there supposes to exist an "almost topos" explaining these geometric phenomena; i.e., according to Bhatt, there supposed to be a "open subscheme" complementary to the closed $Z$, the localization on which results in $\operatorname{Mod}_{R/I}$.
What is the "almost topos"?
Actually, localizing a topos should result in another topos? But how do I get a handle on the site, or specifically on what $\bar{U}$ is?