Let (X, O_{X}) be a ringed space. Let π : *C* —> X be a stack over X where we use the topology on X to view X as a site. Endow *C* with the topology inherited from X (see Definition 06NV). This (roughly) means that the fibre categories *C*_{U} where U ⊂ X is open are endowed with the chaotic topology. Denote B = π^{ -1}O_{X} and think of *C* as a ringed site and π as a morphism of ringed sites

π : (

C, B) —-> (X, O_{X})

The functor π^{*} = π^{ -1} : Mod(O_{X}) —> Mod(B) commutes with all limits and colimits on modules and hence has a left adjoint π_{!} : Mod(B) —> Mod(O_{X}). In fact, if F is a sheaf of B-modules on *C*, then we can describe π_{!}F as the sheaf associated to the presheaf

U |—> colim

_{ξ in opposite of CU}F(ξ)

on the topological space X. (Colimit taken in category O_{X}(U)-modules.) Actually, it turns out that the situation above is a special case of this section of the Stacks project and we obtain a left derived extension Lπ_{!} : D(B) —> D(O_{X}) for free (note there are no boundedness assumptions).

In fact, the construction shows a little bit more. Namely, let ξ be an object of *C* lying over the open U ⊂ X. Then we can consider the localization morphism j_{ξ} : *C*/ξ —> *C* and the sheaf O_{ξ} = j_{ξ, !}B|_{ξ}. Any B-module is a quotient of a direct sum of these O_{ξ} and we have

Lπ

_{!}O_{ξ}= π_{!}O_{ξ}

Cool, so this gives us a bit of control in trying to compute Lπ_{!}.

Let x be a point of X. Let *C*_{x} denote the category

colim

_{x ∈ U ⊂ X}C_{U}

This makes sense as *C* is a stack over X so we can think of it as a sheaf of categories. If F is a sheaf of B-modules on *C*, then the stalk of π_{!}F is just the colimit of the “values” of F over *C*_{x}. Since taking stalks is exact, I think this should mean that we can compute the stalk of Lπ_{!}F at x by taking the corresponding construction over the category *C*_{x} with its chaotic topology.

Another tool to compute Lπ_{!} should be that if *C* is given as the stackification of a category *C*‘ fibred over X, then it should be sufficient to compute with *C*‘. Going back to the discussion and especially the example in this post we have to replace our choice of *C* there. We should start with the fibred category *C*‘ of immersions φ : U —> **A**^{n}_{B} (not necessarily closed) and commutative diagrams over B. Then *C* should be the stackification of that. Then with all of the above you’d get the cotangent complex of X/B by doing the same construction as in the affine case. The key is that affine locally *C*‘ has a good co-simplicial object computing the derived lower shriek functor. You use the localization of sheaves of algebras construction to provide *C* with a sheaf of rings surjecting onto the pullback of the structure sheaf of X (and not to change the underlying category).

A similar procedure is going to define the base change *C*_{S} given a morphism of schemes S —> B, i.e., as underlying fibred category start with some category of diagrams of schemes and use the localization of sheaves of algebras construction to endow this with a structure sheaf.

I think this will just work and in fact it simplifies the original idea I had for the stacks *C* and *C*_{S}. We’ll see.