A simplicial ring A• is just a simplicial object in the category of rings. What is a simplicial module over A•? Well it is a simplicial object in the category of systems (A, M, +, *, +, *) where A is a ring and M is an A-module (so the + and * are multiplication and addition on A and M respectively) such that forgetful functor to the category of rings gives back A•.
Of course this is annoying. Better: A simplicial ring A• is a sheaf on Δ (the category of finite ordered sets endowed with the chaotic topology). Then a simplicial module over A• is just a sheaf of modules.
You can extend this to simplicial sheaves of rings over a site C. Namely, consider the category C x Δ together with the projection C x Δ —> C. This is a fibred category hence we get a topology on C x Δ inherited from C. Then a simplicial sheaf of rings A• is just a sheaf of rings on C x Δ and we define a simplicial module over A• as a sheaf of modules on C x Δ over this sheaf of rings. There is a derived category D(A*) and a derived lower shriek functor
Lπ! : D(A•) ———-> D(C)
as discussed in Tag 08RV. Moreover, a map A• —> B• of simplicial rings on C gives rise to a morphism of ringed topoi, and hence a derived base change functor
D(A•) ———-> D(B•)
as well as a restriction functor the other way.
Why am I pointing this out? The reason is to use it for the following. If A —> B is a map of sheaves of rings and M is a B-module, then a priori the Atiyah class “is” the extension of principal parts
0 —> ΩP•/A ⊗ M —> E —> M —> 0
over the polynomial simplicial resolution P• of B over A. To get it in D(B) Illusie uses the base change along the map P• —> B. I was worried that we’d have to introduce lots of new stuff in the Stacks project to even define this, but all the nuts and bolts are already there. Cool!
PS: Warning! The category D(A•) is not the same as the category D•(A•) defined in Illusie.