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 —> Ω_{P•/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.