In which category does Illusies derived deRham complex live?

Question

In "Complexe cotangent et déformations II", Illusie introduces the derived deRham complex as the pro-completion of the total complex associated to the double complex $$\Omega_{P_j^A(B)/A}^i$$

where $P_j^A(B)$ is the standard simplicial resolution of a ringmap $A\to B$.

More precisely we equip $L\Omega^\bullet_{B/A}:=Tot \Omega_{P_j^A(B)/A}^i$ with the Hodge-filtration $F$ and define $$ L\Omega^\bullet_{B/A}:= "\varprojlim" L\Omega^\bullet_{B/A}/F^pL\Omega^\bullet_{B/A} $$

he then claims that this lives in the category of complexs of pro-$A$-modules.

My problem is that this seems from the definition as it should be in the category of pro-complex of $A$-modules. But this should matter or are the categories $pro(CoChain(A-Mod))$ and $CoChain(pro(A-Mod))$ equivalent? I tried to prove it but the Hom-Sets don't seem to agree.

Answer 1

I think the answer is in the set of results known as reindexing lemmas. For quite a class of small categories, I, there is an equivalence of categories between $pro(C)^I$ and $pro(C^I)$. There is a set of useful results in Dan Isaksen's paper:

D. C. Isaksen, Calculating limits and colimits in pro-categories, Fundamenta Mathematicae, 175, (2002), 175 – 194.

There are earlier results in the thesis of Meyer:

C. V. Meyer, 1983, Completion of categories under certain limits, Ph.D. thesis, McGill University.

Whether these answer your question completely depends, I think, on whether the de Rham complexes are of finite length or not and without Illusie's thesis in front of me I cannot remember . I hope this helps.