References/clarifications for the Mittag-Leffler argument

Question

While reading the paper MC elements in pronilpotent dg Lie algebras by Yekutieli, I stumbled across the folowing lines:

According to lemma 3.10(1) the complexes $M_j$ are acyclic. Threfore, using the Mittag-Leffler argument, the complex $M$ is also acyclic.

(page 15, in the proof of lemma 4.1(1))

Here, $M,M_j$ are complexes of $R$-modules, $R$ a ring, and $M\cong \lim_{\leftarrow j}M_j$. I would like some clarification about this Mittag-Leffler argument. I think I can reverse-engineer something close enough to the correct statement in this case, but what is the precise statement? What assumptions do we need, and how to prove it?

I tried looking it up on google, but I couldn't find any (elementary) exposition of it. Any insight or reference is welcome.

Answer 1

A. Grothendieck, Éléments de Géométrie Algébrique, $0_{III}$ Proposition 13.2.3 (on page 66 of the first volume of EGA III), since in your case $\{M_*\}$ and $\{H(M_*)\}$ are Mittag-Leffler.