Let \(I\) denote an ideal of a commutative ring \(R\). The \(I\)-adic completion \(M^{\wedge}_I\) of an \(R\)-module \(M\) is \(\varprojlim M/I^nM\). It defines a functor on the category of \(R\)-modules. Its left derived functors \(L^I_n\) were systematically studied at first by J. P. C. Greenlees and J. P. May [J. Algebra 149, No. 2, 438–453 (1992; Zbl 0774.18007)] and A.-M. Simon [Math. Proc. Camb. Philos. Soc. 108, No. 2, 231–246 (1990; Zbl 0719.13007)]. Then \(M\) is called \(I\)-adically complete if the natural map \(M \to M^{\wedge}_I\) is an isomorphism. It is called \(L_0^I\)-complete, whenever \(M \to L_0^I(M)\) is an isomorphism. Finally, for a complex \(M\) the authors define the derived completion of \(M\) by \({\mathbb{R}} \operatorname{Hom}_R(\check{C}_{x_1,\ldots,x_r},M)\), where \(I = x_1,\ldots,x_r\) is finitely generated and \(\check{C}_{x_1,\ldots,x_r}\) denotes the (local) Čech complex. As the main result it is shown that if \(I\) is generated by a weakly pro-regular sequence then there are symmetric monoidal Quillen equivalences between all the three notions of completeness. Then they investigate how base change along a map of commutative rings interacts with the Quillen equivalences of their result.
(Reviewer’s Remark: Several of the technical details of the paper may be found also in the book [P. Schenzel and A.-M. Simon, Completion, Čech and local homology and cohomology. Interactions between them. Cham: Springer (2018; Zbl 1402.13001)].)