Remarks on derived complete modules and complexes

Let R be a commutative ring and $I?R$ a finitely generated ideal. We discuss two definitions of derived I-adically complete (also derived I-torsion) complexes of R-modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I-adically flat R-modules.