Completion by Derived Double Centralizer

Let A be a commutative ring, and let $mathfrak{a}$ be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically $mathfrak{a}$ -torsion complexes. We prove that the derived double centralizer of M is isomorphic to the $mathfrak{a}$ -adic completion of A. The proof relies on the MGM equivalence from Porta et al. (Algebr Represent Theor, 2013) and on derived Morita equivalence. Our result extends earlier work of Dwyer et al. (Adv Math 200:357–402, 2006) and Efimov (2010).