It seems to me that in slight paraphrase the central result of the article

- Marco Porta, Liran Shaul, Amnon Yekutieli,
*On the Homology of Completion and Torsion*(arXiv:1010.4386)

(theorems 6.11 and 6.12) means that for $\mathfrak{a} \subset A$ a suitably nice ideal inside a commutative ring $A$, then the total derived functors of

1) adic completion of modules at $\mathfrak{a}$

and

2) of taking $\mathfrak{a}$-torsion submodules

form an adjoint pair of (co-)reflections of homotopy theories (i.e. an adjoint pair of idempotent $\infty$-(co-)monads on the $\infty$-category of chain complexes of $A$-modules).

I am wondering if an analogous result would not also hold for spectra in the case that $\mathfrak{a} = (p)$ is a prime. If so that would yield a nice enhancement of the story of the arithmetic fracture square.

Is forming $p$-completion of spectra adjoint to forming universal $\mathbb{Z}[p^{-1}]$-acyclic spectra (hence adic completion to $\mathbb{Q}$-acyclification), maybe at least on suitably small spectra?

And how about lifting either statement to commutative monoids, i.e. to dg-algebras and further to $E_\infty$-rings, is anything known?