# AdicCompletion$\dashv$Torsion adjunction on spectra?

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?

• You might want to take a look at Section 4 in Lurie's DAG XII about completion of modules over ring spectra. Aug 12 '14 at 11:47
• Thanks! I had missed that. This is excellent, just what I was hoping for. Thanks again. Aug 12 '14 at 15:03

If I understand what you are asking, then yes. p-completion of p-local spectra is $X \mapsto F(M, X)$, where $M=$ fiber of $S\to S\mathbb{Q}$, while the p-torsion approximation is $X\mapsto X\wedge M$.

The same story holds for any "smashing" localization. Added. A "smashing localization" $L$ gives a map of spectra $\eta\colon S\to T:=LS$ such that $T\wedge \eta$ is an equivalence. Consider the cofiber sequence $$M\xrightarrow{\epsilon} S\xrightarrow{\eta} T.$$ Then we obtain a couple of idempotent monads $T\wedge-$ and $F(M,-)$ on spectra, and a couple of idempotent comonads $M\wedge-$ and $F(T,-)$ on spectra. Clearly, these come as two adjoint pairs of functors on spectra.

Then $T\wedge -$ is just the original smashing localization $L$, and $F(M,-)$ is a "cosmashing localization". The other two functors are the corresponding acyclizations.

In the case of $$M=\Sigma^{-1} S\mathbb{Q}_p/\mathbb{Z}_p \to S \to S\mathbb{Z}[\tfrac{1}{p}]=T$$ we get the situation you described, where $M\wedge -$ is the $p$-torsion-approximation idempotent-comonad, and $F(M,-)$ is the $p$-completion idempotent-monad. (Note: as a functor from spectra to spectra, $p$-torsion approximation $M\wedge-$ is a left adjoint, but is also right adjoint to the inclusion functor of $p$-torsion spectra into all spectra.)

• Thanks, Charles. I am behind the curve here, please bear with me. I realize that the statement you refer to is prop. 2.5 in Bousfield 79. Two questions, though, I still have: First regarding "holds for any smashing localization": for what I am after I suppose I'd have to read that as "holds whenever one of the two localizations is smashing and the other's acyclification is the 'co-smashing' co-localization of the former"? Aug 12 '14 at 10:18
• Second question, related to that: is there an issue with variance here? maybe I am mixed up, sorry. In what you write p-torsion approximation is left adjoint, but in that story of "Greenlees-May duality" on the level of chain complexes it is right adjoint, no? Do we have a dual statement? Aug 12 '14 at 10:21
• Secondly, just to sanity check: so then for every smashing localization there is in fact canonically a pair of fracture squares which -- when regarded in the opposite $\mathrm{Spectra}^{\mathrm{op}}$ -- fit into an exact hexagon of just the "differential cohomology hexagon"-form discussed here: ncatlab.org/nlab/show/… -- where $\flat$ is the given localization and $\Pi$ the given co-localization. Aug 12 '14 at 14:27