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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?

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

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  • $\begingroup$ 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"? $\endgroup$ – Urs Schreiber Aug 12 '14 at 10:18
  • $\begingroup$ 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? $\endgroup$ – Urs Schreiber Aug 12 '14 at 10:21
  • $\begingroup$ @UrsSchreiber I tried to answer your questions $\endgroup$ – Charles Rezk Aug 12 '14 at 13:39
  • $\begingroup$ Thank you, Charles, this is very much appreciated. Please allow me to follow up with two more questions, which will be no less trivial than the previous ones must have been: First: so will the functors of p-completion and p-torsion approximation as functors from spectra to spectra not be both left and right adjoint to each other? $\endgroup$ – Urs Schreiber Aug 12 '14 at 14:27
  • $\begingroup$ 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. $\endgroup$ – Urs Schreiber Aug 12 '14 at 14:27

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