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I suppose we were lacking an entry on p-localization (?)
The page p-localization implies that its definitions make sense for unstable spaces, but I’m unsure of that. The cited notes by Lurie are only about spectra. In the unstable case, usually (e.g. in May-Ponto) one defines a local object to be right orthogonal to $p$-local equivalences rather than to $p$-acyclic objects. In the stable case one can usually replace morphisms by objects (their cofibers), but not usually in the unstable case. Am I wrong?
I’m also confused a bit by the rings appearing here. May-Ponto define $T$-localization, for a set of primes $T$, with reference to homology with coefficients in the ring $\mathbb{Z}_T$ of integers localized at $T$, i.e. with primes not in $T$ inverted. In particular, $p$-localization has to do with homology with $\mathbb{Z}_{(p)}$-coefficients, where $\mathbb{Z}_{(p)}$ has all primes other than $p$ inverted. The ring $\mathbb{Z}/p\mathbb{Z} = \mathbb{F}_p$ appears only in the discussion of $p$-completion.
Lurie’s notes seem to be consistent with this, except that he refers to $p$-completion as “$\mathbb{Z}/p\mathbb{Z}$-localization” (which makes sense from a general perspective, as long as we don’t confuse it with $p$-localization, and indeed May-Ponto also do it when they get to model categories, e.g. Example 19.2.3).
However, the pages p-localization and Bousfield localization of spectra don’t seem to match this. The definitions at the former seems to be about $p$-completion, i.e. $\mathbb{Z}/p\mathbb{Z}$-localization, rather than what is classically called $p$-localization (i.e. $\mathbb{Z}_{(p)}$-localization). And the latter claims (in the section “$p$-Localization”) that the local objects in this sense are those whose homotopy/homology groups are $\mathbb{Z}[\frac{1}{p}]$-modules, which doesn’t seem right to me for either definition. Am I confused?
I suppose you are right. Should fix this. Need to remind myself when I am back online.
Okay, I have edited a good bit at p-localization in an attempt to correct and clarify. Have made explicit “localization away from $p$” and “localization at $p$” and highlighted how the former corresponds to $\mathbb{Z}/p\mathbb{Z}$-acyclicity, whose corresponding localization in turn is $p$-completion.
This deserves more work still, but I am out of time now. Please don’t hesitate to further edit as need be.
Thanks. I would be happy to do editing myself, but I’m still figuring these things out so I don’t feel too confident. Is the characterization of local objects in terms of homotopy groups still valid for unstable, non-nilpotent spaces?
This is the original definition of local space. Eg def 2.1 in the 1970 MIT notes by Sullivan.
I’d need to think to check how this harmonizes with Bousfield localization. Now changing trains, though…
Am now at the airport and so briefly: the note with that def. 2.1 that I mean is
Okay — but he only constructs a localization in the case of simple spaces (which also works for nilpotent ones). So that definition may not be very useful in the non-nilpotent case.
I tried to unify and focus the discussion at p-localization, and clarify the different behavior for spectra, nilpotent spaces, and general spaces and its relationship to completion. I also renamed it to localization of a space, since I thought that just “$p$-localization” could equally well refer to the algebraic notion (localization of a commutative ring).
Thanks!
To come back to an old question that David Corfield used to ask:
so the classical fracture theorem for suitable spectra $X$ asserts exact diagrams of the form
$\array{ && && X_p^{\wedge} \\ && & \nearrow && \searrow \\ && X && && (X_p^\wedge)_{\mathbb{Q}} \\ & \nearrow & & \searrow && \nearrow \\ G_{\mathbb{Z}/p\mathbb{Z}}(X) && && X_{\mathbb{Q}} }$where $G_{(-)}$ is the co-monadic acyclification functor as in e.g. Lurie 10, lecture 20.
May this be completed to more of an exact hexagon?
I have edited at localization of a space a bit more. Mike had produced (#10) a vastly and vastly better content than there was before, and I didn’t change any content. But in reading through it again I did various cosmetic changes that help me, the reader, to read the entry. So there shouldn’t be anything controversial, but of course please have a look to check.
Then I have wanted to sort out, for myself, better that sticking issue of $p$-localization and $p$-completion. Since this maybe deserves more chit-chat, I have moved the existing remark (of Mike) to a dedicated subsection
and expanded a bit more. Now this edit (this subsection) is in need of attention. :-)
First, to help deconfusion, I have added a table
$E$ | $E$-acyclic | $E$-local |
---|---|---|
$H \mathbb{Z}_{(p)}$ | $p$-local | |
$H \mathbb{F}_p$ | $\not\{p\}$-local | $p$-complete |
Then staring at this I thought this should have a good geometric explanation. The issue here is one of comparing points, their formal neighbourhoods, and their open neighbourhoods, I suppose. I started writing into the entry:
In terms of arithmetic geometry this may be understood as follows:
$\mathbb{F}_p = \mathbb{Z}/(p \mathbb{Z})$ is the ring of functions exactly on the point $(p)\in$ Spec(Z)
$\mathbb{Z}_p$ is the functions on the formal neighbourhood of $(p)$.
$\mathbb{Z}_{(p)}$ is the ring of functions defined not on any point $(p^\prime)$ with $p^\prime \neq p$, hence on the complement of the complement of $(p)$;
So therefore…
But that’s how far I got. What’s the right way to say it?
Probably the right way to say it is this:
localization at some “subspace” of $Spec(\mathbb{Z})$ always localizes on a formal neighbourhood of that subspace. If the original subspace is closed, then we see this explicitly in that the localization is localization at the subspace plus the formal completion around the subspace. But if the original subspace is open, then it contains its own formal neighbourhood, and so the localization is just that, the localization on the subspace.
(Sorry, this is elementary, please excuse but ignorance.)
When I am back online, need to add the keyword for this: mod p Whitehead theorem
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