# E_\infty spectrum corresponding to Z_p

First of the questions about derived algebraic geometry from a noobie.

The way I understand it, every discrete ring R corresponds to some ring spectrum whose \pi_0 is R. Now consider p-adic numbers. They are a limit of discrete rings — what should correspond to them? How to generalize?

(+ what could be good tags for derived algebraic geometry? I was considering: e-infinity, infty-structures, math-0703204, derived-alggeom, derived-spaces, infty-topology, a-infinity-algebras (2 currently tagged))

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The word "spectrum" doesn't need infinity attached to it. Also, I recommend "derived-stuff" –  S. Carnahan Oct 13 '09 at 19:25
(Simpsons quote: Garbage goes to the garbage bin... Makes sense.) Derived-stuff it is. –  Ilya Nikokoshev Oct 13 '09 at 19:43
The E_\infty says that it has a highly structured multiplication lifting the multiplication on R - properly it's an E_\infty ring spectrum. –  Tyler Lawson Oct 15 '09 at 15:55

There is indeed an Eilenberg-Maclane spectrum HZ_p. It is equivalent to the p-completion of HZ and the (homotopy) limit of the HZ/p^k. However, it doesn't remember the topology.

If you want to remember the topology on the p-adics you need to do something more complicated, such as view the inverse system {HZ/p^k} as a pro-object in spectra rather than actually taking the limit. This is something like what you would do if you wanted to talk about formal schemes.

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There is a perfectly functional Eilenberg-Maclane spectrum HZ_p, although it returns the discrete ring. It looks like you want a way to attach spectra to cohomology theories with topologized coefficient groups, and I don't know if that's possible (in particular the wedge axiom in Brown representability smells funny).

Random ideas that may or may not work:

• Take a limit of Eilenberg-Maclane spectra HZ/p^n
• Take a p-completion of HZ
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I don't know whether this actually helps answer your question, but Gereon Quick has done some work on profinite simplicial objects that may be relevant. You might want to poke around his papers a bit.

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