$E_\infty$ spectrum corresponding to $\Bbb Z_p$

Question

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))

Answer 1

There is indeed an Eilenberg-Maclane spectrum $H\Bbb Z_p$. It is equivalent to the $p$-completion of $H\Bbb Z$ and the (homotopy) limit of the $H\Bbb Z/p^k$. 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 $\{H\Bbb Z/p^k\}$ 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.

Answer 2

I would argue that looking for a ring spectrum is not the right thing to do. What you should be looking for is the category of modules over that possibly non-existent ring spectrum, as an infinity-category or just as a triangulated category. If you think about it this way, an obvious answer presents itself.

Begin with the category of $H\Bbb Z$-modules, or the derived category of $\Bbb Z$, or its infinity-category version. Now take the Bousfield localization with respect to the object $Fp$ (thought of as a complex in degree $0$). This is not a smashing localization, so this category is not equivalent to modules over $H\Bbb Z_p$ . As a triangulated category, it is compactly generated, but by $Fp$ itself. The sphere is not small. So this category is equivalent to modules over a DGA, the endomorphism DGA of $Fp$, but it is not commutative. It is more like a DG Hopf algebra, I suspect, so that its homotopy category has a tensor product even though it is not commutative. I have always thought this example needs more investigation, though it might be in Dwyer-Greenlees-Iyengar somewhere. It is a toy version of the $K(n)$-local stable homotopy category.

Mark

Answer 3

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.

Answer 4

There is a perfectly functional Eilenberg-Maclane spectrum $H\Bbb Z_p$, 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 $H\Bbb Z/p^n$
• Take a $p$-completion of $H\Bbb Z$